**Meet the editor**

Auteliano Antunes dos Santos Junior is an Associate Professor and Head of the Mechanical Design Department at the University of Campinas – Unicamp. He obtained his bachelor degree in Mechanical Engineering from the University of the State of São Paulo in 1987, his MS in Mechanical Design in 1992, and PhD in Mechanical Engineering in 1996, both from Unicamp. From 1998

to 1999, he worked as a Visiting Scholar at the Texas A&M University, USA researching acoustoelasticity. His main subjects of research are stress measurement using ultrasound (acoustoelasticity) and ESPI (electronic speckle pattern interferometry), tribology and brake design modeling of elasto-plastic contact between rolling bodies and multi-body dynamic analysis of railroad freight cars.

He developed partnerships for research and service projects with companies like VALE, Petrobras, Faiveley Transport, Thyssenkrup Metalúrgica Campo Limpo, Fras-le and Union Park Limited, as well as with research foundations like São Paulo Research Foundation (FAPESP), National Council for Scientific and Technological Development (CNPq) and Air Force Office for Scientific Research – USA (AFOSR).

Contents

**Preface IX** 

Chapter 2 **Ultrasonic Projection 29**  Krzysztof J. Opieliński

Hassina Khelladi

Hikaru Miura

Chapter 8 **Ultrasonic Thruster 147** 

Chapter 1 **Modelling the Generation and Propagation** 

Fernando Seco and Antonio R. Jiménez

Chapter 3 **3-D Modelings of an Ultrasonic Phased Array** 

Kazuyuki Nakahata and Naoyuki Kono

Chapter 4 **Goldberg's Number Influence on the Validity** 

Gaowei Hu and Yuguang Ye

Chapter 6 **Intense Aerial Ultrasonic Source and** 

**Domain of the Quasi-Linear Approximation of Finite Amplitude Acoustic Waves 81** 

Chapter 5 **Ultrasonic Waves on Gas Hydrates Experiments 87** 

Chapter 7 **Application of Pulsed Ultrasonic Doppler Velocimetry** 

N. Sad Chemloul, K. Chaib and K. Mostefa

Chapter 9 **Real-Time Distance Measurement for Indoor Positioning** 

**System Using Spread Spectrum Ultrasonic Waves 173**  Akimasa Suzuki, Taketoshi Iyota and Kazuhiro Watanabe

Alfred C. H. Tan and Franz S. Hover

**Removal of Unnecessary Gas by the Source 107** 

**to the Simultaneous Measurement of Velocity and Concentration Profiles in Two Phase Flow 133** 

**of Ultrasonic Signals in Cylindrical Waveguides 1** 

**Transducer and Its Radiation Properties in Solid 59** 

## Contents

## **Preface XI**


Akimasa Suzuki, Taketoshi Iyota and Kazuhiro Watanabe


## Preface

The idea of using ultrasonic waves in medical and engineering fields came from military applications of sound waves in sonar systems. Sokolov, Firestone, Simons, and several other researchers are among the ones who studied this high frequency kind of wave in the first half of twentieth century.

Ultrasonic waves are defined as those with frequencies higher than 20 kHz. The constant development of data acquisition systems and probes seems to work, removing the upper limit of the wave frequency that can be acquired. The phenomenal improvement of this technique, allied with its simplicity and the low cost of the required equipment transformed the initially focused technique to one of the most accepted ways to inspect biological structures, fluid flow, and mineral bodies.

Today, medical applications of ultrasonic techniques (UT) allow physicians to see virtually all parts and details of organic bodies, permitting them to detect abnormalities. The same happens with the inspection of non-organic bodies, for which non-destructive and semi destructive ultrasonic methods have been standardized as the principal way to evaluate safety and quality. Fluid flow and its contents and uniformity are among the important applications, mainly in oil industry and environmental studies.

This book is part of the effort to spread the knowledge about ultrasonic applications and the theory involved. In it, a group of outstanding researchers present insights into their respectivebranches of study related to ultrasonic waves. The book covers topics from different areas of study without losing the focus on innovative developments.

Chapter 1 presents modeling of the generation and propagation of ultrasonic waves in cylindrical waveguides, based on Pochhammer-Chree equations and modal analysis. The authors show the theoretical background used in the development of a code named PCDISP, and application of the methodology to study the wave propagation in a hollow cylinder.

Chapter 2 describes the application of ultrasonic pulses to create images to be used in medical diagnosis. 3D projection techniques are discussed and 2D ultrasonic arrays are presented as an evolution of the technique to be utilized with soft tissue. Tomographic

#### X Preface

reconstruction is one of the benefits reached with rotation of pairs of probes, and 3D reconstruction of internal structures is possible.

Preface XI

based on the method used for evaluation of concrete. The simplicity, low cost, and

The main subject of Chapter 11 is the design of an ultrasonic system for Tomography based on an array of transducers. The authors developed a method able to identify the size and position of gas bubbles in measurement columns. Thresholding techniques were used to improve the resulting images, making the methodology useful to

The corrosion products that cover the pits in stainless steel are responsible for the growth of these pits. Chapter 12 addressed the ultrasonic method to remove such corrosion products, resulting in extending life of underwater structures. The authors studied the effect of power of the UT transducer, as well as its frequency and distance where it is placed. From those parameters, the best setting of the system was achieved.

Concluding the book, Chapter 13 presents the new trends in material characterization using ultrasound. The authors concentrate their analysis on concrete samples and aerospace composites. They show that it is possible to characterize mortar specimens with inhomogeneities (small flakey inclusions). Besides that, they show that ultrasonic microscopy and nonlinear acoustics can be used to inspect aerospace materials. The chapter also shows the limits of the techniques and presents suggestions for improving

The editor would like to congratulate the authors for their contribution to this book. It's a privilege to work with researchers filled with such enthusiasm for scientific work. The quality of the book is a natural result of the competency and dedication of each of the participants. The editor trusts that the reader will find this book to be a

**Prof. Auteliano Antunes dos Santos Júnior** 

School of Mechanical Engineering University of Campinas - Unicamp

Brazil

adequate results are pointed out and confirmed by experiments.

measure liquid/gas concentrations.

valuable and up to date source of knowledge.

them.

Chapter 3 shows the study of the radiated beam from phased array transducers (PA) in solids. The authors present the principles involved in using PA transducers and the way to effectively set the delay time required to avoid grating and side lobes. Frequency and time delay evaluations are presented. The technique is used to evaluate a welded T-joint with adequate results.

Chapter 4 discusses the influence of Goldberg's number on the validity domain for quasi-linear approach used with finite amplitude acoustic waves. The parameter measures the relative importance of nonlinear and dissipative phenomena that appears when ultrasonic waves propagate inside biological media.

Chapter 5 is an investigation into the acoustic properties of gas hydrate-bearing sediments. Two kinds of ultrasonic methods, namely the flat-plate transducers and the one based on a new kind of bender elements, have been successfully used in measuring such properties. The study of gas hydrates is becoming highly important due to their application as a possible source of energy.

Chapter 6 introduces a new probe design to generate intense ultrasonic waves in air. The fixture is based on a flexural vibrating plate and is used to remove unnecessary gas. The author shows successful experiments with a low-hydrophilicity gas in water particles of two sizes, proving the efficiency of the technique.

In Chapter 7, ultrasonic Doppler technique is applied to measure velocity and concentration in two-phase flows. The authors propose a new method, expanding the characteristics of the conventional PDUV, which only gives the velocity as result. Their approach was tested in an experimental set up specially built for this purpose, and the results proved that the method is effective for small concentrations of the phase under analysis.

The application of ultrasonic thrusters as an alternative small-scale propulsion system for underwater robotic devices is the problem explored in Chapter 8. The authors present a comparison among the main technologies for thrusters and describe the characteristics of UT thrusters. They also present a detailed outline of the construction of a prototype and an analysis of the behavior of UT transducers in such devices.

Chapter 9 addresses real time positioning of robots using spread spectrum ultrasonic waves. The authors built a system that is able to evaluate positioning, receive the signal from sensors, calculate correlation, detect peak, and estimate the distance from a known position. All tasks were accomplished in less than 6 microseconds, with resolution of 1.5 cm.

Application of ultrasonic waves to inspect rocks is the main theme studied by the authors in Chapter 10. They present a method to find discontinuities in marble rocks based on the method used for evaluation of concrete. The simplicity, low cost, and adequate results are pointed out and confirmed by experiments.

X Preface

analysis.

resolution of 1.5 cm.

reconstruction is one of the benefits reached with rotation of pairs of probes, and 3D

Chapter 3 shows the study of the radiated beam from phased array transducers (PA) in solids. The authors present the principles involved in using PA transducers and the way to effectively set the delay time required to avoid grating and side lobes. Frequency and time delay evaluations are presented. The technique is used to evaluate

Chapter 4 discusses the influence of Goldberg's number on the validity domain for quasi-linear approach used with finite amplitude acoustic waves. The parameter measures the relative importance of nonlinear and dissipative phenomena that

Chapter 5 is an investigation into the acoustic properties of gas hydrate-bearing sediments. Two kinds of ultrasonic methods, namely the flat-plate transducers and the one based on a new kind of bender elements, have been successfully used in measuring such properties. The study of gas hydrates is becoming highly important

Chapter 6 introduces a new probe design to generate intense ultrasonic waves in air. The fixture is based on a flexural vibrating plate and is used to remove unnecessary gas. The author shows successful experiments with a low-hydrophilicity gas in water

In Chapter 7, ultrasonic Doppler technique is applied to measure velocity and concentration in two-phase flows. The authors propose a new method, expanding the characteristics of the conventional PDUV, which only gives the velocity as result. Their approach was tested in an experimental set up specially built for this purpose, and the results proved that the method is effective for small concentrations of the phase under

The application of ultrasonic thrusters as an alternative small-scale propulsion system for underwater robotic devices is the problem explored in Chapter 8. The authors present a comparison among the main technologies for thrusters and describe the characteristics of UT thrusters. They also present a detailed outline of the construction of a prototype and an analysis of the behavior of UT transducers in such devices.

Chapter 9 addresses real time positioning of robots using spread spectrum ultrasonic waves. The authors built a system that is able to evaluate positioning, receive the signal from sensors, calculate correlation, detect peak, and estimate the distance from a known position. All tasks were accomplished in less than 6 microseconds, with

Application of ultrasonic waves to inspect rocks is the main theme studied by the authors in Chapter 10. They present a method to find discontinuities in marble rocks

appears when ultrasonic waves propagate inside biological media.

due to their application as a possible source of energy.

particles of two sizes, proving the efficiency of the technique.

reconstruction of internal structures is possible.

a welded T-joint with adequate results.

The main subject of Chapter 11 is the design of an ultrasonic system for Tomography based on an array of transducers. The authors developed a method able to identify the size and position of gas bubbles in measurement columns. Thresholding techniques were used to improve the resulting images, making the methodology useful to measure liquid/gas concentrations.

The corrosion products that cover the pits in stainless steel are responsible for the growth of these pits. Chapter 12 addressed the ultrasonic method to remove such corrosion products, resulting in extending life of underwater structures. The authors studied the effect of power of the UT transducer, as well as its frequency and distance where it is placed. From those parameters, the best setting of the system was achieved.

Concluding the book, Chapter 13 presents the new trends in material characterization using ultrasound. The authors concentrate their analysis on concrete samples and aerospace composites. They show that it is possible to characterize mortar specimens with inhomogeneities (small flakey inclusions). Besides that, they show that ultrasonic microscopy and nonlinear acoustics can be used to inspect aerospace materials. The chapter also shows the limits of the techniques and presents suggestions for improving them.

The editor would like to congratulate the authors for their contribution to this book. It's a privilege to work with researchers filled with such enthusiasm for scientific work. The quality of the book is a natural result of the competency and dedication of each of the participants. The editor trusts that the reader will find this book to be a valuable and up to date source of knowledge.

> **Prof. Auteliano Antunes dos Santos Júnior**  School of Mechanical Engineering University of Campinas - Unicamp Brazil

**1. Introduction**

waveguides.

to consider this problem.

Elongated cylindrical structures like rods, pipes, cable strands or fibers, support the propagation of mechanical waves at ultrasonic frequencies along their axes. This waveguide behaviour is used in a number of scientific and engineering applications: the Non Destructive Evaluation (NDE) of the structural health of civil engineering elements for safety purposes (Rose, 2000), in linear displacement sensors (Seco et al., 2009) for high accuracy absolute linear position estimation, in the evaluation of material properties of metal wires, optical fibers or composites (Nayfeh & Nagy, 1996), and as fluid sensors in pipes transporting liquids (Ma et al., 2007). These applications demand exact quantitative models of the processes of wave generation, propagation and reception of the ultrasonic signals in the

**Modelling the Generation and Propagation of Ultrasonic Signals in Cylindrical Waveguides** 

> Fernando Seco and Antonio R. Jiménez *Centro de Automática y Robótica (CAR)*

*Consejo Superior de Investigaciones Científicas (CSIC)-UPM* 

*Ctra. de Campo Real, Madrid* 

**1**

*Spain* 

The mathematical treatment of mechanical wave propagation in cylindrical structures was provided by J. Pochhammer and C. Chree at the end of the XIX century, but its complexity prevented researchers from obtaining quantitative results until the advent of computers. D. Gazis (Gazis, 1959) reported the first exact solutions of the Pochhammer-Chree frequency equation, as well as a complete description of propagation modes and displacement and stress distributions for an isotropic elastic tube, found with an IBM 704 computer. Since then, the literature on the topic has grown steadily, and references are too numerous for this book chapter. We will only mention a few landmark developments: the study of multilayered waveguides beginning with a composite (two-layer) cylinder by H. D. McNiven in 1963; the extension of Gazis' results to anisotropic waveguides, initiated by I. Mirsky in 1965; the consideration of fluids and media with losses surrounding, or contained in the waveguides, beginning with V. A. Del Grosso in 1968; and finally, the demonstration of ultrasonic guided waves generated with electromagnetic transducers by W. Mohr and P. Holler in 1976, and piezoelectrically by M. Silk and K. Bainton in 1979, for the nondestructive testing of pipes.

Of particular importance for transducer design is the determination of the mechanical response of a waveguide when subjected to an external excitation. Several approaches exist

**Integral transform methods** (Graff, 1991) convert the differential equations that physically model the excitation forces and the behaviour of the waveguide into a set of algebraic

**1.1 Modelling the response of the waveguide to external excitation**

## **Modelling the Generation and Propagation of Ultrasonic Signals in Cylindrical Waveguides**

Fernando Seco and Antonio R. Jiménez

*Centro de Automática y Robótica (CAR) Ctra. de Campo Real, Madrid Consejo Superior de Investigaciones Científicas (CSIC)-UPM Spain* 

## **1. Introduction**

Elongated cylindrical structures like rods, pipes, cable strands or fibers, support the propagation of mechanical waves at ultrasonic frequencies along their axes. This waveguide behaviour is used in a number of scientific and engineering applications: the Non Destructive Evaluation (NDE) of the structural health of civil engineering elements for safety purposes (Rose, 2000), in linear displacement sensors (Seco et al., 2009) for high accuracy absolute linear position estimation, in the evaluation of material properties of metal wires, optical fibers or composites (Nayfeh & Nagy, 1996), and as fluid sensors in pipes transporting liquids (Ma et al., 2007). These applications demand exact quantitative models of the processes of wave generation, propagation and reception of the ultrasonic signals in the waveguides.

The mathematical treatment of mechanical wave propagation in cylindrical structures was provided by J. Pochhammer and C. Chree at the end of the XIX century, but its complexity prevented researchers from obtaining quantitative results until the advent of computers. D. Gazis (Gazis, 1959) reported the first exact solutions of the Pochhammer-Chree frequency equation, as well as a complete description of propagation modes and displacement and stress distributions for an isotropic elastic tube, found with an IBM 704 computer. Since then, the literature on the topic has grown steadily, and references are too numerous for this book chapter. We will only mention a few landmark developments: the study of multilayered waveguides beginning with a composite (two-layer) cylinder by H. D. McNiven in 1963; the extension of Gazis' results to anisotropic waveguides, initiated by I. Mirsky in 1965; the consideration of fluids and media with losses surrounding, or contained in the waveguides, beginning with V. A. Del Grosso in 1968; and finally, the demonstration of ultrasonic guided waves generated with electromagnetic transducers by W. Mohr and P. Holler in 1976, and piezoelectrically by M. Silk and K. Bainton in 1979, for the nondestructive testing of pipes.

#### **1.1 Modelling the response of the waveguide to external excitation**

Of particular importance for transducer design is the determination of the mechanical response of a waveguide when subjected to an external excitation. Several approaches exist to consider this problem.

**Integral transform methods** (Graff, 1991) convert the differential equations that physically model the excitation forces and the behaviour of the waveguide into a set of algebraic

circumstances. The main features of the PCDISP software will be introduced in this chapter

Modelling the Generation and Propagation of Ultrasonic Signals in Cylindrical Waveguides 3

The purpose of PCDISP is freeing the researchers from the numerically delicate, time consuming issues arising in the solution of the PC equations, such as the creation of the waveguide matrix, the numerical instabilities encountered when the thickness of the waveguide or the operating frequency are high, the determination of proper modes and the tracing of the dispersive wavenumber-frequency curves. In this way, the researcher can

As far as we are aware of, only two other software suites specifically designed for modelling elastic wave propagation in cylindrical waveguides exist. Disperse (Pavlakovic & Lowe, 1999) is a commercial package, based on matrix techniques, capable of analyzing cylindrical or plate waveguides made of perfectly elastic or damped solids, as well as fluids. GUIGUW (Bocchini et al., 2011) is a Matlab-based software which utilizes a SAFE-based approach to model ultrasonic propagation in cylindrical, plate, and arbitrary cross section waveguides. However, none of these computer solutions permit to model the waveguide response to

The organization of this chapter is detailed next. Section 2 briefly reminds the mathematical background of the PC theory. Section 3 properly describes the main features of our methodology and how it is implemented in the PCDISP package. Two common transducer setups for the generation of ultrasonic waves are studied in section 4 with the help of PCDISP. Finally, we will offer some conclusions and point to lines in which this research could be

In this section we present a summarized theoretical background on wave propagation in cylindrical waveguides, treating such aspects as relevant for our purposes; standard references can be consulted for further information (Graff, 1991; Meeker & Meitzler, 1972;

A **waveguide** is a physical structure which supports the propagation of mechanical waves along its elongated direction *z*, and modifies the behaviour of such waves with respect to free propagation in the bulk material. There are two fundamental characteristics of waveguide propagation. The first is the discretization of waves into **propagating modes**, of which only a finite number are permitted for a given frequency, and whose properties are determined by the shape of the cross section and boundary conditions of the waveguide. The second is the existence of **dispersion**, which is the nonlinear relationship between wavenumber and frequency. As a consequence, signals with a significant bandwidth are distorted as they travel along the waveguide, because their spectral components propagate at different phase speeds. The solutions of the wave equation in a cylindrical material are readily found by the use of potentials and the technique of separation of variables, arriving at the following general form

*<sup>u</sup>*(*r*, *<sup>θ</sup>*, *<sup>z</sup>*) = *<sup>u</sup>*(*r*, *<sup>θ</sup>*)*ejkz* <sup>=</sup> *<sup>u</sup>*(*r*)*ejnθejkz <sup>σ</sup>*(*r*, *<sup>θ</sup>*, *<sup>z</sup>*) = *<sup>σ</sup>*(*r*, *<sup>θ</sup>*)*ejkz* <sup>=</sup> *<sup>σ</sup>*(*r*)*ejn<sup>θ</sup> <sup>e</sup>jkz*, (1)

where the cylindrical system is used (with coordinates (*r*, *θ*, *z*), and unit vectors (*er*,*eθ*,*ez*)), harmonic time variation *e*−*jω<sup>t</sup>* is assumed, and *ω* is the angular frequency, *k* the wavenumber,

alongside with the theoretical concepts upon which it is based.

external excitations.

further extended.

Rose, 1999).

**2. Background and nomenclature**

for the displacement vector (*<sup>u</sup>*) and stress tensor (*<sup>σ</sup>*):

concentrate in the study of the waveguide/transducer interaction as such.

equations, which are more easily solvable. However, in order to find the actual distribution of the elastic field excited in the waveguide, inverse contour integration in the complex plane has to be performed, which is usually complicated. Due to the complexity of the Pochhammer-Chree equations, this procedure is only practical with simplified versions of the wave equation, which in general are not accurate enough for ultrasonic frequencies. See for example, Folk's solution for the transient response of a semi-infinite rod to a step pressure applied to its end (Folk et al., 1958).

The **Semi-Analytical Finite Element (SAFE)** method is a modification of Finite Element Methods (FEM) in which the elastic field is expanded as a superposition of harmonic waves in the azimuthal-axial (*θ*-*z*) plane, while discretized mechanical equations are used in the radial (*r*) direction of the waveguide. This reduction of the number of dimensions permits a much higher efficiency in the computation of the elastic fields (Hayashi et al., 2003). Waveguides surrounded by infinite media (like a pipe submerged in soil) can be handled by SAFE techniques with proper discretized elements (Jia et al., 2011), as well as waveguides with arbitrary profiles: for example, a railroad rail in (Damljanovic & Weaver, 2004). Although finite element methods are powerful and flexible, they have the shortcoming of great requirements on computer memory and processing time when large structures or high frequencies of operation are considered, and the difficulty encountered in the parameterization of transducer designs (for example, the determination of the transfer function of the transducer-waveguide coupling).

**Spectral methods** are another numerical technique which approximate the differential elastic equations of the waveguide (Helmholtz equations) by differentiation operators, turning the problem of finding the wavenumber-frequency roots into a matrix eigenvalues determination (Doyle, 1997). This numerical method, which is computationally simple and reportedly does not suffer from the problems associated with large diameter waveguides at high frequencies, has been recently applied to model multi-layered cylindrical waveguides (Karpfinger et al., 2008).

**Modal analysis** is an analytical method based on the expansion of the forcing terms acting in the waveguide into the set of its proper modes (Auld, 1973). In (Ditri & Rose, 1992), modal analysis is employed to model the loading of a waveguide by a transducer array. This treatment is extended to more general transducers and antisymmetric modes by (Li & Rose, 2001). Modal analysis is a mathematically exact technique that leads to a closed form integral equation for the elastic fields in the waveguide, and which incorporates in a natural way the issue of mode selectivity, offering insight on the physics of waveguide behaviour. For these reasons, modal analysis will be the approach used in this work.

#### **1.2 Intention and scope of the research**

With this book chapter we contribute a numerical simulation treatment of the ultrasonic behaviour of cylindrical waveguides, based on the Pochhammer-Chree (PC) theory, and covering the aspects of assembly of the description matrix of the waveguide, tracing of the frequency-wavenumber curves, computation of mode shapes, use of modal analysis to determine the response of the waveguide to external excitations, and the dispersive propagation of signals.

The work described here has resulted in a software package, named PCDISP, written in the Matlab environment (*Matlab*, 2004), and freely available<sup>1</sup> to be adapted to particular

<sup>1</sup> PCDISP webpage: http://www.car.upm-csic.es/lopsi/people/fernando.seco/pcdisp

circumstances. The main features of the PCDISP software will be introduced in this chapter alongside with the theoretical concepts upon which it is based.

The purpose of PCDISP is freeing the researchers from the numerically delicate, time consuming issues arising in the solution of the PC equations, such as the creation of the waveguide matrix, the numerical instabilities encountered when the thickness of the waveguide or the operating frequency are high, the determination of proper modes and the tracing of the dispersive wavenumber-frequency curves. In this way, the researcher can concentrate in the study of the waveguide/transducer interaction as such.

As far as we are aware of, only two other software suites specifically designed for modelling elastic wave propagation in cylindrical waveguides exist. Disperse (Pavlakovic & Lowe, 1999) is a commercial package, based on matrix techniques, capable of analyzing cylindrical or plate waveguides made of perfectly elastic or damped solids, as well as fluids. GUIGUW (Bocchini et al., 2011) is a Matlab-based software which utilizes a SAFE-based approach to model ultrasonic propagation in cylindrical, plate, and arbitrary cross section waveguides. However, none of these computer solutions permit to model the waveguide response to external excitations.

The organization of this chapter is detailed next. Section 2 briefly reminds the mathematical background of the PC theory. Section 3 properly describes the main features of our methodology and how it is implemented in the PCDISP package. Two common transducer setups for the generation of ultrasonic waves are studied in section 4 with the help of PCDISP. Finally, we will offer some conclusions and point to lines in which this research could be further extended.

#### **2. Background and nomenclature**

2 Will-be-set-by-IN-TECH

equations, which are more easily solvable. However, in order to find the actual distribution of the elastic field excited in the waveguide, inverse contour integration in the complex plane has to be performed, which is usually complicated. Due to the complexity of the Pochhammer-Chree equations, this procedure is only practical with simplified versions of the wave equation, which in general are not accurate enough for ultrasonic frequencies. See for example, Folk's solution for the transient response of a semi-infinite rod to a step pressure

The **Semi-Analytical Finite Element (SAFE)** method is a modification of Finite Element Methods (FEM) in which the elastic field is expanded as a superposition of harmonic waves in the azimuthal-axial (*θ*-*z*) plane, while discretized mechanical equations are used in the radial (*r*) direction of the waveguide. This reduction of the number of dimensions permits a much higher efficiency in the computation of the elastic fields (Hayashi et al., 2003). Waveguides surrounded by infinite media (like a pipe submerged in soil) can be handled by SAFE techniques with proper discretized elements (Jia et al., 2011), as well as waveguides with arbitrary profiles: for example, a railroad rail in (Damljanovic & Weaver, 2004). Although finite element methods are powerful and flexible, they have the shortcoming of great requirements on computer memory and processing time when large structures or high frequencies of operation are considered, and the difficulty encountered in the parameterization of transducer designs (for example, the determination of the transfer

**Spectral methods** are another numerical technique which approximate the differential elastic equations of the waveguide (Helmholtz equations) by differentiation operators, turning the problem of finding the wavenumber-frequency roots into a matrix eigenvalues determination (Doyle, 1997). This numerical method, which is computationally simple and reportedly does not suffer from the problems associated with large diameter waveguides at high frequencies, has been recently applied to model multi-layered cylindrical waveguides (Karpfinger et al.,

**Modal analysis** is an analytical method based on the expansion of the forcing terms acting in the waveguide into the set of its proper modes (Auld, 1973). In (Ditri & Rose, 1992), modal analysis is employed to model the loading of a waveguide by a transducer array. This treatment is extended to more general transducers and antisymmetric modes by (Li & Rose, 2001). Modal analysis is a mathematically exact technique that leads to a closed form integral equation for the elastic fields in the waveguide, and which incorporates in a natural way the issue of mode selectivity, offering insight on the physics of waveguide behaviour. For these

With this book chapter we contribute a numerical simulation treatment of the ultrasonic behaviour of cylindrical waveguides, based on the Pochhammer-Chree (PC) theory, and covering the aspects of assembly of the description matrix of the waveguide, tracing of the frequency-wavenumber curves, computation of mode shapes, use of modal analysis to determine the response of the waveguide to external excitations, and the dispersive

The work described here has resulted in a software package, named PCDISP, written in the Matlab environment (*Matlab*, 2004), and freely available<sup>1</sup> to be adapted to particular

<sup>1</sup> PCDISP webpage: http://www.car.upm-csic.es/lopsi/people/fernando.seco/pcdisp

applied to its end (Folk et al., 1958).

function of the transducer-waveguide coupling).

reasons, modal analysis will be the approach used in this work.

**1.2 Intention and scope of the research**

propagation of signals.

2008).

In this section we present a summarized theoretical background on wave propagation in cylindrical waveguides, treating such aspects as relevant for our purposes; standard references can be consulted for further information (Graff, 1991; Meeker & Meitzler, 1972; Rose, 1999).

A **waveguide** is a physical structure which supports the propagation of mechanical waves along its elongated direction *z*, and modifies the behaviour of such waves with respect to free propagation in the bulk material. There are two fundamental characteristics of waveguide propagation. The first is the discretization of waves into **propagating modes**, of which only a finite number are permitted for a given frequency, and whose properties are determined by the shape of the cross section and boundary conditions of the waveguide. The second is the existence of **dispersion**, which is the nonlinear relationship between wavenumber and frequency. As a consequence, signals with a significant bandwidth are distorted as they travel along the waveguide, because their spectral components propagate at different phase speeds.

The solutions of the wave equation in a cylindrical material are readily found by the use of potentials and the technique of separation of variables, arriving at the following general form for the displacement vector (*<sup>u</sup>*) and stress tensor (*<sup>σ</sup>*):

$$
\hat{u}(r,\theta,z) = \tilde{u}(r,\theta)e^{j kz} = u(r)e^{j n \theta}e^{j kz} \qquad \hat{\sigma}(r,\theta,z) = \tilde{\sigma}(r,\theta)e^{j kz} = \sigma(r)e^{j n \theta}e^{j kz}, \tag{1}
$$

where the cylindrical system is used (with coordinates (*r*, *θ*, *z*), and unit vectors (*er*,*eθ*,*ez*)), harmonic time variation *e*−*jω<sup>t</sup>* is assumed, and *ω* is the angular frequency, *k* the wavenumber,

*D<sup>u</sup>*

*D<sup>u</sup>*

*D<sup>u</sup>*

*D<sup>u</sup>*

*D<sup>u</sup>*

*D<sup>u</sup>*

*D<sup>u</sup>*

*D<sup>u</sup>*

*D<sup>u</sup>*

*D<sup>u</sup>*

*D<sup>u</sup>*

*D<sup>u</sup>*

*D<sup>u</sup>*

*D<sup>u</sup>*

*D<sup>u</sup>*

*D<sup>u</sup>*

*D<sup>u</sup>* <sup>35</sup> = 0 *D<sup>u</sup>* <sup>36</sup> = 0 Table 1. Coefficients of the displacement matrix *D<sup>u</sup>* of equation 2 (all *D<sup>u</sup>*

**3. Methodology for the simulation of the waveguide behaviour**

modes, and modelling of the signal propagation along the waveguide.

pcmat) to refer to programs of the package.

matrix itself is built in pcmatdet.

**3.1 Assembling the waveguide description matrix**

multiplied by 1/*r*).

<sup>11</sup> = *nWn*(*αr*) − *αrWn*+1(*αr*)

Modelling the Generation and Propagation of Ultrasonic Signals in Cylindrical Waveguides 5

<sup>12</sup> = *nZn*(*αr*) − *λ*1*αrZn*+1(*αr*)

<sup>13</sup> = *krWn*+1(*βr*)

<sup>14</sup> = *krZn*+1(*βr*)

<sup>15</sup> = *nWn*(*βr*)

<sup>16</sup> = *nZn*(*βr*)

<sup>21</sup> = *jnWn*(*αr*)

<sup>22</sup> = *jnZn*(*αr*)

<sup>23</sup> = −*jkrWn*+1(*βr*)

<sup>24</sup> = −*jkrZn*+1(*βr*)

<sup>31</sup> = *jkrWn*(*αr*)

<sup>32</sup> = *jkrZn*(*αr*)

<sup>34</sup> = *jβrZn*(*βr*)

<sup>33</sup> = *jλ*2*βrWn*(*βr*)

<sup>25</sup> = *jnWn*(*βr*) − *jβrWn*+1(*βr*)

<sup>26</sup> = *jnZn*(*βr*) − *jλ*2*βrZn*+1(*βr*)

In this section we describe the methodology used to study waveguide generation and propagation of ultrasonic signals, discuss the numerical issues encountered, and the approach used in the PCDISP package. The full process consists in four stages: assembly of the waveguide description matrix, tracing of the dispersion curves, modal analysis of the excited

While dealing with these topics, we will introduce the relevant PCDISP routines that should be used for the computations. Table 5 shows the components of the PCDISP software, arranged by their functionality. Throughout this chapter, we will use monospace fonts (like

The waveguide description matrix contains the necessary information to study the mechanical behaviour of the waveguide. It is built by matching the displacement vector and stress tensor between adjacent layers, and applying the external boundary conditions. In PCDISP, the physical data of the waveguide is provided in routine pcwaveguide, and the description

*ij* coefficients must be

and integer *n* is a separation constant called the circumferential order, which determines the symmetry of the solutions in the azimuthal direction.

The radial dependent part of the displacement vector and stress tensor is expressed in matrix form as (Gazis, 1959):

$$\boldsymbol{u}(\boldsymbol{r}) = \begin{bmatrix} \boldsymbol{u}\_{\boldsymbol{r}}(\boldsymbol{r}) \\ \boldsymbol{u}\_{\boldsymbol{\theta}}(\boldsymbol{r}) \\ \boldsymbol{u}\_{\boldsymbol{z}}(\boldsymbol{r}) \end{bmatrix} = \boldsymbol{D}^{\boldsymbol{u}}(\boldsymbol{r}) \cdot \begin{bmatrix} \mathcal{L}\_{+} \\ \mathcal{L}\_{-} \\ \text{SV}\_{+} \\ \text{SV}\_{-} \\ \text{SH}\_{+} \\ \text{SH}\_{-} \end{bmatrix}, \qquad \boldsymbol{\sigma}(\boldsymbol{r}) = \begin{bmatrix} \boldsymbol{\sigma}\_{\rm{rr}}(\boldsymbol{r}) \\ \boldsymbol{\sigma}\_{\theta\theta}(\boldsymbol{r}) \\ \boldsymbol{\sigma}\_{\rm{zz}}(\boldsymbol{r}) \\ \boldsymbol{\sigma}\_{\theta\boldsymbol{z}}(\boldsymbol{r}) \\ \boldsymbol{\sigma}\_{\rm{rz}}(\boldsymbol{r}) \\ \boldsymbol{\sigma}\_{\theta\boldsymbol{\theta}}(\boldsymbol{r}) \end{bmatrix} = \boldsymbol{D}^{\boldsymbol{\sigma}}(\boldsymbol{r}) \cdot \begin{bmatrix} \mathcal{L}\_{+} \\ \mathcal{L}\_{-} \\ \text{SV}\_{+} \\ \text{SV}\_{-} \\ \text{SH}\_{+} \\ \text{SH}\_{-} \end{bmatrix}. \tag{2}$$

The amplitude coefficient vector *A* = [L<sup>+</sup> L<sup>−</sup> SV<sup>+</sup> SV<sup>−</sup> SH<sup>+</sup> SH−] *<sup>T</sup>* consists of longitudinal (L), shear vertical (SV), and horizontal (SH) deformation components, and the + and − terms stand for perturbations propagating in the direction of increasing and decreasing radius, respectively.

The coefficients of matrices *D<sup>u</sup>* and *D<sup>σ</sup>* are of the general form *Dij*(*r*; *n*, *k*, *ω*, *c*), with *c* being the elastic compliance tensor of the solid. These matrices are given explicitly in tables 1 and 2, for the case of a mechanically isotropic material. They have been obtained from the equations of motion with a symbolic computation program (*Maple*, 2007), and match those found in (Gazis, 1959), except for some typographical errors in the paper, also propagated to later works as (Graff, 1991). Matrices *D<sup>u</sup>* and *D<sup>σ</sup>* are implemented in the PCDISP package in function pcmat.

In tables 1 and 2, *α*<sup>2</sup> = *ω*2/*c*<sup>2</sup> vol <sup>−</sup> *<sup>k</sup>*<sup>2</sup> and *<sup>β</sup>*<sup>2</sup> <sup>=</sup> *<sup>ω</sup>*2/*c*<sup>2</sup> rot <sup>−</sup> *<sup>k</sup>*2, where *<sup>c</sup>*vol and *<sup>c</sup>*rot are respectively the volumetric and rotational speeds of the solid (Rose, 1999). Functions *Zn*(*x*) and *Wn*(*x*) are two independent solutions of Bessel's differential equation, with, in general, complex arguments *x* = *αr*, *βr*. Of the possible choices for *Zn*(*x*) and *Wn*(*x*), the numerical stability of the frequency equation determinant is increased when Bessel's ordinary functions *Jn*(*x*) and *Yn*(*x*) are employed for real arguments, and the modified Bessel functions *In*(*x*) and *Kn*(*x*) for purely imaginary arguments. With this choice, the contributions to the elastic field from the standing waves propagating towards increasing (+) and decreasing (-) radius are separated (see section 3.1.4 for more on the stability of the frequency equation determinant). To cope with the fact that the recurrence relationships between Bessel's ordinary functions are different from those of the modified functions (Abramowitz & Stegun, 1964), signs *λ*<sup>1</sup> and *λ*<sup>2</sup> are introduced, following the scheme of table 3. For complex wavenumbers, PCDISP uses the ordinary Bessel functions *Jn*(*x*), *Yn*(*x*) with complex values, and *λ*<sup>1</sup> = *λ*<sup>2</sup> = 1.

The solutions of the wave equation are classified in family modes according to their **symmetry properties**, which depend on the circumferential index *n* of equation 1. Modes for which *n* = 0 have no dependence on the azimuthal coordinate *θ* and are labelled axisymmetric. They are further divided into **torsional** modes T(0, *m*) (which only involve the azimuthal component, and can be thought of as waves which twist the waveguide), and **longitudinal** modes L(0, *m*) (with both radial and axial components). Antisymmetric modes (*n* ≥ 1) are labelled **flexural** F(*n*, *m*), and involve all three components of the displacement vector. In general, multiple propagating modes exist for a given circumferential order and frequency, so a second index *m* is used to order them. Table 4 summarizes this information.

4 Will-be-set-by-IN-TECH

and integer *n* is a separation constant called the circumferential order, which determines the

The radial dependent part of the displacement vector and stress tensor is expressed in matrix

, *σ*(*r*) =

(L), shear vertical (SV), and horizontal (SH) deformation components, and the + and − terms stand for perturbations propagating in the direction of increasing and decreasing radius,

The coefficients of matrices *D<sup>u</sup>* and *D<sup>σ</sup>* are of the general form *Dij*(*r*; *n*, *k*, *ω*, *c*), with *c* being the elastic compliance tensor of the solid. These matrices are given explicitly in tables 1 and 2, for the case of a mechanically isotropic material. They have been obtained from the equations of motion with a symbolic computation program (*Maple*, 2007), and match those found in (Gazis, 1959), except for some typographical errors in the paper, also propagated to later works as (Graff, 1991). Matrices *D<sup>u</sup>* and *D<sup>σ</sup>* are implemented in the PCDISP package in function pcmat.

the volumetric and rotational speeds of the solid (Rose, 1999). Functions *Zn*(*x*) and *Wn*(*x*) are two independent solutions of Bessel's differential equation, with, in general, complex arguments *x* = *αr*, *βr*. Of the possible choices for *Zn*(*x*) and *Wn*(*x*), the numerical stability of the frequency equation determinant is increased when Bessel's ordinary functions *Jn*(*x*) and *Yn*(*x*) are employed for real arguments, and the modified Bessel functions *In*(*x*) and *Kn*(*x*) for purely imaginary arguments. With this choice, the contributions to the elastic field from the standing waves propagating towards increasing (+) and decreasing (-) radius are separated (see section 3.1.4 for more on the stability of the frequency equation determinant). To cope with the fact that the recurrence relationships between Bessel's ordinary functions are different from those of the modified functions (Abramowitz & Stegun, 1964), signs *λ*<sup>1</sup> and *λ*<sup>2</sup> are introduced, following the scheme of table 3. For complex wavenumbers, PCDISP uses the

The solutions of the wave equation are classified in family modes according to their **symmetry properties**, which depend on the circumferential index *n* of equation 1. Modes for which *n* = 0 have no dependence on the azimuthal coordinate *θ* and are labelled axisymmetric. They are further divided into **torsional** modes T(0, *m*) (which only involve the azimuthal component, and can be thought of as waves which twist the waveguide), and **longitudinal** modes L(0, *m*) (with both radial and axial components). Antisymmetric modes (*n* ≥ 1) are labelled **flexural** F(*n*, *m*), and involve all three components of the displacement vector. In general, multiple propagating modes exist for a given circumferential order and frequency, so a second index *m*

vol <sup>−</sup> *<sup>k</sup>*<sup>2</sup> and *<sup>β</sup>*<sup>2</sup> <sup>=</sup> *<sup>ω</sup>*2/*c*<sup>2</sup>

ordinary Bessel functions *Jn*(*x*), *Yn*(*x*) with complex values, and *λ*<sup>1</sup> = *λ*<sup>2</sup> = 1.

is used to order them. Table 4 summarizes this information.

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

*σrr*(*r*) *σθθ* (*r*) *σzz*(*r*) *σθz*(*r*) *σrz*(*r*) *σrθ*(*r*)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

<sup>=</sup> *<sup>D</sup>σ*(*r*) ·

rot <sup>−</sup> *<sup>k</sup>*2, where *<sup>c</sup>*vol and *<sup>c</sup>*rot are respectively

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

L+ L− SV+ SV<sup>−</sup> SH+ SH<sup>−</sup>

*<sup>T</sup>* consists of longitudinal

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

. (2)

symmetry of the solutions in the azimuthal direction.

<sup>⎦</sup> <sup>=</sup> *<sup>D</sup>u*(*r*) ·

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

The amplitude coefficient vector *A* = [L<sup>+</sup> L<sup>−</sup> SV<sup>+</sup> SV<sup>−</sup> SH<sup>+</sup> SH−]

L+ L− SV+ SV<sup>−</sup> SH+ SH<sup>−</sup> ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

form as (Gazis, 1959):

*u*(*r*) =

respectively.

⎡ ⎢ ⎣

In tables 1 and 2, *α*<sup>2</sup> = *ω*2/*c*<sup>2</sup>

*ur*(*r*) *u<sup>θ</sup>* (*r*) *uz*(*r*)

⎤ ⎥

$$\begin{aligned} D\_{11}^{\underline{u}} &= nW\_{n}(\beta r) - \alpha r W\_{n+1}(\alpha r) \\ D\_{22}^{\underline{u}} &= nZ\_{n}(\alpha r) - \lambda\_{1}\alpha r Z\_{n+1}(\alpha r) \\ D\_{13}^{\underline{u}} &= krW\_{n+1}(\beta r) \\ D\_{14}^{\underline{u}} &= krZ\_{n+1}(\beta r) \\ D\_{15}^{\underline{u}} &= nW\_{n}(\beta r) \\ D\_{16}^{\underline{u}} &= j\alpha W\_{n}(\alpha r) \\ D\_{21}^{\underline{u}} &= j\alpha Z\_{n}(\alpha r) \\ D\_{22}^{\underline{u}} &= -jkrW\_{n+1}(\beta r) \\ D\_{23}^{\underline{u}} &= -jkrZ\_{n+1}(\beta r) \\ D\_{24}^{\underline{u}} &= -jkrZ\_{n+1}(\beta r) \\ D\_{25}^{\underline{u}} &= j\alpha W\_{n}(\beta r) - j\beta rW\_{n+1}(\beta r) \\ D\_{26}^{\underline{u}} &= j\alpha Z\_{n}(\beta r) - j\lambda\_{2}\beta rZ\_{n+1}(\beta r) \\ D\_{31}^{\underline{u}} &= j\alpha Z\_{n}(\alpha r) \\ D\_{32}^{\underline{u}} &= j\beta Z\_{n}(\beta r) \\ D\_{34}^{\underline{u}} &= j\beta rZ\_{n}(\beta r) \\ D\_{34}^{\underline{u}} &= 0 \\ D\_{34}^{\underline{u}} &= 0 \\ D\_{36}^{\underline{u}} &= 0 \\ \end{aligned}$$

Table 1. Coefficients of the displacement matrix *D<sup>u</sup>* of equation 2 (all *D<sup>u</sup> ij* coefficients must be multiplied by 1/*r*).

#### **3. Methodology for the simulation of the waveguide behaviour**

In this section we describe the methodology used to study waveguide generation and propagation of ultrasonic signals, discuss the numerical issues encountered, and the approach used in the PCDISP package. The full process consists in four stages: assembly of the waveguide description matrix, tracing of the dispersion curves, modal analysis of the excited modes, and modelling of the signal propagation along the waveguide.

While dealing with these topics, we will introduce the relevant PCDISP routines that should be used for the computations. Table 5 shows the components of the PCDISP software, arranged by their functionality. Throughout this chapter, we will use monospace fonts (like pcmat) to refer to programs of the package.

#### **3.1 Assembling the waveguide description matrix**

The waveguide description matrix contains the necessary information to study the mechanical behaviour of the waveguide. It is built by matching the displacement vector and stress tensor between adjacent layers, and applying the external boundary conditions. In PCDISP, the physical data of the waveguide is provided in routine pcwaveguide, and the description matrix itself is built in pcmatdet.

Wavenumber Frequency range Coefficients Bessel functions

*α*2, *β*<sup>2</sup> < 0 *λ*1, *λ*<sup>2</sup> = −1

Modelling the Generation and Propagation of Ultrasonic Signals in Cylindrical Waveguides 7

*α*<sup>2</sup> < 0, *β*<sup>2</sup> > 0 *λ*<sup>1</sup> = −1, *λ*<sup>2</sup> = 1

*α*2, *β*<sup>2</sup> complex *λ*1, *λ*<sup>2</sup> = 1 Table 3. Choice of Bessel functions in the solution of the Pochhammer-Chree's equations.

Modes Coefficients Displacement Stress Torsional T(0, *m*) SH<sup>±</sup> *u<sup>θ</sup> σθz*, *σr<sup>θ</sup>*

Longitudinal L(0, *m*) L±, SV<sup>±</sup> *ur*, *uz σrr*, *σθθ*, *σzz σrz*

vector, and stress tensor, for the three family modes of a cylindrical waveguide.

*σ<sup>t</sup>* = *σ* · *er* = [*σrr*, *σrθ*, *σrz*]

which leads to the following matrix determinant equation:

*Dσ<sup>t</sup>* (*r*int)

*Dσ<sup>t</sup>* (*r*ext)

following homogeneous system of equations:

Flexural F(*n*, *m*) L±, SV±, SH<sup>±</sup> *ur*, *uz*, *u<sup>θ</sup> σrr*, *σθθ*, *σzz*, *σrz*, *σθz*, *σr<sup>θ</sup>*

Consider an isotropic tube of inner radius *r*int and outer radius *r*ext in vacuum or air. The boundary conditions specify that the traction part of the stress tensor is null in both surfaces

Equation 4 is called the frequency or characteristic equation of the waveguide, and its roots (*ω*, *k*) determine the proper modes supported by it. Once these roots are known, the vector of amplitude coefficients *A* is determined (up to a multiplicative constant) by solving the

Since matrix *D*(*ω*, *k*) is singular at the mode's frequency-wavenumber roots, a robust method for computing the amplitude, like the singular value decomposition (SVD), is recommended (Press et al., 1992). With *A* determined, the distribution of *u*(*r*) and *σ*(*r*) is computed by

The original Pochhammer-Chree formulation was developed for the simple case of a one-layer isotropic waveguide in vacuum; this, however, represents just a fraction of the waveguides of practical importance. Waveguides may be constituted by several layers, might be built with

= 0, where *Dσ<sup>t</sup>* = *D<sup>σ</sup>*

Table 4. Notation and non-null components of the amplitude coefficients, displacement

*Zn*(*αr*) = *In*(*αr*) *Wn*(*αr*) = *Kn*(*αr*) *Zn*(*βr*) = *In*(*βr*) *Wn*(*βr*) = *Kn*(*βr*)

*Zn*(*αr*) = *In*(*αr*) *Wn*(*αr*) = *Kn*(*αr*) *Zn*(*βr*) = *Jn*(*βr*) *Wn*(*βr*) = *Yn*(*βr*)

*Zn*(*βr*) = *Jn*(*βr*) *Wn*(*βr*) = *Yn*(*βr*)

*<sup>T</sup>* = 0 for *r* = *r*int,*r*ext, (3)

*D*(*ω*, *k*) · *A* = 0. (5)

*ij* with *i* = 1, 5, 6. (4)

*λ*1, *λ*<sup>2</sup> = 1 *Zn*(*αr*) = *Jn*(*αr*) *Wn*(*αr*) = *Yn*(*αr*)

Real *ω*/*k* < *c*vol, *c*rot

Real *c*rot < *ω*/*k* < *c*vol

Imaginary any

Complex any

**3.1.1 Single layer waveguides**

of the tube (Gazis, 1959), so:

det *<sup>D</sup>*(*ω*, *<sup>k</sup>*) = det

routine pcmatdet.

Real *ω*/*k* > *c*vol, *c*rot *α*2, *β*<sup>2</sup> > 0

*D<sup>σ</sup>* <sup>11</sup> = ((*k*<sup>2</sup> <sup>−</sup> *<sup>β</sup>*2)*r*<sup>2</sup> <sup>+</sup> <sup>2</sup>(*<sup>n</sup>* <sup>−</sup> <sup>1</sup>))*Wn*(*αr*) + <sup>2</sup>*αrWn*+1(*αr*) *D<sup>σ</sup>* <sup>12</sup> = ((*k*<sup>2</sup> <sup>−</sup> *<sup>β</sup>*2)*r*<sup>2</sup> <sup>+</sup> <sup>2</sup>(*<sup>n</sup>* <sup>−</sup> <sup>1</sup>))*Zn*(*αr*) + <sup>2</sup>*λ*1*αrZn*+1(*αr*) *D<sup>σ</sup>* <sup>13</sup> <sup>=</sup> <sup>2</sup>*λ*2*βkr*2*Wn*(*βr*) <sup>−</sup> <sup>2</sup>(*<sup>n</sup>* <sup>+</sup> <sup>1</sup>)*krWn*+1(*βr*) *D<sup>σ</sup>* <sup>14</sup> <sup>=</sup> <sup>2</sup>*βkr*2*Zn*(*βr*) <sup>−</sup> <sup>2</sup>(*<sup>n</sup>* <sup>+</sup> <sup>1</sup>)*krZn*+1(*βr*) *D<sup>σ</sup>* <sup>15</sup> = 2*n*(*n* − 1)*Wn*(*βr*) − 2*nβrWn*+1(*βr*) *D<sup>σ</sup>* <sup>16</sup> = 2*n*(*n* − 1)*Zn*(*βr*) − 2*nλ*2*βrZn*+1(*βr*) *D<sup>σ</sup>* <sup>21</sup> = ((2*α*<sup>2</sup> <sup>−</sup> *<sup>β</sup>*<sup>2</sup> <sup>+</sup> *<sup>k</sup>*2)*r*<sup>2</sup> <sup>−</sup> <sup>2</sup>*n*(*<sup>n</sup>* <sup>−</sup> <sup>1</sup>))*Wn*(*αr*) <sup>−</sup> <sup>2</sup>*αrWn*+1(*αr*) *D<sup>σ</sup>* <sup>22</sup> = ((2*α*<sup>2</sup> <sup>−</sup> *<sup>β</sup>*<sup>2</sup> <sup>+</sup> *<sup>k</sup>*2)*r*<sup>2</sup> <sup>−</sup> <sup>2</sup>*n*(*<sup>n</sup>* <sup>−</sup> <sup>1</sup>))*Zn*(*αr*) <sup>−</sup> <sup>2</sup>*λ*1*αrZn*+1(*αr*) *D<sup>σ</sup>* <sup>23</sup> = 2(*n* + 1)*krWn*+1(*βr*) *D<sup>σ</sup>* <sup>24</sup> = 2(*n* + 1)*krZn*+1(*βr*) *D<sup>σ</sup>* <sup>25</sup> = −2*n*(*n* − 1)*Wn*(*βr*) + 2*nβrWn*+1(*βr*) *D<sup>σ</sup>* <sup>26</sup> = −2*n*(*n* − 1)*Zn*(*βr*) + 2*nλ*2*βrZn*+1(*βr*) *D<sup>σ</sup>* <sup>31</sup> = (2*α*<sup>2</sup> <sup>−</sup> *<sup>β</sup>*<sup>2</sup> <sup>−</sup> *<sup>k</sup>*2)*r*2*Wn*(*αr*) *D<sup>σ</sup>* <sup>32</sup> = (2*α*<sup>2</sup> <sup>−</sup> *<sup>β</sup>*<sup>2</sup> <sup>−</sup> *<sup>k</sup>*2)*r*2*Zn*(*αr*) *D<sup>σ</sup>* <sup>33</sup> <sup>=</sup> <sup>−</sup>2*λ*2*βkr*2*Wn*(*βr*) *D<sup>σ</sup>* <sup>34</sup> <sup>=</sup> <sup>−</sup>2*βkr*2*Zn*(*βr*) *D<sup>σ</sup>* <sup>35</sup> = 0 *D<sup>σ</sup>* <sup>36</sup> = 0 *D<sup>σ</sup>* <sup>41</sup> = −2*nkrWn*(*αr*) *D<sup>σ</sup>* <sup>42</sup> = −2*nkrZn*(*αr*) *D<sup>σ</sup>* <sup>43</sup> <sup>=</sup> *<sup>k</sup>*2*r*<sup>2</sup>*Wn*+1(*βr*) <sup>−</sup> *<sup>λ</sup>*2*nβrWn*(*βr*) *D<sup>σ</sup>* <sup>44</sup> <sup>=</sup> *<sup>k</sup>*2*r*<sup>2</sup>*Zn*+1(*βr*) <sup>−</sup> *<sup>n</sup>βrZn*(*βr*) *D<sup>σ</sup>* <sup>45</sup> <sup>=</sup> <sup>−</sup>*nkrWn*(*βr*) + *<sup>β</sup>kr*<sup>2</sup>*Wn*+1(*βr*) *D<sup>σ</sup>* <sup>46</sup> <sup>=</sup> <sup>−</sup>*nkrZn*(*βr*) + *<sup>λ</sup>*2*βkr*<sup>2</sup>*Zn*+1(*βr*) *D<sup>σ</sup>* <sup>51</sup> <sup>=</sup> <sup>2</sup>*jnkrWn*(*αr*) <sup>−</sup> <sup>2</sup>*jkαr*<sup>2</sup>*Wn*+1(*αr*) *D<sup>σ</sup>* <sup>52</sup> <sup>=</sup> <sup>2</sup>*jnkrZn*(*αr*) <sup>−</sup> <sup>2</sup>*jλ*1*kαr*<sup>2</sup>*Zn*+1(*αr*) *D<sup>σ</sup>* <sup>53</sup> <sup>=</sup> *<sup>j</sup>λ*2*nβrWn*(*βr*) <sup>−</sup> *<sup>j</sup>*(*β*<sup>2</sup> <sup>−</sup> *<sup>k</sup>*2)*r*<sup>2</sup>*Wn*+1(*βr*) *D<sup>σ</sup>* <sup>54</sup> <sup>=</sup> *jnβrZn*(*βr*) <sup>−</sup> *<sup>j</sup>*(*β*<sup>2</sup> <sup>−</sup> *<sup>k</sup>*2)*r*<sup>2</sup>*Zn*+1(*βr*) *D<sup>σ</sup>* <sup>55</sup> = *jnkrWn*(*βr*) *D<sup>σ</sup>* <sup>56</sup> = *jnkrZn*(*βr*) *D<sup>σ</sup>* <sup>61</sup> = 2*jn*(*n* − 1)*Wn*(*αr*) − 2*jnαrWn*+1(*αr*) *D<sup>σ</sup>* <sup>62</sup> = 2*jn*(*n* − 1)*Zn*(*αr*) − 2*jnλ*1*αrZn*+1(*αr*) *D<sup>σ</sup>* <sup>63</sup> <sup>=</sup> <sup>−</sup>*jλ*2*βkr*2*Wn*(*βr*) + <sup>2</sup>*jkr*(*<sup>n</sup>* <sup>+</sup> <sup>1</sup>)*Wn*+1(*βr*) *D<sup>σ</sup>* <sup>64</sup> <sup>=</sup> <sup>−</sup>*jβkr*2*Zn*(*βr*) + <sup>2</sup>*jkr*(*<sup>n</sup>* <sup>+</sup> <sup>1</sup>)*Zn*+1(*βr*) *D<sup>σ</sup>* <sup>65</sup> <sup>=</sup> *<sup>j</sup>*(2*n*(*<sup>n</sup>* <sup>−</sup> <sup>1</sup>) <sup>−</sup> *<sup>β</sup>*2*r*2)*Wn*(*βr*) + <sup>2</sup>*jβrWn*+1(*βr*) *D<sup>σ</sup>* <sup>66</sup> <sup>=</sup> *<sup>j</sup>*(2*n*(*<sup>n</sup>* <sup>−</sup> <sup>1</sup>) <sup>−</sup> *<sup>β</sup>*2*r*2)*Zn*(*βr*) + <sup>2</sup>*jλ*2*βrZn*+1(*βr*)

Table 2. Coefficients of the stress matrix *D<sup>σ</sup>* of equation 2 (all *D<sup>σ</sup> ij* coefficients must be multiplied by *G*/*r*2, where *G* is the shear modulus of the material).


Table 3. Choice of Bessel functions in the solution of the Pochhammer-Chree's equations.


Table 4. Notation and non-null components of the amplitude coefficients, displacement vector, and stress tensor, for the three family modes of a cylindrical waveguide.

#### **3.1.1 Single layer waveguides**

6 Will-be-set-by-IN-TECH

<sup>21</sup> = ((2*α*<sup>2</sup> <sup>−</sup> *<sup>β</sup>*<sup>2</sup> <sup>+</sup> *<sup>k</sup>*2)*r*<sup>2</sup> <sup>−</sup> <sup>2</sup>*n*(*<sup>n</sup>* <sup>−</sup> <sup>1</sup>))*Wn*(*αr*) <sup>−</sup> <sup>2</sup>*αrWn*+1(*αr*)

<sup>22</sup> = ((2*α*<sup>2</sup> <sup>−</sup> *<sup>β</sup>*<sup>2</sup> <sup>+</sup> *<sup>k</sup>*2)*r*<sup>2</sup> <sup>−</sup> <sup>2</sup>*n*(*<sup>n</sup>* <sup>−</sup> <sup>1</sup>))*Zn*(*αr*) <sup>−</sup> <sup>2</sup>*λ*1*αrZn*+1(*αr*)

<sup>11</sup> = ((*k*<sup>2</sup> <sup>−</sup> *<sup>β</sup>*2)*r*<sup>2</sup> <sup>+</sup> <sup>2</sup>(*<sup>n</sup>* <sup>−</sup> <sup>1</sup>))*Wn*(*αr*) + <sup>2</sup>*αrWn*+1(*αr*)

<sup>13</sup> <sup>=</sup> <sup>2</sup>*λ*2*βkr*2*Wn*(*βr*) <sup>−</sup> <sup>2</sup>(*<sup>n</sup>* <sup>+</sup> <sup>1</sup>)*krWn*+1(*βr*)

<sup>14</sup> <sup>=</sup> <sup>2</sup>*βkr*2*Zn*(*βr*) <sup>−</sup> <sup>2</sup>(*<sup>n</sup>* <sup>+</sup> <sup>1</sup>)*krZn*+1(*βr*)

<sup>15</sup> = 2*n*(*n* − 1)*Wn*(*βr*) − 2*nβrWn*+1(*βr*)

<sup>23</sup> = 2(*n* + 1)*krWn*+1(*βr*)

<sup>24</sup> = 2(*n* + 1)*krZn*+1(*βr*)

<sup>31</sup> = (2*α*<sup>2</sup> <sup>−</sup> *<sup>β</sup>*<sup>2</sup> <sup>−</sup> *<sup>k</sup>*2)*r*2*Wn*(*αr*)

<sup>32</sup> = (2*α*<sup>2</sup> <sup>−</sup> *<sup>β</sup>*<sup>2</sup> <sup>−</sup> *<sup>k</sup>*2)*r*2*Zn*(*αr*)

<sup>43</sup> <sup>=</sup> *<sup>k</sup>*2*r*<sup>2</sup>*Wn*+1(*βr*) <sup>−</sup> *<sup>λ</sup>*2*nβrWn*(*βr*)

<sup>45</sup> <sup>=</sup> <sup>−</sup>*nkrWn*(*βr*) + *<sup>β</sup>kr*<sup>2</sup>*Wn*+1(*βr*)

<sup>46</sup> <sup>=</sup> <sup>−</sup>*nkrZn*(*βr*) + *<sup>λ</sup>*2*βkr*<sup>2</sup>*Zn*+1(*βr*)

<sup>51</sup> <sup>=</sup> <sup>2</sup>*jnkrWn*(*αr*) <sup>−</sup> <sup>2</sup>*jkαr*<sup>2</sup>*Wn*+1(*αr*)

<sup>52</sup> <sup>=</sup> <sup>2</sup>*jnkrZn*(*αr*) <sup>−</sup> <sup>2</sup>*jλ*1*kαr*<sup>2</sup>*Zn*+1(*αr*)

<sup>53</sup> <sup>=</sup> *<sup>j</sup>λ*2*nβrWn*(*βr*) <sup>−</sup> *<sup>j</sup>*(*β*<sup>2</sup> <sup>−</sup> *<sup>k</sup>*2)*r*<sup>2</sup>*Wn*+1(*βr*)

<sup>54</sup> <sup>=</sup> *jnβrZn*(*βr*) <sup>−</sup> *<sup>j</sup>*(*β*<sup>2</sup> <sup>−</sup> *<sup>k</sup>*2)*r*<sup>2</sup>*Zn*+1(*βr*)

<sup>61</sup> = 2*jn*(*n* − 1)*Wn*(*αr*) − 2*jnαrWn*+1(*αr*)

<sup>62</sup> = 2*jn*(*n* − 1)*Zn*(*αr*) − 2*jnλ*1*αrZn*+1(*αr*)

<sup>64</sup> <sup>=</sup> <sup>−</sup>*jβkr*2*Zn*(*βr*) + <sup>2</sup>*jkr*(*<sup>n</sup>* <sup>+</sup> <sup>1</sup>)*Zn*+1(*βr*)

Table 2. Coefficients of the stress matrix *D<sup>σ</sup>* of equation 2 (all *D<sup>σ</sup>*

multiplied by *G*/*r*2, where *G* is the shear modulus of the material).

<sup>63</sup> <sup>=</sup> <sup>−</sup>*jλ*2*βkr*2*Wn*(*βr*) + <sup>2</sup>*jkr*(*<sup>n</sup>* <sup>+</sup> <sup>1</sup>)*Wn*+1(*βr*)

<sup>65</sup> <sup>=</sup> *<sup>j</sup>*(2*n*(*<sup>n</sup>* <sup>−</sup> <sup>1</sup>) <sup>−</sup> *<sup>β</sup>*2*r*2)*Wn*(*βr*) + <sup>2</sup>*jβrWn*+1(*βr*)

<sup>66</sup> <sup>=</sup> *<sup>j</sup>*(2*n*(*<sup>n</sup>* <sup>−</sup> <sup>1</sup>) <sup>−</sup> *<sup>β</sup>*2*r*2)*Zn*(*βr*) + <sup>2</sup>*jλ*2*βrZn*+1(*βr*)

*ij* coefficients must be

<sup>44</sup> <sup>=</sup> *<sup>k</sup>*2*r*<sup>2</sup>*Zn*+1(*βr*) <sup>−</sup> *<sup>n</sup>βrZn*(*βr*)

<sup>33</sup> <sup>=</sup> <sup>−</sup>2*λ*2*βkr*2*Wn*(*βr*)

<sup>34</sup> <sup>=</sup> <sup>−</sup>2*βkr*2*Zn*(*βr*)

<sup>41</sup> = −2*nkrWn*(*αr*)

<sup>42</sup> = −2*nkrZn*(*αr*)

<sup>55</sup> = *jnkrWn*(*βr*)

<sup>56</sup> = *jnkrZn*(*βr*)

<sup>16</sup> = 2*n*(*n* − 1)*Zn*(*βr*) − 2*nλ*2*βrZn*+1(*βr*)

<sup>25</sup> = −2*n*(*n* − 1)*Wn*(*βr*) + 2*nβrWn*+1(*βr*)

<sup>26</sup> = −2*n*(*n* − 1)*Zn*(*βr*) + 2*nλ*2*βrZn*+1(*βr*)

<sup>12</sup> = ((*k*<sup>2</sup> <sup>−</sup> *<sup>β</sup>*2)*r*<sup>2</sup> <sup>+</sup> <sup>2</sup>(*<sup>n</sup>* <sup>−</sup> <sup>1</sup>))*Zn*(*αr*) + <sup>2</sup>*λ*1*αrZn*+1(*αr*)

*D<sup>σ</sup>*

*D<sup>σ</sup>*

*D<sup>σ</sup>*

*D<sup>σ</sup>*

*D<sup>σ</sup>*

*D<sup>σ</sup>*

*D<sup>σ</sup>*

*D<sup>σ</sup>*

*D<sup>σ</sup>*

*D<sup>σ</sup>*

*D<sup>σ</sup>*

*D<sup>σ</sup>*

*D<sup>σ</sup>*

*D<sup>σ</sup>*

*D<sup>σ</sup>*

*D<sup>σ</sup>*

*D<sup>σ</sup>* <sup>35</sup> = 0 *D<sup>σ</sup>* <sup>36</sup> = 0 *D<sup>σ</sup>*

*D<sup>σ</sup>*

*D<sup>σ</sup>*

*D<sup>σ</sup>*

*D<sup>σ</sup>*

*D<sup>σ</sup>*

*D<sup>σ</sup>*

*D<sup>σ</sup>*

*D<sup>σ</sup>*

*D<sup>σ</sup>*

*D<sup>σ</sup>*

*D<sup>σ</sup>*

*D<sup>σ</sup>*

*D<sup>σ</sup>*

*D<sup>σ</sup>*

*D<sup>σ</sup>*

*D<sup>σ</sup>*

*D<sup>σ</sup>*

Consider an isotropic tube of inner radius *r*int and outer radius *r*ext in vacuum or air. The boundary conditions specify that the traction part of the stress tensor is null in both surfaces of the tube (Gazis, 1959), so:

$$
\sigma\_t = \sigma \cdot e\_r = [\sigma\_{\tau\prime}, \sigma\_{r\theta}, \sigma\_{r\mathbb{Z}}]^T = 0 \qquad \text{for } r = r\_{\text{int}\prime} r\_{\text{ext}\prime} \tag{3}
$$

which leads to the following matrix determinant equation:

$$\det D(\omega, k) = \det \begin{bmatrix} D^{\sigma t}(r\_{\text{int}}) \\ D^{\sigma t}(r\_{\text{ext}}) \end{bmatrix} = 0, \qquad \text{where} \qquad D^{\sigma t} = D\_{ij}^{\sigma} \qquad \text{with} \qquad i = 1, 5, 6. \tag{4}$$

Equation 4 is called the frequency or characteristic equation of the waveguide, and its roots (*ω*, *k*) determine the proper modes supported by it. Once these roots are known, the vector of amplitude coefficients *A* is determined (up to a multiplicative constant) by solving the following homogeneous system of equations:

$$D(\omega, k) \cdot A = 0.\tag{5}$$

Since matrix *D*(*ω*, *k*) is singular at the mode's frequency-wavenumber roots, a robust method for computing the amplitude, like the singular value decomposition (SVD), is recommended (Press et al., 1992). With *A* determined, the distribution of *u*(*r*) and *σ*(*r*) is computed by routine pcmatdet.

The original Pochhammer-Chree formulation was developed for the simple case of a one-layer isotropic waveguide in vacuum; this, however, represents just a fraction of the waveguides of practical importance. Waveguides may be constituted by several layers, might be built with

l=1

Modelling the Generation and Propagation of Ultrasonic Signals in Cylindrical Waveguides 9

by a tube of inner radius *r*<sup>1</sup> and outer radius *r*2, in turn surrounded by an infinite medium.

<sup>3</sup>+(*r*2)

⎤ ⎥ ⎥ ⎥ ⎦ ·

⎡ ⎢ ⎣

*A*1<sup>−</sup> *A*2<sup>±</sup> *A*3<sup>+</sup> ⎤ ⎥

<sup>⎦</sup> <sup>=</sup> 0. (6)

<sup>3</sup>+(*r*2)

l=2

l=3

**r**

Fig. 1. Example of a three-layer cylindrical waveguide.

The corresponding system of equations is:

⎡ ⎢ ⎢ ⎢ ⎣

**3.1.3 Anisotropic and fluid-loaded waveguides**

waveguides is found in (Grigorenko, 2005).

*D<sup>u</sup>*

*Dσ<sup>t</sup>*

<sup>1</sup>−(*r*1) <sup>−</sup>*D<sup>u</sup>*

<sup>1</sup>−(*r*1) <sup>−</sup>*Dσ<sup>t</sup>*

*D<sup>u</sup>*

*Dσ<sup>t</sup>*

2

<sup>2</sup>±(*r*1)

<sup>2</sup>±(*r*1)

amplitude vector, in the same way as the single layer waveguide of section 3.1.1.

<sup>2</sup>±(*r*2) <sup>−</sup>*D<sup>u</sup>*

<sup>2</sup>±(*r*2) <sup>−</sup>*Dσ<sup>t</sup>*

Note that the radiation conditions are used to simplify the system matrix, leading to discard the + terms in region 1, since no waves can emanate from *r* = 0; similarly, the - terms in region 3 are not considered, as no energy comes from *r* = ∞; however, both outgoing and incoming terms are allowed in the middle region. In each case the unneeded columns of matrix *Dσ<sup>t</sup>* are removed. Next, this equation is solved by the SVD method to find the mode's

Materials with mechanical **anisotropy** are routinely employed to build waveguides, and their behaviour is modelled by taking into account the adequate compliance tensor for the material (Pollard, 1977). The important case of hexagonal symmetry is found in waveguides with transverse isotropy (with respect to the propagation axis *z*), like beryllium, or in fiber reinforced composite cylinders. Although in this case there are five elastic constants (up from two for an isotropic material), the mechanical fields are still separable with the treatment of Gazis for isotropic waveguides, with different coefficients for the *D<sup>u</sup>* and *D<sup>σ</sup>* matrices, obviously (Mirsky, 1965). In the case of purely orthotropic symmetry (three orthogonal planes of symmetry), the solution of the wave equation is *not* separable; still, closed solutions can be achieved in the form of a Frobenius power series (obtained in (Mirsky, 1964) for the axisymmetric case, and extended later in (Markus & Mead, 1995) to the asymmetric problem). A recent state of the art in the theory of mechanical waves in anisotropic cylindrical

Materials with **elastic losses** (damping) are treated by the Kelvin-Voigt viscoelastic model (Lowe, 1995), which replaces the elements of the compliance tensor *c* of the material by

**r**

1


Table 5. Components of the PCDISP software.

anisotropic materials, be embedded in the ground, or transport (or be surrounded by, or both) fluids. We will consider next the extensions of the PC theory which permit to model these situations.

#### **3.1.2 Multilayered waveguides**

Some waveguides are formed by several layers: for example, a metallic rod with external insulation, or a tube embedded in rock. The Pochhammer-Chree approach was first used to analyze a two-layer waveguide in (McNiven et al., 1963; Whittier & Jones, 1967), and later extended to laminated waveguides (formed by an arbitrary number of layers) in (Nelson et al., 1971). The modern technique to simulate multilayered waveguides is called the **Multiple Layer Matrix (MLM)** (Lowe, 1995), and is adapted from the transfer matrix and global matrix techniques developed by W.T. Thomson and L. Knopoff in the period 1950-1964 to study wave propagation in stratified media in seismology.

Following the MLM approach, we assemble a system of linear equations for the complete waveguide, which includes the equations of the elastic waves for each individual layer (obtained with pcmat), the equations which match the displacement and traction stresses at the interface between adjacent layers, and the boundary conditions. Consider the example of a multilayered waveguide shown in figure 1, where a solid cylinder of radius *r*<sup>1</sup> is enclosed

Fig. 1. Example of a three-layer cylindrical waveguide.

8 Will-be-set-by-IN-TECH

**Core routines**

**Plotters and solvers of the frequency equation** pcplotmatdet1D One-dimensional plot of the freq. eq. determinant vs. *k*, *f* , or *c*ph pcplotmatdet2D Two-dimensional plot of the freq. eq. determinant in *k*-*f* space pcsolvebisection Bisection method to find roots of the freq. eq. vs *k*, *f* , or *c*ph

**Field computing and wave propagation** pcwaveform Finds the displacement vector *u*(*r*) and the stress tensor *σ*(*r*)

anisotropic materials, be embedded in the ground, or transport (or be surrounded by, or both) fluids. We will consider next the extensions of the PC theory which permit to model these

Some waveguides are formed by several layers: for example, a metallic rod with external insulation, or a tube embedded in rock. The Pochhammer-Chree approach was first used to analyze a two-layer waveguide in (McNiven et al., 1963; Whittier & Jones, 1967), and later extended to laminated waveguides (formed by an arbitrary number of layers) in (Nelson et al., 1971). The modern technique to simulate multilayered waveguides is called the **Multiple Layer Matrix (MLM)** (Lowe, 1995), and is adapted from the transfer matrix and global matrix techniques developed by W.T. Thomson and L. Knopoff in the period 1950-1964 to study wave

Following the MLM approach, we assemble a system of linear equations for the complete waveguide, which includes the equations of the elastic waves for each individual layer (obtained with pcmat), the equations which match the displacement and traction stresses at the interface between adjacent layers, and the boundary conditions. Consider the example of a multilayered waveguide shown in figure 1, where a solid cylinder of radius *r*<sup>1</sup> is enclosed

pcmat Computes matrices *Du*(*r*) and *Dσ*(*r*) (tables 1 and 2) pcmatdet Assembles and solves the waveguide description matrix

pcwaveguide Physical description of the waveguide

pcviewmatdet View the entries of the matrix determinant

pcdisp Plots the phase and group speeds vs. frequency pckfcurves Traces *k*-*f* curves for real, imaginary, and complex *k* pcsolverandom Random solutions of the freq. eq. for complex *k*

pcorthogonalcheck Checks the orthogonality of modes in the waveguide pcsignalpropagation Simulates the propagation of a signal along the waveguide **Modal analysis**

pcextsurfacestress External traction stresses *σe* acting on the waveguide pcextvolumforce External volumetric forces *fe* acting on the waveguide pcplotexcitation Plots the excitation volumetric force and surface stress pcmodalanalysis Finds the amplitudes of modes excited in the waveguide

Table 5. Components of the PCDISP software.

propagation in stratified media in seismology.

**3.1.2 Multilayered waveguides**

situations.

by a tube of inner radius *r*<sup>1</sup> and outer radius *r*2, in turn surrounded by an infinite medium. The corresponding system of equations is:

$$
\begin{bmatrix} D\_{1-}^{\mu}(r\_1) \ -D\_{2\pm}^{\mu}(r\_1) \\ D\_{1-}^{\sigma t}(r\_1) \ -D\_{2\pm}^{\sigma t}(r\_1) \\ D\_{2\pm}^{\mu}(r\_2) \ -D\_{3+}^{\mu}(r\_2) \\ D\_{2\pm}^{\sigma t}(r\_2) \ -D\_{3+}^{\sigma t}(r\_2) \end{bmatrix} \cdot \begin{bmatrix} A\_{1-} \\ A\_{2\pm} \\ A\_{3+} \end{bmatrix} = 0. \tag{6}
$$

Note that the radiation conditions are used to simplify the system matrix, leading to discard the + terms in region 1, since no waves can emanate from *r* = 0; similarly, the - terms in region 3 are not considered, as no energy comes from *r* = ∞; however, both outgoing and incoming terms are allowed in the middle region. In each case the unneeded columns of matrix *Dσ<sup>t</sup>* are removed. Next, this equation is solved by the SVD method to find the mode's amplitude vector, in the same way as the single layer waveguide of section 3.1.1.

#### **3.1.3 Anisotropic and fluid-loaded waveguides**

Materials with mechanical **anisotropy** are routinely employed to build waveguides, and their behaviour is modelled by taking into account the adequate compliance tensor for the material (Pollard, 1977). The important case of hexagonal symmetry is found in waveguides with transverse isotropy (with respect to the propagation axis *z*), like beryllium, or in fiber reinforced composite cylinders. Although in this case there are five elastic constants (up from two for an isotropic material), the mechanical fields are still separable with the treatment of Gazis for isotropic waveguides, with different coefficients for the *D<sup>u</sup>* and *D<sup>σ</sup>* matrices, obviously (Mirsky, 1965). In the case of purely orthotropic symmetry (three orthogonal planes of symmetry), the solution of the wave equation is *not* separable; still, closed solutions can be achieved in the form of a Frobenius power series (obtained in (Mirsky, 1964) for the axisymmetric case, and extended later in (Markus & Mead, 1995) to the asymmetric problem). A recent state of the art in the theory of mechanical waves in anisotropic cylindrical waveguides is found in (Grigorenko, 2005).

Materials with **elastic losses** (damping) are treated by the Kelvin-Voigt viscoelastic model (Lowe, 1995), which replaces the elements of the compliance tensor *c* of the material by

r r r

l

L-1

r=r <sup>L</sup> ext

1

u(r ) (r )

l

Fig. 2. Partition of the waveguide's cross section for solving the large *f d* problem.

can ultimately be ignored, and the solutions are numerically stable.

frequency, and its solution will eventually become unstable.

 *u*(*r*) *σ*(*r*)

where the vector of amplitude coefficients *Al* = [L*<sup>l</sup>*

and scale this matrix by columns for each layer as:

purely imaginary wavenumber).

are given by:

to be different for each layer. We use the shorthand notation:

behaviour is different depending on the frequency range:

l

Modelling the Generation and Propagation of Ultrasonic Signals in Cylindrical Waveguides 11

r =r <sup>0</sup> int

use pcviewmatdet to see the individual entries of the waveguide description matrix). The

• For *c*ph < *c*rot, both *α* and *β* are imaginary, and there exist two solutions of the frequency equation, corresponding to two Rayleigh modes propagating close to the outer and inner surfaces of the waveguide, and with amplitudes decaying exponentially from them. As the frequency increases, these two modes become decoupled, the opposite waveguide surface

• For *c*rot < *c*ph < *c*vol, *α* is imaginary and *β* real. While the *SV*<sup>±</sup> and *SH*<sup>±</sup> terms remain bounded, the terms *L*<sup>+</sup> and *L*<sup>−</sup> decrease and increase, respectively, with the radius, as described above. The condition number of the waveguide matrix grows with the

• For *c*ph > *c*vol, both *α* and *β* are real, and the solution is stable (this is also the case with

Since the detection of the large *f d* problem in 1965, several solutions have been proposed to increase the numerical stability in plane waveguides (Lowe, 1995). We have not been able to locate similar studies for cylindrical waveguides, so, for the PCDISP software, we have developed an algorithm adapted from the transfer matrix and global matrix approaches and

The cross section of the pipe is divided into *L* layers of equal thickness, where the *l*-th layer is given by *rl*−<sup>1</sup> < *<sup>r</sup>* < *rl*, and *<sup>r</sup>*<sup>0</sup> = *<sup>r</sup>*int and *rL* = *<sup>r</sup>*ext are the inner and outer radii of the pipe (see figure 2). In the *l*-th layer, the displacement vector and traction part of the stress tensor

<sup>+</sup> <sup>L</sup>*<sup>l</sup>* <sup>−</sup> SV*<sup>l</sup>*

> ,

<sup>+</sup> SV*<sup>l</sup>*

<sup>−</sup> SH*<sup>l</sup>*

*<sup>l</sup>* = *Dl* · *Gl*, (9)

· *Al*, (8)

<sup>+</sup> SH*<sup>l</sup>* −]

*<sup>T</sup>* is permitted

discussed here. We consider this method as a new contribution to the literature.

*l* = *Du*(*r*) *Dσ<sup>t</sup>* (*r*)

*Dl* =

*Ds*

where *Gl* is a diagonal matrix whose entries are taken as 1/ max col|*Dl*|.

*Du*(*rl*) *Dσ<sup>t</sup>* (*rl*)

operators which contain time derivatives, therefore modifying Hooke's law:

$$
\sigma = \mathfrak{c} \cdot \mathfrak{e} \qquad \Rightarrow \qquad \sigma = \mathfrak{c}' \cdot \mathfrak{e} + \mathfrak{c}'' \cdot \frac{d\mathfrak{e}}{dt} = (\mathfrak{c}' - j\omega \mathfrak{c}'')\mathfrak{e},
$$

where *σ* and *�* are the stress and strain tensors, and component *c*�� of the compliance tensor models the viscoelastic losses. The solutions of the frequency equation for a waveguide with viscoelastic losses, are, by default, complex wavenumbers.

A case of particular practical importance is that of waveguides including a fluid layer (for example, a pipe carrying a fluid, submerged in a fluid, or both). The theoretical treatment depends on the viscosity of the fluid.

The influence of **inviscid fluids** (which do not support shear stresses) on cylindrical wave propagation is treated theoretically and experimentally in (Sinha et al., 1992). If the waveguide is submerged in a liquid, propagating modes with complex wavenumber appear, which radiate (leak) energy into the surrounding fluid. That does not happen for waveguides containing fluids and surrounded by vacuum, although the propagating modes themselves are modified from the unloaded situation.

A treatment for Newtonian **viscous fluids** which is compatible with the PC based formulation of wave propagation in cylinders is introduced in (Nagy & Nayfeh, 1996). In a form similar to the Kelvin-Voigt model, the viscous liquid is modelled as an isotropic solid whose compliance tensor includes complex elements:

$$c\_{11} = \lambda + \frac{4}{3}c\_{44} \qquad c\_{12} = \lambda - \frac{2}{3}c\_{44} \qquad c\_{44} = -j\omega\eta,\tag{7}$$

where *λ* is the compressibility of the fluid, and *η* its viscosity. This simple model has shown a good accuracy in predicting propagation in waveguides with viscous fluids (Aristegui et al., 2001). Indeed, the changes in the propagation of ultrasonic waves in a pipe caused by the presence of a fluid in its interior can be used to measure the longitudinal wave speed and the viscosity of the fluid (Ma et al., 2007).

## **3.1.4 The case of large frequency** × **thickness product**

Solutions of the frequency equation become numerically unstable when the product frequency times thickness of the waveguide is high. Physically, this phenomenon arises because the standing waves established in the radial direction of the waveguide are formed by a combination of terms which increase exponentially with the radius *r* and others that decrease exponentially with it. Since it is the sum of both terms which must match the boundary conditions at *r* = *r*int and *r* = *r*ext, the dynamic range of positive and negative exponentials in the frequency equation will eventually overflow the numerical capacity of the machine if the radius or frequency increase. With the 64 bits double precision arithmetic of the IEEE 754 standard, we have found that a direct implementation of the frequency equation determinant fails when the product *f* · (*r*ext − *r*int) is higher than approximately 30 MHz·mm. This threshold is easily reached in the NDE of piping with ultrasonic waves, where frequencies of a few megahertz are common (Rose, 2000).

From table 3, we can see that the problem arises when a mode's phase velocity falls below the volumetric (*c*vol) or rotational (*c*rot) speeds of the solid, making parameters *α* or *β*, respectively, become imaginary. This changes the radial dependence of the mode amplitude from the bounded Bessel functions *J* and *Y*, to the exponentially varying *I* and *K* (you can

Fig. 2. Partition of the waveguide's cross section for solving the large *f d* problem.

use pcviewmatdet to see the individual entries of the waveguide description matrix). The behaviour is different depending on the frequency range:


Since the detection of the large *f d* problem in 1965, several solutions have been proposed to increase the numerical stability in plane waveguides (Lowe, 1995). We have not been able to locate similar studies for cylindrical waveguides, so, for the PCDISP software, we have developed an algorithm adapted from the transfer matrix and global matrix approaches and discussed here. We consider this method as a new contribution to the literature.

The cross section of the pipe is divided into *L* layers of equal thickness, where the *l*-th layer is given by *rl*−<sup>1</sup> < *<sup>r</sup>* < *rl*, and *<sup>r</sup>*<sup>0</sup> = *<sup>r</sup>*int and *rL* = *<sup>r</sup>*ext are the inner and outer radii of the pipe (see figure 2). In the *l*-th layer, the displacement vector and traction part of the stress tensor are given by:

$$
\begin{bmatrix} u(r) \\ \sigma(r) \end{bmatrix}\_l = \begin{bmatrix} D^u(r) \\ D^{\sigma t}(r) \end{bmatrix} \cdot A\_{l\prime} \tag{8}
$$

where the vector of amplitude coefficients *Al* = [L*<sup>l</sup>* <sup>+</sup> <sup>L</sup>*<sup>l</sup>* <sup>−</sup> SV*<sup>l</sup>* <sup>+</sup> SV*<sup>l</sup>* <sup>−</sup> SH*<sup>l</sup>* <sup>+</sup> SH*<sup>l</sup>* −] *<sup>T</sup>* is permitted to be different for each layer.

We use the shorthand notation:

10 Will-be-set-by-IN-TECH

where *σ* and *�* are the stress and strain tensors, and component *c*�� of the compliance tensor models the viscoelastic losses. The solutions of the frequency equation for a waveguide with

A case of particular practical importance is that of waveguides including a fluid layer (for example, a pipe carrying a fluid, submerged in a fluid, or both). The theoretical treatment

The influence of **inviscid fluids** (which do not support shear stresses) on cylindrical wave propagation is treated theoretically and experimentally in (Sinha et al., 1992). If the waveguide is submerged in a liquid, propagating modes with complex wavenumber appear, which radiate (leak) energy into the surrounding fluid. That does not happen for waveguides containing fluids and surrounded by vacuum, although the propagating modes themselves

A treatment for Newtonian **viscous fluids** which is compatible with the PC based formulation of wave propagation in cylinders is introduced in (Nagy & Nayfeh, 1996). In a form similar to the Kelvin-Voigt model, the viscous liquid is modelled as an isotropic solid whose compliance

where *λ* is the compressibility of the fluid, and *η* its viscosity. This simple model has shown a good accuracy in predicting propagation in waveguides with viscous fluids (Aristegui et al., 2001). Indeed, the changes in the propagation of ultrasonic waves in a pipe caused by the presence of a fluid in its interior can be used to measure the longitudinal wave speed and the

Solutions of the frequency equation become numerically unstable when the product frequency times thickness of the waveguide is high. Physically, this phenomenon arises because the standing waves established in the radial direction of the waveguide are formed by a combination of terms which increase exponentially with the radius *r* and others that decrease exponentially with it. Since it is the sum of both terms which must match the boundary conditions at *r* = *r*int and *r* = *r*ext, the dynamic range of positive and negative exponentials in the frequency equation will eventually overflow the numerical capacity of the machine if the radius or frequency increase. With the 64 bits double precision arithmetic of the IEEE 754 standard, we have found that a direct implementation of the frequency equation determinant fails when the product *f* · (*r*ext − *r*int) is higher than approximately 30 MHz·mm. This threshold is easily reached in the NDE of piping with ultrasonic waves, where frequencies

From table 3, we can see that the problem arises when a mode's phase velocity falls below the volumetric (*c*vol) or rotational (*c*rot) speeds of the solid, making parameters *α* or *β*, respectively, become imaginary. This changes the radial dependence of the mode amplitude from the bounded Bessel functions *J* and *Y*, to the exponentially varying *I* and *K* (you can

3

*<sup>c</sup>*<sup>44</sup> *<sup>c</sup>*<sup>12</sup> <sup>=</sup> *<sup>λ</sup>* <sup>−</sup> <sup>2</sup>

*d�*

*dt* = (*c*� <sup>−</sup> *<sup>j</sup>ωc*��)*�*,

*c*<sup>44</sup> *c*<sup>44</sup> = −*jωη*, (7)

operators which contain time derivatives, therefore modifying Hooke's law:

*σ* = *c* · *�* ⇒ *σ* = *c*� · *�* + *c*�� ·

viscoelastic losses, are, by default, complex wavenumbers.

depends on the viscosity of the fluid.

are modified from the unloaded situation.

*c*<sup>11</sup> = *λ* +

**3.1.4 The case of large frequency** × **thickness product**

of a few megahertz are common (Rose, 2000).

4 3

tensor includes complex elements:

viscosity of the fluid (Ma et al., 2007).

$$D\_l = \begin{bmatrix} D^u(r\_l) \\ D^{\sigma t}(r\_l) \end{bmatrix} \prime$$

and scale this matrix by columns for each layer as:

$$D\_l^s = D\_l \cdot G\_{l\nu} \tag{9}$$

where *Gl* is a diagonal matrix whose entries are taken as 1/ max col|*Dl*|.

Material Aluminium Standard DN 25, SCH 80 Inner radius (*r*int) 12.15 mm Outer radius (*r*ext) 16.70 mm Poisson's ratio (*ν*) 0.35 Bar velocity (*c*0) 5000 m/s Density (*ρ*) 2700 kg/m3 Shear modulus (*G*) 25.5 GPa

Modelling the Generation and Propagation of Ultrasonic Signals in Cylindrical Waveguides 13

For a given frequency *f* , the roots of the frequency equation are in general complex wavenumbers *k* = *kr* + *jki*. Wavenumbers with *ki* = 0 correspond to the **propagating** or **proper** modes of the waveguide; those with *ki* � 0 represent **evanescent** modes which attenuate along the axial distance *z*. Due to the symmetry of the coefficients of the frequency equation, purely real or imaginary solutions appear in pairs (±*kr* and ±*jki*), while complex solutions do so in quartets (±*kr* ± *jki*). The principle of the conservation of energy dictates which solutions are valid in a waveguide problem. For example, waves which propagate towards the *z*+ axis of an infinite waveguide, will necessarily have Im{*k*} ≥ 0. Although signal propagation does not occur for imaginary or complex wavenumbers, those solutions are needed to fulfill the condition of completeness which will be stated in section 3.3. Furthermore, stationary waves formed by combination of the two wavenumbers *kr* + *jki* and −*kr* + *jki* (provided that *ki* > 0), can exist locally, storing but not dissipating energy (Meeker & Meitzler, 1972). In ultrasonic applications, these waves are important in the region of generation of ultrasonic waves, at waveguide discontinuities (like defects), and at the

Solutions of the frequency equation can be traced like continuous curves in the *k*-*f* space, where *k* = *kr* + *jki*. As an example, we show the dispersion curves of the longitudinal modes L(0,m) of a sample waveguide with the data shown in table 6 (this waveguide will be further used in the examples of section 4). The complete spectrum, up to 3 MHz, has been computed with the pckfcurves routine of PCDISP, and is shown in figure 3. As it can be seen, the wavenumber curve of a given mode changes from a real value (shown in blue colour) to purely imaginary (coloured red, and projected into the negative wavenumber axis), or complex (plotted in green, with the real part on the positive *k* axis, and the imaginary part

After finding all possible roots of the frequency equation, branches L(0,m) have to be ordered in such a way that each mode is assigned a unique, continuous wavenumber between the maximum frequency *f*max and zero frequency. Part (b) of figure 3 shows a reduced frequency range of the spectrum in part (a), and illustrates the convention for labelling modes. PCDISP uses the following rules about the behaviour of dispersion curves (Meeker & Meitzler, 1972): • Only the first torsional T(0,1), longitudinal L(0,1), and first flexural F(1,1), modes propagate

• Higher order modes switch from real to imaginary wavenumber at the **cutoff frequencies** when the wavenumber becomes null (*k* = 0, *f* = *f*cutoff), and the phase speed infinite. • Miminum points (*dω*/*dk* = 0, *d*2*ω*/*dk*<sup>2</sup> > 0) of either real or imaginary wavenumber branches are also cutoff frequencies (with finite phase speed), from which complex

Table 6. Physical data for the aluminium tube used for demonstration of the PCDISP

**3.2.1 Nature and ordering of the solutions of the frequency equation**

software in this book chapter.

waveguide ends.

on the negative *k* axis), in a complicated fashion.

down to zero frequency with real wavenumber.

wavenumber branches start.

The elastic field [*u σ*] *<sup>T</sup>* is propagated from the inner to the outer part of a layer by the following equation:

$$
\begin{bmatrix} u(r\_l) \\ \sigma(r\_l) \end{bmatrix} - P\_l \cdot \begin{bmatrix} u(r\_{l-1}) \\ \sigma(r\_{l-1}) \end{bmatrix} = 0,\tag{10}
$$

where *Pl* is the propagator matrix of layer *l*, and is given by:

$$P\_l = D\_l^s \cdot \left(\mathbf{G}\_l^{-1} \cdot \mathbf{G}\_{l-1}\right) \cdot \left(D\_{l-1}^s\right)^{-1}.\tag{11}$$

Applying equation 10 to all the layers of the waveguide, we can assemble a global matrix:

$$\begin{bmatrix} P\_1 \ -I\_6 \\ & P\_2 \ -I\_6 \\ & & \ddots \\ & & & P\_{L-1} \ -I\_6 \\ & & & P\_{L-1} \ -I\_6 \end{bmatrix} \begin{bmatrix} u\_0 \\ \sigma\_0 \\ u\_1 \\ \sigma\_1 \\ \vdots \\ u\_{L-1} \\ u\_{L-1} \\ \sigma\_{L-1} \\ u\_L \\ \sigma\_L \end{bmatrix} = 0,\tag{12}$$

where *I*<sup>6</sup> is the identity matrix of size 6 × 6.

The boundary conditions to match the wave fields to the surrounding medium (or to other layers in multilayered waveguides) are introduced at radii *r*int and *r*ext. In the case of stress free boundaries, the terms *σ*<sup>0</sup> and *σ<sup>L</sup>* are zero, and their corresponding columns are simply removed from the global matrix.

Once equation 12 has been solved, and we have determined the displacements and traction stresses at each layer boundary [*ul*, *σl*], the vector of amplitude coefficients for each layer is found by solving:

$$
\begin{bmatrix} D\_{l-1}^s \\\\ D\_l^s \cdot (G\_l^{-1} \cdot G\_{l-1}) \end{bmatrix} \cdot A\_l^s = \begin{bmatrix} u\_{l-1} \\ \sigma\_{l-1} \\ u\_l \\ \sigma\_l \end{bmatrix} \tag{13}
$$

for *<sup>l</sup>* <sup>=</sup> 1, . . . *<sup>L</sup>*. The unscaled amplitude vector is simply: *Al* <sup>=</sup> *Gl* · *<sup>A</sup><sup>s</sup> l* .

The algorithm described increases the *f d* stability limit by a factor proportional to the number of layers *L*, at the expense of larger matrices and longer computational time. In PCIDSP, the algorithm described is written into routine pcmatdet, and is activated automatically if the waveguide defined in pcwaveguide consists of *NL* > 1 layers of the same material.

#### **3.2 Computation of the dispersion curves**

The roots of the waveguide's frequency equation represent the mechanical modes which satisfy the boundary conditions. The procedure for computing such solutions is described in this section.


Table 6. Physical data for the aluminium tube used for demonstration of the PCDISP software in this book chapter.

#### **3.2.1 Nature and ordering of the solutions of the frequency equation**

12 Will-be-set-by-IN-TECH

�

*<sup>u</sup>*(*rl*−1) *<sup>σ</sup>*(*rl*−1)

*<sup>l</sup>* · *Gl*−1) · (*D<sup>s</sup>*

� *u*(*rl*) *σ*(*rl*)

*Pl* = *D<sup>s</sup>*

where *Pl* is the propagator matrix of layer *l*, and is given by:

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

where *I*<sup>6</sup> is the identity matrix of size 6 × 6.

**3.2 Computation of the dispersion curves**

removed from the global matrix.

found by solving:

in this section.

*P*<sup>1</sup> −*I*<sup>6</sup>

⎡ ⎢ ⎣

*Ds <sup>l</sup>* · (*G*−<sup>1</sup>

for *<sup>l</sup>* <sup>=</sup> 1, . . . *<sup>L</sup>*. The unscaled amplitude vector is simply: *Al* <sup>=</sup> *Gl* · *<sup>A</sup><sup>s</sup>*

*Ds l*−1

*<sup>l</sup>* · *Gl*−1)

waveguide defined in pcwaveguide consists of *NL* > 1 layers of the same material.

*P*<sup>2</sup> −*I*<sup>6</sup>

...

�

*<sup>l</sup>* · (*G*−<sup>1</sup>

− *Pl* ·

Applying equation 10 to all the layers of the waveguide, we can assemble a global matrix:

*PL*−<sup>1</sup> −*I*<sup>6</sup>

The boundary conditions to match the wave fields to the surrounding medium (or to other layers in multilayered waveguides) are introduced at radii *r*int and *r*ext. In the case of stress free boundaries, the terms *σ*<sup>0</sup> and *σ<sup>L</sup>* are zero, and their corresponding columns are simply

Once equation 12 has been solved, and we have determined the displacements and traction stresses at each layer boundary [*ul*, *σl*], the vector of amplitude coefficients for each layer is

> ⎤ ⎥ <sup>⎦</sup> · *<sup>A</sup><sup>s</sup> <sup>l</sup>* =

The algorithm described increases the *f d* stability limit by a factor proportional to the number of layers *L*, at the expense of larger matrices and longer computational time. In PCIDSP, the algorithm described is written into routine pcmatdet, and is activated automatically if the

The roots of the waveguide's frequency equation represent the mechanical modes which satisfy the boundary conditions. The procedure for computing such solutions is described

⎡ ⎢ ⎢ ⎢ ⎣ *ul*−<sup>1</sup> *σl*−<sup>1</sup> *ul σl*

⎤ ⎥ ⎥ ⎥

> *l* .

*PL* −*I*<sup>6</sup>

*<sup>T</sup>* is propagated from the inner to the outer part of a layer by the following

�

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

*u*0 *σ*0 *u*1 *σ*1 . . . *uL*−<sup>1</sup> *σL*−<sup>1</sup> *uL σL*

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

= 0, (10)

*<sup>l</sup>*−1)<sup>−</sup>1. (11)

= 0, (12)

<sup>⎦</sup> , (13)

The elastic field [*u σ*]

equation:

For a given frequency *f* , the roots of the frequency equation are in general complex wavenumbers *k* = *kr* + *jki*. Wavenumbers with *ki* = 0 correspond to the **propagating** or **proper** modes of the waveguide; those with *ki* � 0 represent **evanescent** modes which attenuate along the axial distance *z*. Due to the symmetry of the coefficients of the frequency equation, purely real or imaginary solutions appear in pairs (±*kr* and ±*jki*), while complex solutions do so in quartets (±*kr* ± *jki*). The principle of the conservation of energy dictates which solutions are valid in a waveguide problem. For example, waves which propagate towards the *z*+ axis of an infinite waveguide, will necessarily have Im{*k*} ≥ 0. Although signal propagation does not occur for imaginary or complex wavenumbers, those solutions are needed to fulfill the condition of completeness which will be stated in section 3.3. Furthermore, stationary waves formed by combination of the two wavenumbers *kr* + *jki* and −*kr* + *jki* (provided that *ki* > 0), can exist locally, storing but not dissipating energy (Meeker & Meitzler, 1972). In ultrasonic applications, these waves are important in the region of generation of ultrasonic waves, at waveguide discontinuities (like defects), and at the waveguide ends.

Solutions of the frequency equation can be traced like continuous curves in the *k*-*f* space, where *k* = *kr* + *jki*. As an example, we show the dispersion curves of the longitudinal modes L(0,m) of a sample waveguide with the data shown in table 6 (this waveguide will be further used in the examples of section 4). The complete spectrum, up to 3 MHz, has been computed with the pckfcurves routine of PCDISP, and is shown in figure 3. As it can be seen, the wavenumber curve of a given mode changes from a real value (shown in blue colour) to purely imaginary (coloured red, and projected into the negative wavenumber axis), or complex (plotted in green, with the real part on the positive *k* axis, and the imaginary part on the negative *k* axis), in a complicated fashion.

After finding all possible roots of the frequency equation, branches L(0,m) have to be ordered in such a way that each mode is assigned a unique, continuous wavenumber between the maximum frequency *f*max and zero frequency. Part (b) of figure 3 shows a reduced frequency range of the spectrum in part (a), and illustrates the convention for labelling modes. PCDISP uses the following rules about the behaviour of dispersion curves (Meeker & Meitzler, 1972):


dr(i-2)

Parts (c) and (d) of figure 3 show the reason why a robust algorithm for curve tracing of the dispersion branches is needed, in order to avoid the apparent crossings between branches, such as modes L(0,7) and L(0,8), with real wavenumbers, and modes L(0,9) and L(0,10), with imaginary wavenumbers. The curve tracing method used in pckfcurves is shown in figure 4. The dispersion curve being traced is extrapolated from the three last computed points {*i* − 2, *i* − 1, *i*} to define an angular interval of width *dθ* in a circle of radius *dr*(*i*) centered in the last correctly determined point (*i*) (*dr* is the step size of the algorithm, in normalized coordinates of the *k*-*f* space). If a sign change is found in this interval, the algorithm proceeds with a bisection method to accurately estimate the position of point *i* + 1 (this is the normal situation). Otherwise, the dispersion curve might have undergone a sudden change of curvature, or another mode might have come very close to the one being traced, provoking multiple sign changes. In this case, the step *dr*(*i*) is decreased, or, if needed, points *i* − 1, *i* − 2, etc, are recomputed with a smaller step *dr*. Summarizing, the tracing algorithm of PCDISP keeps track of the curvature of the branch and the proximity of neighbouring

It must be pointed out that the frequency equation has spurious solutions at the lines with slope equal to the volumetric and rotational speeds of the solid (*ω*/*k* = *c*vol and *ω*/*k* = *c*rot), which have to be removed by the root finding algorithm. In the case of multilayered

The dispersion curves are completed by tracing the complex wavenumber branches *k* = *kr* + *jki*, starting from the extrema points of the real/imaginary wavenumber branches. Tracing the complex wavenumber branches needs more computational effort, since it is required to solve simultaneously for the real and imaginary parts of the wavenumber; we have obtained satisfactory results with Muller's method (Press et al., 1992) using also an adaptive step.

Finally, the rules enumerated at the end of section 3.2.1 are used to convert the initially obtained branches into continuous dispersion curves *k* = *k*(*f*) for each mode. An example of this procedure is shown in part (b) of figure 3. First, note that all longitudinal modes except L(0,1) exhibit cutoff. Mode L(0,2) is cut off at (*k* = 0, *f* = 59.7 kHz), where it switches to a branch with imaginary wavenumber which goes on until (*k* = 67*j* m−1, 58.5 kHz),

Current branch

i-2

branches, adjusting the interval step between consecutive points accordingly.

waveguides, the same phenomenon happens for the speeds of each layer.

i-1

Neighbouring branch

Fig. 4. Illustrates the root tracing algorithm used by PCDISP.

dr(i-1)

i

Modelling the Generation and Propagation of Ultrasonic Signals in Cylindrical Waveguides 15

dr(i)

d

i+1

Search interval

Fig. 3. Wavenumber-frequency plot for the L(0,m) modes of the aluminum pipe of table 6, for frequencies up to 3 MHz (part a). Part b shows the low-frequency spectrum, and the labelling scheme for the modes. Parts c and d show near crosses of the branches of two different modes; they have to be largely magnified to be visible.


#### **3.2.2 Algorithm for curve tracing in PCDISP**

In order to generate continuous curves *k* = *k*(*f*), from *f* = 0 to *f* = *f*max, for each mode, pckfcurves first traces the real wavenumber branches, which start at solutions found with pcsolvebisection at the *f* = *f*max and *k* = *k*max axes, and finish at the *k* = 0 axis. If one is only interested in propagating modes, this finishes the procedure, and the pcdisp routine can be used to plot the phase and group speeds of propagating modes in the specified frequency range. Otherwise, the next step consists in tracing branches with purely imaginary wavenumber, which start at points (0, *f*cutoff) on the vertical axis of zero wavenumber, and finish when they reach the *f* = *f*max, *k* = *jk*max axes, or at another cutoff frequency in the *k* = 0 axis.

Fig. 4. Illustrates the root tracing algorithm used by PCDISP.

14 Will-be-set-by-IN-TECH

1524.9

1524.95

1525

L(0,10)

1525.05

f (kHz)

Fig. 3. Wavenumber-frequency plot for the L(0,m) modes of the aluminum pipe of table 6, for

• Complex branches terminate either at zero frequency or at the maximum points (*dω*/*dk* =

• The branches are ordered such that *ki* is positive for modes propagating in the *z*+ direction, and that the sign of the group speed does not change along the curve (although it becomes

In order to generate continuous curves *k* = *k*(*f*), from *f* = 0 to *f* = *f*max, for each mode, pckfcurves first traces the real wavenumber branches, which start at solutions found with pcsolvebisection at the *f* = *f*max and *k* = *k*max axes, and finish at the *k* = 0 axis. If one is only interested in propagating modes, this finishes the procedure, and the pcdisp routine can be used to plot the phase and group speeds of propagating modes in the specified frequency range. Otherwise, the next step consists in tracing branches with purely imaginary wavenumber, which start at points (0, *f*cutoff) on the vertical axis of zero wavenumber, and finish when they reach the *f* = *f*max, *k* = *jk*max axes, or at another cutoff frequency in the

frequencies up to 3 MHz (part a). Part b shows the low-frequency spectrum, and the labelling scheme for the modes. Parts c and d show near crosses of the branches of two

null at the cutoff frequencies, and at the purely imaginary branches).

1525.1

1525.15

L(0,7)

f (kHz)

L(0,4) & L(0,5)

> L(0,3) L(0,2) & L(0,3)

−1000 −500 <sup>0</sup> <sup>500</sup> <sup>1000</sup> <sup>1500</sup> <sup>0</sup>

(b) First longitudinal modes L(0,m) up to 850 kHz

L(0,5)

L(0,4) L(0,5)

−Im{k} (1/m) Re{k}

(d) Zoom of near cross of imaginary branches

−1416.5 −1416.48−1416.46−1416.44−1416.42 −1416.4 −1416.38

−Im{k} (1/m)

L(0,10)

L(0,9)

L(0,1)

L(0,10)

L(0,9) L(0,9) &

L(0,2)

L(0,3)

−6000 −4000 −2000 <sup>0</sup> <sup>2000</sup> <sup>4000</sup> <sup>6000</sup> <sup>0</sup>

1719.5 <sup>1720</sup> 1720.5 <sup>1721</sup> 1868.2

0, *d*2*ω*/*dk*<sup>2</sup> < 0) of imaginary branches.

**3.2.2 Algorithm for curve tracing in PCDISP**

Re{k} (1/m)

different modes; they have to be largely magnified to be visible.

(b)

(d)

(a) Spectrum of longitudinal modes L(0,m) up to 3 MHz

(c)

−Im{k} (1/m) Re{k}

(c) Zoom of near cross of real branches

L(0,8)

500

1868.3 1868.4 1868.5 1868.6 1868.7 1868.8 1868.9 1869 1869.1 1869.2

*k* = 0 axis.

f (kHz)

1000

1500

f (kHz)

2000

2500

3000

Parts (c) and (d) of figure 3 show the reason why a robust algorithm for curve tracing of the dispersion branches is needed, in order to avoid the apparent crossings between branches, such as modes L(0,7) and L(0,8), with real wavenumbers, and modes L(0,9) and L(0,10), with imaginary wavenumbers. The curve tracing method used in pckfcurves is shown in figure 4. The dispersion curve being traced is extrapolated from the three last computed points {*i* − 2, *i* − 1, *i*} to define an angular interval of width *dθ* in a circle of radius *dr*(*i*) centered in the last correctly determined point (*i*) (*dr* is the step size of the algorithm, in normalized coordinates of the *k*-*f* space). If a sign change is found in this interval, the algorithm proceeds with a bisection method to accurately estimate the position of point *i* + 1 (this is the normal situation). Otherwise, the dispersion curve might have undergone a sudden change of curvature, or another mode might have come very close to the one being traced, provoking multiple sign changes. In this case, the step *dr*(*i*) is decreased, or, if needed, points *i* − 1, *i* − 2, etc, are recomputed with a smaller step *dr*. Summarizing, the tracing algorithm of PCDISP keeps track of the curvature of the branch and the proximity of neighbouring branches, adjusting the interval step between consecutive points accordingly.

It must be pointed out that the frequency equation has spurious solutions at the lines with slope equal to the volumetric and rotational speeds of the solid (*ω*/*k* = *c*vol and *ω*/*k* = *c*rot), which have to be removed by the root finding algorithm. In the case of multilayered waveguides, the same phenomenon happens for the speeds of each layer.

The dispersion curves are completed by tracing the complex wavenumber branches *k* = *kr* + *jki*, starting from the extrema points of the real/imaginary wavenumber branches. Tracing the complex wavenumber branches needs more computational effort, since it is required to solve simultaneously for the real and imaginary parts of the wavenumber; we have obtained satisfactory results with Muller's method (Press et al., 1992) using also an adaptive step.

Finally, the rules enumerated at the end of section 3.2.1 are used to convert the initially obtained branches into continuous dispersion curves *k* = *k*(*f*) for each mode. An example of this procedure is shown in part (b) of figure 3. First, note that all longitudinal modes except L(0,1) exhibit cutoff. Mode L(0,2) is cut off at (*k* = 0, *f* = 59.7 kHz), where it switches to a branch with imaginary wavenumber which goes on until (*k* = 67*j* m−1, 58.5 kHz),

so equation 14 becomes:

 *∂D*

as shown in figure 5.

*<sup>P</sup>*<sup>12</sup> <sup>=</sup> <sup>−</sup> *<sup>j</sup><sup>ω</sup>* 4 *D* (*<sup>u</sup>*<sup>1</sup> · *<sup>σ</sup>*<sup>∗</sup>

∇ · (*<sup>u</sup>*<sup>1</sup> · *<sup>σ</sup>*<sup>∗</sup>

<sup>2</sup> <sup>−</sup> *<sup>u</sup>*<sup>∗</sup>

Discarding the common factor *ej*(*k*1−*k*<sup>∗</sup>

(*<sup>u</sup>*<sup>1</sup> · *<sup>σ</sup>*<sup>∗</sup>

<sup>2</sup> · *<sup>σ</sup>*1)*ej*(*k*1−*k*<sup>∗</sup>

<sup>2</sup> <sup>−</sup> *<sup>u</sup>*<sup>∗</sup>

by the mode. For nonpropagating modes with *ki* �= 0, *P*<sup>11</sup> is zero.

and applying the divergence theorem, we find that:

<sup>2</sup> <sup>−</sup> *<sup>u</sup>*<sup>∗</sup>

With this notation, equation 16 reduces to:

which implies that *P*<sup>12</sup> = 0 unless *k*<sup>1</sup> = *k*<sup>∗</sup>

*<sup>u</sup>*<sup>1</sup>(*r*, *<sup>θ</sup>*, *<sup>z</sup>*) = ∑

*p*

waveguide is under an arbitrary excitation composed of:

stress is defined in pcextsurfacestress.

where *f*1,2(*r*, *θ*, *z*) represent the forcing terms.

generalized to (Auld, 1973):

this vector force field is defined in pcextvolumforce.

∇ · (*<sup>u</sup>*<sup>1</sup> · *<sup>σ</sup>*<sup>∗</sup>

<sup>2</sup> <sup>−</sup> *<sup>u</sup>*<sup>∗</sup>

with routine pcorthogonalcheck.

<sup>2</sup>)*<sup>z</sup>* <sup>+</sup> *<sup>j</sup>*(*k*<sup>1</sup> <sup>−</sup> *<sup>k</sup>*<sup>∗</sup>

Modelling the Generation and Propagation of Ultrasonic Signals in Cylindrical Waveguides 17

<sup>2</sup> · *<sup>σ</sup>*1) · *en dl* <sup>+</sup> *<sup>j</sup>*(*k*<sup>1</sup> <sup>−</sup> *<sup>k</sup>*<sup>∗</sup>

<sup>2</sup> )*ej*(*k*1−*k*<sup>∗</sup>

2) *D* (*<sup>u</sup>*<sup>1</sup> · *<sup>σ</sup>*<sup>∗</sup>

In equation 16, *∂D* = *∂D*int ∪ *∂D*ext represents the inner and outer surfaces of the waveguide, and the normal unit vector *en* is taken on each surface pointing out of the waveguide's interior,

Because for proper modes the surface traction stress is null (*<sup>σ</sup><sup>t</sup>* <sup>=</sup> *<sup>σ</sup>* · *en* <sup>=</sup> 0 in *<sup>∂</sup>D*), the first

In the right part of equation 17 we have assumed that the circumferential order *n* of modes (1) and (2) is the same; otherwise, *P*<sup>12</sup> is zero automatically due to the integration over the *θ* coordinate. The factor −*jω*/4 is introduced so that the quantity *P*<sup>11</sup> equals to the integral of the acoustic Poynting vector in the cross section of the waveguide, i.e., the power transported

The second condition for modal analysis is **completeness**, which is based on the premise that an arbitrary perturbation in the waveguide can be expanded in the set of normal modes:

*ap*(*z*)*<sup>u</sup><sup>p</sup>*(*r*, *<sup>θ</sup>*) *<sup>σ</sup>*1(*r*, *<sup>θ</sup>*, *<sup>z</sup>*) = ∑

where the modes are indexed by *p*, and no distinction has been made between propagating and evanescent modes. The problem lies in computing the set of coefficients *ap*(*z*), when the

1. A vector force field *fe*(*r*, *θ*, *z*) acting on the bulk material of the tube (region *D*). In PCDISP,

2. A traction stress *<sup>σ</sup>e*(*r*, *<sup>θ</sup>*, *<sup>z</sup>*) applied to the tube surfaces (region *<sup>∂</sup>D*). In PCDISP, this surface

In the case of existence of external fields, the orthogonality relationship, equation 14, must be

<sup>2</sup> · *<sup>σ</sup>*1) = <sup>−</sup>*<sup>u</sup>*<sup>1</sup> · *<sup>f</sup>*

∗ <sup>2</sup> <sup>+</sup> *<sup>u</sup>*<sup>∗</sup> <sup>2</sup> · *f* 

2

 *r*ext *r*int

integral of equation 16 is zero. Then a suitable scalar product of modes (1) and (2) is:

<sup>2</sup> · *<sup>σ</sup>*1) · *ez dS* <sup>=</sup> <sup>−</sup> *<sup>j</sup>πω*

(*k*<sup>1</sup> − *k*<sup>∗</sup>

<sup>2</sup> )*z*(*<sup>u</sup>*<sup>1</sup> · *<sup>σ</sup>*<sup>∗</sup>

<sup>2</sup>)*z*, integrating over the cross section *D* of the waveguide,

<sup>2</sup> <sup>−</sup> *<sup>u</sup>*<sup>∗</sup>

(*u*<sup>1</sup> · *σ*<sup>∗</sup>

<sup>2</sup> − *u*<sup>∗</sup>

<sup>2</sup>)*P*<sup>12</sup> = 0, (18)

<sup>2</sup>. In PCDISP, mode orthogonality can be verified

*p*

<sup>2</sup> <sup>−</sup> *<sup>u</sup>*<sup>∗</sup>

<sup>2</sup> · *<sup>σ</sup>*1) · *ez* <sup>=</sup> 0. (15)

<sup>2</sup> · *<sup>σ</sup>*1) · *ez dS* <sup>=</sup> 0. (16)

<sup>2</sup> · *σ*1) · *ez* · *r dr*. (17)

*ap*(*z*)*<sup>σ</sup>p*(*r*, *<sup>θ</sup>*). (19)

1, (20)

Fig. 5. Cross section of the waveguide and definition of the regions of integration for modal analysis.

the minimum point of the imaginary branch. At that point, it is joined by the branch corresponding to mode L(0,3), and both go down to zero frequency with negative complex conjugate wavenumbers *kr* + *jki* and −*kr* + *jki*.

Similarly, mode L(0,4) is cut off at the minimum point of the real wavenumber branch (*k* = 345 m−1, *f* = 621.5 kHz). Mode L(0,5) is cut off at (*k* = 0, *f* = 698.6 kHz), changes to imaginary wavenumber until it reaches point (*k* = 0, *f* = 669.4 kHz), and then again to real, *but negative*, wavenumber down to (*k* = 345 m−1, *f* = 621.5 kHz), where it joins mode L(0,4). Mode L(0,5) has a negative wavenumber (and consequently, negative phase speed *ω*/*k*), in order to maintain a positive group velocity *dω*/*dk*, since this is a propagating mode in the *z*+ direction of the waveguide, according to the last rule of section 3.2.1. Below 621.5 kHz, the dispersion curves of modes L(0,4) and L(0,5) descend to zero frequency with negative complex conjugate wavenumbers, in the same way as modes L(0,2) and L(0,3).

#### **3.3 Modal analysis**

Modal analysis is a mathematical technique which permits to compute the dynamic response of a waveguide subject to arbitrary external forces, as an expansion of the excited wave over the set of normal modes of the waveguide, as defined in section 2 (Auld, 1973). Modal analysis is based upon two properties of normal modes: orthogonality, the existence of a scalar product which is null for any two different modes; and completeness, the capacity of the set of normal modes to span arbitrary waveforms in the waveguide.

For two different modes of the waveguide (1) and (2) of the form given in equation 1, the **orthogonality relationship** (Auld, 1973) establishes that:

$$\widehat{\nabla} \cdot \left( \widehat{\mathfrak{u}}\_1 \cdot \widehat{\sigma}\_2^\* - \widehat{\mathfrak{u}}\_2^\* \cdot \widehat{\sigma}\_1 \right) = 0,\tag{14}$$

which is applicable to linear elastic materials and also to piezoelectric or magnetostrictive linear materials, assuming no elastic or dielectric losses. Later on this result will be generalized to include external forces and stresses.

For separable vector fields in the *z* coordinate, the tridimensional divergence operator can be written as:

$$
\hat{\nabla} \cdot \{\} = \tilde{\nabla} \cdot \{\} + \frac{\partial}{\partial z} \{\} \cdot e\_{z\nu}
$$

so equation 14 becomes:

16 Will-be-set-by-IN-TECH

**e**

**int**

Fig. 5. Cross section of the waveguide and definition of the regions of integration for modal

the minimum point of the imaginary branch. At that point, it is joined by the branch corresponding to mode L(0,3), and both go down to zero frequency with negative complex

Similarly, mode L(0,4) is cut off at the minimum point of the real wavenumber branch (*k* = 345 m−1, *f* = 621.5 kHz). Mode L(0,5) is cut off at (*k* = 0, *f* = 698.6 kHz), changes to imaginary wavenumber until it reaches point (*k* = 0, *f* = 669.4 kHz), and then again to real, *but negative*, wavenumber down to (*k* = 345 m−1, *f* = 621.5 kHz), where it joins mode L(0,4). Mode L(0,5) has a negative wavenumber (and consequently, negative phase speed *ω*/*k*), in order to maintain a positive group velocity *dω*/*dk*, since this is a propagating mode in the *z*+ direction of the waveguide, according to the last rule of section 3.2.1. Below 621.5 kHz, the dispersion curves of modes L(0,4) and L(0,5) descend to zero frequency with negative complex

Modal analysis is a mathematical technique which permits to compute the dynamic response of a waveguide subject to arbitrary external forces, as an expansion of the excited wave over the set of normal modes of the waveguide, as defined in section 2 (Auld, 1973). Modal analysis is based upon two properties of normal modes: orthogonality, the existence of a scalar product which is null for any two different modes; and completeness, the capacity of the set of normal

For two different modes of the waveguide (1) and (2) of the form given in equation 1, the

<sup>2</sup> <sup>−</sup> *<sup>u</sup>*<sup>∗</sup>

which is applicable to linear elastic materials and also to piezoelectric or magnetostrictive linear materials, assuming no elastic or dielectric losses. Later on this result will be generalized

For separable vector fields in the *z* coordinate, the tridimensional divergence operator can be

+

*∂ ∂z*

{}· *ez*,

<sup>2</sup> · *<sup>σ</sup>*1) = 0, (14)

**r**

**int**

**r**

conjugate wavenumbers *kr* + *jki* and −*kr* + *jki*.

analysis.

**3.3 Modal analysis**

written as:

**ext**

conjugate wavenumbers, in the same way as modes L(0,2) and L(0,3).

∇ · (*<sup>u</sup>*<sup>1</sup> · *<sup>σ</sup>*<sup>∗</sup>

∇·{}

= ∇·{}

modes to span arbitrary waveforms in the waveguide.

**orthogonality relationship** (Auld, 1973) establishes that:

to include external forces and stresses.

D D

**int ext**

**e**

**ext**

D

$$\widetilde{\nabla} \cdot (\widetilde{\boldsymbol{\mu}}\_1 \cdot \widetilde{\boldsymbol{\sigma}}\_2^\* - \widetilde{\boldsymbol{\mu}}\_2^\* \cdot \widetilde{\boldsymbol{\sigma}}\_1) \boldsymbol{e}^{\mathrm{j}(k\_1 - k\_2^\*)z} + \boldsymbol{j}(k\_1 - k\_2^\*) \boldsymbol{e}^{\mathrm{j}(k\_1 - k\_2^\*)z} (\widetilde{\boldsymbol{\mu}}\_1 \cdot \widetilde{\boldsymbol{\sigma}}\_2^\* - \widetilde{\boldsymbol{\mu}}\_2^\* \cdot \widetilde{\boldsymbol{\sigma}}\_1) \cdot \boldsymbol{e}\_z = 0. \tag{15}$$

Discarding the common factor *ej*(*k*1−*k*<sup>∗</sup> <sup>2</sup>)*z*, integrating over the cross section *D* of the waveguide, and applying the divergence theorem, we find that:

$$\oint\_{\partial D} \left( \widetilde{\boldsymbol{u}}\_{1} \cdot \widetilde{\boldsymbol{\sigma}}\_{2}^{\*} - \widetilde{\boldsymbol{u}}\_{2}^{\*} \cdot \widetilde{\boldsymbol{\sigma}}\_{1} \right) \cdot \boldsymbol{e}\_{n} \, dl + j(k\_{1} - k\_{2}^{\*}) \iint\_{D} \left( \widetilde{\boldsymbol{u}}\_{1} \cdot \widetilde{\boldsymbol{\sigma}}\_{2}^{\*} - \widetilde{\boldsymbol{u}}\_{2}^{\*} \cdot \widetilde{\boldsymbol{\sigma}}\_{1} \right) \cdot \boldsymbol{e}\_{2} \, dS = 0. \tag{16}$$

In equation 16, *∂D* = *∂D*int ∪ *∂D*ext represents the inner and outer surfaces of the waveguide, and the normal unit vector *en* is taken on each surface pointing out of the waveguide's interior, as shown in figure 5.

Because for proper modes the surface traction stress is null (*<sup>σ</sup><sup>t</sup>* <sup>=</sup> *<sup>σ</sup>* · *en* <sup>=</sup> 0 in *<sup>∂</sup>D*), the first integral of equation 16 is zero. Then a suitable scalar product of modes (1) and (2) is:

$$P\_{12} = -\frac{j\omega}{4} \iint\_{D} \left(\tilde{\boldsymbol{u}}\_{1} \cdot \tilde{\boldsymbol{\sigma}}\_{2}^{\*} - \tilde{\boldsymbol{u}}\_{2}^{\*} \cdot \tilde{\boldsymbol{\sigma}}\_{1}\right) \cdot \boldsymbol{e}\_{z} \, d\mathcal{S} = -\frac{j\pi\omega}{2} \int\_{r\_{\text{int}}}^{r\_{\text{out}}} \left(\boldsymbol{u}\_{1} \cdot \boldsymbol{\sigma}\_{2}^{\*} - \boldsymbol{u}\_{2}^{\*} \cdot \boldsymbol{\sigma}\_{1}\right) \cdot \boldsymbol{e}\_{z} \cdot \boldsymbol{r} \, d\boldsymbol{r}.\tag{17}$$

In the right part of equation 17 we have assumed that the circumferential order *n* of modes (1) and (2) is the same; otherwise, *P*<sup>12</sup> is zero automatically due to the integration over the *θ* coordinate. The factor −*jω*/4 is introduced so that the quantity *P*<sup>11</sup> equals to the integral of the acoustic Poynting vector in the cross section of the waveguide, i.e., the power transported by the mode. For nonpropagating modes with *ki* �= 0, *P*<sup>11</sup> is zero.

With this notation, equation 16 reduces to:

$$(k\_1 - k\_2^\*)P\_{12} = 0,\tag{18}$$

which implies that *P*<sup>12</sup> = 0 unless *k*<sup>1</sup> = *k*<sup>∗</sup> <sup>2</sup>. In PCDISP, mode orthogonality can be verified with routine pcorthogonalcheck.

The second condition for modal analysis is **completeness**, which is based on the premise that an arbitrary perturbation in the waveguide can be expanded in the set of normal modes:

$$
\hat{u}\_1(r,\theta,z) = \sum\_p a\_p(z)\tilde{u}\_p(r,\theta) \qquad \qquad \hat{\sigma}\_1(r,\theta,z) = \sum\_p a\_p(z)\tilde{\sigma}\_p(r,\theta). \tag{19}
$$

where the modes are indexed by *p*, and no distinction has been made between propagating and evanescent modes. The problem lies in computing the set of coefficients *ap*(*z*), when the waveguide is under an arbitrary excitation composed of:


In the case of existence of external fields, the orthogonality relationship, equation 14, must be generalized to (Auld, 1973):

$$
\widehat{\nabla} \cdot (\widehat{\boldsymbol{\mu}}\_1 \cdot \widehat{\sigma}\_2^\* - \widehat{\boldsymbol{\mu}}\_2^\* \cdot \widehat{\sigma}\_1) = -\widehat{\boldsymbol{\mu}}\_1 \cdot \widehat{f}\_2^\* + \widehat{\boldsymbol{\mu}}\_2^\* \cdot \widehat{f}\_1. \tag{20}
$$

where *f*1,2(*r*, *θ*, *z*) represent the forcing terms.

and

*f v*

from region *Rg*.

of the system.

are given by:

section 3.2.1.

*<sup>p</sup>* (*z*) = −*jω*

of variables, resulting in:

 *D* [*u*∗

*<sup>p</sup>*(*r*, *θ*) · *f*

*ap*(*z*) = *<sup>e</sup>jkpz*

equations are used by routine pcmodalanalysis of PCDISP .

However, *Pp*,*p*<sup>∗</sup> �= 0, and we can set *q* = *p*∗, *kq* = *k*<sup>∗</sup>

**3.4 Propagation of waveforms in the waveguide**

recovered by the inverse Fourier transform:

4*Pp*

 *Rg*

*<sup>e</sup>*(*r*, *θ*, *z*)] *dS* = −*jω*

equation 25 is changed into an ordinary differential equation, solvable by a standard change

Modelling the Generation and Propagation of Ultrasonic Signals in Cylindrical Waveguides 19

*e*−*jkpz*� [ *f s p*(*z*�

where the integration takes place in the region *Rg* where the generating terms *f <sup>s</sup>* and *f <sup>v</sup>* are not null, and *z* is the point where the ultrasonic signal is observed, in the direction of increasing *z*

If *p* is a non-propagating mode, our computation method is changed slightly, since *Pp* = 0.

As a summary, we have established the equations that permit to find the amplitude of the proper modes excited in the waveguide by an arbitrary set of external driving forces. These

The modal analysis equations discussed in section 3.3 permit to obtain the frequency response of a transducer exciting the waveguide. In many applications we want to predict what ultrasonic waveforms will be obtained at a certain distance *z* from the excitation source, when the transducer is excited by a finite length time signal, i.e., to model the transient behaviour

The method to study the propagation of signal waveforms is relatively straightforward (Doyle, 1997). Let *u*(0, *t*) be the input signal in the transducer (placed in region *Rg*), and *U*(0, *ω*) = F[*u*(0,*t*)] its Fourier transform. When this signal excites the waveguide, we determine the corresponding volumetric forces and surface stresses, and evaluate terms *ap*(*z*) from equation 28 for each significant frequency component of *U* and all normal modes of the waveguide. Note that the terms *ap*(*z*) incorporate both the frequency response of the transducer itself (inside the integral term) and the effect caused by signal propagation (exp(*jkp*(*ω*)*z*)). Thus, the frequency components at a distance *z* from the generating region

*U*(*z*, *ω*) = *U*(0, *ω*) · *ap*(*z*, *ω*)

where the values of the Fourier transform for negative frequencies are taken as complex conjugate of the positive ones, in order to obtain a real time signal. The waveform at *z* is

The imaginary and complex wavenumber parts of the spectrum are required in equation 29 if the exciting signal has significant frequency content below cutoff of the propagating mode, and the measurement point is not far away from the transducer, as was described in

*<sup>U</sup>*(*z*, <sup>−</sup>*ω*) = *<sup>U</sup>*∗(*z*, *<sup>ω</sup>*), (29)

*<sup>u</sup>*(*z*, *<sup>t</sup>*) = <sup>F</sup> <sup>−</sup>1[*U*(*z*, *<sup>ω</sup>*)]. (30)

 *D e*

> ) + *f <sup>v</sup> <sup>p</sup>* (*z*� )] *dz*�

<sup>−</sup>*jnp <sup>θ</sup>* · [*u*<sup>∗</sup>

*<sup>p</sup>*(*r*) · *f*

*<sup>p</sup>*, and modify equations 26-28 accordingly.

*<sup>e</sup>*(*r*, *θ*, *z*)] *dS*, (27)

, (28)

If we take subscript (1) for the wave existing in the waveguide (equation 19) and (2) as the *q*-th proper mode of the waveguide:

$$
\hat{u}\_2(r,\theta,z) = \tilde{u}\_q(r,\theta)e^{\vec{k}\_q z} \qquad \hat{\sigma}\_2(r,\theta,z) = \tilde{\sigma}\_\emptyset(r,\theta)e^{\vec{k}\_q z} \tag{21}
$$

we can insert both expressions into equation 20, and, letting *f* <sup>2</sup> = 0 since it corresponds to a normal mode in the waveguide, we obtain that:

$$\widehat{\nabla} \cdot \left[ \sum\_{p} a\_p(z) (\widetilde{u}\_p \cdot \widetilde{\sigma}\_q^\* - \widetilde{u}\_q^\* \cdot \widetilde{\sigma}\_p) e^{-jk\_q z} \right] = \widetilde{u}\_q^\* \cdot \widehat{f}\_1 e^{-jk\_q z}.\tag{22}$$

Operating with the divergence operator:

$$\sum\_{p} a\_p(z) \widetilde{\nabla} \cdot (\widetilde{\boldsymbol{u}}\_p \cdot \widetilde{\boldsymbol{\sigma}}\_q^\* - \widetilde{\boldsymbol{u}}\_q^\* \cdot \widetilde{\boldsymbol{\sigma}}\_p) + \sum\_{p} \left[ -jk\_q a\_p(z) + \frac{da\_p(z)}{dz} \right] (\widetilde{\boldsymbol{u}}\_p \cdot \widetilde{\boldsymbol{\sigma}}\_q^\* - \widetilde{\boldsymbol{u}}\_q^\* \cdot \widetilde{\boldsymbol{\sigma}}\_p) \cdot \boldsymbol{e}\_z = \widetilde{\boldsymbol{u}}\_q^\* \cdot \widehat{f}\_1. \tag{23}$$

which, when integrating across the transversal section of the tube, with the divergence theorem, reduces to:

$$\underbrace{\sum\_{p} a\_{p}(z) \oint\_{\partial D} (\tilde{u}\_{p} \cdot \tilde{\sigma}\_{q}^{\*} - \tilde{u}\_{q}^{\*} \cdot \tilde{\sigma}\_{p}) \cdot e\_{n} \, dl}\_{(A)} +$$

$$\underbrace{\sum\_{p} \left[ -jk\_{q} + \frac{d}{dz} \right] a\_{p}(z) \iint\_{D} (\tilde{u}\_{p} \cdot \tilde{\sigma}\_{q}^{\*} - \tilde{u}\_{q}^{\*} \cdot \tilde{\sigma}\_{p}) \cdot e\_{z} \, dS}\_{(B)} = \underbrace{\iint\_{D} \tilde{u}\_{q}^{\*} \cdot \hat{f}\_{1} \, dS}\_{(C)}.\tag{24}$$

The first term simplifies as *<sup>σ</sup><sup>q</sup>* · *en* <sup>=</sup> 0 in *<sup>∂</sup>D*, and we can combine the sum over the index *<sup>p</sup>* to find:

$$\mathcal{I}(A) = -\oint\_{\partial D} (\widetilde{\mu}\_q^\* \cdot \widehat{\sigma}\_1) \cdot e\_n \, dl \, A$$

As for the second term, if we assume that *p* is a propagating mode, we recall the property of orthogonality of the propagating modes of the waveguide, and the definition of acoustic power *Pp* in equation 17, to eliminate all the terms of the sum except the one for which *p* = *q*:

$$\delta(B) = -\frac{4P\_p}{j\omega} \left( -jk\_p + \frac{d}{dz} \right) a\_p(z).$$

Inserting this result in the preceding equation:

$$-\frac{4P\_p}{j\omega}\left(-jk\_p + \frac{d}{dz}\right)a\_p(z) = \oint\_{\partial D} (\widetilde{u}\_p^\* \cdot \widehat{v}\_1) \cdot e\_n \, dl + \iint\_D \widetilde{u}\_p^\* \cdot \widehat{f}\_1 \, dS\_\star \tag{25}$$

where the contributions to the mode amplitude due to the volumetric forces (*f* <sup>1</sup> = *f <sup>e</sup>*) and the surface tractions (*<sup>σ</sup>*<sup>1</sup> <sup>=</sup> *<sup>σ</sup>e*) appear clearly separated. With the following definitions:

$$\delta f\_p^s(z) = -j\omega \oint\_{\partial D} \left[ \overleftarrow{u}\_p^\*(r, \theta) \cdot \widehat{\boldsymbol{\sigma}}\_\ell(r, \theta, z) \right] \cdot \boldsymbol{\varepsilon}\_\mathcal{n} \, dl = -j\omega \oint\_{\partial D} e^{-j\boldsymbol{\eta}\_p \theta} \cdot \left[ \boldsymbol{u}\_p^\*(r) \cdot \widehat{\boldsymbol{\sigma}}\_\ell(r, \theta, z) \right] \cdot \boldsymbol{\varepsilon}\_\mathcal{n} \, dl,\tag{26}$$

and

18 Will-be-set-by-IN-TECH

If we take subscript (1) for the wave existing in the waveguide (equation 19) and (2) as the

*<sup>u</sup>*<sup>2</sup>(*r*, *<sup>θ</sup>*, *<sup>z</sup>*) = *<sup>u</sup><sup>q</sup>*(*r*, *<sup>θ</sup>*)*ejkqz <sup>σ</sup>*2(*r*, *<sup>θ</sup>*, *<sup>z</sup>*) = *<sup>σ</sup>q*(*r*, *<sup>θ</sup>*)*ejkqz*, (21)

(*<sup>u</sup><sup>p</sup>* · *<sup>σ</sup>*<sup>∗</sup>

*<sup>q</sup>* · *<sup>σ</sup>p*) · *ez dS*

*<sup>q</sup>* · *<sup>σ</sup>*1) · *en dl*.

*d dz ap*(*z*).

 *∂D*

*<sup>p</sup>* · *<sup>σ</sup>*1) · *en dl* <sup>+</sup>

*<sup>e</sup>*−*jnp <sup>θ</sup>* · [*u*<sup>∗</sup>

 *D u*∗ *<sup>p</sup>* · *f* 

 (*A*)

<sup>=</sup> *<sup>u</sup>*<sup>∗</sup> *<sup>q</sup>* · *f* 1*e*−*jkqz*

(*<sup>u</sup><sup>p</sup>* · *<sup>σ</sup>*<sup>∗</sup>

*<sup>q</sup>* <sup>−</sup> *<sup>u</sup>*<sup>∗</sup>

= *D u*∗ *<sup>q</sup>* · *f* <sup>1</sup> *dS*

*<sup>q</sup>* <sup>−</sup> *<sup>u</sup>*<sup>∗</sup>

*<sup>q</sup>* · *<sup>σ</sup>p*) · *en dl*

 (*C*)

<sup>2</sup> = 0 since it corresponds to a

*<sup>q</sup>* · *<sup>σ</sup>p*)·*ez* <sup>=</sup> *<sup>u</sup>*<sup>∗</sup>

+

. (22)

*<sup>q</sup>* · *f*1, (23)

. (24)

<sup>1</sup> *dS*, (25)

*<sup>e</sup>*) and the

 <sup>1</sup> = *f*

*<sup>p</sup>*(*r*) · *<sup>σ</sup>e*(*r*, *<sup>θ</sup>*, *<sup>z</sup>*)] · *en dl*, (26)

*q*-th proper mode of the waveguide:

we can insert both expressions into equation 20, and, letting *f*

*<sup>q</sup>* · *<sup>σ</sup>p*) + ∑ *p*

*ap*(*z*)(*<sup>u</sup><sup>p</sup>* · *<sup>σ</sup>*<sup>∗</sup>

 (*B*)

(*A*) = −

(*B*) = <sup>−</sup>4*Pp*

*jω*

*ap*(*z*) =

where the contributions to the mode amplitude due to the volumetric forces (*f*

*<sup>p</sup>*(*r*, *<sup>θ</sup>*) · *<sup>σ</sup>e*(*r*, *<sup>θ</sup>*, *<sup>z</sup>*)] · *en dl* <sup>=</sup> <sup>−</sup>*j<sup>ω</sup>*

surface tractions (*<sup>σ</sup>*<sup>1</sup> <sup>=</sup> *<sup>σ</sup>e*) appear clearly separated. With the following definitions:

 *∂D* (*u*∗

*<sup>q</sup>* <sup>−</sup> *<sup>u</sup>*<sup>∗</sup>

*<sup>q</sup>* · *<sup>σ</sup>p*)*e*−*jkqz*

*dz*

<sup>−</sup>*jkqap*(*z*) + *dap*(*z*)

which, when integrating across the transversal section of the tube, with the divergence

*ap*(*z*) *∂D*

*<sup>q</sup>* <sup>−</sup> *<sup>u</sup>*<sup>∗</sup>

The first term simplifies as *<sup>σ</sup><sup>q</sup>* · *en* <sup>=</sup> 0 in *<sup>∂</sup>D*, and we can combine the sum over the index *<sup>p</sup>* to

As for the second term, if we assume that *p* is a propagating mode, we recall the property of orthogonality of the propagating modes of the waveguide, and the definition of acoustic power *Pp* in equation 17, to eliminate all the terms of the sum except the one for which *p* = *q*:

−*jkp* +

 *∂D* (*u*∗

∑ *p*

normal mode in the waveguide, we obtain that:

∇ · ∑ *p*

*<sup>q</sup>* <sup>−</sup> *<sup>u</sup>*<sup>∗</sup>

Operating with the divergence operator:

(*<sup>u</sup><sup>p</sup>* · *<sup>σ</sup>*<sup>∗</sup>

∑ *p*

find:

*f s*

*<sup>p</sup>*(*z*) = −*jω*

*ap*(*z*)∇ ·

theorem, reduces to:

∑ *p*

−*jkq* +

*d dz ap*(*z*) *D* (*<sup>u</sup><sup>p</sup>* · *<sup>σ</sup>*<sup>∗</sup>

Inserting this result in the preceding equation:

−*jkp* +

*d dz* 

<sup>−</sup> <sup>4</sup>*Pp jω*

 *∂D* [*u*∗

$$f\_p^\mathbb{F}(z) = -j\omega \iint\_D \left[ \overleftarrow{u}\_p^\*(r,\theta) \cdot \widehat{f}\_\varepsilon(r,\theta,z) \right] dS = -j\omega \iint\_D e^{-j\eta\_p\theta} \cdot \left[ u\_p^\*(r) \cdot \widehat{f}\_\varepsilon(r,\theta,z) \right] dS,\tag{27}$$

equation 25 is changed into an ordinary differential equation, solvable by a standard change of variables, resulting in:

$$a\_p(z) = \frac{e^{\mathbb{j}k\_p z}}{4P\_p} \int\_{\mathcal{R}\_\mathcal{g}} e^{-\mathbb{j}k\_p z'} \left[ f\_p^s(z') + f\_p^v(z') \right] dz',\tag{28}$$

where the integration takes place in the region *Rg* where the generating terms *f <sup>s</sup>* and *f <sup>v</sup>* are not null, and *z* is the point where the ultrasonic signal is observed, in the direction of increasing *z* from region *Rg*.

If *p* is a non-propagating mode, our computation method is changed slightly, since *Pp* = 0. However, *Pp*,*p*<sup>∗</sup> �= 0, and we can set *q* = *p*∗, *kq* = *k*<sup>∗</sup> *<sup>p</sup>*, and modify equations 26-28 accordingly.

As a summary, we have established the equations that permit to find the amplitude of the proper modes excited in the waveguide by an arbitrary set of external driving forces. These equations are used by routine pcmodalanalysis of PCDISP .

#### **3.4 Propagation of waveforms in the waveguide**

The modal analysis equations discussed in section 3.3 permit to obtain the frequency response of a transducer exciting the waveguide. In many applications we want to predict what ultrasonic waveforms will be obtained at a certain distance *z* from the excitation source, when the transducer is excited by a finite length time signal, i.e., to model the transient behaviour of the system.

The method to study the propagation of signal waveforms is relatively straightforward (Doyle, 1997). Let *u*(0, *t*) be the input signal in the transducer (placed in region *Rg*), and *U*(0, *ω*) = F[*u*(0,*t*)] its Fourier transform. When this signal excites the waveguide, we determine the corresponding volumetric forces and surface stresses, and evaluate terms *ap*(*z*) from equation 28 for each significant frequency component of *U* and all normal modes of the waveguide. Note that the terms *ap*(*z*) incorporate both the frequency response of the transducer itself (inside the integral term) and the effect caused by signal propagation (exp(*jkp*(*ω*)*z*)). Thus, the frequency components at a distance *z* from the generating region are given by:

$$\begin{aligned} \mathcal{U}(z,\omega) &= \mathcal{U}(0,\omega) \cdot a\_p(z,\omega) \\ \mathcal{U}(z,-\omega) &= \mathcal{U}^\*(z,\omega), \end{aligned} \tag{29}$$

where the values of the Fourier transform for negative frequencies are taken as complex conjugate of the positive ones, in order to obtain a real time signal. The waveform at *z* is recovered by the inverse Fourier transform:

$$
\mu(z,t) = \mathcal{F}^{-1}[\mathcal{U}(z,\omega)].\tag{30}
$$

The imaginary and complex wavenumber parts of the spectrum are required in equation 29 if the exciting signal has significant frequency content below cutoff of the propagating mode, and the measurement point is not far away from the transducer, as was described in section 3.2.1.

**Part A**: Input data of the waveguide (pcwaveguide)

and pcsignalpropagation)

Loop over propagating modes *p*

Compute *f <sup>s</sup>*

Compute *f <sup>v</sup>*

End loop over frequency

End loop over propagating modes

routines.

table 6.

Phase speed (m/s)

L(0,1)

L(0,2)

f 1

TDPRA load line 1

*p*(*z*�

*<sup>p</sup>* (*z*�

End of integration in region *Rg*

Sum over propagating modes *u*(*z*, *t*) = ∑*<sup>p</sup> up*(*z*, *t*)

(a) Phase speed of the L(0,m) modes of the waveguide

TDPRA load line 2

<sup>0</sup> <sup>100</sup> <sup>200</sup> <sup>300</sup> <sup>400</sup> <sup>500</sup> <sup>600</sup> <sup>700</sup> <sup>800</sup> <sup>0</sup>

Freq (kHz)

Assembling of the waveguide description matrix (pcmatdet) Compute the dispersion curves *k* = *k*(*ω*) (pckfcurves)

Fourier transform of excitation signal: *U*(0, *ω*) = FFT[*u*(0, *t*)]

Loop over frequencies *ω* of the signal's spectrum

Integration in the region of generation *z*� ∈ *Rg*

) by integrating *f*

Computation of mode amplitude *ap*(*z*) (equation 28)

Inverse Fourier transform: *up*(*z*, *t*) = IFFT[*Up*(*z*, *ω*)] (eq. 30)

L(0,3)

f 2 L(0,4)

Set negative frequencies of the FFT: *Up*(*z*, −*ω*) = *U*<sup>∗</sup>

Determine gain for frequency *ω*: *Up*(*z*, *ω*) = *U*(0, *ω*) · *ap*(*z*)

Table 7. General method used for modal analysis computations, and required PCDISP

1000

2000

3000

L(0,1)

L(0,2)

f 1

Group speed (m/s)

Fig. 7. (a) Phase and (b) group speeds of the first axisymmetric modes of the waveguide of

4000

5000

6000

**Part B**: Frequency / transient response of the waveguide (pcmodalanalysis

Modelling the Generation and Propagation of Ultrasonic Signals in Cylindrical Waveguides 21

Compute amplitude vector *Ap* by SVD solution of *D*(*ω*, *kp*) · *Ap* = 0 Computation of ultrasonic fields of *<sup>p</sup>*-th mode: *<sup>u</sup><sup>p</sup>*, *<sup>σ</sup><sup>p</sup>* (pcmatdet)

) by integrating *<sup>σ</sup>e*(*r*, *<sup>θ</sup>*, *<sup>z</sup>*) (pcextsurfacestress) in *<sup>∂</sup><sup>D</sup>*

*<sup>e</sup>*(*r*, *θ*, *z*) (pcextvolumforce) in *D*

*<sup>p</sup>* (*z*, *ω*)

<sup>0</sup> <sup>100</sup> <sup>200</sup> <sup>300</sup> <sup>400</sup> <sup>500</sup> <sup>600</sup> <sup>700</sup> <sup>800</sup> <sup>0</sup>

(b) Group speed of the L(0,m) modes of the waveguide

L(0,3)

f 2 L(0,4)

Freq (kHz)

Fig. 6. Transducers used for generation of ultrasonic waves in cylindrical waveguides: (a) Electromagnetic Acoustic Transducer (EMAT) for Lorentz force excitation; (b) Time-Delay Periodic Ring Array (TDPRA) for piezoelectric excitation.

The dispersive effect characteristic of waveguide propagation is frequently undesired in applications, since it implies a distortion of the original signal. One practical way of minimizing dispersion is to employ signals with narrow spectral content (typically by windowing a sine pulse train) with a central frequency in the region where the curve *c*ph(*ω*) = *ω*/*k*(*ω*) is relatively flat (Lowe et al., 1998). Where this is not feasible, compensation methods based on inverting the nonlinear *k* = *k*(*ω*) dependence have been developed (Wilcox, 2003).

The propagation of ultrasonic signals in the waveguide is simulated in PCDISP with routine pcsignalpropagation. When customising this routine, the user must be careful to zero pad the excitation signal *u*(0, *t*) at the end such that the resulting time window has enough duration to allow propagation of the signal for the distance between transducer and receiving point. Likewise, a sampling frequency high enough to cover all the spectral content of the signal should be used; in practical applications, oversampling the signal above the Nyquist rate is advantageous since it enhances the signal to noise ratio.

#### **4. Demonstration of the methodology**

In this section we will illustrate the use of the methodology described in this chapter and the developed software, to model the performance of a given transducer. The complete procedure is summarized in table 7, along with the needed routines of the PCDISP package.

Two transducer setups commonly found in ultrasonic guided wave applications will be analyzed (see figure 6). The first one is an electromagnetic acoustic transducer (EMAT) used to generate ultrasound in metallic waveguides without physical contact between the transducer and the sample; the second, an array of piezoelectric rings which generates ultrasound by mechanically loading the external tube surface. We begin by considering the mechanical behaviour of the sample waveguide.

#### **4.1 Dispersive curves of longitudinal modes**

We will continue to use the waveguide described in table 6. The complete signal spectrum of the longitudinal modes was already shown in figure 3; in figure 7 we plot the phase and group speeds in the range of 0 to 800 kHz. An important requisite for guided waves applications of ultrasound is the selection and exploitation of a single propagating mode, in a region where dispersive effects are minimum, since, as a general principle, an external force will excite all propagating modes existing within its bandwidth (Lowe et al., 1998). We will consider two possibilities: excitation of mode L(0,2) at frequency *f*<sup>1</sup> = 250 kHz, and use of mode L(0,3) at frequency *f*<sup>2</sup> = 565 kHz (see figure 7 b). The dispersion curves of these modes are relatively flat at these frequencies, and their group speeds are higher than those of other coexisting modes.

20 Will-be-set-by-IN-TECH

Fig. 6. Transducers used for generation of ultrasonic waves in cylindrical waveguides: (a) Electromagnetic Acoustic Transducer (EMAT) for Lorentz force excitation; (b) Time-Delay

The dispersive effect characteristic of waveguide propagation is frequently undesired in applications, since it implies a distortion of the original signal. One practical way of minimizing dispersion is to employ signals with narrow spectral content (typically by windowing a sine pulse train) with a central frequency in the region where the curve *c*ph(*ω*) = *ω*/*k*(*ω*) is relatively flat (Lowe et al., 1998). Where this is not feasible, compensation methods based on inverting the nonlinear *k* = *k*(*ω*) dependence have been developed (Wilcox, 2003). The propagation of ultrasonic signals in the waveguide is simulated in PCDISP with routine pcsignalpropagation. When customising this routine, the user must be careful to zero pad the excitation signal *u*(0, *t*) at the end such that the resulting time window has enough duration to allow propagation of the signal for the distance between transducer and receiving point. Likewise, a sampling frequency high enough to cover all the spectral content of the signal should be used; in practical applications, oversampling the signal above the Nyquist

In this section we will illustrate the use of the methodology described in this chapter and the developed software, to model the performance of a given transducer. The complete procedure

Two transducer setups commonly found in ultrasonic guided wave applications will be analyzed (see figure 6). The first one is an electromagnetic acoustic transducer (EMAT) used to generate ultrasound in metallic waveguides without physical contact between the transducer and the sample; the second, an array of piezoelectric rings which generates ultrasound by mechanically loading the external tube surface. We begin by considering the mechanical

We will continue to use the waveguide described in table 6. The complete signal spectrum of the longitudinal modes was already shown in figure 3; in figure 7 we plot the phase and group speeds in the range of 0 to 800 kHz. An important requisite for guided waves applications of ultrasound is the selection and exploitation of a single propagating mode, in a region where dispersive effects are minimum, since, as a general principle, an external force will excite all propagating modes existing within its bandwidth (Lowe et al., 1998). We will consider two possibilities: excitation of mode L(0,2) at frequency *f*<sup>1</sup> = 250 kHz, and use of mode L(0,3) at frequency *f*<sup>2</sup> = 565 kHz (see figure 7 b). The dispersion curves of these modes are relatively flat at these frequencies, and their group speeds are higher than those of other coexisting

is summarized in table 7, along with the needed routines of the PCDISP package.

Periodic Ring Array (TDPRA) for piezoelectric excitation.

rate is advantageous since it enhances the signal to noise ratio.

**4. Demonstration of the methodology**

behaviour of the sample waveguide.

modes.

**4.1 Dispersive curves of longitudinal modes**

tail head period (B)

**Part A**: Input data of the waveguide (pcwaveguide) Assembling of the waveguide description matrix (pcmatdet) Compute the dispersion curves *k* = *k*(*ω*) (pckfcurves) **Part B**: Frequency / transient response of the waveguide (pcmodalanalysis and pcsignalpropagation) Fourier transform of excitation signal: *U*(0, *ω*) = FFT[*u*(0, *t*)] Loop over propagating modes *p* Loop over frequencies *ω* of the signal's spectrum Compute amplitude vector *Ap* by SVD solution of *D*(*ω*, *kp*) · *Ap* = 0 Computation of ultrasonic fields of *<sup>p</sup>*-th mode: *<sup>u</sup><sup>p</sup>*, *<sup>σ</sup><sup>p</sup>* (pcmatdet) Integration in the region of generation *z*� ∈ *Rg* Compute *f <sup>s</sup> p*(*z*� ) by integrating *<sup>σ</sup>e*(*r*, *<sup>θ</sup>*, *<sup>z</sup>*) (pcextsurfacestress) in *<sup>∂</sup><sup>D</sup>* Compute *f <sup>v</sup> <sup>p</sup>* (*z*� ) by integrating *f <sup>e</sup>*(*r*, *θ*, *z*) (pcextvolumforce) in *D* End of integration in region *Rg* Computation of mode amplitude *ap*(*z*) (equation 28) Determine gain for frequency *ω*: *Up*(*z*, *ω*) = *U*(0, *ω*) · *ap*(*z*) End loop over frequency Set negative frequencies of the FFT: *Up*(*z*, −*ω*) = *U*<sup>∗</sup> *<sup>p</sup>* (*z*, *ω*) Inverse Fourier transform: *up*(*z*, *t*) = IFFT[*Up*(*z*, *ω*)] (eq. 30) End loop over propagating modes Sum over propagating modes *u*(*z*, *t*) = ∑*<sup>p</sup> up*(*z*, *t*)

Table 7. General method used for modal analysis computations, and required PCDISP routines.

Fig. 7. (a) Phase and (b) group speeds of the first axisymmetric modes of the waveguide of table 6.

Previously to computing the Lorentz force, we must determine the distribution of the electromagnetic field in the waveguide. Although an exact solution exists, it is complicated (Dodd & Deeds, 1968), so, for the purposes of this example, we will consider a simplified model in which the penetration depth of the EM field in the metal (*δ* = (2/*ωμ*0*σe*)1/2, with *μ*<sup>0</sup> being the magnetic permeability of vacuum and *σ<sup>e</sup>* = 38 MS/m the electrical conductivity of aluminum) is small compared with the thickness *r*ext − *r*int of the tube (in our case *δ* < (*r*ext − *r*int)/10 for *f* > 30 kHz). Then the magnetic field in the tube is mainly axial and can be

Modelling the Generation and Propagation of Ultrasonic Signals in Cylindrical Waveguides 23

where *H*zext(*z*) is the axial field at the outer surface (*r* = *r*ext) of the tube. We will further

*<sup>s</sup>* <sup>+</sup> *<sup>z</sup>*<sup>2</sup> <sup>+</sup>

where *Ls* and *Rs* are the solenoid's length and radius, *Ns* the number of turns, and *Is* the

Computing the eddy current with Ampère's law, *J*(*r*, *z*) = ∇ × *H*(*r*, *z*), and since the bias field

And using equation 27, we can compute the volumetric forcing term for mode *p* in the

� *r*ext *r*int *u*∗ *pr*(*r*)*<sup>f</sup>* em

If we consider an EMAT with a solenoid length *Ls* = 30 mm and *Rs* = *r*ext = 16.70 mm we obtain the dependence of transducer gain with frequency shown in figure 9 (a), where the radial component of displacement at the surface, *ur*(*r*ext), is plotted on a log scale on the vertical axis. As we can see, the EMAT exhibits high gain in the low frequency region, which unfortunately coincides with the zone where the dispersive behaviour of mode L(0,1) is maximum, making it difficult for guided waves applications. In parts (b) and (c) of the same figure we show the transient response of the waveguide when excited with the pulse train described in section 4.1, for central frequencies of 250 kHz and 565 kHz, respectively, when the radial component of surface displacement (*ur*) is measured at a point *z* = 1.5 m from the EMAT. For 250 kHz, the faster propagating L(0,2) mode is excited with lower amplitude than the mode L(0,1), while for 565 kHz all modes are excited with approximately equal

Summarizing, the basic EMAT described in this section shows poor mode selectivity control, exciting all modes within the bandwidth of the source signal with relatively equal amplitudes,

In this section we consider a Time-Delay Periodic Ring Array (TDPRA), an ultrasonic transducer with very good mode selectivity and also capable of achieving one directional emission, emitting much more ultrasonic energy from its enhanced side than in the direction

*z* �*R*<sup>2</sup>

assume that *H*zext(*z*) can be obtained by the elementary formula:

2*Ls*

1 + *j*

*<sup>p</sup>* (*z*) = −2*πjω*

⎡ ⎣

*<sup>H</sup>*zext(*z*) = *NsIs*

*<sup>f</sup>* em(*r*, *<sup>z</sup>*) = <sup>−</sup>*μ*<sup>0</sup>

*f v*

amplitudes, appearing also very close in the time domain.

**4.3 Generation of ultrasonic wave with piezoelectric surface loading**

which makes it a poor choice for this waveguide.

*H*(*r*, *z*) = *H*zext(*z*) exp[−(1 + *j*)(*r*ext − *r*)/*δ*]*ez r*int ≤ *r* ≤ *r*ext, (32)

� *R*2 *Ls* − *z*

*<sup>s</sup>* + (*Ls* − *<sup>z</sup>*)<sup>2</sup>

*<sup>δ</sup> <sup>H</sup>*zext(*z*)*H*<sup>0</sup> exp[−(<sup>1</sup> <sup>+</sup> *<sup>j</sup>*)(*r*ext <sup>−</sup> *<sup>r</sup>*)/*δ*]*er*. (34)

⎤

*<sup>r</sup>* (*r*)*r dr*. (35)

⎦ , (33)

written as:

current through one turn.

is *H*<sup>0</sup> = *H*0*ez*, we obtain:

waveguide as :

Fig. 8. Displacement and stress profiles for mode L(0,2) at f = 250 kHz (upper row), and mode L(0,3) at f = 565 kHz (bottom row), computed with pcwaveform.

The displacement and stress profiles of the selected modes are shown in figure 8. Their determination is important in NDE applications since the sensitivity of a propagating mode to a waveguide defect depends on the matching between the defect's shape and the mode profile (Ditri, 1994).

In order to minimize the influence of dispersion in the propagation of signals, we use as excitation signal a tone burst consisting of *n*cyc = 16 cycles of a central frequency modulated by a raised cosine window. This waveform does a good job in exciting a single frequency of the waveguide with a finite length signal and minimum sidelobes (Oppenheim et al., 1999).

#### **4.2 Generation of ultrasonic waves with an electromagnetic acoustic transducer**

Electromagnetic acoustic transducers (EMATs) are used to excite ultrasonic waves in metallic waveguides by non contact means (Cawley et al., 2004). Basically, an EMAT consists of a generating coil, which creates a dynamic field *H*(*t*) at ultrasonic frequencies, and a bias magnet providing a constant magnetic field *H*0, as shown in figure 6 (a). The physical phenomenon that couples the magnetic field with the elastic field in our non-ferromagnetic aluminium waveguide is the Lorentz force resulting from the interaction between the eddy currents *J*(*t*) induced in the tube by the dynamic field and the bias field *H*0, which create a volumetric force given by:

$$f^{\rm em} = \mu\_0 I \times H\_0. \tag{31}$$

22 Will-be-set-by-IN-TECH

12.5 13 13.5 14 14.5 15 15.5 16 16.5

12.5 13 13.5 14 14.5 15 15.5 16 16.5

Radius (mm)

ur uz

Fig. 8. Displacement and stress profiles for mode L(0,2) at f = 250 kHz (upper row), and

The displacement and stress profiles of the selected modes are shown in figure 8. Their determination is important in NDE applications since the sensitivity of a propagating mode to a waveguide defect depends on the matching between the defect's shape and the mode profile

In order to minimize the influence of dispersion in the propagation of signals, we use as excitation signal a tone burst consisting of *n*cyc = 16 cycles of a central frequency modulated by a raised cosine window. This waveform does a good job in exciting a single frequency of the waveguide with a finite length signal and minimum sidelobes (Oppenheim et al., 1999).

Electromagnetic acoustic transducers (EMATs) are used to excite ultrasonic waves in metallic waveguides by non contact means (Cawley et al., 2004). Basically, an EMAT consists of a generating coil, which creates a dynamic field *H*(*t*) at ultrasonic frequencies, and a bias magnet providing a constant magnetic field *H*0, as shown in figure 6 (a). The physical phenomenon that couples the magnetic field with the elastic field in our non-ferromagnetic aluminium waveguide is the Lorentz force resulting from the interaction between the eddy currents *J*(*t*) induced in the tube by the dynamic field and the bias field *H*0, which create a

**4.2 Generation of ultrasonic waves with an electromagnetic acoustic transducer**

mode L(0,3) at f = 565 kHz (bottom row), computed with pcwaveform.

Radius (mm)

−1 −0.5 0 0.5 1

−1 −0.5 0 0.5 1

*<sup>f</sup>* em <sup>=</sup> *<sup>μ</sup>*<sup>0</sup> *<sup>J</sup>* <sup>×</sup> *<sup>H</sup>*0. (31)

Stress

Stress

(d) Stresses, mode L(0,3), f = 565 kHz

(b) Stresses, mode L(0,2), f = 250 kHz

σ rr σθθ σ zz σ rz

σ rr σθθ σ zz σ rz

−1 −0.5 0 0.5 1

−1 −0.5 0 0.5 1

Displacement

Displacement

(c) Displacements, mode L(0,3), f = 565 kHz

(a) Displacements, mode L(0,2), f = 250 kHz

12.5 13 13.5 14 14.5 15 15.5 16 16.5

12.5 13 13.5 14 14.5 15 15.5 16 16.5

(Ditri, 1994).

volumetric force given by:

Radius (mm)

Radius (mm)

ur uz Previously to computing the Lorentz force, we must determine the distribution of the electromagnetic field in the waveguide. Although an exact solution exists, it is complicated (Dodd & Deeds, 1968), so, for the purposes of this example, we will consider a simplified model in which the penetration depth of the EM field in the metal (*δ* = (2/*ωμ*0*σe*)1/2, with *μ*<sup>0</sup> being the magnetic permeability of vacuum and *σ<sup>e</sup>* = 38 MS/m the electrical conductivity of aluminum) is small compared with the thickness *r*ext − *r*int of the tube (in our case *δ* < (*r*ext − *r*int)/10 for *f* > 30 kHz). Then the magnetic field in the tube is mainly axial and can be written as:

$$H(r, z) = H\_{\text{next}}(z) \exp[- (1 + j)(r\_{\text{ext}} - r) / \delta] e\_2 \qquad \qquad r\_{\text{int}} \le r \le r\_{\text{ext}} \tag{32}$$

where *H*zext(*z*) is the axial field at the outer surface (*r* = *r*ext) of the tube. We will further assume that *H*zext(*z*) can be obtained by the elementary formula:

$$H\_{\text{2ext}}(z) = \frac{N\_s I\_s}{2L\_s} \left[ \frac{z}{\sqrt{R\_s^2 + z^2}} + \frac{L\_s - z}{\sqrt{R\_s^2 + (L\_s - z)^2}} \right],\tag{33}$$

where *Ls* and *Rs* are the solenoid's length and radius, *Ns* the number of turns, and *Is* the current through one turn.

Computing the eddy current with Ampère's law, *J*(*r*, *z*) = ∇ × *H*(*r*, *z*), and since the bias field is *H*<sup>0</sup> = *H*0*ez*, we obtain:

$$f^{\rm em}(r,z) = -\mu\_0 \frac{1+j}{\delta} H\_{\rm zext}(z) H\_0 \exp[-(1+j)(r\_{\rm ext}-r)/\delta] e\_{\rm r} \tag{34}$$

And using equation 27, we can compute the volumetric forcing term for mode *p* in the waveguide as :

$$f\_p^v(z) = -2\pi j \omega \int\_{r\_{\rm int}}^{r\_{\rm ext}} u\_{pr}^\*(r) f\_r^{\rm em}(r) \, r \, dr. \tag{35}$$

If we consider an EMAT with a solenoid length *Ls* = 30 mm and *Rs* = *r*ext = 16.70 mm we obtain the dependence of transducer gain with frequency shown in figure 9 (a), where the radial component of displacement at the surface, *ur*(*r*ext), is plotted on a log scale on the vertical axis. As we can see, the EMAT exhibits high gain in the low frequency region, which unfortunately coincides with the zone where the dispersive behaviour of mode L(0,1) is maximum, making it difficult for guided waves applications. In parts (b) and (c) of the same figure we show the transient response of the waveguide when excited with the pulse train described in section 4.1, for central frequencies of 250 kHz and 565 kHz, respectively, when the radial component of surface displacement (*ur*) is measured at a point *z* = 1.5 m from the EMAT. For 250 kHz, the faster propagating L(0,2) mode is excited with lower amplitude than the mode L(0,1), while for 565 kHz all modes are excited with approximately equal amplitudes, appearing also very close in the time domain.

Summarizing, the basic EMAT described in this section shows poor mode selectivity control, exciting all modes within the bandwidth of the source signal with relatively equal amplitudes, which makes it a poor choice for this waveguide.

#### **4.3 Generation of ultrasonic wave with piezoelectric surface loading**

In this section we consider a Time-Delay Periodic Ring Array (TDPRA), an ultrasonic transducer with very good mode selectivity and also capable of achieving one directional emission, emitting much more ultrasonic energy from its enhanced side than in the direction

while the volumetric term is null. The total pressure *p*(*z*) is a sum over the *Np* periods of *Nr*

Modelling the Generation and Propagation of Ultrasonic Signals in Cylindrical Waveguides 25

�( *zci zw*

where *zci* = (*zw* + *zs*)*i* + *zw*/2 is the position of the center of each ring, and �(*x*) = 1 for

The parameters of the TDPRA must be tuned to the frequency to be excited. The load line of the TDPRA, given by *c*ph = *Nr*(*zw* + *zs*)*f* , is shown in figure 7 (a), along with the phase speed curves of the aluminum tube given in section 4.1, for two different designs: the first one with *Np* = 4, *Nr* = 8, *zw* = 2.2 mm and *zs* = 0.4 mm, intended to excite mode L(0,2) at 250 kHz, and the second with *Np* = 5, *Nr* = 6, *zw* = 1.5 mm, *zs* = 0.4 mm for excitation of mode L(0,3) at 565 kHz. The intersection points of these lines with the phase speed curves correspond to the frequencies for which the TDPRA achieves maximum efficiency in mode coupling.

The transducer gain of the first TDPRA is shown in figure 10 (a). For 250 kHz, the mode L(0,2) is effectively excited, and the excitation frequency can be fine tuned to make it coincide with

> −1.5 −1 −0.5 0 0.5 1 1.5 2

> −1.5 −1 −0.5 0 0.5 1 1.5 2

Fig. 10. Plots of the (a) frequency and (b) transient response of the first design of the TDPRA at *f* = 250 kHz; plots of the (c) frequency and (d) transient response of the second design of the TDPRA at *f* = 565 kHz. In the gain plots, the dashed lines correspond to the opposite ("tail") direction. The signals in the transient plots have been normalized to unit amplitude.

L(0,4)

head

L(0,1) L(0,2)

tail

head

L(0,3)

tail

)*ej*2*π*mod(*i*,*Nr*)/*Nr*, (37)

<sup>0</sup> <sup>200</sup> <sup>400</sup> <sup>600</sup> <sup>800</sup> <sup>1000</sup> −2

(b) TDPRA transient (f = 250 kHz)

Time (μs)

(d) TDPRA transient (f = 565 kHz)

L(0,1) L(0,2)

<sup>0</sup> <sup>200</sup> <sup>400</sup> <sup>600</sup> <sup>800</sup> <sup>1000</sup> −2

Time (μs)

*NpNr*−1 ∑ *i*=0

*p*(*z*) = *p*<sup>0</sup> ·


<sup>150</sup> <sup>200</sup> <sup>250</sup> <sup>300</sup> <sup>350</sup> −50

L(0,2) L(0,3)

<sup>500</sup> <sup>550</sup> <sup>600</sup> <sup>650</sup> −50

L(0,1)

Frequency (kHz)

Frequency (kHz)

(c) TDPRA frequency response (radial disp ur)

(a) TDPRA frequency response (radial disp ur)

L(0,2)

L(0,1)

rings each:

−45 −40 −35 −30 −25 −20 −15 −10 −5 0

−45 −40 −35 −30 −25 −20 −15 −10 −5 0

System gain, |ur(r=b)| (dB)

System gain, |ur(r=b)| (dB)

Fig. 9. (a) Computed frequency response of the EMAT for solenoid length *Ls* = 30 mm; transient waveforms at (b) 250 kHz and (c) 565 kHz. The signals in the transient plots have been normalized to unit amplitude.

of its weakened side (Zhu, 2001). A TDPRA, shown in figure 6 (b), consists in a number of piezoelectric rings capable of exerting a pressure loading on the outer surface of the tube. The rings are organized into *Np* identical periods of *Nr* rings each, with the length of a period matched to the wavelength *λ* of the mode to be excited. The rings of a period are connected to the same excitation source, but with a relative delay between them proportional to their position within the period of the TDPRA; this scheme is repeated throughout the TDPRA. This reinforces the wavelength matching of a single period and also creates constructive interference in the enhanced direction (the "head" of the array) and destructive in the other (the "tail").

To model numerically the TDPRA, we assume that each ring has width *zw*, with a separation *zs* between adjacent rings and vibrates in its thickness mode, exerting a pressure loading, *σrr* = −*p* over the outer surface of the waveguide, constant for all frequencies. This leads to an axisymmetric loading (no *θ* dependence). In this case, the term corresponding to the surface loading (equation 26) is:

$$f\_p^s(z) = 2\pi j \omega r\_{\text{ext}} u\_r^\*(r\_{\text{ext}}) p(z), \tag{36}$$

24 Will-be-set-by-IN-TECH

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

(c) Transient response of the EMAT (f = 565 kHz)

L(0,1) L(0,2) L(0,3)

<sup>0</sup> <sup>200</sup> <sup>400</sup> <sup>600</sup> <sup>800</sup> <sup>1000</sup> −1

Time (μs)

of its weakened side (Zhu, 2001). A TDPRA, shown in figure 6 (b), consists in a number of piezoelectric rings capable of exerting a pressure loading on the outer surface of the tube. The rings are organized into *Np* identical periods of *Nr* rings each, with the length of a period matched to the wavelength *λ* of the mode to be excited. The rings of a period are connected to the same excitation source, but with a relative delay between them proportional to their position within the period of the TDPRA; this scheme is repeated throughout the TDPRA. This reinforces the wavelength matching of a single period and also creates constructive interference in the enhanced direction (the "head" of the array) and destructive in the other

To model numerically the TDPRA, we assume that each ring has width *zw*, with a separation *zs* between adjacent rings and vibrates in its thickness mode, exerting a pressure loading, *σrr* = −*p* over the outer surface of the waveguide, constant for all frequencies. This leads to an axisymmetric loading (no *θ* dependence). In this case, the term corresponding to the

*<sup>p</sup>*(*z*) = 2*πjωr*ext*u*<sup>∗</sup>

*f s*

Fig. 9. (a) Computed frequency response of the EMAT for solenoid length *Ls* = 30 mm; transient waveforms at (b) 250 kHz and (c) 565 kHz. The signals in the transient plots have

<sup>0</sup> <sup>200</sup> <sup>400</sup> <sup>600</sup> <sup>800</sup> <sup>1000</sup> −1

(b) Transient response of the EMAT (f = 250 kHz)

L(0,1)

L(0,2)

Time (μs)

*<sup>r</sup>* (*r*ext)*p*(*z*), (36)

= 30 mm)

L(0,4)

<sup>100</sup> <sup>200</sup> <sup>300</sup> <sup>400</sup> <sup>500</sup> <sup>600</sup> <sup>700</sup> <sup>800</sup> −90

(a) EMAT frequency response (Ls

Frequency (kHz)

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

been normalized to unit amplitude.

surface loading (equation 26) is:

(the "tail").

L(0,3)

−80 −70 −60 −50 −40 −30 −20 −10

L(0,1)

L(0,2)

System gain, |ur(r=b)| (dB)

while the volumetric term is null. The total pressure *p*(*z*) is a sum over the *Np* periods of *Nr* rings each:

$$p(z) = p\_0 \cdot \sum\_{i=0}^{N\_p N\_r - 1} \sqcap (\frac{z\_{ci}}{z\_{w}}) e^{j2\pi \text{mod}(i, N\_r)/N\_r} \, \tag{37}$$

where *zci* = (*zw* + *zs*)*i* + *zw*/2 is the position of the center of each ring, and �(*x*) = 1 for |*x*| < 1/2, �(*x*) = 0 for |*x*| > 1/2, is the rectangular function.

The parameters of the TDPRA must be tuned to the frequency to be excited. The load line of the TDPRA, given by *c*ph = *Nr*(*zw* + *zs*)*f* , is shown in figure 7 (a), along with the phase speed curves of the aluminum tube given in section 4.1, for two different designs: the first one with *Np* = 4, *Nr* = 8, *zw* = 2.2 mm and *zs* = 0.4 mm, intended to excite mode L(0,2) at 250 kHz, and the second with *Np* = 5, *Nr* = 6, *zw* = 1.5 mm, *zs* = 0.4 mm for excitation of mode L(0,3) at 565 kHz. The intersection points of these lines with the phase speed curves correspond to the frequencies for which the TDPRA achieves maximum efficiency in mode coupling.

The transducer gain of the first TDPRA is shown in figure 10 (a). For 250 kHz, the mode L(0,2) is effectively excited, and the excitation frequency can be fine tuned to make it coincide with

Fig. 10. Plots of the (a) frequency and (b) transient response of the first design of the TDPRA at *f* = 250 kHz; plots of the (c) frequency and (d) transient response of the second design of the TDPRA at *f* = 565 kHz. In the gain plots, the dashed lines correspond to the opposite ("tail") direction. The signals in the transient plots have been normalized to unit amplitude.

Aristegui, C., Lowe, M. J. S. & Cawley, P. (2001). Guided waves in fluid-filled pipes

Modelling the Generation and Propagation of Ultrasonic Signals in Cylindrical Waveguides 27

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a minimum of the amplitude of the L(0,1) mode, improving the dynamic range between the two modes. The simulation of the propagated wave (at a distance *z* = 1.5 m from the TDPRA) gives the expected results (part (b)). The results obtained with the second TDPRA design are shown in parts (c) and (d) of figure 10. Mode L(0,3) dominates in this case at 565 kHz, with a higher relative amplitude over the L(0,1) and L(0,2) modes. In this case, however, as the group velocities are similar, the received signals appear closer in the time view of part (d). In all cases, ultrasonic generation in the "head" side has higher amplitude than in the "tail" side.

As a conclusion, the TDPRA is an efficient transducer for generating ultrasonic signals in cylindrical waveguides, and its feature of phase and wavelength matching permits to excite modes selectively, fine tune the system gain to a desired frequency, and direct the generated signal in only one direction. Although in this communication we have concerned ourselves only with axisymmetric transducers, PCDISP can also be used to study excitation of nonsymmetric modes by piezoelectric arrays (see reference (Li & Rose, 2001) for an example).

## **5. Conclusions**

In this chapter we have presented a methodology to model the dynamic response of a waveguide of cylindrical symmetry when subject to an arbitrary set of external forces acting at ultrasonic frequencies, based on the combination of the mechanical Pochhammer-Chree equations and modal analysis techniques. Furthermore, a software package (named PCDISP), created in the Matlab environment, is offered freely with the intention of saving other researchers from the time needed for implementation of the PC theory equations, permitting them to focus on their particular problems.

Throughout this communication, we have paid special attention to the numerical issues of stability of the matrix determinant for large frequency thickness products, provided algorithms for robust root solving and tracing of the dispersion curves, and modelled the dispersive effect of the waveguide on signal propagation. The methods described in this chapter are valid for waveguides formed by any number of layers as long as they have cylindrical symmetry. The PCDISP software can be further extended to consider materials with anisotropy (transversely isotropic and orthotropic), as well as materials with elastic damping and waveguides surrounded by, or containing, fluids. Guidelines for such extensions are given in the text.

The performance of modal analysis is illustrated by studying two common transducers employed in guided wave ultrasonic applications: an electromagnetic-acoustic transducer and a time-delay piezoelectric ring array. We believe that transducer analysis with quantitative results is achieved comparatively easier and faster than with other competing techniques like spectral or finite element methods, obtaining significant time savings in the design stage of ultrasonic transducers.

#### **6. Acknowledgments**

The financial support for this work was provided by the Spanish Ministerio de Ciencia e Innovación through project Lemur (TIN2009-14114-C04-03).

### **7. References**

Abramowitz, M. & Stegun, I. A. (1964). *Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables*, 9 edn, Dover.


26 Will-be-set-by-IN-TECH

a minimum of the amplitude of the L(0,1) mode, improving the dynamic range between the two modes. The simulation of the propagated wave (at a distance *z* = 1.5 m from the TDPRA) gives the expected results (part (b)). The results obtained with the second TDPRA design are shown in parts (c) and (d) of figure 10. Mode L(0,3) dominates in this case at 565 kHz, with a higher relative amplitude over the L(0,1) and L(0,2) modes. In this case, however, as the group velocities are similar, the received signals appear closer in the time view of part (d). In all cases, ultrasonic generation in the "head" side has higher amplitude than in the "tail" side. As a conclusion, the TDPRA is an efficient transducer for generating ultrasonic signals in cylindrical waveguides, and its feature of phase and wavelength matching permits to excite modes selectively, fine tune the system gain to a desired frequency, and direct the generated signal in only one direction. Although in this communication we have concerned ourselves only with axisymmetric transducers, PCDISP can also be used to study excitation of nonsymmetric modes by piezoelectric arrays (see reference (Li & Rose, 2001) for an example).

In this chapter we have presented a methodology to model the dynamic response of a waveguide of cylindrical symmetry when subject to an arbitrary set of external forces acting at ultrasonic frequencies, based on the combination of the mechanical Pochhammer-Chree equations and modal analysis techniques. Furthermore, a software package (named PCDISP), created in the Matlab environment, is offered freely with the intention of saving other researchers from the time needed for implementation of the PC theory equations, permitting

Throughout this communication, we have paid special attention to the numerical issues of stability of the matrix determinant for large frequency thickness products, provided algorithms for robust root solving and tracing of the dispersion curves, and modelled the dispersive effect of the waveguide on signal propagation. The methods described in this chapter are valid for waveguides formed by any number of layers as long as they have cylindrical symmetry. The PCDISP software can be further extended to consider materials with anisotropy (transversely isotropic and orthotropic), as well as materials with elastic damping and waveguides surrounded by, or containing, fluids. Guidelines for such

The performance of modal analysis is illustrated by studying two common transducers employed in guided wave ultrasonic applications: an electromagnetic-acoustic transducer and a time-delay piezoelectric ring array. We believe that transducer analysis with quantitative results is achieved comparatively easier and faster than with other competing techniques like spectral or finite element methods, obtaining significant time savings in the

The financial support for this work was provided by the Spanish Ministerio de Ciencia e

Abramowitz, M. & Stegun, I. A. (1964). *Handbook of Mathematical Functions with Formulas,*

**5. Conclusions**

them to focus on their particular problems.

extensions are given in the text.

design stage of ultrasonic transducers.

Innovación through project Lemur (TIN2009-14114-C04-03).

*Graphs, and Mathematical Tables*, 9 edn, Dover.

**6. Acknowledgments**

**7. References**


**1. Introduction** 

**2** 

*Poland* 

**Ultrasonic Projection** 

*Wroclaw University of Technology* 

*Institute of Telecommunications, Teleinformatics and Acoustics,* 

Ultrasonic technique of imaging serves an increasingly important role in medical diagnostics. In most of applications, echographic methods are used (ultrasonography, ultrasonic microscopy). Using such methods, an image presenting changes of a reflection coefficient in the interior of analysed structure is being constructed. This chapter presents possibilities of utilising information included in ultrasonic pulses, which penetrate an object in order to create images presenting the projection of analysed structure (Opielinski & Gudra, 2004b, 2004c, 2005, 2006, 2008, 2010a, 2010b, 2010c; Opielinski et al., 2009, 2010a, 2010b) in the form of a distribution of mean values of a measured acoustic parameter, for one or numerous planes, perpendicular to the direction of ultrasonic waves incidence (analogical as in X-ray radiography). Due to the possibility of obtaining images in pseudoreal time, the device using this method was named the ultrasonic transmission camera (UTC) (Ermert et al., 2000). Only some centres in the world work on this issue and there are a low number of laboratory research setups, which enable pseudo-real time visualisation of biological structures using UTC: Stanford Research Institute in USA (Green et al., 1974; Green et al., 1976), Gesellschaft für Strahlen- und Umweltforschung at Neuherberg in Germany (Brettel et al., 1981; Brettel et al., 1987), Siemens Corporate Technology in Germany (Ermert et al., 2000; Granz & Oppelt, 1987; Keitmann et al., 2002), Wroclaw University of Technology in Poland (Opielinski & Gudra, 2000; Opielinski & Gudra, 2005, Opielinski et al., 2010a, 2010b), University of California in San Diego and University of Washington Medical Center in Seattle (Lehmann et al., 1999). In most of studies, the projection parameter is the signal amplitude, not its transition time, which seems to more attractive due to the simplicity and precision of measurements. The majority of 2-D ultrasonic multi-element matrices are designed for miniature 3-D volumetric medical endoscopic imaging as intracavital probes provided unique opportunities for guiding surgeries or minimally invasive therapeutic procedures (Eames & Hossack, 2008; Karaman et al., 2009; Wygant et al., 2006a; Wygant et al., 2006b). It can be concluded based on worldwide literature review that most of the 2-D ultrasonic matrices are assigned to work of echo method (Drinkwater & Wilcox, 2006). Moreover, the commercial devices (e.g. Submersible Ultrasonic Scanning Camera made of Matec Micro Electronics, AcoustoCam produced by Imperium Inc.) work using reflection method and are designed for nondestructive inspection (NDI), most of all. It allows manufacturers to instantly visualize a

variety of material subsurface faults (voids, delaminating, cracks and corrosion).

Krzysztof J. Opieliński


## **Ultrasonic Projection**

Krzysztof J. Opieliński

*Institute of Telecommunications, Teleinformatics and Acoustics, Wroclaw University of Technology Poland* 

## **1. Introduction**

28 Will-be-set-by-IN-TECH

28 Ultrasonic Waves

McNiven, H., Sackman, J. & Shah, A. (1963). Dispersion of axially symmetric waves in composite, elastic rods, *Journal of the Acoustical Society of America* 35(10): 1602–1609. Meeker, T. R. & Meitzler, A. H. (1972). Guided wave propagation in elongated cylinders

Mirsky, I. (1964). Axisymmetric vibrations of orthotropic cylinders, *Journal of the Acoustical*

Mirsky, I. (1965). Wave propagation in transversely isotropic circular cylinders: Part I: theory;

Nagy, P. B. & Nayfeh, A. H. (1996). Viscosity-induced attenuation of longitudinal guided

Nayfeh, A. H. & Nagy, P. B. (1996). General study of axisymmetric waves in layered anisotropic fibers and their composites, *J. Acoust. Soc. Am.* 99(2): 931–941. Nelson, R., Dong, S. & Kalra, R. (1971). Vibrations and waves in laminated orthotropic circular

Oppenheim, A. V., Schafer, R. W. & Buck, J. R. (1999). *Discrete-Time Signal Processing*, 2 edn,

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Press, W. H., Teukolsky, S. A., Vetterling, W. T. & Flannery, B. P. (1992). *Numerical Recipes in C.*

Rose, J. L. (2000). Guided wave nuances for ultrasonic nondestructive evaluation, *IEEE Trans.*

Seco, F., Martín, J. M. & Jiménez, A. R. (2009). Improving the accuracy of magnetostrictive

Sinha, B. K., Plona, T. J., Kostek, S. & Chang, S.-K. (1992). Axisymmetric wave propagation in fluid-loaded cylindrical shells. i: Theory, *J. Acoust. Soc. Am.* 92(2): 1132–1143. Whittier, J. & Jones, J. (1967). Axially symmetric wave propagation in a two-layered cylinder,

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linear position sensors, *IEEE Trans. on Instrumentation and Measurement* 58(3): 722–729.

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Prentice Hall.

and plates, *in* W. P. Mason & R. N. Thurston (eds), *Physical Acoustics, Principles and*

part II: numerical results, *Journal of the Acoustical Society of America* 37(6): 1016–1026.

Ultrasonic technique of imaging serves an increasingly important role in medical diagnostics. In most of applications, echographic methods are used (ultrasonography, ultrasonic microscopy). Using such methods, an image presenting changes of a reflection coefficient in the interior of analysed structure is being constructed. This chapter presents possibilities of utilising information included in ultrasonic pulses, which penetrate an object in order to create images presenting the projection of analysed structure (Opielinski & Gudra, 2004b, 2004c, 2005, 2006, 2008, 2010a, 2010b, 2010c; Opielinski et al., 2009, 2010a, 2010b) in the form of a distribution of mean values of a measured acoustic parameter, for one or numerous planes, perpendicular to the direction of ultrasonic waves incidence (analogical as in X-ray radiography). Due to the possibility of obtaining images in pseudoreal time, the device using this method was named the ultrasonic transmission camera (UTC) (Ermert et al., 2000). Only some centres in the world work on this issue and there are a low number of laboratory research setups, which enable pseudo-real time visualisation of biological structures using UTC: Stanford Research Institute in USA (Green et al., 1974; Green et al., 1976), Gesellschaft für Strahlen- und Umweltforschung at Neuherberg in Germany (Brettel et al., 1981; Brettel et al., 1987), Siemens Corporate Technology in Germany (Ermert et al., 2000; Granz & Oppelt, 1987; Keitmann et al., 2002), Wroclaw University of Technology in Poland (Opielinski & Gudra, 2000; Opielinski & Gudra, 2005, Opielinski et al., 2010a, 2010b), University of California in San Diego and University of Washington Medical Center in Seattle (Lehmann et al., 1999). In most of studies, the projection parameter is the signal amplitude, not its transition time, which seems to more attractive due to the simplicity and precision of measurements. The majority of 2-D ultrasonic multi-element matrices are designed for miniature 3-D volumetric medical endoscopic imaging as intracavital probes provided unique opportunities for guiding surgeries or minimally invasive therapeutic procedures (Eames & Hossack, 2008; Karaman et al., 2009; Wygant et al., 2006a; Wygant et al., 2006b). It can be concluded based on worldwide literature review that most of the 2-D ultrasonic matrices are assigned to work of echo method (Drinkwater & Wilcox, 2006). Moreover, the commercial devices (e.g. Submersible Ultrasonic Scanning Camera made of Matec Micro Electronics, AcoustoCam produced by Imperium Inc.) work using reflection method and are designed for nondestructive inspection (NDI), most of all. It allows manufacturers to instantly visualize a variety of material subsurface faults (voids, delaminating, cracks and corrosion).

Ultrasonic Projection 31

Phenomena related with propagation of ultrasonic waves in biological structures (diffusion, diffraction, interference, refraction and reflection) induce small distortions of a projection image, if the local values of acoustic impedance in an examined structure are not significantly diversified (Opielinski & Gudra, 2000). The greatest advantage of ultrasonic diagnostics is its non-invasiveness, by dint of which, it is possible to attain a multiple projection of an examined biological structure at numerous different directions *in vivo*, what enables a three-dimensional reconstruction of heterogeneity borders in its interior

By the means of using the transmission method, it is also possible to obtain a twice higher level of amplitude of ultrasonic receiving pulses in comparison with the echo method. In this case, the subject of imaging can simultaneously be several acoustic parameters, digitally determined on the basis of information directly contained in ultrasonic pulses, which penetrate a biological structure (e.g. amplitude, transition time, mid-frequency down shift, spectrum of receiving pulse). Such method enables obtaining various projection images, every of which characterises slightly different traits of a structure (e.g. distribution of mean (projective) values of attenuation coefficient and propagation velocity of ultrasonic waves, frequency derivative of attenuation coefficient, nonlinear acoustic parameter *B*/*A* (Greenleaf & Sehgal, 1992; Kak & Slaney, 1988). What is more, if projective measurements of a distribution of a specified acoustic parameter of an examined projection plane of a biological medium, are recorded from numerous directions around the medium, it is possible to reconstruct a distribution of local values of this parameter in 3-D space by determining the inverse Radon transform (Kak & Slaney, 1988; Opielinski & Gudra, 2010b) (ultrasonic transmission tomography). Such complex tomographic characteristics may have a key importance, for example at detecting and

According to the definition of function projection, projective values of an acoustic parameter, measured at projections, are an integral of their local values, in the path of an ultrasonic beam, transmitted from a source to a detector. Moreover, point sizes of the transmitter and the receiver, the rectilinear path between them and an infinitely narrow ultrasonic wave beam are assumed. It is easy to prove that transition time of an ultrasonic wave is an integer of an inverse of local values of sound velocity through the propagation

*p*

where *c*(*x*,*y*,*T*) denotes the sound speed local value at point (*x*,*y*) of an object's cross-section on the propagation path, at set temperature *T,* (*dl*)2 = *x*2 + *y*2. By a direct measurement of values of transition time of an ultrasonic wave, it is not difficult to image a distribution of projective values of sound velocity *cp* = *L*/*tp* in determined plane of projection of a biological

By the means of measuring the amplitude of an ultrasonic pulse after a transition through the structure of a biological medium, it is possible to obtain information about the projective

*LL L dt t dt = = dl = dl* 

1 ( ,)

*dl c x,y T* (1)

diagnosing cancerous changes in soft tissues (e.g. in female breast).

*p p*

medium, immersed in water, assuming the linear propagation path of wave.

**3. Projection of acoustic parameters** 

path *L* (Opielinski & Gudra, 2010b):

(Opielinski & Gudra, 2004a).

This chapter, with the use of a computer simulation and real measurements, includes the complex analysis of the precision of images obtained using visualisation of mean values of sound speed and a frequency derivative of the ultrasonic wave amplitude attenuation coefficient (by the means of measurements of the transition time and the frequency down shift projection values of transmitted ultrasonic pulses) from the point of view of possibilities of using UTC for visualisation of biological structures, especially for soft tissue examinations (*in vivo* female breast). The principles of ultrasonic projection methods are described at the beginning (Section 2), and then the projections of acoustic parameters are clearly defined (Section 3). Section 4 contains theoretical analysis of ultrasonic projection method accuracy. By the means of elaborated software, there was done a simulation of the ultrasonic projection data for several three-dimensional objects immersed in water, what is presented in Section 5, as well as obtained projection images of these objects. The simple measurement setup for examining biological structures by the means of ultrasonic projection, in the set of single-element ultrasonic sending and receiving probes and projection measurement results are presented in Section 6. Next, the construction, parameters and operating way of three different types of ultrasonic 2-D flat multi-element matrices (standard, passive and active one) elaborated by the author and his team, are described (Section 7). The models of ultrasonic transmission camera were constructed in the result, what is presented in Section 8. The conclusion (Section 9) contains a summary and the plane for the future to improve the quality of ultrasonic projection images and increase scanning resolution.

## **2. Ultrasonic projection methods**

Analogically as in case of X-ray pictures (RTG - roentgenography), for visualisation of biological structures, it is possible to use the projection method with utilisation of ultrasonic waves. In case of generating of ultrasonic plane waves (parallel beam rays), we shall obtain an image in the parallel projection and in case of generating of ultrasonic spherical waves (divergent beam rays), it shall be an image in the central projection; it is also possible to use a source of cylindrical waves (beam rays are divergent on one plane and parallel in perpendicular plane) – central-parallel projection (Fig.1) (Opielinski & Gudra, 2004c).

Fig. 1. The way of visualization of a biological structure by means of the ultrasonic projection: a) parallel, b) central (divergent), c) central–parallel (divergent–parallel)

This chapter, with the use of a computer simulation and real measurements, includes the complex analysis of the precision of images obtained using visualisation of mean values of sound speed and a frequency derivative of the ultrasonic wave amplitude attenuation coefficient (by the means of measurements of the transition time and the frequency down shift projection values of transmitted ultrasonic pulses) from the point of view of possibilities of using UTC for visualisation of biological structures, especially for soft tissue examinations (*in vivo* female breast). The principles of ultrasonic projection methods are described at the beginning (Section 2), and then the projections of acoustic parameters are clearly defined (Section 3). Section 4 contains theoretical analysis of ultrasonic projection method accuracy. By the means of elaborated software, there was done a simulation of the ultrasonic projection data for several three-dimensional objects immersed in water, what is presented in Section 5, as well as obtained projection images of these objects. The simple measurement setup for examining biological structures by the means of ultrasonic projection, in the set of single-element ultrasonic sending and receiving probes and projection measurement results are presented in Section 6. Next, the construction, parameters and operating way of three different types of ultrasonic 2-D flat multi-element matrices (standard, passive and active one) elaborated by the author and his team, are described (Section 7). The models of ultrasonic transmission camera were constructed in the result, what is presented in Section 8. The conclusion (Section 9) contains a summary and the plane for the future to improve the quality of ultrasonic projection images and increase

Analogically as in case of X-ray pictures (RTG - roentgenography), for visualisation of biological structures, it is possible to use the projection method with utilisation of ultrasonic waves. In case of generating of ultrasonic plane waves (parallel beam rays), we shall obtain an image in the parallel projection and in case of generating of ultrasonic spherical waves (divergent beam rays), it shall be an image in the central projection; it is also possible to use a source of cylindrical waves (beam rays are divergent on one plane and parallel in

(a) (b) (c)

Fig. 1. The way of visualization of a biological structure by means of the ultrasonic projection: a) parallel, b) central (divergent), c) central–parallel (divergent–parallel)

perpendicular plane) – central-parallel projection (Fig.1) (Opielinski & Gudra, 2004c).

scanning resolution.

**2. Ultrasonic projection methods** 

Phenomena related with propagation of ultrasonic waves in biological structures (diffusion, diffraction, interference, refraction and reflection) induce small distortions of a projection image, if the local values of acoustic impedance in an examined structure are not significantly diversified (Opielinski & Gudra, 2000). The greatest advantage of ultrasonic diagnostics is its non-invasiveness, by dint of which, it is possible to attain a multiple projection of an examined biological structure at numerous different directions *in vivo*, what enables a three-dimensional reconstruction of heterogeneity borders in its interior (Opielinski & Gudra, 2004a).

By the means of using the transmission method, it is also possible to obtain a twice higher level of amplitude of ultrasonic receiving pulses in comparison with the echo method. In this case, the subject of imaging can simultaneously be several acoustic parameters, digitally determined on the basis of information directly contained in ultrasonic pulses, which penetrate a biological structure (e.g. amplitude, transition time, mid-frequency down shift, spectrum of receiving pulse). Such method enables obtaining various projection images, every of which characterises slightly different traits of a structure (e.g. distribution of mean (projective) values of attenuation coefficient and propagation velocity of ultrasonic waves, frequency derivative of attenuation coefficient, nonlinear acoustic parameter *B*/*A* (Greenleaf & Sehgal, 1992; Kak & Slaney, 1988). What is more, if projective measurements of a distribution of a specified acoustic parameter of an examined projection plane of a biological medium, are recorded from numerous directions around the medium, it is possible to reconstruct a distribution of local values of this parameter in 3-D space by determining the inverse Radon transform (Kak & Slaney, 1988; Opielinski & Gudra, 2010b) (ultrasonic transmission tomography). Such complex tomographic characteristics may have a key importance, for example at detecting and diagnosing cancerous changes in soft tissues (e.g. in female breast).

## **3. Projection of acoustic parameters**

According to the definition of function projection, projective values of an acoustic parameter, measured at projections, are an integral of their local values, in the path of an ultrasonic beam, transmitted from a source to a detector. Moreover, point sizes of the transmitter and the receiver, the rectilinear path between them and an infinitely narrow ultrasonic wave beam are assumed. It is easy to prove that transition time of an ultrasonic wave is an integer of an inverse of local values of sound velocity through the propagation path *L* (Opielinski & Gudra, 2010b):

$$\mathbf{t}\_p = \int\_L dt\_p = \int\_L \frac{dt\_p}{dl} dl = \int\_L \frac{1}{c \left(\mathbf{x}\_r \mathbf{y}\_r T\right)} dl\tag{1}$$

where *c*(*x*,*y*,*T*) denotes the sound speed local value at point (*x*,*y*) of an object's cross-section on the propagation path, at set temperature *T,* (*dl*)2 = *x*2 + *y*2. By a direct measurement of values of transition time of an ultrasonic wave, it is not difficult to image a distribution of projective values of sound velocity *cp* = *L*/*tp* in determined plane of projection of a biological medium, immersed in water, assuming the linear propagation path of wave.

By the means of measuring the amplitude of an ultrasonic pulse after a transition through the structure of a biological medium, it is possible to obtain information about the projective

Ultrasonic Projection 33

ultrasonic wave propagation velocity (mean sound velocity on a rectilinear section of the beam path from the source to the detector), obtained after wave's transition through the

Fig. 2. Theoretical model of a heterogeneous sphere (because of ultrasound propagation speed)

It was assumed that a sphere of the diameter *Ds*1 and sound speed *cs*1 = 1500 m/s contains a spherical heterogeneity of the diameter *Ds*2 and sound speed *cs*2. It was also assumed that the sound speed in water where the sphere is immersed *cmed* = 1485 m/s (for ~21C) and the distance between surfaces of the sending and receiving transducers is *Dmed* = 20 cm (simulation of female breast examination). The mean value (projection) of sound speed *cp* between surfaces of the transmitter and the receiver was determined using the formula

*cc D D c c D D c cD*

(a) (b) Fig. 3. Dependence of *cp* on |*cs*2 - *cs*1| (for *cs*2 > *cs*1) with parameter *Ds*2 /*Ds*1 (formula (5)), for

1 2 1 2 1 2 12 *med med s s*

*s s med s med s s s med s s D c cc*

1 2

(5)

(Opielinski & Gudra, 2004b, 2006):

*p*

different ratios of *Ds*1 /*Dmed* = 0.9 (a), 0.5 (b)

*c*

model of a heterogeneous sphere in its axis (Fig.2) (Opielinski & Gudra, 2004b, 2006).

value of an amplitude attenuation coefficient *αp = ln(AN/AL)/L*, measured at the set frequency of a transmitting pulse *fN* and temperature *T* (Opielinski & Gudra, 2010b):

$$\ln \frac{A\_N}{A\_L} = \int\_L \frac{1}{dl} \ln \frac{A(l\_i)}{A(l\_{i+1})} dl = \int\_L \alpha(\mathbf{x}, y, f\_{N'}, T) dl\tag{2}$$

where *AN* – amplitude of an ultrasonic pulse before transition (near the source), *AL* – amplitude of an ultrasonic pulse after transition through an object (dipped in water) on the path of *L*, *A*(*li*) and *A*(*li*+1) – amplitudes of an ultrasonic pulse after transition through an object on segments *li* and *li*+1, respectively, of the path *L*, and distance *dl* = *li*+1 – *li*. Due to difficulties related with measuring the amplitude *AN*, it is possible to directly determine the difference of projective values of the amplitude attenuation coefficients (*αp* – *αw*) = ln(*Aw*/*AL*)/*L*, where *αw* – attenuation coefficient in water, *Aw* – amplitude of an pulse after passing the path *L* in water without any biological medium.

Assuming a linear change of attenuation with frequency (as a certain approximation for soft tissues (Kak & Slaney, 1988)), it is possible to obtain an amplitude attenuation coefficient, independent from frequency *αo*(*x*,*y*,*T*):

$$a(\mathbf{x}, \mathbf{y}, T, f) \equiv \left(\alpha\_o(\mathbf{x}, \mathbf{y}, T)\right) \cdot f \tag{3}$$

On this basis, it is possible to use in ultrasonic projection measurements the method of measuring down shifting of mid-frequency of a receiving pulse. After transition of an ultrasonic signal through an object, there is a slight change of its mid-frequency *fr*, which can be measured by FFT or by a zero-crossing counter (Opielinski & Gudra, 2010b). Then, the projective value of a frequency derivative of an amplitude attenuation coefficient *αo*(*x*,*y*,*T*) can be determined in the following form (Kak & Slaney, 1988):

$$\frac{f\_N \cdot f\_r}{2\sigma^2} = \int\_L \alpha\_o(\mathbf{x}, \mathbf{y}, T) dl \tag{4}$$

where variance *σ*2 is the measure of the power spectrum bandwidth of receiving signal after a transition through water. More complex models, e.g. with considering the relationship *α* = *αo*·*fn*, where *n* ≠ 1 (in case of soft tissues 1 ≤ *n* ≤ 2), can be found in references (Narayana & Ophir, 1983).

The development of computer technologies enables now also projective measurements of acoustic parameters of biological media, which require time-consuming calculations. One of such parameters is non-linear acoustic parameter *B*/*A*, which characterises non-linear response of a measured tissue structure on propagation of an ultrasonic wave (Opielinski & Gudra, 2010b). Parameter *B*/*A* can be determined by the means of measuring transition times for different static pressures or by measurements of higher harmonics as a distance function (Greenleaf & Sehgal, 1992; Zhang et al., 1997).

#### **4. Theoretical analysis of ultrasonic projection method accuracy**

The theoretical analysis, which enables estimating of the accuracy of the projective visualisation of heterogeneities in a tissue structure was conducted for a projection of

value of an amplitude attenuation coefficient *αp = ln(AN/AL)/L*, measured at the set frequency

1 <sup>1</sup> ( ) ln ln ( ,) ( )

where *AN* – amplitude of an ultrasonic pulse before transition (near the source), *AL* – amplitude of an ultrasonic pulse after transition through an object (dipped in water) on the path of *L*, *A*(*li*) and *A*(*li*+1) – amplitudes of an ultrasonic pulse after transition through an object on segments *li* and *li*+1, respectively, of the path *L*, and distance *dl* = *li*+1 – *li*. Due to difficulties related with measuring the amplitude *AN*, it is possible to directly determine the difference of projective values of the amplitude attenuation coefficients (*αp* – *αw*) = ln(*Aw*/*AL*)/*L*, where *αw* – attenuation coefficient in water, *Aw* – amplitude of an pulse after

Assuming a linear change of attenuation with frequency (as a certain approximation for soft tissues (Kak & Slaney, 1988)), it is possible to obtain an amplitude attenuation coefficient,

> ( ,) ( ,) *x,y,T f = x,*

On this basis, it is possible to use in ultrasonic projection measurements the method of measuring down shifting of mid-frequency of a receiving pulse. After transition of an ultrasonic signal through an object, there is a slight change of its mid-frequency *fr*, which can be measured by FFT or by a zero-crossing counter (Opielinski & Gudra, 2010b). Then, the projective value of a frequency derivative of an amplitude attenuation coefficient *αo*(*x*,*y*,*T*)

> <sup>2</sup> (,, ) <sup>2</sup> *N r <sup>o</sup> L f f - = x y T dl*

where variance *σ*2 is the measure of the power spectrum bandwidth of receiving signal after a transition through water. More complex models, e.g. with considering the relationship *α* = *αo*·*fn*, where *n* ≠ 1 (in case of soft tissues 1 ≤ *n* ≤ 2), can be found in references (Narayana &

The development of computer technologies enables now also projective measurements of acoustic parameters of biological media, which require time-consuming calculations. One of such parameters is non-linear acoustic parameter *B*/*A*, which characterises non-linear response of a measured tissue structure on propagation of an ultrasonic wave (Opielinski & Gudra, 2010b). Parameter *B*/*A* can be determined by the means of measuring transition times for different static pressures or by measurements of higher harmonics as a distance

The theoretical analysis, which enables estimating of the accuracy of the projective visualisation of heterogeneities in a tissue structure was conducted for a projection of

**4. Theoretical analysis of ultrasonic projection method accuracy** 

*<sup>A</sup> <sup>A</sup> <sup>l</sup> <sup>=</sup> dl = x,y, f T dl <sup>A</sup> dl <sup>A</sup> <sup>l</sup>*

*L L L i+*

*N*

(2)

*<sup>o</sup> y T f* (3)

(4)

of a transmitting pulse *fN* and temperature *T* (Opielinski & Gudra, 2010b):

*N i*

passing the path *L* in water without any biological medium.

can be determined in the following form (Kak & Slaney, 1988):

function (Greenleaf & Sehgal, 1992; Zhang et al., 1997).

independent from frequency *αo*(*x*,*y*,*T*):

Ophir, 1983).

ultrasonic wave propagation velocity (mean sound velocity on a rectilinear section of the beam path from the source to the detector), obtained after wave's transition through the model of a heterogeneous sphere in its axis (Fig.2) (Opielinski & Gudra, 2004b, 2006).

Fig. 2. Theoretical model of a heterogeneous sphere (because of ultrasound propagation speed)

It was assumed that a sphere of the diameter *Ds*1 and sound speed *cs*1 = 1500 m/s contains a spherical heterogeneity of the diameter *Ds*2 and sound speed *cs*2. It was also assumed that the sound speed in water where the sphere is immersed *cmed* = 1485 m/s (for ~21C) and the distance between surfaces of the sending and receiving transducers is *Dmed* = 20 cm (simulation of female breast examination). The mean value (projection) of sound speed *cp* between surfaces of the transmitter and the receiver was determined using the formula (Opielinski & Gudra, 2004b, 2006):

$$\mathcal{L}\_p = \frac{D\_{med} \, c\_{med} \, c\_{s1} \, c\_{s2}}{c\_{s1} \, c\_{s2} \left(D\_{med} - D\_{s1}\right) + c\_{med} \, c\_{s2} \left(D\_{s1} - D\_{s2}\right) + c\_{med} \, c\_{s1} D\_{s2}} \tag{5}$$

Fig. 3. Dependence of *cp* on |*cs*2 - *cs*1| (for *cs*2 > *cs*1) with parameter *Ds*2 /*Ds*1 (formula (5)), for different ratios of *Ds*1 /*Dmed* = 0.9 (a), 0.5 (b)

Ultrasonic Projection 35

For the diameter of measurement area *Dmed* = 20 cm, *fs* = 200 MHz (*Ts* = 5 ns), *cp* = 1500 m/s, resolution Δ*cp* ≈ 0.06 m/s. Using burst type ultrasonic pulses of the frequency of 5 MHz, 20

It means that at digital measuring of transition time, using the method of determining zerocrossing by the means of linear interpolation between samples of negative and positive amplitude values, it is possible to achieve the resolution of transition time measurement better than Δ*tp* ≈ *Tp*/2 = 2.5 ns (Δ*cp* ≈ 0.03 m/s); at a significant interval between a signal and noise (e.g. using algorithms of noise reduction), uncertainty of such measurements can be

Calculations show (Fig.3, Fig.4) that by the means of projection of ultrasonic wave propagation speed it is possible to detect differences of its local values in a biological structure. In case of a difference of speed in heterogeneity and its surrounding |*cs*2 - *cs*1| ≈ 0 (homogeneous sphere), all values of projection of speed *cp* tend to the determined value, marked by a dotted line in Fig.3a,b and Fig.4a,b. In order to conduct an interpretation of charts obtained from calculations, it is necessary to determine the resolution of measuring transition time of a wave through a structure. If we assume that using good class devices for digital measurements of transition time with interpolation it is possible to detect changes of sound speed of minimum about 0.005 m/s (dotted line in Fig.3a,b and Fig.4a,b) it can mean e.g. that for biological media of size of about 20 cm (in case of adjusting a distance between sending and receiving transducers to external dimensions of that media), by the means of ultrasound speed projection it is possible to detect heterogeneity of a local value of this

considering that there still is an obligatory limitation due to the diffraction of an ultrasonic wave (a wave penetrates heterogeneities of sizes larger than a half of wave's length). Resolution of a detection of differences of local sound speed values in heterogeneity and its surrounding strongly depends on the ratio of their dimensions D*s*2/*Ds*1 and the ratio of the size of an analysed object in the axis of ultrasonic transducers – sending and receiving – to the distance between the transducers D*s*1/*Dmed* (the less ratios, the worse resolution –

A similar analysis was conducted for the projection of a frequency derivative of the ultrasonic wave amplitude attenuation coefficient, obtained after a transition of a wave through the model of a heterogeneous sphere in its axis (Fig.2). In case of projective measurements of the value of a frequency derivative of the amplitude attenuation coefficient of an ultrasonic wave by the means of detecting the frequency of a pulse of an ultrasonic wave after the transition, it is possible to assume a minimal uncertainty of the measurement of the projective value of frequency of about 2 kHz, what at assuming the ultrasonic wave frequency of 2 MHz, results in the uncertainty of determining the projective value of a frequency derivative of attenuation coefficient of about 0.001 dB/(cm·MHz). It means that in the measured projective values of receiving signal frequencies, it is possible to distinguish an influence of a heterogeneity for its dimensions proper correlated with a difference of a local value of the amplitude coefficient of attenuation of an ultrasonic wave in the structure of this heterogeneity and in the structure of

samples accrues to a half of wave period.

a. 1 m/s for heterogeneity size *Ds*<sup>2</sup> ≈ 1 mm, b. 0.6 m/s for heterogeneity size *Ds*<sup>2</sup> ≈ 2 mm, c. 0.2 m/s for heterogeneity size *Ds*<sup>2</sup> ≈ 5 mm, d. 0.1 m/s for heterogeneity size *Ds*<sup>2</sup> ≈ 10 mm,

compare: Figures: 3a with 3b, and 4a with 4b).

even better than about ±Δ*cp*/6 ≈ ±0.005 m/s (formula (6)).

speed in a structure (diversification) at the minimal level of about (Fig.3a):

Figures 3a and 3b (for *cs*2 > *cs*1) and Figures 4a and 4b (for *cs*1 > *cs*2) present dependences between mean (projective) sound speed value, determined using formula (5) and the absolute value of the speed difference in heterogeneity and its spherical surrounding (Fig.2) for 29 various ratios of their diameters (0.001, 0.002, ..., 0.01, 0.02, ..., 0.1, 0.2, ..., 0.9) and for two relations between sphere diameter and distance between transducers (0.9 and 0.5) (Opielinski & Gudra, 2006).

Fig. 4. Dependence of *cp* on |*cs*2 - *cs*1| (for *cs*1 > *cs*2) with parameter *Ds*2 /*Ds*1 (formula (5)), for different ratios of *Ds*1 /*Dmed* = 0.9 (a), 0.5 (b)

If we assume that the resolution of digital measurement of transition time at sampling frequency *fs*, is 1/*fs*, then depending on the distance between surfaces of sending and receiving transducers and depending on the value of measured mean speed, the resolution of measurement of mean propagation speed of an ultrasonic wave (projection) can be determined using the formula (Fig.5) (Opielinski & Gudra, 2004b):

$$
\Delta c\_p = \frac{D\_{med}}{D\_{med} \;/\ c\_p - 1 \;/\ f\_s} - c\_p \tag{6}
$$

Fig. 5. Dependence of Δ*cp* on *Dmed* (formula (6)) for different values of *cp* in a measurement area

Figures 3a and 3b (for *cs*2 > *cs*1) and Figures 4a and 4b (for *cs*1 > *cs*2) present dependences between mean (projective) sound speed value, determined using formula (5) and the absolute value of the speed difference in heterogeneity and its spherical surrounding (Fig.2) for 29 various ratios of their diameters (0.001, 0.002, ..., 0.01, 0.02, ..., 0.1, 0.2, ..., 0.9) and for two relations between sphere diameter and distance between transducers (0.9 and 0.5)

(a) (b) Fig. 4. Dependence of *cp* on |*cs*2 - *cs*1| (for *cs*1 > *cs*2) with parameter *Ds*2 /*Ds*1 (formula (5)), for

If we assume that the resolution of digital measurement of transition time at sampling frequency *fs*, is 1/*fs*, then depending on the distance between surfaces of sending and receiving transducers and depending on the value of measured mean speed, the resolution of measurement of mean propagation speed of an ultrasonic wave (projection) can be

> / 1/ *med p p med p s D c c Dc f*

Fig. 5. Dependence of Δ*cp* on *Dmed* (formula (6)) for different values of *cp* in a measurement area

(6)

(Opielinski & Gudra, 2006).

different ratios of *Ds*1 /*Dmed* = 0.9 (a), 0.5 (b)

determined using the formula (Fig.5) (Opielinski & Gudra, 2004b):

For the diameter of measurement area *Dmed* = 20 cm, *fs* = 200 MHz (*Ts* = 5 ns), *cp* = 1500 m/s, resolution Δ*cp* ≈ 0.06 m/s. Using burst type ultrasonic pulses of the frequency of 5 MHz, 20 samples accrues to a half of wave period.

It means that at digital measuring of transition time, using the method of determining zerocrossing by the means of linear interpolation between samples of negative and positive amplitude values, it is possible to achieve the resolution of transition time measurement better than Δ*tp* ≈ *Tp*/2 = 2.5 ns (Δ*cp* ≈ 0.03 m/s); at a significant interval between a signal and noise (e.g. using algorithms of noise reduction), uncertainty of such measurements can be even better than about ±Δ*cp*/6 ≈ ±0.005 m/s (formula (6)).

Calculations show (Fig.3, Fig.4) that by the means of projection of ultrasonic wave propagation speed it is possible to detect differences of its local values in a biological structure. In case of a difference of speed in heterogeneity and its surrounding |*cs*2 - *cs*1| ≈ 0 (homogeneous sphere), all values of projection of speed *cp* tend to the determined value, marked by a dotted line in Fig.3a,b and Fig.4a,b. In order to conduct an interpretation of charts obtained from calculations, it is necessary to determine the resolution of measuring transition time of a wave through a structure. If we assume that using good class devices for digital measurements of transition time with interpolation it is possible to detect changes of sound speed of minimum about 0.005 m/s (dotted line in Fig.3a,b and Fig.4a,b) it can mean e.g. that for biological media of size of about 20 cm (in case of adjusting a distance between sending and receiving transducers to external dimensions of that media), by the means of ultrasound speed projection it is possible to detect heterogeneity of a local value of this speed in a structure (diversification) at the minimal level of about (Fig.3a):


considering that there still is an obligatory limitation due to the diffraction of an ultrasonic wave (a wave penetrates heterogeneities of sizes larger than a half of wave's length). Resolution of a detection of differences of local sound speed values in heterogeneity and its surrounding strongly depends on the ratio of their dimensions D*s*2/*Ds*1 and the ratio of the size of an analysed object in the axis of ultrasonic transducers – sending and receiving – to the distance between the transducers D*s*1/*Dmed* (the less ratios, the worse resolution – compare: Figures: 3a with 3b, and 4a with 4b).

A similar analysis was conducted for the projection of a frequency derivative of the ultrasonic wave amplitude attenuation coefficient, obtained after a transition of a wave through the model of a heterogeneous sphere in its axis (Fig.2). In case of projective measurements of the value of a frequency derivative of the amplitude attenuation coefficient of an ultrasonic wave by the means of detecting the frequency of a pulse of an ultrasonic wave after the transition, it is possible to assume a minimal uncertainty of the measurement of the projective value of frequency of about 2 kHz, what at assuming the ultrasonic wave frequency of 2 MHz, results in the uncertainty of determining the projective value of a frequency derivative of attenuation coefficient of about 0.001 dB/(cm·MHz). It means that in the measured projective values of receiving signal frequencies, it is possible to distinguish an influence of a heterogeneity for its dimensions proper correlated with a difference of a local value of the amplitude coefficient of attenuation of an ultrasonic wave in the structure of this heterogeneity and in the structure of

Ultrasonic Projection 37

Figure 7 presents images of ultrasonic projection for objects *C* and *D* obtained on the basis of computer simulated measurements of mean values of sound speed in an orthogonal projection (Opielinski & Gudra, 2004c). These values were linearly imaged in greyscale, from black to white. Shapes of projected objects, together with edges, are clearly visible in images. The object *C* interior seems to be homogeneous (Fig.7a) and spherical structures inside object *D* (Fig.7b) are hardly visible against the background of the ellipsoid projection

(a) (b)

(a) (b)

Fig. 8. Ultrasonic projection images of sphere – object *C* (a) and ellipsoid – object *D* (b),

Fig. 7. Ultrasonic projection images of objects shown in Fig.6, obtained on the basis of computer-simulated measurements of mean values of ultrasonic wave propagation velocity:

(bright, round spots).

a) object *C*, b) object *D*

shown in pseudo-3-D

its surroundings Δ*α*. If the presence of a heterogeneity changes measured projective values of a receiving pulse frequency, this heterogeneity is able to be detected. Calculations prove that at assuming the total uncertainty of projective measurements of values of the receiving pulse frequency of 2 kHz, it shall be possible to detect in a projective image heterogeneities, which differ from surrounding tissue by the minimum value of sound attenuation about: 0.1 dB/cm of the size > 9 mm, 0.2 dB/cm of the size > 4 mm, 0.5 dB/cm of the size > 2 mm, 1 dB/cm of the size > 900 μm, 2 dB/cm of the size > 400 μm, 5 dB/cm of the size > 200 μm. Calculated contrast resolution increases at higher spectrum width of the receiving pulse and a change of sound attenuation in water around a sphere has a negligible influence on this resolution.

In case of heterogeneities of diversified sizes, it can happen that in projective images for several projection planes, heterogeneities can be visible, while in others can be not.

## **5. Computer simulation of ultrasonic projection data**

By the means of elaborated software, there was done a simulation of a distribution of local values of propagation velocity of an ultrasonic wave inside several three-dimensional objects immersed in water (Fig.6), and next, for a selected projection plane, there was done a calculation (using the Radon transform (Kak & Slaney, 1988)) of distributions of mean values (projections) in a parallel-ray geometry, at the set distance *Dmed* = 20 cm between surfaces of the transmitter and the receiver (Opielinski & Gudra, 2004c, 2006). In the space surrounding each of the objects, the speed of ultrasonic wave propagation was assumed at *cmed* = 1485 m/s, what corresponds to the speed of sound in water of the temperature of about 21°C. In case of ultrasonic projection, using water as a coupling medium is obligatory due to a matching to the acoustic impedance of an examined biological structure. The Section 5 presents simulations of projective data for exemplary virtual objects *C* and *D*. Object *C* is a sphere of the diameter of 14 cm, containing 7 homogeneous balls of 0.5 cm diameters each, situated on the major axis, at 2 cm intervals between their centres, apart from two terminal ones - 1.85 cm intervals. Object *D* is an ellipsoid of the semi-major axis of 7 cm and the 1 cm semi-minor axis, containing 7 homogeneous balls of 0.5 cm diameters each, located on the diameter in the same way as in the object *C*. Local values of ultrasonic wave propagation speed are 1500 m/s for each point of the interior of objects *C* and *D*, apart from small balls – 1499 m/s.

Fig. 6. Shape of created 3-D virtual objects in cross-section: a) object *C* – small balls in sphere, b) object *D* – small balls in ellipsoid

its surroundings Δ*α*. If the presence of a heterogeneity changes measured projective values of a receiving pulse frequency, this heterogeneity is able to be detected. Calculations prove that at assuming the total uncertainty of projective measurements of values of the receiving pulse frequency of 2 kHz, it shall be possible to detect in a projective image heterogeneities, which differ from surrounding tissue by the minimum value of sound attenuation about: 0.1 dB/cm of the size > 9 mm, 0.2 dB/cm of the size > 4 mm, 0.5 dB/cm of the size > 2 mm, 1 dB/cm of the size > 900 μm, 2 dB/cm of the size > 400 μm, 5 dB/cm of the size > 200 μm. Calculated contrast resolution increases at higher spectrum width of the receiving pulse and a change of sound attenuation in water around a sphere has a negligible influence on this resolution.

In case of heterogeneities of diversified sizes, it can happen that in projective images for

By the means of elaborated software, there was done a simulation of a distribution of local values of propagation velocity of an ultrasonic wave inside several three-dimensional objects immersed in water (Fig.6), and next, for a selected projection plane, there was done a calculation (using the Radon transform (Kak & Slaney, 1988)) of distributions of mean values (projections) in a parallel-ray geometry, at the set distance *Dmed* = 20 cm between surfaces of the transmitter and the receiver (Opielinski & Gudra, 2004c, 2006). In the space surrounding each of the objects, the speed of ultrasonic wave propagation was assumed at *cmed* = 1485 m/s, what corresponds to the speed of sound in water of the temperature of about 21°C. In case of ultrasonic projection, using water as a coupling medium is obligatory due to a matching to the acoustic impedance of an examined biological structure. The Section 5 presents simulations of projective data for exemplary virtual objects *C* and *D*. Object *C* is a sphere of the diameter of 14 cm, containing 7 homogeneous balls of 0.5 cm diameters each, situated on the major axis, at 2 cm intervals between their centres, apart from two terminal ones - 1.85 cm intervals. Object *D* is an ellipsoid of the semi-major axis of 7 cm and the 1 cm semi-minor axis, containing 7 homogeneous balls of 0.5 cm diameters each, located on the diameter in the same way as in the object *C*. Local values of ultrasonic wave propagation speed are 1500 m/s for each point of the

(a) (b) Fig. 6. Shape of created 3-D virtual objects in cross-section: a) object *C* – small balls in sphere,

several projection planes, heterogeneities can be visible, while in others can be not.

**5. Computer simulation of ultrasonic projection data** 

interior of objects *C* and *D*, apart from small balls – 1499 m/s.

b) object *D* – small balls in ellipsoid

Figure 7 presents images of ultrasonic projection for objects *C* and *D* obtained on the basis of computer simulated measurements of mean values of sound speed in an orthogonal projection (Opielinski & Gudra, 2004c). These values were linearly imaged in greyscale, from black to white. Shapes of projected objects, together with edges, are clearly visible in images. The object *C* interior seems to be homogeneous (Fig.7a) and spherical structures inside object *D* (Fig.7b) are hardly visible against the background of the ellipsoid projection (bright, round spots).

Fig. 7. Ultrasonic projection images of objects shown in Fig.6, obtained on the basis of computer-simulated measurements of mean values of ultrasonic wave propagation velocity: a) object *C*, b) object *D*

Fig. 8. Ultrasonic projection images of sphere – object *C* (a) and ellipsoid – object *D* (b), shown in pseudo-3-D

Ultrasonic Projection 39

probe by a digital oscilloscope are sent through RS232 bus to a computer and stored on a hard disk. Projective values of proper acoustic parameters, determined from particular receiving pulses, are imaged in the contour form in colour or grey scale and also in pseudo-3-D, using special software, which enables also advanced image processing (Opielinski & Gudra, 2005).

Fig. 9. Block scheme of the measurement setup for biological media structure imaging by

On the developed research setup, measurements of several 3-D biological objects were conducted in various scanning planes (Opielinski & Gudra, 2004b, 2004c, 2005, 2006). One of real biological media that was subjected to projective measurements was a hard-boiled chicken egg without a shell. Proteins composed of amino-acids are present in all living organisms, both animals and plants, and they are the most essential elements, being the basic structural material of tissues. A chicken egg is an easy accessible bio-molecular sample for ultrasonic examinations. Due to its oval shape (a possibility to transmit ultrasonic waves from numerous directions around it), structure and acoustic parameters (a relatively low attenuation and a slight refraction of beam rays of an ultrasonic waves at boundaries water/white/yolk), a boiled chicken egg without a shell is a great object, enabling testing of visualisation of biological structures by the means of ultrasonic projection and the method of ultrasonic transmission tomography (Opielinski, 2007). Figure 10 presents obtained projective images (the transmission method, *f* = 5 MHz) in comparison to an image obtained from an ultrasonography (the reflection method, *f* = 3.5 MHz) of a hard-boiled chicken egg without a shell. On the basis of the set of recorded receiving pulses in one scanning plane, with the step of 1.5 mm x 1.5 mm, images presenting the projection of a distribution of three various acoustic parameters in the object structure were obtained: propagation velocity, frequency derivative of ultrasonic waves attenuation and ultrasonic wave amplitude after transition (Opielinski & Gudra, 2004b, 2004c, 2005). In the images, distributions of particular parameters are marked by a solid line for pixels along marked broken lines. Negative values of a derivative of the ultrasonic wave attenuation on frequency in the image in Fig.10b result

means of the ultrasonic projection method using one-element ultrasonic probes

In order to improve the contrast of visualisation of heterogeneous structures, Figure 8 presents ultrasonic projection images of objects *C* and *D* in pseudo-three-dimensional way with an lighting, imaging mean values of propagation speed of an ultrasonic wave for the plane parallel to the axis, along which small balls are arranged in the Cartesian co-ordinate system. Here, projections of spherical heterogeneities are shown in the form of characteristic oval concavities in the object's structure. Difficulties in imaging inclusions in the object *D* structure result from the projective method of averaging values of ultrasonic wave speed on the propagation path. In the projection images (Fig.7), objects' shapes and their edges are clearly visible in the parallel projection. Small spherical heterogeneities inside the sphere are hardly evident on the projective pseudo-three-dimensional image (Fig.8) due to scant sizes in comparison to the diameter of surrounding sphere.

Conducted simulative calculations enable an initial estimation of the accuracy of the ultrasonic projective imaging in respect of detecting heterogeneities in the internal structure. In the contour image of object *C* (Fig.7a), structure heterogeneities are imperceptible and in the contour image of object *D* (Fig.7b) – hardly visible. The reason of such imaging in ultrasonic projection is mainly a small difference of propagation velocity of an ultrasonic wave in heterogeneities in comparison to propagation in their surrounding (1 m/s). Projection values of sound velocity of propagation of an ultrasonic wave, corresponding to central pixels of the image from Figures 7a and 7b, are equal to 1495.4185 m/s and 1486.4374 m/s, respectively. If structures of objects *C* and *D* were homogeneous, these values would be equal to 1495.4683 m/s and 1486.4865 m/s, what means that an inclusion in the form of a central ball changes the velocity by about 0.05 m/s in both cases. Thus, such small change is not visible in contour images, due to a limited human eye's ability of distinguishing shades; however, it is noticeable in pseudo-3-D images (Fig.8). It seems that in ultrasonic projection it is possible to image even smaller differences in velocity values, thanks to using additional operations of data processing in the form of algorithms of compression, expansion, gating, filtration and limitation of measurement data values (Opielinski & Gudra, 2000).

The change of mean speed caused by an inclusion is approximately the same for objects *C*  and *D*, however, their widths are significantly various. It means that a deterioration of the dynamics of velocity values simulated for object *C* is caused by too big distance between sending and receiving transducers – sound speed in the measurement medium (water) has a negative influence on measurement results. Due to the above, such distance should be possibly slightly higher from the size of an object in the analysed projection.

## **6. Projection measurements**

The developed, computer-assisted measurement setup for examining biological structures by the means of ultrasonic projection, in the set of single-element ultrasonic probes - sending and receiving - is presented in Fig.9 (Opielinski & Gudra, 2004c, 2005).

For the aims of scanning of objects immersed in water, there were used two ultrasonic probes of 5 mm diameter, which played roles of a source and a detector of ultrasonic waves, of work frequency of 5 MHz (Opielinski & Gudra, 2005). Probes, mounted on the axis opposite each other, are shifted with a meander move with a set step in a selected plane of an object. Movements of probes are controlled by the means of a computer software through RS232 bus, using mechanisms of shifting *XYZ* (Opielinski & Gudra, 2005). The sending probe is supplied by a burst-type sinusoidal signal and pulses received from the receiving

In order to improve the contrast of visualisation of heterogeneous structures, Figure 8 presents ultrasonic projection images of objects *C* and *D* in pseudo-three-dimensional way with an lighting, imaging mean values of propagation speed of an ultrasonic wave for the plane parallel to the axis, along which small balls are arranged in the Cartesian co-ordinate system. Here, projections of spherical heterogeneities are shown in the form of characteristic oval concavities in the object's structure. Difficulties in imaging inclusions in the object *D* structure result from the projective method of averaging values of ultrasonic wave speed on the propagation path. In the projection images (Fig.7), objects' shapes and their edges are clearly visible in the parallel projection. Small spherical heterogeneities inside the sphere are hardly evident on the projective pseudo-three-dimensional image (Fig.8) due to scant sizes

Conducted simulative calculations enable an initial estimation of the accuracy of the ultrasonic projective imaging in respect of detecting heterogeneities in the internal structure. In the contour image of object *C* (Fig.7a), structure heterogeneities are imperceptible and in the contour image of object *D* (Fig.7b) – hardly visible. The reason of such imaging in ultrasonic projection is mainly a small difference of propagation velocity of an ultrasonic wave in heterogeneities in comparison to propagation in their surrounding (1 m/s). Projection values of sound velocity of propagation of an ultrasonic wave, corresponding to central pixels of the image from Figures 7a and 7b, are equal to 1495.4185 m/s and 1486.4374 m/s, respectively. If structures of objects *C* and *D* were homogeneous, these values would be equal to 1495.4683 m/s and 1486.4865 m/s, what means that an inclusion in the form of a central ball changes the velocity by about 0.05 m/s in both cases. Thus, such small change is not visible in contour images, due to a limited human eye's ability of distinguishing shades; however, it is noticeable in pseudo-3-D images (Fig.8). It seems that in ultrasonic projection it is possible to image even smaller differences in velocity values, thanks to using additional operations of data processing in the form of algorithms of compression, expansion, gating,

filtration and limitation of measurement data values (Opielinski & Gudra, 2000).

possibly slightly higher from the size of an object in the analysed projection.

and receiving - is presented in Fig.9 (Opielinski & Gudra, 2004c, 2005).

**6. Projection measurements** 

The change of mean speed caused by an inclusion is approximately the same for objects *C*  and *D*, however, their widths are significantly various. It means that a deterioration of the dynamics of velocity values simulated for object *C* is caused by too big distance between sending and receiving transducers – sound speed in the measurement medium (water) has a negative influence on measurement results. Due to the above, such distance should be

The developed, computer-assisted measurement setup for examining biological structures by the means of ultrasonic projection, in the set of single-element ultrasonic probes - sending

For the aims of scanning of objects immersed in water, there were used two ultrasonic probes of 5 mm diameter, which played roles of a source and a detector of ultrasonic waves, of work frequency of 5 MHz (Opielinski & Gudra, 2005). Probes, mounted on the axis opposite each other, are shifted with a meander move with a set step in a selected plane of an object. Movements of probes are controlled by the means of a computer software through RS232 bus, using mechanisms of shifting *XYZ* (Opielinski & Gudra, 2005). The sending probe is supplied by a burst-type sinusoidal signal and pulses received from the receiving

in comparison to the diameter of surrounding sphere.

probe by a digital oscilloscope are sent through RS232 bus to a computer and stored on a hard disk. Projective values of proper acoustic parameters, determined from particular receiving pulses, are imaged in the contour form in colour or grey scale and also in pseudo-3-D, using special software, which enables also advanced image processing (Opielinski & Gudra, 2005).

Fig. 9. Block scheme of the measurement setup for biological media structure imaging by means of the ultrasonic projection method using one-element ultrasonic probes

On the developed research setup, measurements of several 3-D biological objects were conducted in various scanning planes (Opielinski & Gudra, 2004b, 2004c, 2005, 2006). One of real biological media that was subjected to projective measurements was a hard-boiled chicken egg without a shell. Proteins composed of amino-acids are present in all living organisms, both animals and plants, and they are the most essential elements, being the basic structural material of tissues. A chicken egg is an easy accessible bio-molecular sample for ultrasonic examinations. Due to its oval shape (a possibility to transmit ultrasonic waves from numerous directions around it), structure and acoustic parameters (a relatively low attenuation and a slight refraction of beam rays of an ultrasonic waves at boundaries water/white/yolk), a boiled chicken egg without a shell is a great object, enabling testing of visualisation of biological structures by the means of ultrasonic projection and the method of ultrasonic transmission tomography (Opielinski, 2007). Figure 10 presents obtained projective images (the transmission method, *f* = 5 MHz) in comparison to an image obtained from an ultrasonography (the reflection method, *f* = 3.5 MHz) of a hard-boiled chicken egg without a shell. On the basis of the set of recorded receiving pulses in one scanning plane, with the step of 1.5 mm x 1.5 mm, images presenting the projection of a distribution of three various acoustic parameters in the object structure were obtained: propagation velocity, frequency derivative of ultrasonic waves attenuation and ultrasonic wave amplitude after transition (Opielinski & Gudra, 2004b, 2004c, 2005). In the images, distributions of particular parameters are marked by a solid line for pixels along marked broken lines. Negative values of a derivative of the ultrasonic wave attenuation on frequency in the image in Fig.10b result

Ultrasonic Projection 41

These images can be properly processed and correlated by the means of special software, what enables recognising structures, which are not visible in single images. The image of the distribution of the sound speed projection values clearly visualises constant changes of heterogeneity (Fig.10a), while the image of the distribution of sound attenuation frequency derivative projection values better visualises discrete changes (Fig.10b). In the image of the distribution of sound velocity projection values, it is also visible that sound speed in a yolk is higher than in water and lower than in white of an egg (Fig.10a). In the image of the distribution of the amplitude projection values, it is visible that attenuation in yolk is larger than in egg white and much higher than in water. The image of the distribution of amplitudes of receiving pulses is characterized by a large dynamics of value changes and in a similar rate visualises both continuous and step changes (Fig.10c). The ultrasonographic image of an egg (Fig.10d) visualises clearly only boundaries of yolk and white structures.

(a)

(b) Fig. 12. Ultrasonic projection images of CIRS model 052 breast biopsy phantom in the range of measured altitudes *h* = 3.50 ÷ 5.25 mm along its longest dimension – length: a) image in

gray scale, b) image in pseudo-3-D

from errors of the measurement of a shift of mid-frequency of pulse after transition at edges of structures, where a signal is weakened or faded most often.

Fig. 10. Ultrasonic projection images of a hard-boiled hen's egg without shell, obtained from the following measurements of mean values: a) ultrasonic wave propagation velocity, b) derivative of the ultrasonic wave amplitude attenuation coefficient on frequency, c) ultrasonic wave pulse amplitude, in comparison to the ultrasonogram of the same egg (d)

In order to verify the correctness of imaging the internal structure of the analysed object, Fig.11 presents its optical image in the cross-section for the analysed plane, obtained after cutting an egg into half and scanning the examined section using an optical scanner. Comparing images presented in Figures 10 and 11, it can be unequivocally stated that a computer-assisted ultrasonic projection enables proper recognising of biological structures (a yolk is clearly visible in the egg structure). One advantage of the ultrasonic projection method is an availability to obtain several different images from a single measurement set, every of which characterises some other features of an object.

Fig. 11. The optical image (scan) of the measured egg cross-section structure

from errors of the measurement of a shift of mid-frequency of pulse after transition at edges

(a) (b)

(c) (d) Fig. 10. Ultrasonic projection images of a hard-boiled hen's egg without shell, obtained from the following measurements of mean values: a) ultrasonic wave propagation velocity, b) derivative of the ultrasonic wave amplitude attenuation coefficient on frequency, c) ultrasonic wave pulse amplitude, in comparison to the ultrasonogram of the same egg (d)

In order to verify the correctness of imaging the internal structure of the analysed object, Fig.11 presents its optical image in the cross-section for the analysed plane, obtained after cutting an egg into half and scanning the examined section using an optical scanner. Comparing images presented in Figures 10 and 11, it can be unequivocally stated that a computer-assisted ultrasonic projection enables proper recognising of biological structures (a yolk is clearly visible in the egg structure). One advantage of the ultrasonic projection method is an availability to obtain several different images from a single measurement set,

every of which characterises some other features of an object.

Fig. 11. The optical image (scan) of the measured egg cross-section structure

of structures, where a signal is weakened or faded most often.

These images can be properly processed and correlated by the means of special software, what enables recognising structures, which are not visible in single images. The image of the distribution of the sound speed projection values clearly visualises constant changes of heterogeneity (Fig.10a), while the image of the distribution of sound attenuation frequency derivative projection values better visualises discrete changes (Fig.10b). In the image of the distribution of sound velocity projection values, it is also visible that sound speed in a yolk is higher than in water and lower than in white of an egg (Fig.10a). In the image of the distribution of the amplitude projection values, it is visible that attenuation in yolk is larger than in egg white and much higher than in water. The image of the distribution of amplitudes of receiving pulses is characterized by a large dynamics of value changes and in a similar rate visualises both continuous and step changes (Fig.10c). The ultrasonographic image of an egg (Fig.10d) visualises clearly only boundaries of yolk and white structures.

Fig. 12. Ultrasonic projection images of CIRS model 052 breast biopsy phantom in the range of measured altitudes *h* = 3.50 ÷ 5.25 mm along its longest dimension – length: a) image in gray scale, b) image in pseudo-3-D

Ultrasonic Projection 43

obtaining 100 projection images in *XZ* planes (of resolutions 101 x 8 pixels each), from all sides of the object flank, in the angle range of 0° ÷ 178.2°. Figures 12 and 13 present images of sound speed projections for parallel ultrasonic rays, penetrating lateral surfaces of the CIRS phantom in the range of measured heights of *h* = 3.50 ÷ 5.25 mm, in greyscale and

These images were obtained by temperature scaling and assembling all measurement projections (sets of values for full shifts of the probes pair along the object) for rotation angles of 0° (Fig.12) and 90° (Fig.13), extracted from projection measurement sets of cross-

Figure 14 presents 2 geometrical projections of the CIRS breast phantom structure from the sides of its length and width, obtained by the means of 3-D reconstruction from tomographic images (Opielinski & Gudra, 2004a, 2004c). It can be observed that there is a good conformity of geometrical projections of the phantom with projection images obtained by the means of ultrasound transmission (compare Fig.12 and Fig.13 with Fig.14a,b). An interpretation of projection images requires a spatial intelligence and basic knowledge in spatial geometry. Particular images present projections of the distribution of local values of a measured acoustic parameter in the plane parallel to the scan surface and are a mirror reflection for projection planes from the back and front sides of the object. Projection imaging is not a full quantitative imaging; however, on the basis of values of particular pixels of an image it is possible to determine the diversification of parameters of an examined structure. At a proper resolution, step changes of values of an examined acoustic parameter in the projection plane are noticeable in an image, while continuous changes are, in general, not easy detectable. A step change of values of an examined parameter is visible in a contour image in the form of a discrete change of contrast. Pseudo-3-D images provide a better dynamics of projection visualisation. On the basis of several projection images of an examined structure from numerous directions, it is possible to undertake an attempt of a 3- D reconstruction of heterogeneity borders in its interior (Opielinski & Gudra, 2004a).

(a)

(b) Fig. 14. Geometric projections of phantom CIRS structure along its length (a) and width (b), obtained on the basis of three-dimensional reconstruction from tomographic images

Ultrasonic projection imaging can be also used for non-invasive *in vivo* visualisation of injuries and lesions of human upper and lower limbs (Ermert et al., 2000) for the aims of e.g. diagnosing osteoporosis degree. However, it is necessary to consider specific distortions, which occur in an image due to refraction of ultrasonic wave rays in case of biological

pseudo-3-D (Opielinski & Gudra, 2004c).

sections of the phantom for particular heights.

Fig. 13. Ultrasonic projection images of CIRS model 052 breast biopsy phantom in the range of measured altitudes *h* = 3.50 ÷ 5.25 mm, along its shortest dimension – width: a) image in gray scale, b) image in pseudo-3-D

A subject of projection studies was also a breast biopsy phantom of the American CIRS company, model 052, which simulates acoustic parameters of tissues that are present in a female breast. Ultrasonic projection measurements of the CIRS phantom were conducted on the research setup for ultrasonic transmission tomography (Opielinski & Gudra, 2010b), measuring mean values of transition times of an ultrasonic pulse in the geometry of parallelray projections, by the means of single-element ultrasonic probes of 5 mm diameters and work frequency of 5 MHz, located centrally opposite each other, in the distance of 160 mm (Opielinski & Gudra, 2004a, 2004c). There were used 161 steps of probes pair shift along the phantom, with the 1 mm step (161 measurement rays) for each of 100 turns around the phantom with the step of 1.8° (100 measurement projections) and for each of eight positions of probes pair in vertical direction, in distances from the phantom base of 35 mm, 37.5 mm, 40 mm, 42.5 mm, 45 mm, 47.5 mm 50 mm, 52.5 mm, respectively. Such data set enables

(a)

(b) Fig. 13. Ultrasonic projection images of CIRS model 052 breast biopsy phantom in the range of measured altitudes *h* = 3.50 ÷ 5.25 mm, along its shortest dimension – width: a) image in

A subject of projection studies was also a breast biopsy phantom of the American CIRS company, model 052, which simulates acoustic parameters of tissues that are present in a female breast. Ultrasonic projection measurements of the CIRS phantom were conducted on the research setup for ultrasonic transmission tomography (Opielinski & Gudra, 2010b), measuring mean values of transition times of an ultrasonic pulse in the geometry of parallelray projections, by the means of single-element ultrasonic probes of 5 mm diameters and work frequency of 5 MHz, located centrally opposite each other, in the distance of 160 mm (Opielinski & Gudra, 2004a, 2004c). There were used 161 steps of probes pair shift along the phantom, with the 1 mm step (161 measurement rays) for each of 100 turns around the phantom with the step of 1.8° (100 measurement projections) and for each of eight positions of probes pair in vertical direction, in distances from the phantom base of 35 mm, 37.5 mm, 40 mm, 42.5 mm, 45 mm, 47.5 mm 50 mm, 52.5 mm, respectively. Such data set enables

gray scale, b) image in pseudo-3-D

obtaining 100 projection images in *XZ* planes (of resolutions 101 x 8 pixels each), from all sides of the object flank, in the angle range of 0° ÷ 178.2°. Figures 12 and 13 present images of sound speed projections for parallel ultrasonic rays, penetrating lateral surfaces of the CIRS phantom in the range of measured heights of *h* = 3.50 ÷ 5.25 mm, in greyscale and pseudo-3-D (Opielinski & Gudra, 2004c).

These images were obtained by temperature scaling and assembling all measurement projections (sets of values for full shifts of the probes pair along the object) for rotation angles of 0° (Fig.12) and 90° (Fig.13), extracted from projection measurement sets of crosssections of the phantom for particular heights.

Figure 14 presents 2 geometrical projections of the CIRS breast phantom structure from the sides of its length and width, obtained by the means of 3-D reconstruction from tomographic images (Opielinski & Gudra, 2004a, 2004c). It can be observed that there is a good conformity of geometrical projections of the phantom with projection images obtained by the means of ultrasound transmission (compare Fig.12 and Fig.13 with Fig.14a,b). An interpretation of projection images requires a spatial intelligence and basic knowledge in spatial geometry. Particular images present projections of the distribution of local values of a measured acoustic parameter in the plane parallel to the scan surface and are a mirror reflection for projection planes from the back and front sides of the object. Projection imaging is not a full quantitative imaging; however, on the basis of values of particular pixels of an image it is possible to determine the diversification of parameters of an examined structure. At a proper resolution, step changes of values of an examined acoustic parameter in the projection plane are noticeable in an image, while continuous changes are, in general, not easy detectable. A step change of values of an examined parameter is visible in a contour image in the form of a discrete change of contrast. Pseudo-3-D images provide a better dynamics of projection visualisation. On the basis of several projection images of an examined structure from numerous directions, it is possible to undertake an attempt of a 3- D reconstruction of heterogeneity borders in its interior (Opielinski & Gudra, 2004a).

Fig. 14. Geometric projections of phantom CIRS structure along its length (a) and width (b), obtained on the basis of three-dimensional reconstruction from tomographic images

Ultrasonic projection imaging can be also used for non-invasive *in vivo* visualisation of injuries and lesions of human upper and lower limbs (Ermert et al., 2000) for the aims of e.g. diagnosing osteoporosis degree. However, it is necessary to consider specific distortions, which occur in an image due to refraction of ultrasonic wave rays in case of biological

Ultrasonic Projection 45

imaging errors associated with side lobes. However there are a number of mechanisms that suppress grating side lobes (a few wavelengths short excitation pulses, apodization weighting functions, a different transmit and receive geometries, random element spacing) and hence allow this criterion to be somewhat relaxed in practice (Drinkwater & Wilcox, 2006; Karaman et al., 2009; Kim & Song, 2006; Lockwood & Foster, 1996; Thomenius, 1996; Yen & Smith, 2004). This method of focusing makes it also necessary to develop synchronised delaying systems and sophisticated technologies of attaching a large number of electrodes to the surface of minute piezoceramic transducers by integration of some of the electronics with the transducer matrix enabling for miniaturization of the front-end and funnelling the electrical connections of a 2-D matrix consisted with hundreds of elements into reduced number of channels (Eames & Hossack,

2008; Opielinski et al., 2010a, 2010b; Wygant et al., 2006a; 2006b; Yen, Smith, 2004).

transmission is more time consuming.

ultrasonic wave generated in the medium.

(Lockwood & Foster, 1996; Yen & Smith, 2004).

The experimental results have shown that using double ultrasonic pulse transmission of short coded sequences based on well-known Golay complementary codes allows considerably suppressing the noise level (Drinkwater & Wilcox, 2006). However this type of

Increase of directivity and intensity of the wave generated by the multi-element ultrasonic probe can be achieved by simultaneous in phase powering (no delays) of sequences of many elementary transducers (using one generator) in the sending system (which will however result in the probe's input impedance decrease) or by simultaneous receiving ultrasonic wave by means of sequences of many elementary transducers (Chiao & Thomas, 1996; Hoctor & Kassam, 1990; Karaman et al., 2009; Lockwood & Foster, 1996; Opielinski & Gudra, 2010c; Wygant et al., 2006a; Yen & Smith, 2004). Such sending probes are usually activated by low power generators of sinusoidal burst type pulses, the generated voltage values of which are low (a couple of tens of volts peak-to-peak) and output resistance of about 50 Ω. If the probe's impedance is close to output impedance of the generator, it results in a decrease of amplitude of activating pulse voltage which in turn causes decrease of intensity of the

Recently in the medical imaging field a number of authors have suggested the use of matrices with large numbers of elements and investigated methods of selecting the optimal numbers and distribution of elements for the transmit and receive apertures (Drinkwater & Wilcox, 2006). The development of 2-D matrices for clinical ultrasound imaging could greatly improve the detection of small or low contrast structures. Using 2-D matrices, the ultrasound beam could be symmetrically focused and scanned throughout a volume

This Section (7) presents an idea of minimising the number of connections between individual piezoelectric transducers in a row-column multi-element ultrasonic matrix system used for imaging of biological media structure by means of ultrasonic projection (Opielinski & Gudra, 2010c; Opielinski et al., 2010a, 2010b). It allows achieving significant directivity and increased wave intensity with acceptable matrix input impedance decrease without any complicated and expensive focusing or 2-D beamforming systems, what is a great advantage. This concept results in the necessity of creating several small models (e.g. square matrices of 16 transducers in 4 x 4 configuration) at the designing stage, in order to

structures of a large refraction index (the ratio of velocity in water to velocity in an examined structure (Opielinski & Gudra, 2008)). In case of bones examinations, it is possible not only to record projection values of acoustic parameters described in Section 3, but also to determine images of the distribution o two main parameters of ultrasonic waves, used in medicine for diagnosing various stages of osteoporosis: SOS (speed of sound) and BUA (broadband ultrasound attenuation). On their basis, it is also to determine the distribution of the stiffness factor, which defines the state of osseous tissues in relation to a healthy female population, in the age of so called peak bone mass.

## **7. 2-D ultrasonic matrices**

At designing multi-element ultrasonic matrices – sending and receiving one – for the aims of projective imaging of biological media, it is essential to achieve a compromise between the resolution (it depends on the type and size of elementary transducers, their work frequency, distance between them) and the efficiency and sensitivity, which grows along with the surface size of an elementary transducer. Thus, it is very important to specify the sizes of elementary transducers and distances between them already at the designing stage. The technique of mounting transducers on a matrix and the way of conducting electrodes has also a substantial influence on the way of transducers' work. The ongoing search for improved ultrasonic imaging performance will continue to introduce new challenges for beamforming design (Thomenius, 1996). The goal of beamformer is to create as narrow and uniform a beam with as low sidelobes over as long a depth as possible. Among already proposed imaging methods and techniques are elevation focusing (1.25-D, 1.5-D and 1.75-D arrays) (Wildes et al., 1997), beam steering, synthetic apertures, 2-D and sparse matrices, configurable matrices, parallel beamforming, microbeamformers, rectilinear scanning, coded excitation, phased subarray processing, phase aberration correction, and others (Drinkwater & Wilcox, 2006; Johnson et al., 2005; Karaman et al., 2009; Kim & Song, 2006; Lockwood & Foster, 1996; Nowicki et al., 2009). The most common complication introduced by these is a significant increase in channel count. Generating narrow ultrasonic wave beams in biological media by multi-element probes, built as matrices of elementary piezoceramic transducers in a rectangular configuration, can be realised using transducers having a spherical surface, ultrasonic lenses, mechanical elements (e.g. complex system of properly rotated prisms), focusing devices or electronic devices, which control the system of activating and powering individual matrix elements in a proper manner (Drinkwater & Wilcox, 2006; Ermert et al., 2000; Granz & Oppelt, 1987; Green et al., 1974; Nowicki, 1995; Opielinski et al., 2009; Opielinski & Gudra, 2010a, 2010c; Opielinski et al. 2010a, 2010b; Ramm & Smith, 1983; Thomenius, 1996). Exciting individual transducers of the multi-element probe using pulses with various delay is a universal method of focusing and deflecting a beam (Johnson et al., 2005; Thomenius, 1996). Adequate delays between activations of each successive elementary transducer allow shaping of the wave front and the direction of its propagation. In case of multi-element ultrasonic matrices (Opielinski et al., 2009, 2010a, 2010b) used for projection imaging of internal structure of biological media (Ermert et al., 2000; Granz & Oppelt, 1987; Green et al., 1974; Opielinski & Gudra, 2005), introduction of delays of propagation of pulse ultrasonic wave for individual transducers can result in

structures of a large refraction index (the ratio of velocity in water to velocity in an examined structure (Opielinski & Gudra, 2008)). In case of bones examinations, it is possible not only to record projection values of acoustic parameters described in Section 3, but also to determine images of the distribution o two main parameters of ultrasonic waves, used in medicine for diagnosing various stages of osteoporosis: SOS (speed of sound) and BUA (broadband ultrasound attenuation). On their basis, it is also to determine the distribution of the stiffness factor, which defines the state of osseous tissues in relation to a healthy female

At designing multi-element ultrasonic matrices – sending and receiving one – for the aims of projective imaging of biological media, it is essential to achieve a compromise between the resolution (it depends on the type and size of elementary transducers, their work frequency, distance between them) and the efficiency and sensitivity, which grows along with the surface size of an elementary transducer. Thus, it is very important to specify the sizes of elementary transducers and distances between them already at the designing stage. The technique of mounting transducers on a matrix and the way of conducting electrodes has also a substantial influence on the way of transducers' work. The ongoing search for improved ultrasonic imaging performance will continue to introduce new challenges for beamforming design (Thomenius, 1996). The goal of beamformer is to create as narrow and uniform a beam with as low sidelobes over as long a depth as possible. Among already proposed imaging methods and techniques are elevation focusing (1.25-D, 1.5-D and 1.75-D arrays) (Wildes et al., 1997), beam steering, synthetic apertures, 2-D and sparse matrices, configurable matrices, parallel beamforming, microbeamformers, rectilinear scanning, coded excitation, phased subarray processing, phase aberration correction, and others (Drinkwater & Wilcox, 2006; Johnson et al., 2005; Karaman et al., 2009; Kim & Song, 2006; Lockwood & Foster, 1996; Nowicki et al., 2009). The most common complication introduced by these is a significant increase in channel count. Generating narrow ultrasonic wave beams in biological media by multi-element probes, built as matrices of elementary piezoceramic transducers in a rectangular configuration, can be realised using transducers having a spherical surface, ultrasonic lenses, mechanical elements (e.g. complex system of properly rotated prisms), focusing devices or electronic devices, which control the system of activating and powering individual matrix elements in a proper manner (Drinkwater & Wilcox, 2006; Ermert et al., 2000; Granz & Oppelt, 1987; Green et al., 1974; Nowicki, 1995; Opielinski et al., 2009; Opielinski & Gudra, 2010a, 2010c; Opielinski et al. 2010a, 2010b; Ramm & Smith, 1983; Thomenius, 1996). Exciting individual transducers of the multi-element probe using pulses with various delay is a universal method of focusing and deflecting a beam (Johnson et al., 2005; Thomenius, 1996). Adequate delays between activations of each successive elementary transducer allow shaping of the wave front and the direction of its propagation. In case of multi-element ultrasonic matrices (Opielinski et al., 2009, 2010a, 2010b) used for projection imaging of internal structure of biological media (Ermert et al., 2000; Granz & Oppelt, 1987; Green et al., 1974; Opielinski & Gudra, 2005), introduction of delays of propagation of pulse ultrasonic wave for individual transducers can result in

population, in the age of so called peak bone mass.

**7. 2-D ultrasonic matrices** 

imaging errors associated with side lobes. However there are a number of mechanisms that suppress grating side lobes (a few wavelengths short excitation pulses, apodization weighting functions, a different transmit and receive geometries, random element spacing) and hence allow this criterion to be somewhat relaxed in practice (Drinkwater & Wilcox, 2006; Karaman et al., 2009; Kim & Song, 2006; Lockwood & Foster, 1996; Thomenius, 1996; Yen & Smith, 2004). This method of focusing makes it also necessary to develop synchronised delaying systems and sophisticated technologies of attaching a large number of electrodes to the surface of minute piezoceramic transducers by integration of some of the electronics with the transducer matrix enabling for miniaturization of the front-end and funnelling the electrical connections of a 2-D matrix consisted with hundreds of elements into reduced number of channels (Eames & Hossack, 2008; Opielinski et al., 2010a, 2010b; Wygant et al., 2006a; 2006b; Yen, Smith, 2004).

The experimental results have shown that using double ultrasonic pulse transmission of short coded sequences based on well-known Golay complementary codes allows considerably suppressing the noise level (Drinkwater & Wilcox, 2006). However this type of transmission is more time consuming.

Increase of directivity and intensity of the wave generated by the multi-element ultrasonic probe can be achieved by simultaneous in phase powering (no delays) of sequences of many elementary transducers (using one generator) in the sending system (which will however result in the probe's input impedance decrease) or by simultaneous receiving ultrasonic wave by means of sequences of many elementary transducers (Chiao & Thomas, 1996; Hoctor & Kassam, 1990; Karaman et al., 2009; Lockwood & Foster, 1996; Opielinski & Gudra, 2010c; Wygant et al., 2006a; Yen & Smith, 2004). Such sending probes are usually activated by low power generators of sinusoidal burst type pulses, the generated voltage values of which are low (a couple of tens of volts peak-to-peak) and output resistance of about 50 Ω. If the probe's impedance is close to output impedance of the generator, it results in a decrease of amplitude of activating pulse voltage which in turn causes decrease of intensity of the ultrasonic wave generated in the medium.

Recently in the medical imaging field a number of authors have suggested the use of matrices with large numbers of elements and investigated methods of selecting the optimal numbers and distribution of elements for the transmit and receive apertures (Drinkwater & Wilcox, 2006). The development of 2-D matrices for clinical ultrasound imaging could greatly improve the detection of small or low contrast structures. Using 2-D matrices, the ultrasound beam could be symmetrically focused and scanned throughout a volume (Lockwood & Foster, 1996; Yen & Smith, 2004).

This Section (7) presents an idea of minimising the number of connections between individual piezoelectric transducers in a row-column multi-element ultrasonic matrix system used for imaging of biological media structure by means of ultrasonic projection (Opielinski & Gudra, 2010c; Opielinski et al., 2010a, 2010b). It allows achieving significant directivity and increased wave intensity with acceptable matrix input impedance decrease without any complicated and expensive focusing or 2-D beamforming systems, what is a great advantage. This concept results in the necessity of creating several small models (e.g. square matrices of 16 transducers in 4 x 4 configuration) at the designing stage, in order to

Ultrasonic Projection 47

Fig. 15. Photo of the developed 512-element standard matrix for ultrasonic projection

wave intensity at acceptable drop of input impedance of the matrix.

So called passive matrix is the simplest solution for the matrix type controlling. This matrix includes two sets of keys (electronic switches), of which one *Ky* selects an active column and

An increase of directivity and intensity of the wave generated by the multi-element standard 2-D ultrasonic matrix (Section 7.1) can be obtained by simultaneous in phase supply (without delays) of a sequence of numerous elementary transducers, using one generator in the transmitting set-up (an aperture in transmitting and/or in receiving set-up - (Opielinski & Gudra, 2010c)), what causes a drop of the input impedance of the matrix. Such matrices are, usually, powered by low-power generators of burst-type sinusoidal pulses of low values of generated voltages (max. 20 Vpp) and output resistance of about 50 Ω. If the matrix sending transducer group impedance is close to the output impedance of the generator, a drop of the amplitude of an exciting pulse voltage occurs, what is followed by a drop of the intensity of the ultrasonic wave, which is generated to a medium. In the far field, in the plane perpendicular to axis *Z*, a distribution of the acoustic field generated by multielement matrices with simultaneous supply to the group of elementary transducers has consecutive maxima and minima. Locations of maxima and minima and the acoustic pressure amplitude of the major and side lobes depend on the sizes of matrix's elementary transducers, distance between them and the length of the wave radiated into the medium. If the distance between adjacent transducers is small enough (*d* < *λ*/2), there are no side lobes in the area of 90°. This criterion is very difficult to be achieved as for the frequency of 2 MHz, the length of a wave in tissue is about 0.75 mm, what forces the use of elementary transducers of sizes below 0.375 mm. So, this Section (7.2) presents the concept of minimising the number of connections of particular piezoelectric transducers in the rowcolumn arrangement of the multi-element ultrasonic passive matrix, assigned for imaging of biological media structures, by the means of the ultrasonic projection method (Opielinski et al., 2010a), what simultaneously enables achieving a large directivity and an increase of the

examinations

**7.2 Ultrasonic passive matrices** 

optimally select all essential parameters, together with developing a proper production technology (Opielinski & Gudra, 2010a). Such technology can be easily copied later with proper modifications and improvements, developing the matrix model by adding a greater number of elementary transducers. The following assumptions of constructing multielement ultrasonic matrices were adopted (Opielinski & Gudra, 2010a):


Desired dimensions of elementary piezoceramic transducers at possibly least material losses were achieved thanks to mechanical cutting with a diamond saw of 0.2 mm thickness. Cut plates were selected from a larger group due to repeatability of work frequency and electric conductance values.

#### **7.1 Ultrasonic standard matrices**

First of all, ultrasonic 512-element standard matrices (with separated electrodes connections) assigned for work in the sending and receiving character were constructed from SONOX P2 piezoceramic transducers, of the size of 1.5 x 1.5 mm (Opielinski & Gudra, 2010a). For a precise mounting of transducers, a mask made of engraving laminate of 0.8 mm thickness, with square-shaped holes for elementary transducers, cut using laser, at accuracy of 10 μm was used. At the back side (ground), transducers were stuck using conducting glue, to properly etched paths of a printed-circuit board, located at distances of 1 mm each other, in the layout of 16 rows and 32 columns. In order to connect a signal lead (electrode) to the active surface of each elementary transducer of the matrix, contact elastic connection was used. The lead, bent in the form of a hook, elastically adhering to the transducer surface, is conducted to the other side through the hole in the printedcircuit board, where it is stuck to its surface. The matching layer of matrices, which also serves the roles of isolation and stabilisation of contacts, was made by spraying a proper lacquer over their surfaces. A picture of the developed standard ultrasonic matrix is shown in Fig.15 (Opielinski & Gudra, 2010a). Detailed measurements of electromechanical parameters of elementary ultrasonic transducers of the developed standard matrix showed a presence of three resonance frequencies *fr*<sup>1</sup> ≈ 1 MHz, *fr*<sup>2</sup> ≈ 1.8 MHz and *fr*<sup>3</sup> ≈ 2 MHz. The reasons of occurrence of so many resonances is the assumed way of mounting transducers using conducting glue to proper patch of the board, without a back attenuation layer. The average efficiency of elementary transducers of the standard matrix, in the distance of 2.5 cm from the surface can be estimated on average at about 750 Pa/V for *fr*1 = 1.1 MHz and at about 400 Pa/V for *fr*2 = 2.1 MHz. Study results show that the developed standard matrices are suitable for projective imaging of biological media of parameters close to parameters of soft tissues, using the scanning method through switching proper pairs of sending and receiving elementary transducers (or proper synthetic apertures (Opielinski & Gudra, 2010c)).

optimally select all essential parameters, together with developing a proper production technology (Opielinski & Gudra, 2010a). Such technology can be easily copied later with proper modifications and improvements, developing the matrix model by adding a greater number of elementary transducers. The following assumptions of constructing multi-

a. transducer work frequency in a biological medium in the range of *fr* = 1 MHz ÷ 2 MHz, in order to achieve a proper depth resolution (measurement accuracy) and relatively

b. dimensions of matrix piezoceramic transducers of *a* ≈ 1.5 mm, *b* ≈ 1.5 mm, in order to

c. distance between transducers of *d* ≈ 1 mm, in order to enable conducting electrodes and

Desired dimensions of elementary piezoceramic transducers at possibly least material losses were achieved thanks to mechanical cutting with a diamond saw of 0.2 mm thickness. Cut plates were selected from a larger group due to repeatability of work frequency and electric

First of all, ultrasonic 512-element standard matrices (with separated electrodes connections) assigned for work in the sending and receiving character were constructed from SONOX P2 piezoceramic transducers, of the size of 1.5 x 1.5 mm (Opielinski & Gudra, 2010a). For a precise mounting of transducers, a mask made of engraving laminate of 0.8 mm thickness, with square-shaped holes for elementary transducers, cut using laser, at accuracy of 10 μm was used. At the back side (ground), transducers were stuck using conducting glue, to properly etched paths of a printed-circuit board, located at distances of 1 mm each other, in the layout of 16 rows and 32 columns. In order to connect a signal lead (electrode) to the active surface of each elementary transducer of the matrix, contact elastic connection was used. The lead, bent in the form of a hook, elastically adhering to the transducer surface, is conducted to the other side through the hole in the printedcircuit board, where it is stuck to its surface. The matching layer of matrices, which also serves the roles of isolation and stabilisation of contacts, was made by spraying a proper lacquer over their surfaces. A picture of the developed standard ultrasonic matrix is shown in Fig.15 (Opielinski & Gudra, 2010a). Detailed measurements of electromechanical parameters of elementary ultrasonic transducers of the developed standard matrix showed a presence of three resonance frequencies *fr*<sup>1</sup> ≈ 1 MHz, *fr*<sup>2</sup> ≈ 1.8 MHz and *fr*<sup>3</sup> ≈ 2 MHz. The reasons of occurrence of so many resonances is the assumed way of mounting transducers using conducting glue to proper patch of the board, without a back attenuation layer. The average efficiency of elementary transducers of the standard matrix, in the distance of 2.5 cm from the surface can be estimated on average at about 750 Pa/V for *fr*1 = 1.1 MHz and at about 400 Pa/V for *fr*2 = 2.1 MHz. Study results show that the developed standard matrices are suitable for projective imaging of biological media of parameters close to parameters of soft tissues, using the scanning method through switching proper pairs of sending and receiving elementary transducers (or proper

element ultrasonic matrices were adopted (Opielinski & Gudra, 2010a):

achieve a proper scanning resolution, identical all over matrix plane,

low attenuation of an ultrasonic wave,

conductance values.

**7.1 Ultrasonic standard matrices** 

mounting matrices in laboratory conditions.

synthetic apertures (Opielinski & Gudra, 2010c)).

Fig. 15. Photo of the developed 512-element standard matrix for ultrasonic projection examinations

#### **7.2 Ultrasonic passive matrices**

An increase of directivity and intensity of the wave generated by the multi-element standard 2-D ultrasonic matrix (Section 7.1) can be obtained by simultaneous in phase supply (without delays) of a sequence of numerous elementary transducers, using one generator in the transmitting set-up (an aperture in transmitting and/or in receiving set-up - (Opielinski & Gudra, 2010c)), what causes a drop of the input impedance of the matrix. Such matrices are, usually, powered by low-power generators of burst-type sinusoidal pulses of low values of generated voltages (max. 20 Vpp) and output resistance of about 50 Ω. If the matrix sending transducer group impedance is close to the output impedance of the generator, a drop of the amplitude of an exciting pulse voltage occurs, what is followed by a drop of the intensity of the ultrasonic wave, which is generated to a medium. In the far field, in the plane perpendicular to axis *Z*, a distribution of the acoustic field generated by multielement matrices with simultaneous supply to the group of elementary transducers has consecutive maxima and minima. Locations of maxima and minima and the acoustic pressure amplitude of the major and side lobes depend on the sizes of matrix's elementary transducers, distance between them and the length of the wave radiated into the medium. If the distance between adjacent transducers is small enough (*d* < *λ*/2), there are no side lobes in the area of 90°. This criterion is very difficult to be achieved as for the frequency of 2 MHz, the length of a wave in tissue is about 0.75 mm, what forces the use of elementary transducers of sizes below 0.375 mm. So, this Section (7.2) presents the concept of minimising the number of connections of particular piezoelectric transducers in the rowcolumn arrangement of the multi-element ultrasonic passive matrix, assigned for imaging of biological media structures, by the means of the ultrasonic projection method (Opielinski et al., 2010a), what simultaneously enables achieving a large directivity and an increase of the wave intensity at acceptable drop of input impedance of the matrix.

So called passive matrix is the simplest solution for the matrix type controlling. This matrix includes two sets of keys (electronic switches), of which one *Ky* selects an active column and

Ultrasonic Projection 49

which is simultaneously a mask for positioning transducers. It is a single-sided circuit, composed of sixteen especially designed horizontal paths. Each path has square areas deprived of copper. These areas mark locations of ultrasonic transducers and were cut out by the means of a special punch. Such square openings enable a precise placing of transducers. Electrical connection of ultrasonic transducers with the back base board were realised by hot-air soldering. The upper board (mask) thickness of about 0.8 mm was selected in order to locate transducer surface evenly with the board surface. A small amount of conducting glue enabled connection of transducer metallization with the path around a hole. There were developed two 512-element passive matrices with Pz37 Ferroperm ceramic transducers of the size 1.6 x 1.6 mm, arranged at the distances of 0.9 mm. One matrix is assigned for sending work and the second for receiving. Figure 17 presents the view of the developed 512-element ultrasonic passive matrix. Large round holes at the sides of matrix surface are designed for mounting semi-conducting lasers in the sending matrix for

positioning the sending matrix with the receiving one.

Fig. 17. The view of the developed 512-element ultrasonic passive matrix

*x*

of *N* x *M* transducers was derived:

One characteristic feature of the passive matrix is occurrence of crosstalk (Opielinski et al., 2010a), which causes a presence of some signal voltage, even on inactive transducers. This voltage is most often substantially lower than the voltage on a switched transducer but has an influence of the shape of obtained resultant characteristics of the matrix directivity. On the basis of conducted simulations and calculations, the following formula that describes the distribution of an excitement voltage on the passive 512-element matrix in the arrangement

> 1 1 11

The switched on transducer of the passive matrix is supplied by voltage +*U*, all transducers in this (switched on) row are supplied by voltage +*U*x(*N*–1), all transducers in this (switched on) column are supplied by +*Ux*(*M*–1), and all the other transducers (in switched

*M N*

(7)

the second *Wx* – an active row. In this way, transducer *Pxy* is activated, which is located on the cross-cut of the selected row and column (Fig.16a). Piezoceramic transducers of the matrix are supplied with an alternating signal of high frequency (several MHz). Their structure (dielectric with sublimated electrodes) makes that, from the electronic circuit point of view, they have the capacitive character and an exciting signal can pass through inactive transducers to other rows and columns, despite disconnection of keys (Fig.16b). Directivity and efficiency/sensitivity of such matrix shall, therefore, depend to a high degree on a distribution of voltages of an exciting/receiving signal on all elementary transducers of the matrix, which in such way (with crosstalk) shall be excited to oscillations (Opielinski et al., 2010a).

Fig. 16. Diagram of connections of transducers in a passive matrix (a) and a phenomenon of crosstalk formation (b)

Calculations and measurements of distributions of an acoustic field of the developed model of the 16-element passive matrix, in the 4 x 4 transducers layout, showed that after making a switch of one elementary transducer, all matrix transducers are excited by voltages, according to a specified pattern, thanks to which the matrix generated a directional beam. Exciting elementary transducers in a row-columns arrangement in a multi-element ultrasonic passive matrix enables a substantial minimisation of the number of connections of particular piezoelectric transducers. For example, for a 1024-element matrix, it is enough to use 64 paths, etched at a printed-circuit board, which conduct a signal exciting elementary transducers, instead of soldering 1024 separate electrodes in the form of thin, isolated leads to transducers surface and mounting multi-pin slots. The concept of a passive matrix enables also achieving a large directivity and an increase the wave intensity at an acceptable drop of the input impedance of the matrix. The developed concept of a passive matrix enabled designing of a full-sized matrix of 512 transducers, arranged in the structure of 16 rows and 32 columns. Due to a compact construction of the system, it was decided to put transducers and all electronic systems on a one printed circuit. Elements were arranged on the board with a division into two areas. The lower part is the proper matrix of transducers and the upper one includes switching circuits. Such strict division enables immersing the lower part in water, without the risk of damaging electronic circuits. Printed circuits include connections only for columns of the passive matrix. Connections of rows were realised using an additional upper board (microwave teflon-ceramic laminate) with a printed circuit,

the second *Wx* – an active row. In this way, transducer *Pxy* is activated, which is located on the cross-cut of the selected row and column (Fig.16a). Piezoceramic transducers of the matrix are supplied with an alternating signal of high frequency (several MHz). Their structure (dielectric with sublimated electrodes) makes that, from the electronic circuit point of view, they have the capacitive character and an exciting signal can pass through inactive transducers to other rows and columns, despite disconnection of keys (Fig.16b). Directivity and efficiency/sensitivity of such matrix shall, therefore, depend to a high degree on a distribution of voltages of an exciting/receiving signal on all elementary transducers of the matrix, which in such way (with

(a) (b) Fig. 16. Diagram of connections of transducers in a passive matrix (a) and a phenomenon of

Calculations and measurements of distributions of an acoustic field of the developed model of the 16-element passive matrix, in the 4 x 4 transducers layout, showed that after making a switch of one elementary transducer, all matrix transducers are excited by voltages, according to a specified pattern, thanks to which the matrix generated a directional beam. Exciting elementary transducers in a row-columns arrangement in a multi-element ultrasonic passive matrix enables a substantial minimisation of the number of connections of particular piezoelectric transducers. For example, for a 1024-element matrix, it is enough to use 64 paths, etched at a printed-circuit board, which conduct a signal exciting elementary transducers, instead of soldering 1024 separate electrodes in the form of thin, isolated leads to transducers surface and mounting multi-pin slots. The concept of a passive matrix enables also achieving a large directivity and an increase the wave intensity at an acceptable drop of the input impedance of the matrix. The developed concept of a passive matrix enabled designing of a full-sized matrix of 512 transducers, arranged in the structure of 16 rows and 32 columns. Due to a compact construction of the system, it was decided to put transducers and all electronic systems on a one printed circuit. Elements were arranged on the board with a division into two areas. The lower part is the proper matrix of transducers and the upper one includes switching circuits. Such strict division enables immersing the lower part in water, without the risk of damaging electronic circuits. Printed circuits include connections only for columns of the passive matrix. Connections of rows were realised using an additional upper board (microwave teflon-ceramic laminate) with a printed circuit,

crosstalk) shall be excited to oscillations (Opielinski et al., 2010a).

crosstalk formation (b)

which is simultaneously a mask for positioning transducers. It is a single-sided circuit, composed of sixteen especially designed horizontal paths. Each path has square areas deprived of copper. These areas mark locations of ultrasonic transducers and were cut out by the means of a special punch. Such square openings enable a precise placing of transducers. Electrical connection of ultrasonic transducers with the back base board were realised by hot-air soldering. The upper board (mask) thickness of about 0.8 mm was selected in order to locate transducer surface evenly with the board surface. A small amount of conducting glue enabled connection of transducer metallization with the path around a hole. There were developed two 512-element passive matrices with Pz37 Ferroperm ceramic transducers of the size 1.6 x 1.6 mm, arranged at the distances of 0.9 mm. One matrix is assigned for sending work and the second for receiving. Figure 17 presents the view of the developed 512-element ultrasonic passive matrix. Large round holes at the sides of matrix surface are designed for mounting semi-conducting lasers in the sending matrix for positioning the sending matrix with the receiving one.

Fig. 17. The view of the developed 512-element ultrasonic passive matrix

One characteristic feature of the passive matrix is occurrence of crosstalk (Opielinski et al., 2010a), which causes a presence of some signal voltage, even on inactive transducers. This voltage is most often substantially lower than the voltage on a switched transducer but has an influence of the shape of obtained resultant characteristics of the matrix directivity. On the basis of conducted simulations and calculations, the following formula that describes the distribution of an excitement voltage on the passive 512-element matrix in the arrangement of *N* x *M* transducers was derived:

$$\infty = \frac{1}{\left(M - 1\right) + \left(N - 1\right) + 1} \tag{7}$$

The switched on transducer of the passive matrix is supplied by voltage +*U*, all transducers in this (switched on) row are supplied by voltage +*U*x(*N*–1), all transducers in this (switched on) column are supplied by +*Ux*(*M*–1), and all the other transducers (in switched

Ultrasonic Projection 51

Key *Ky* in the signal path conducts voltage to a particular column, while a closing of keys *Wx* determines an activated row. Field-effect transistors play the role of individual keys *Txy*. Their advantage is a small housing size, what enables a miniaturisation of connections.

On the basis of earlier conducted experiment, similarly as in case of the passive matrix, an active matrix, including 512 ultrasonic transducers was developed. Its construction assumes the use of two keys switching signal on for each of elementary transducer. Such solution complicates the matrix construction but enables elimination of crosstalk, present in the passive matrix. Matrix control is realised in a similar way as in the passive matrix (section 7.2), however, circuits of switching matrix columns were more developed. Only an active column is connected to the signal source and all other are shorted to ground. It eliminates a possibility of signal slips into inactive columns. Thanks to utilisation of individual transistors at each piezoceramic transducer, there is no need to key signal in matrix rows (Fig.20). This way, using additional keys *K*z*<sup>y</sup>* (Fig.20), the active matrix is working as a standard matrix (see Section 7.1). Without using additional keys *K*z*<sup>y</sup>* (Fig.20), after switching electrodes to a proper row (key *Wx*) and column (key *Ky*) of the matrix and after switching a transistor key for a certain transducer (key *Txy*) (see Fig.19), it shall be excited by fed voltage *U*, and other transducers in this column shall be excited with a voltage below 0.3*U*

(a) (b)

additional switches *K*z*<sup>y</sup>*: (a) short-circuiting an inactive transducer, (b) open-circuiting when

Fig. 20. The method of crosstalk elimination in a column of the active matrix using

Fig. 19. Diagram of transducer connections in an active matrix

(Opielinski et al., 2010b).

activating a transducer

off rows and columns) are supplied by voltage –*Ux*. The simulated in this way distribution of voltages for the passive matrix of 16 x 32 sizes is presented in Fig.18. Elementary transducers of the passive matrix, in the frequency range of 1.4 ÷ 2.4 MHz, exhibit a possibility of working at three resonance frequencies *fr*<sup>1</sup> ≈ 1.6 MHz, *fr*<sup>2</sup> ≈ 1.8 MHz, *fr*<sup>3</sup> ≈ 2 MHz. The imaginary part of the amplitude-phase electrical admittance characteristics of transducers of the passive matrix reveals their capacitive character (*Co* ≈ 160 pF, including the capacity of connection leads of about 100 pF, *Ro* ≈ 3 kΩ |*Z*(*fr*)| ≈ 2 kΩ).


Fig. 18. Simulated distribution of values of voltages exciting elementary transducers of the passive matrix of 16 x 32 transducers arrangement ("–" marks a inversed phase)

Single transducers of the earlier developed multi-element matrices with separated electrode connections (standard matrices) exhibited electrical capacity *Co* ≈ 10 pF, resistance of electric losses *Ro* ≈ 30 kΩ and electric impedance in resonance |*Z*(*fr*)| ≈ 7 kΩ (Opielinski & Gudra, 2010a) (for a comparison, at simultaneously excitement of 16 transducers of the standard matrix with the same voltage *U*, impedance of the circuit would be |*Z*(*fr*)| ≈ 440 Ω). These values suggest that in case of the passive matrix, the phenomenon of increasing electric capacity of transducers and reducing their losses resistance occurs due to a parallel connection of all matrix elements, what is confirmed by the presence of crosstalk in such arrangement. Measurements and calculations show that the developed passive matrix enable achieving much larger directivity and almost three time higher amplitude of an ultrasonic wave generated in a medium than in case of a single supply, electrically separated transducer of the standard matrix (Opielinski & Gudra, 2010a). The divergence angle of the beam generated by the developed 16-element model of a passive matrix in the 4 x 4 arrangement, with an activation of 1 element is about 6 ÷ 8°.

#### **7.3 Ultrasonic active matrices**

A large number of elementary transducer of a projection matrix forces a high number of electrical connections and substantially makes miniaturisation of the whole unit difficult. A solution to such inconvenience is using a row-column selection of active transducers (so called passive matrix), presented in Section 7.2. For the matrix of 512 elements, arranged in 16 rows and 32 columns, it is enough to connect 48 leads (16+32) this way, instead of 512. A solution that can enable elimination of crosstalk between transducers in the passive matrix (Fig.16) with the maintenance of the electrode minimisation is the use of active elements (keys) in matrix nodes (active matrix) (Opielinski et al., 2010b). In this solution, each transducer is switched by its own individual key *Txy* (Fig.19).

off rows and columns) are supplied by voltage –*Ux*. The simulated in this way distribution of voltages for the passive matrix of 16 x 32 sizes is presented in Fig.18. Elementary transducers of the passive matrix, in the frequency range of 1.4 ÷ 2.4 MHz, exhibit a possibility of working at three resonance frequencies *fr*<sup>1</sup> ≈ 1.6 MHz, *fr*<sup>2</sup> ≈ 1.8 MHz, *fr*<sup>3</sup> ≈ 2 MHz. The imaginary part of the amplitude-phase electrical admittance characteristics of transducers of the passive matrix reveals their capacitive character (*Co* ≈ 160 pF, including

Fig. 18. Simulated distribution of values of voltages exciting elementary transducers of the

Single transducers of the earlier developed multi-element matrices with separated electrode connections (standard matrices) exhibited electrical capacity *Co* ≈ 10 pF, resistance of electric losses *Ro* ≈ 30 kΩ and electric impedance in resonance |*Z*(*fr*)| ≈ 7 kΩ (Opielinski & Gudra, 2010a) (for a comparison, at simultaneously excitement of 16 transducers of the standard matrix with the same voltage *U*, impedance of the circuit would be |*Z*(*fr*)| ≈ 440 Ω). These values suggest that in case of the passive matrix, the phenomenon of increasing electric capacity of transducers and reducing their losses resistance occurs due to a parallel connection of all matrix elements, what is confirmed by the presence of crosstalk in such arrangement. Measurements and calculations show that the developed passive matrix enable achieving much larger directivity and almost three time higher amplitude of an ultrasonic wave generated in a medium than in case of a single supply, electrically separated transducer of the standard matrix (Opielinski & Gudra, 2010a). The divergence angle of the beam generated by the developed 16-element model of a passive matrix in the 4 x 4

A large number of elementary transducer of a projection matrix forces a high number of electrical connections and substantially makes miniaturisation of the whole unit difficult. A solution to such inconvenience is using a row-column selection of active transducers (so called passive matrix), presented in Section 7.2. For the matrix of 512 elements, arranged in 16 rows and 32 columns, it is enough to connect 48 leads (16+32) this way, instead of 512. A solution that can enable elimination of crosstalk between transducers in the passive matrix (Fig.16) with the maintenance of the electrode minimisation is the use of active elements (keys) in matrix nodes (active matrix) (Opielinski et al., 2010b). In this solution, each

passive matrix of 16 x 32 transducers arrangement ("–" marks a inversed phase)

arrangement, with an activation of 1 element is about 6 ÷ 8°.

transducer is switched by its own individual key *Txy* (Fig.19).

**7.3 Ultrasonic active matrices** 

the capacity of connection leads of about 100 pF, *Ro* ≈ 3 kΩ |*Z*(*fr*)| ≈ 2 kΩ).

Fig. 19. Diagram of transducer connections in an active matrix

Key *Ky* in the signal path conducts voltage to a particular column, while a closing of keys *Wx* determines an activated row. Field-effect transistors play the role of individual keys *Txy*. Their advantage is a small housing size, what enables a miniaturisation of connections.

On the basis of earlier conducted experiment, similarly as in case of the passive matrix, an active matrix, including 512 ultrasonic transducers was developed. Its construction assumes the use of two keys switching signal on for each of elementary transducer. Such solution complicates the matrix construction but enables elimination of crosstalk, present in the passive matrix. Matrix control is realised in a similar way as in the passive matrix (section 7.2), however, circuits of switching matrix columns were more developed. Only an active column is connected to the signal source and all other are shorted to ground. It eliminates a possibility of signal slips into inactive columns. Thanks to utilisation of individual transistors at each piezoceramic transducer, there is no need to key signal in matrix rows (Fig.20). This way, using additional keys *K*z*<sup>y</sup>* (Fig.20), the active matrix is working as a standard matrix (see Section 7.1). Without using additional keys *K*z*<sup>y</sup>* (Fig.20), after switching electrodes to a proper row (key *Wx*) and column (key *Ky*) of the matrix and after switching a transistor key for a certain transducer (key *Txy*) (see Fig.19), it shall be excited by fed voltage *U*, and other transducers in this column shall be excited with a voltage below 0.3*U* (Opielinski et al., 2010b).

Fig. 20. The method of crosstalk elimination in a column of the active matrix using additional switches *K*z*<sup>y</sup>*: (a) short-circuiting an inactive transducer, (b) open-circuiting when activating a transducer

Ultrasonic Projection 53

transition through the object and the lens, are received by proper linear transducers of the receiving probe. One disadvantage in this case is beam focusing in a particular area of the

The device presented in work (Lehmann et al., 1999) is a model of an acoustic holograph used the through-transmission signal. This approach uses coherent sound and coherent light to produce real-time, large field-of-view images with pronounced edge definition in soft

In most of known studies, the projection parameter is the signal amplitude, not its transition time, which seems to more attractive due to the simplicity and precision of measurements. This Section (8) includes a description of the results of author's studies, which aim is a construction of a special measuring setup for visualisation of biological structures by the means of ultrasonic projection, which enables simultaneous measurements of several acoustic parameters in pseudo-real time (multi-parameter ultrasonic transmission camera), using a pair of electronically controlled multi-element matrices of piezoceramic transducers,

object, what causes image spreading and occurrence of artefacts outside the focus.

Fig. 21. Block scheme of the laboratory measurement setup for visualization of the distribution of acoustic parameters of biological media internal structure by means of

Due to diversified ways of conducting electrodes to transducers of matrices and different controlling methods, two laboratory setups were developed, for imaging internal structures of biological media, by the means of ultrasound projection: using a pair of ultrasonic 512 element standard matrix – sending and receiving one - and using a pair of ultrasonic 512 element passive or active matrices (Opielinski & Gudra, 2010a; Opielinski et al., 2010a, 2010b). A general block scheme of the laboratory measurement setup is presented in Fig.21. Elementary piezoceramic transducers of the sending matrix (Opielinski & Gudra, 2010a) are excite by a burst type pulses generator, controlled by a computer through the GPIB

tissues of the body.

described in Section 7.

ultrasonic projection method

Application of such sending and receiving matrix in a projection arrangement is possible in such way that one of them is switched by the angle of 90º in relation to the other one and sequences of switching sending and receiving transducers are so coupled that enables a proper control of scanning of a received beam (Opielinski & Gudra, 2010b).

Active matrices act similarly as earlier developed standard matrices (Section 7.1) but their construction is more simple and improved.

## **8. Models of ultrasonic transmission camera**

Using especially developed multi-element ultrasonic projection matrices with quick electronic switching of elementary transducers (Section 7), it is possible to obtain images in pseudo-real time (e.g. with a slight constant delay, resulting from the need of data buffering and processing), thus, a device using this method can be called an ultrasonic transmission camera (UTC) (Ermert et al., 2000). There are several know works regarding such cameras, conducted at some centres in the world, as described in Introduction (Brettel et al., 1981; Ermert et al., 2000; Granz & Oppelt, 1987; Green et al., 1974; Lehmann et al., 1999).

In the device presented in the work (Green et al., 1974), an incoherent ultrasonic wave was used for sonification of an object and a one-dimensional linear array, mounted in a set position served the role of a receiver. Scanning of an object structure in the second dimension was realised by the means of a complex system of properly rotated prisms. A disadvantage of such system is the necessity to use mechanical elements.

The device presented in (Brettel et al., 1981) uses a similar concept as in work (Green et al., 1974). The difference consists in the fact that a projection image is projected on the water surface of a camera and next, it is detected by a linear array of transducers. What is more, scanning in the second dimension was realised using mechanical movements of an array, instead of rotating prisms. A disadvantage of such system is the necessity to use mechanical elements.

The device presented in work (Granz & Oppelt, 1987) functions in pseudo-real time and its construction is similar to the camera described in (Brettel et al., 1981). An incoherent ultrasonic wave passes through an object in a wide beam in order to reduce the effect of spots. An image is obtained using a spherical mirror, located behind an object and in front of a 2-D, electronically controlled receiving matrix of PVDF foil transducers, in the arrangement of 128 x 128 elements. The camera enables obtaining images in pseudo-real time (with data buffering) at the rate of 25 frames per second. A disadvantage of this system is a high production cost and difficulties related with a faultless construction of such matrix.

The device presented in the work (Ermert et al., 2000) uses a linear sending probe, composed of 128 piezoceramic transducers, of rectangular dimensions (width much larger than height), a linear receiving probe, composed of 128 PVDF transducers (height much larger than width) and a lens, which focuses an ultrasonic wave beam in one direction. Probes work in the frequency range of 2 ÷ 4 MHz. Sweeping of the structure of an object located in water, between probes, is realised by the means of phase focusing of an ultrasonic wave beam in the object plane, in the form of horizontal lines, 1.25 mm wide, which, after a

Application of such sending and receiving matrix in a projection arrangement is possible in such way that one of them is switched by the angle of 90º in relation to the other one and sequences of switching sending and receiving transducers are so coupled that enables a

Active matrices act similarly as earlier developed standard matrices (Section 7.1) but their

Using especially developed multi-element ultrasonic projection matrices with quick electronic switching of elementary transducers (Section 7), it is possible to obtain images in pseudo-real time (e.g. with a slight constant delay, resulting from the need of data buffering and processing), thus, a device using this method can be called an ultrasonic transmission camera (UTC) (Ermert et al., 2000). There are several know works regarding such cameras, conducted at some centres in the world, as described in Introduction (Brettel et al., 1981;

In the device presented in the work (Green et al., 1974), an incoherent ultrasonic wave was used for sonification of an object and a one-dimensional linear array, mounted in a set position served the role of a receiver. Scanning of an object structure in the second dimension was realised by the means of a complex system of properly rotated prisms. A

The device presented in (Brettel et al., 1981) uses a similar concept as in work (Green et al., 1974). The difference consists in the fact that a projection image is projected on the water surface of a camera and next, it is detected by a linear array of transducers. What is more, scanning in the second dimension was realised using mechanical movements of an array, instead of rotating prisms. A disadvantage of such system is the necessity to use mechanical

The device presented in work (Granz & Oppelt, 1987) functions in pseudo-real time and its construction is similar to the camera described in (Brettel et al., 1981). An incoherent ultrasonic wave passes through an object in a wide beam in order to reduce the effect of spots. An image is obtained using a spherical mirror, located behind an object and in front of a 2-D, electronically controlled receiving matrix of PVDF foil transducers, in the arrangement of 128 x 128 elements. The camera enables obtaining images in pseudo-real time (with data buffering) at the rate of 25 frames per second. A disadvantage of this system is a high production cost and difficulties related with a faultless construction of

The device presented in the work (Ermert et al., 2000) uses a linear sending probe, composed of 128 piezoceramic transducers, of rectangular dimensions (width much larger than height), a linear receiving probe, composed of 128 PVDF transducers (height much larger than width) and a lens, which focuses an ultrasonic wave beam in one direction. Probes work in the frequency range of 2 ÷ 4 MHz. Sweeping of the structure of an object located in water, between probes, is realised by the means of phase focusing of an ultrasonic wave beam in the object plane, in the form of horizontal lines, 1.25 mm wide, which, after a

Ermert et al., 2000; Granz & Oppelt, 1987; Green et al., 1974; Lehmann et al., 1999).

disadvantage of such system is the necessity to use mechanical elements.

proper control of scanning of a received beam (Opielinski & Gudra, 2010b).

construction is more simple and improved.

elements.

such matrix.

**8. Models of ultrasonic transmission camera** 

transition through the object and the lens, are received by proper linear transducers of the receiving probe. One disadvantage in this case is beam focusing in a particular area of the object, what causes image spreading and occurrence of artefacts outside the focus.

The device presented in work (Lehmann et al., 1999) is a model of an acoustic holograph used the through-transmission signal. This approach uses coherent sound and coherent light to produce real-time, large field-of-view images with pronounced edge definition in soft tissues of the body.

In most of known studies, the projection parameter is the signal amplitude, not its transition time, which seems to more attractive due to the simplicity and precision of measurements.

This Section (8) includes a description of the results of author's studies, which aim is a construction of a special measuring setup for visualisation of biological structures by the means of ultrasonic projection, which enables simultaneous measurements of several acoustic parameters in pseudo-real time (multi-parameter ultrasonic transmission camera), using a pair of electronically controlled multi-element matrices of piezoceramic transducers, described in Section 7.

Fig. 21. Block scheme of the laboratory measurement setup for visualization of the distribution of acoustic parameters of biological media internal structure by means of ultrasonic projection method

Due to diversified ways of conducting electrodes to transducers of matrices and different controlling methods, two laboratory setups were developed, for imaging internal structures of biological media, by the means of ultrasound projection: using a pair of ultrasonic 512 element standard matrix – sending and receiving one - and using a pair of ultrasonic 512 element passive or active matrices (Opielinski & Gudra, 2010a; Opielinski et al., 2010a, 2010b). A general block scheme of the laboratory measurement setup is presented in Fig.21. Elementary piezoceramic transducers of the sending matrix (Opielinski & Gudra, 2010a) are excite by a burst type pulses generator, controlled by a computer through the GPIB

Ultrasonic Projection 55

(a) (b)

(a) (b)

longitudinal projection of a hard-boiled chicken egg (a) and an image of the distribution of the projection values of frequency of receiving pulses in a longitudinal projection of the

Good image quality without artefacts caused by the beam focusing is a great advantage in that solution in comparison with UTC described in literature (Brettel et al., 1981; Ermert et al., 2000; Granz & Oppelt, 1987; Green et al., 1974). The disadvantage is not so good

The results obtained confirm that the designed multi-element ultrasonic 2-D matrices are suitable for projection imaging of biological media (especially soft tissue) with the use of scanning method through switching of the right pairs (or groups) of sending and receiving elementary transducers. Using appropriate combinations of apertures of the sending and receiving matrix (with accordance to the scanning method) allows to increasing the directivity and the acoustic pressure level of the ultrasonic wave beam and ensures its apodization in the transmission system. Additionally, rotation of a pair of probes around the

Fig. 23. An image of the distribution of the projection values of sound velocity in a

Fig. 22. Lateral projection of the agar phantom *AC\_Blue*: structure (a), image of the

distribution of projection values of sound velocity in grey scale (b)

scanning resolution, at present, what have to be improved.

same egg (b)

**9. Summary** 

connection. A computer is also used for switching transducers of sending and receiving matrices, using developed and software-assisted systems of electronic keys. Signals received by elementary piezoceramic transducers of the receiving matrix (Opielinski & Gudra, 2010a) after an amplification and acquisition by the means of a computer oscilloscope data acquisition (DAQ) card, are recorded on the hard drive. Additionally, a digital oscilloscope is used for visualisation and control of measurement signals. Projection values of acoustic parameters assigned for visualisation (e.g. transition time of an ultrasonic wave or a projective value of sound velocity) are determined from recorded signals, by the means of a specially developed software. Water temperature is controlled by a digital thermometer of the resolution at 0.1 °C. The acceleration of measurements, e.g. by decreasing sampling frequency of receiving signal, enables to obtain images in pseudo-real time (several to tenodd frames per second). It can be used simple square burst pulse generator as integrated electronic circuit instead sinusoidal one, what enables to achieve the amplitude of ultrasonic matrix transducer exciting signal more than 60 Vpp, as well.

One of objects that were subjected to projection examinations on the developed measurement setups, was phantom *AC\_Blue* in the form of a cylinder made of agar gel, of the diameter of 55 mm and the length of 50 mm, made of 3 % (by weight) agar solution, with an additive of propylene glycol (by volume), in order to achieve a greater diversification of sound velocity values in the structure in relation to the values in water (Fig.22a). Two cylinders, made of about 4 % agar solution with an additive of propylene glycol, were decentrally put into the cylinder. Diameters of internal cylinders were 15 mm and 8 mm. Additionally, in the *AC\_Blue* cylinder, a through-and-through round hole, of 6 mm diameter, was made. This hole was filled with water during measurements. Sound velocities in the structure of *AC\_Blue* cylinder are different from the speed in water, by the following values: +10 m/s – surroundings, +13 m/s larger internal cylinder, +12 m/s smaller internal cylinder. Figure 22b presents, in grey scale, an image of the distribution of projection values of sound velocity in a lateral projection of agar phantom *AC\_Blue*, obtained from measurements, made using a pair of 512-element standard matrices (Section 7.1) in the resolution of 32 x 16 pixels.

Figure 23a presents an image (in grey scale) of the distribution of the projection values of sound velocity in longitudinal projection of a hard-boiled chicken egg, without shell, immersed in water, obtained from measurements, made using a pair of 512-element passive matrices, in the resolution of 32 x 16 pixels. A similar image of the distribution of frequency of ultrasonic wave pulses in a longitudinal projection of the same egg is presented in Figure 23b.

Heterogeneities, differing in sound velocities values even by about several m/s are clearly visible in projection images (Fig.22). On the basis of these images, it is possible to estimate sound speeds in the structure of examined objects. The projection image of a hard-boiled egg (Fig.23) distinctly visualises borders of the yolk area. Images obtained from projection measurements of a frequency down shift of receiving pulses (Fig.23b) are of a worse quality than images visualising projections of sound velocity (Fig.23a), however, it is not difficult to clearly recognise heterogeneity boundaries in both images. The higher is attenuation in a medium on the way of an ultrasonic wave beam, the larger is drop of frequency of a receiving pulse (formula (4); see Fig.23b).

connection. A computer is also used for switching transducers of sending and receiving matrices, using developed and software-assisted systems of electronic keys. Signals received by elementary piezoceramic transducers of the receiving matrix (Opielinski & Gudra, 2010a) after an amplification and acquisition by the means of a computer oscilloscope data acquisition (DAQ) card, are recorded on the hard drive. Additionally, a digital oscilloscope is used for visualisation and control of measurement signals. Projection values of acoustic parameters assigned for visualisation (e.g. transition time of an ultrasonic wave or a projective value of sound velocity) are determined from recorded signals, by the means of a specially developed software. Water temperature is controlled by a digital thermometer of the resolution at 0.1 °C. The acceleration of measurements, e.g. by decreasing sampling frequency of receiving signal, enables to obtain images in pseudo-real time (several to tenodd frames per second). It can be used simple square burst pulse generator as integrated electronic circuit instead sinusoidal one, what enables to achieve the amplitude of ultrasonic

One of objects that were subjected to projection examinations on the developed measurement setups, was phantom *AC\_Blue* in the form of a cylinder made of agar gel, of the diameter of 55 mm and the length of 50 mm, made of 3 % (by weight) agar solution, with an additive of propylene glycol (by volume), in order to achieve a greater diversification of sound velocity values in the structure in relation to the values in water (Fig.22a). Two cylinders, made of about 4 % agar solution with an additive of propylene glycol, were decentrally put into the cylinder. Diameters of internal cylinders were 15 mm and 8 mm. Additionally, in the *AC\_Blue* cylinder, a through-and-through round hole, of 6 mm diameter, was made. This hole was filled with water during measurements. Sound velocities in the structure of *AC\_Blue* cylinder are different from the speed in water, by the following values: +10 m/s – surroundings, +13 m/s larger internal cylinder, +12 m/s smaller internal cylinder. Figure 22b presents, in grey scale, an image of the distribution of projection values of sound velocity in a lateral projection of agar phantom *AC\_Blue*, obtained from measurements, made using a pair of 512-element standard matrices (Section 7.1) in the

Figure 23a presents an image (in grey scale) of the distribution of the projection values of sound velocity in longitudinal projection of a hard-boiled chicken egg, without shell, immersed in water, obtained from measurements, made using a pair of 512-element passive matrices, in the resolution of 32 x 16 pixels. A similar image of the distribution of frequency of ultrasonic wave pulses in a longitudinal projection of the same egg is presented in Figure 23b. Heterogeneities, differing in sound velocities values even by about several m/s are clearly visible in projection images (Fig.22). On the basis of these images, it is possible to estimate sound speeds in the structure of examined objects. The projection image of a hard-boiled egg (Fig.23) distinctly visualises borders of the yolk area. Images obtained from projection measurements of a frequency down shift of receiving pulses (Fig.23b) are of a worse quality than images visualising projections of sound velocity (Fig.23a), however, it is not difficult to clearly recognise heterogeneity boundaries in both images. The higher is attenuation in a medium on the way of an ultrasonic wave beam, the larger is drop of frequency of a

matrix transducer exciting signal more than 60 Vpp, as well.

resolution of 32 x 16 pixels.

receiving pulse (formula (4); see Fig.23b).

Fig. 22. Lateral projection of the agar phantom *AC\_Blue*: structure (a), image of the distribution of projection values of sound velocity in grey scale (b)

Fig. 23. An image of the distribution of the projection values of sound velocity in a longitudinal projection of a hard-boiled chicken egg (a) and an image of the distribution of the projection values of frequency of receiving pulses in a longitudinal projection of the same egg (b)

Good image quality without artefacts caused by the beam focusing is a great advantage in that solution in comparison with UTC described in literature (Brettel et al., 1981; Ermert et al., 2000; Granz & Oppelt, 1987; Green et al., 1974). The disadvantage is not so good scanning resolution, at present, what have to be improved.

## **9. Summary**

The results obtained confirm that the designed multi-element ultrasonic 2-D matrices are suitable for projection imaging of biological media (especially soft tissue) with the use of scanning method through switching of the right pairs (or groups) of sending and receiving elementary transducers. Using appropriate combinations of apertures of the sending and receiving matrix (with accordance to the scanning method) allows to increasing the directivity and the acoustic pressure level of the ultrasonic wave beam and ensures its apodization in the transmission system. Additionally, rotation of a pair of probes around the

Ultrasonic Projection 57

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### **10. References**


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**10. References** 

Paris, Tokyo

in a higher amount and arranged more densely.


**0**

**3**

<sup>1</sup>*Ehime University* <sup>2</sup>*Hitachi, Ltd*

*Japan*

**3-D Modelings of an Ultrasonic Phased Array**

Kazuyuki Nakahata1 and Naoyuki Kono<sup>2</sup>

**Transducer and Its Radiation Properties in Solid**

Ultrasonic phased array technology has become popular in the field of industrial nondestructive testing (NDT). The technology is notable for its capability to provide images of the inside of a target in real time. Since the image quality results from rapid steering and focusing of the radiation beam from an array transducer, it is essential to have well understood characteristics of the radiated beam. Even though fundamental concepts of the array transducer in medical fields were introduced over 30 years ago (Macovski, 1979), the beam modeling and optimization of the array transducer for the NDT have only been investigated intensively in the past decade (Azar et al., 2000; Song & Kim, 2002). The characteristics of the radiated beam from the ultrasonic phased array transducer vary according to the transducer parameters such as frequency, aperture size, number of elements, pitch(inter-element spacing), element width, layout dimensions and so forth. If these parameters are not chosen properly, spurious grating lobes or side lobes with high amplitude will exist in the radiation beam field. Consequently, the image quality will be deteriorated

In this chapter, we first show principles of electronic scanning with phased array transducers for the NDT and effective settings of the delay time for the beam steering and focusing. And then mathematical models of linear and matrix array transducers and simulation tools to predict ultrasonic beams from the transducers are outlined. We explain three-dimensional (3-D) numerical simulation tools in both frequency and time domains. This chapter is

• In Section **2**, we show various ultrasonic array transducers for the NDT and electronic control of the array transducer. First the appropriate delay time setting for beam steering and focusing is described, and then causes and prevention of the grating lobe are

• In Section **3**, modelings of the radiated beam field in the frequency domain are introduced. Basically, the Rayleigh Sommerfeld model is used for the expression of the radiated wave field from an array transducer. Here the multi-Gaussian beam (MGB) model (Schmerr, 2000) accelerates the 3-D simulation of the radiated wave field. The MGB model has an ability to calculate the radiated beam by superposing a small number of Gaussian beams. Based on the simulation results, some characteristics about beam steering and focusing of

the linear and matrix array transducers are described.

**1. Introduction**

significantly.

arranged as follows:

explained.

Method, *Acoustics '08*, ISBN 9782952110549 2952110549, Paris, France, June/July 2008


## **3-D Modelings of an Ultrasonic Phased Array Transducer and Its Radiation Properties in Solid**

Kazuyuki Nakahata1 and Naoyuki Kono<sup>2</sup> <sup>1</sup>*Ehime University* <sup>2</sup>*Hitachi, Ltd Japan*

#### **1. Introduction**

58 Ultrasonic Waves

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Ultrasonic phased array technology has become popular in the field of industrial nondestructive testing (NDT). The technology is notable for its capability to provide images of the inside of a target in real time. Since the image quality results from rapid steering and focusing of the radiation beam from an array transducer, it is essential to have well understood characteristics of the radiated beam. Even though fundamental concepts of the array transducer in medical fields were introduced over 30 years ago (Macovski, 1979), the beam modeling and optimization of the array transducer for the NDT have only been investigated intensively in the past decade (Azar et al., 2000; Song & Kim, 2002). The characteristics of the radiated beam from the ultrasonic phased array transducer vary according to the transducer parameters such as frequency, aperture size, number of elements, pitch(inter-element spacing), element width, layout dimensions and so forth. If these parameters are not chosen properly, spurious grating lobes or side lobes with high amplitude will exist in the radiation beam field. Consequently, the image quality will be deteriorated significantly.

In this chapter, we first show principles of electronic scanning with phased array transducers for the NDT and effective settings of the delay time for the beam steering and focusing. And then mathematical models of linear and matrix array transducers and simulation tools to predict ultrasonic beams from the transducers are outlined. We explain three-dimensional (3-D) numerical simulation tools in both frequency and time domains. This chapter is arranged as follows:


Fig. 2. Schematic of the electrical scanning. (a) Transmitting: ultrasonic waves are

synthesized with time-delay signal processing to a single receiving wave.

W

<sup>A</sup> Ax

Fig. 3. Main parameters of array transducers: element pitches *p*, element numbers *N*, transducer apertures *A*, element gaps *g* and element widths *W*; (a) linear array and (b)

array transducers, subscripts of parameters correspond to the *x* and *y* axes.

The main parameters of the linear and matrix arrays are the aperture, the element pitch, and the element number, shown in Fig.3. The apertures *A* are expressed as a function of the element pitch *p*, the element number *N* and the element gap *g* in Equation (1). For the matrix

gx

px

0

1

2

n

*A* = *N p* − *g*. (1)

Ay

gy

N-1

0 1 N-1

p

g

matrix array

transmitted at each transducer element with delayed excitation; each transmitting waves are phased near the focal point. (b) Receiving: received waves at each transducer element are

3-D Modelings of an Ultrasonic Phased Array Transducer and Its Radiation Properties in Solid 61

py <sup>2</sup>

0

1

m

M-1


### **2. Array transducer for NDT**

Ultrasonic phased array techniques have been applied to the NDT. In the phased array techniques, angled beams with various focal lengths are generated by the array transducers composed of small ultrasonic transducer elements. The type of beams is electrically controlled through delay time control for excitation of transmitted waves and signal processing of received waves.

The array transducers have two advantages for NDT in comparison with the conventional single or dual element transducers. Firstly, the various angled beams can cover wider range of an inspection target. Secondly, NDT results can be visualized immediately as a cross section images (B-scan) or 3-D images.

#### **2.1 Principle of electric scanning**

General configurations of the array transducers are classified into one-dimensional (1-D) arrays and two-dimensional (2-D) arrays shown in Fig.1. Linear arrays (Fig.1(a)) and annular arrays (Fig.1(c)) are typical configurations of the 1-D array transducers. The linear arrays provide beam-steering and focusing in a plane. The Annular arrays adjust focal length along an axis. In recent years, the 2-D arrays have been applied to the NDT field for maximization of the advantages of the ultrasonic phased array technology (Drinkwater & Wilcox, 2006). Matrix arrays (Fig.1(b)) and segmented annular arrays (Fig.1(d)) can steer beams in 3-D volume. Ultrasonic beams are electrically scanned by delay-time control for each transducer element composed of array transducers as shown in Fig.2. Patterns of the delay time for transmitting and receiving (hereinafter referred to as "delay law") are stored on the delay-time controller which shifts excitation pulses in transmitting and signals of reflection waves in receiving.

Fig. 1. General element configurations of array transducers: (a) 1-D linear array, (b) 2-D matrix array, (c)1-D annular array, and (d) 2-D segmented annular array.

• In Section **4** the finite integration technique (FIT) (Fellinger et al., 1995) is introduced as a time domain calculation tool. The FIT represents very stable and straightforward schemes to investigate the wave propagation in complex fields such as inhomogeneous and anisotropic media. For example, a dissimilar welding part includes metal grains of different size and orientation, therein the FIT is of assistance to predict the ultrasonic propagation direction as well as scattering and attenuation in the welding part. Here a transient wave simulation of the array ultrasonic testing (UT) for a welded T-joint is

Ultrasonic phased array techniques have been applied to the NDT. In the phased array techniques, angled beams with various focal lengths are generated by the array transducers composed of small ultrasonic transducer elements. The type of beams is electrically controlled through delay time control for excitation of transmitted waves and signal processing of

The array transducers have two advantages for NDT in comparison with the conventional single or dual element transducers. Firstly, the various angled beams can cover wider range of an inspection target. Secondly, NDT results can be visualized immediately as a cross section

General configurations of the array transducers are classified into one-dimensional (1-D) arrays and two-dimensional (2-D) arrays shown in Fig.1. Linear arrays (Fig.1(a)) and annular arrays (Fig.1(c)) are typical configurations of the 1-D array transducers. The linear arrays provide beam-steering and focusing in a plane. The Annular arrays adjust focal length along an axis. In recent years, the 2-D arrays have been applied to the NDT field for maximization of the advantages of the ultrasonic phased array technology (Drinkwater & Wilcox, 2006). Matrix arrays (Fig.1(b)) and segmented annular arrays (Fig.1(d)) can steer beams in 3-D volume. Ultrasonic beams are electrically scanned by delay-time control for each transducer element composed of array transducers as shown in Fig.2. Patterns of the delay time for transmitting and receiving (hereinafter referred to as "delay law") are stored on the delay-time controller which shifts excitation pulses in transmitting and signals of reflection waves in receiving.

(a) (b) (c) (d)

Fig. 1. General element configurations of array transducers: (a) 1-D linear array, (b) 2-D

matrix array, (c)1-D annular array, and (d) 2-D segmented annular array.

x xxx

y y y

• In Section **5**, we summarize our research and discuss prospects for array UT.

demonstrated.

received waves.

**2. Array transducer for NDT**

images (B-scan) or 3-D images.

**2.1 Principle of electric scanning**

y

Fig. 2. Schematic of the electrical scanning. (a) Transmitting: ultrasonic waves are transmitted at each transducer element with delayed excitation; each transmitting waves are phased near the focal point. (b) Receiving: received waves at each transducer element are synthesized with time-delay signal processing to a single receiving wave.

Fig. 3. Main parameters of array transducers: element pitches *p*, element numbers *N*, transducer apertures *A*, element gaps *g* and element widths *W*; (a) linear array and (b) matrix array

The main parameters of the linear and matrix arrays are the aperture, the element pitch, and the element number, shown in Fig.3. The apertures *A* are expressed as a function of the element pitch *p*, the element number *N* and the element gap *g* in Equation (1). For the matrix array transducers, subscripts of parameters correspond to the *x* and *y* axes.

$$A = Np - \text{g.}\tag{1}$$

In the case of the linear array transducers, Equation (3) is simplified to

 1 − 

(sin *θ* − *xn*/*R*)

In the finite limitation of the focal length *R*, Equations (3) and (4) are simplified to the

3-D Modelings of an Ultrasonic Phased Array Transducer and Its Radiation Properties in Solid 63

Δ*τnm* (x; *θ*, *φ*) = Δ*τ<sup>n</sup>* (x; *θ*, *φ*) + Δ*τ<sup>m</sup>* (x; *θ*, *φ*)

Fig. 5. Definition of the coordinate system and focal points F : (a) the 1-D array transducer

Characteristics of the focusing beams , such as a beam width, peak amplitude and focus length, depend on both parameters of the array transducers (shown in Fig.3) and delay laws. As examples, delay laws of normal and angled focused beams for the linear array transducers are plotted in Fig.6. The horizontal axis is position of the transducer element and the vertical axis is delay time. Negative values of the delay time correspond to transmitting ultrasonic beam before the relative origin of the time. Therefore the delay time in an actual ultrasonic phased array equipment is shifted to nonnegative value by adding constant values to the

Beam profiles for the delay laws in Fig.6 are calculated based on a Rayleigh-Sommerfeld model (Kono et al., 2010), as shown in Fig.7. The calculation conditions are as follows: the array configuration is the linear array transducer with 16 elements; element pitches, a half of a wavelength *λ* of 2.95mm; an element width, 8*λ*; center frequency, 2.0MHz; focal lengths, 20-80mm and incident angles, 0–45◦. Peak levels of the amplitude have a propensity to decrease with increased focal length from the results of the normal beams in Fig.7(a)-(c). For the angled beam, peak levels decrease with higher incident angles from the results in

<sup>2</sup> + cos<sup>2</sup> *θ*

= *xn* sin *θ* cos *φ*/*c* + *ym* sin *θ* sin *φ*/*c*, (5)

Δ*τn*(x; *θ*) = *xn* sin *θ*/*c*. (6)

. (4)

Δ*τ<sup>n</sup>* (x; F ) = *R*

following expressions, respectively:

and (b) 2-D array transducer.

**2.3 Effects of beam focusing and steering**

theoretical delay time in Equation (3).

Fig.7(b),(e) and (f).

Fig. 4. Schematics of the electric scanning and B-scan results for electrically discharged notches: (a) sectorial scanning and (b) linear scanning.

Electrical scanning patterns widely-used in the NDT field are sectorial and linear scanning as schematized in Fig.4. B-scan results for electrically discharged notch of the sector and the linear scanning are displayed also in Fig.4. The through-wall depth of the notch is measured as the difference of the *z*-axis between echoes from the root and the tip of the notch. NDT data in the sectorial scanning is displayed as a sector-form B-scan image (cross section) by changing propagation directions. In the sectorial scanning, imaging area can be wider than an aperture of an array transducer. Element number is generally from 15 to 63. The linear scanning provides a parallelogram B-scan image by switching positions of active elements. In the linear scanning, imaging area can be wider by a large number of transducer elements, e.g. 63 to 255. Substituting a motion axis of mechanical scanning for the linear scanning can shorten scanning time of an automated UT.

#### **2.2 Delay setting for electrical scanning**

The coordinate systems for the linear array transducer and the matrix transducer are defined as Fig.5. The delay laws imposed on each transducer element positioned at x = (*xn*, *ym*) for an angled beam with focal point F in a material characterized by velocity *c* is expressed by the difference of the propagation time:

$$
\Delta \tau\_{nm} \left( x; F \right) = \left( \left| F - x\_0 \right| - \left| F - x \right| \right) / c,\tag{2}
$$

where x<sup>0</sup> is the center position of the array transducer. In this section, the center position is set to the origin. The focal point F is expressed by the incident angle (zenith angle) *θ* and rotational angle (azimuth angles) *φ* : F = (*F*1, *F*2, *F*3)=(*R* sin *θ* cos *φ*, *R* sin *θ* sin *φ*, *R* cos *θ*). Therefore the delay law Δ*τnm* is obtained as

$$\Delta \tau\_{\rm mm} \left( x; \mathbf{F} \right) = \mathcal{R} \left[ 1 - \sqrt{\left( \sin \theta \cos \phi - \mathbf{x}\_{\rm n} / \mathcal{R} \right)^2 + \left( \sin \theta \sin \phi - y\_{\rm m} / \mathcal{R} \right)^2 + \cos^2 \theta} \right] / c. \tag{3}$$

In the case of the linear array transducers, Equation (3) is simplified to

4 Ultrasonic Waves

0 N-1 0 N-1

Fig. 4. Schematics of the electric scanning and B-scan results for electrically discharged

Electrical scanning patterns widely-used in the NDT field are sectorial and linear scanning as schematized in Fig.4. B-scan results for electrically discharged notch of the sector and the linear scanning are displayed also in Fig.4. The through-wall depth of the notch is measured as the difference of the *z*-axis between echoes from the root and the tip of the notch. NDT data in the sectorial scanning is displayed as a sector-form B-scan image (cross section) by changing propagation directions. In the sectorial scanning, imaging area can be wider than an aperture of an array transducer. Element number is generally from 15 to 63. The linear scanning provides a parallelogram B-scan image by switching positions of active elements. In the linear scanning, imaging area can be wider by a large number of transducer elements, e.g. 63 to 255. Substituting a motion axis of mechanical scanning for the linear scanning can

The coordinate systems for the linear array transducer and the matrix transducer are defined as Fig.5. The delay laws imposed on each transducer element positioned at x = (*xn*, *ym*) for an angled beam with focal point F in a material characterized by velocity *c* is expressed by

where x<sup>0</sup> is the center position of the array transducer. In this section, the center position is set to the origin. The focal point F is expressed by the incident angle (zenith angle) *θ* and rotational angle (azimuth angles) *φ* : F = (*F*1, *F*2, *F*3)=(*R* sin *θ* cos *φ*, *R* sin *θ* sin *φ*, *R* cos *θ*).

(sin *θ* cos *φ* − *xn*/*R*)

Δ*τnm* (x; F ) = (|F − x0|−|F − x|) /*c*, (2)

<sup>2</sup> <sup>+</sup> (sin *<sup>θ</sup>* sin *<sup>φ</sup>* <sup>−</sup> *ym*/*R*)

<sup>2</sup> + cos2 *θ*

/*c*. (3)

z

notches: (a) sectorial scanning and (b) linear scanning.

shorten scanning time of an automated UT.

**2.2 Delay setting for electrical scanning**

the difference of the propagation time:

Therefore the delay law Δ*τnm* is obtained as

 1 − 

Δ*τnm* (x; F ) = *R*

x x

z

$$\Delta \pi\_{\rm ll}(\mathbf{z}; \mathbf{F}) = \mathbb{R} \left[ 1 - \sqrt{\left( \sin \theta - \mathbf{x}\_{\rm ll} / \mathbb{R} \right)^{2} + \cos^{2} \theta} \right]. \tag{4}$$

In the finite limitation of the focal length *R*, Equations (3) and (4) are simplified to the following expressions, respectively:

$$\begin{split} \Delta \tau\_{\text{ll\%}} (\mathbf{z}; \theta, \phi) &= \Delta \tau\_{\text{ll}} (\mathbf{z}; \theta, \phi) + \Delta \tau\_{\text{ll}} (\mathbf{z}; \theta, \phi) \\ &= \mathbf{x}\_{\text{ll}} \sin \theta \cos \phi / \mathbf{c} + y\_{\text{m}} \sin \theta \sin \phi / \mathbf{c}, \end{split} \tag{5}$$

$$
\Delta \pi\_{\mathbb{H}}(x;\theta) = \mathbb{x}\_{\mathbb{H}} \sin \theta / c. \tag{6}
$$

Fig. 5. Definition of the coordinate system and focal points F : (a) the 1-D array transducer and (b) 2-D array transducer.

#### **2.3 Effects of beam focusing and steering**

Characteristics of the focusing beams , such as a beam width, peak amplitude and focus length, depend on both parameters of the array transducers (shown in Fig.3) and delay laws. As examples, delay laws of normal and angled focused beams for the linear array transducers are plotted in Fig.6. The horizontal axis is position of the transducer element and the vertical axis is delay time. Negative values of the delay time correspond to transmitting ultrasonic beam before the relative origin of the time. Therefore the delay time in an actual ultrasonic phased array equipment is shifted to nonnegative value by adding constant values to the theoretical delay time in Equation (3).

Beam profiles for the delay laws in Fig.6 are calculated based on a Rayleigh-Sommerfeld model (Kono et al., 2010), as shown in Fig.7. The calculation conditions are as follows: the array configuration is the linear array transducer with 16 elements; element pitches, a half of a wavelength *λ* of 2.95mm; an element width, 8*λ*; center frequency, 2.0MHz; focal lengths, 20-80mm and incident angles, 0–45◦. Peak levels of the amplitude have a propensity to decrease with increased focal length from the results of the normal beams in Fig.7(a)-(c). For the angled beam, peak levels decrease with higher incident angles from the results in Fig.7(b),(e) and (f).

Fig. 7. Calculation results of the beam profiles with the delay laws in Fig.6 and calculation region: 100mm square in the *xy*-plane: (a) normal beam with the focal length *R*=20mm, (b) normal beams with *R*=40mm, (c) normal beams with *R*=80mm, (d) 30◦ angled beam with

3-D Modelings of an Ultrasonic Phased Array Transducer and Its Radiation Properties in Solid 65

Equation (13) is consistent with Equation (11) substituted zero for the rotation angles of the main lobe and grating lobes. Grating lobes are generated on the condition that the absolute value of the left side in Equation (13) is less than or equal to one; therefore the grating lobes

Beam profiles for angled beams generated by two kinds of linear array transducers with different element pitch are compared in Fig.8. The position of the focal point F is the same each other. No grating lobe is observed for the linear array transducer with element pitch equal to one-half wavelength. A grating lobe is generated in the case of the array transducer with one-wavelength element pitch. The incident angle of the grating lobe is obtained as

Fundamentally, ultrasonic beam can be modeled by superposing spherical waves according to the Huygens' principle. The Rayleigh–Sommerfeld integral (RSI) model is also based on the Huygens' principle and well known as an expression of wave field in liquid from an immersion piston transducer. On the other hand, the wave field in solid from a contact piston transducer can be expressed with the Vezzetti's model (Vezzetti, 1985). This is a little bit complicated because the Vezzetti's model remains the integral form of the fundamental

can be suppressed when an element pitch is less than one-half wave length.

**3. Modeling of phased array transducer in frequency domain**

*R*=40mm, and (e) 45◦ angled beam with *R*=40mm.

−17.0◦ from Equation (13) for *i*=1 and *θ*=45◦.

#### **2.4 Generation of grating lobes**

Grating lobes can be generated under the condition that difference of the delay time between the adjacent transducer elements is a multiple of a period *T*. For the matrix array transducers, Equation (5) is separated into *x*-dependent and *y*-dependent terms:

$$
\Delta \pi\_{nm} \left( \mathbf{z}; \theta, \phi \right) - \Delta \pi\_{nm} \left( \mathbf{z}; \theta', \phi' \right) = \left( \mathbf{i} + \mathbf{j} \right) T,\tag{7}
$$

$$
\Delta \tau\_{\text{fl}} \left( x; \theta, \phi \right) - \Delta \tau\_{\text{fl}} \left( x; \theta', \phi' \right) = p\_{\text{x}} \left( \sin \theta \cos \phi - \sin \theta' \cos \phi' \right) / c = iT,\tag{8}
$$

$$
\Delta \tau\_{\mathfrak{M}} \left( \mathbf{z}; \theta, \phi \right) - \Delta \tau\_{\mathfrak{M}} \left( \mathbf{z}; \theta', \phi' \right) = p\_{\mathcal{Y}} \left( \sin \theta \sin \phi - \sin \theta' \sin \phi' \right) / \mathfrak{c} = \mathrm{j} \text{T}. \tag{9}
$$

where the position of the transducer element is set to element pitches: x = (*px*, *py*). The prime mark indicates the direction of grating lobes, *m* and *n* are integer indexes for the grating lobe with phase difference 2(*i* + *j*)*π*.

Therefore, the propagation direction of grating lobes for the matrix array transducers are calculated from the delay time differences in Equations (8) and (9) (Kono & Mizota, 2011).

$$\tan\phi' = \frac{p\_x}{p\_y} \left(\frac{p\_y \sin\theta' \sin\phi'/c}{p\_x \sin\theta' \cos\phi'/c}\right) = \frac{p\_x}{p\_y} \left(\frac{p\_y \sin\theta \sin\phi/c - jT}{p\_x \sin\theta \cos\phi/c - iT}\right),\tag{10}$$

$$\sin\theta' = \frac{p\_{\text{x}}\sin\theta\cos\phi/\text{c} - \text{iT}}{p\_{\text{x}}\cos\phi'/\text{c}} = \frac{p\_{\text{y}}\sin\theta\sin\phi/\text{c} - \text{jT}}{p\_{\text{y}}\sin\phi'/\text{c}}.\tag{11}$$

Similarly, the generation conditions of grating lobes for the linear array transducers are derived from Equation (11) substituted *p* for *xn*:

$$
\Delta \tau\_{\hbar} \left( p; \theta \right) - \Delta \tau\_{\hbar} \left( p; \theta' \right) = p \left( \sin \theta - \sin \theta' \right) / c = iT,\tag{12}
$$

where *p* is an element pitch and *i* is an integer index for the grating lobe with phase difference 2*iπ*. The incident angle of grating lobes for the linear array transducer is transformed from Equation (12) and the relation between cycle *T* and wavelength *λ* : *T* = *λ*/*c* :

$$i\sin\theta^{\prime} = \sin\theta - i\lambda/p.\tag{13}$$

Fig. 6. Delay laws for the linear array transducer; number of element *N*=16; element pitch *p* = 1.475mm; velocity *c*=5900m/s (carbon steel): (a) normal beams and (b) angled beams with the focal length *R*=40 mm.

Grating lobes can be generated under the condition that difference of the delay time between the adjacent transducer elements is a multiple of a period *T*. For the matrix array transducers,

> x; *θ*� , *φ*�

where the position of the transducer element is set to element pitches: x = (*px*, *py*). The prime mark indicates the direction of grating lobes, *m* and *n* are integer indexes for the grating lobe

Therefore, the propagation direction of grating lobes for the matrix array transducers are calculated from the delay time differences in Equations (8) and (9) (Kono & Mizota, 2011).

Similarly, the generation conditions of grating lobes for the linear array transducers are

) = *p* 

Fig. 6. Delay laws for the linear array transducer; number of element *N*=16; element pitch *p* = 1.475mm; velocity *c*=5900m/s (carbon steel): (a) normal beams and (b) angled beams

where *p* is an element pitch and *i* is an integer index for the grating lobe with phase difference 2*iπ*. The incident angle of grating lobes for the linear array transducer is transformed from

*px* cos *<sup>φ</sup>*�/*<sup>c</sup>* <sup>=</sup> *py* sin *<sup>θ</sup>* sin *<sup>φ</sup>*/*<sup>c</sup>* <sup>−</sup> *jT*

sin *θ* − sin *θ*�

/*c*

/*c* <sup>=</sup> *px py*

sin *θ* cos *φ* − sin *θ*� cos *φ*�

sin *θ* sin *φ* − sin *θ*� sin *φ*�

 *py* sin *<sup>θ</sup>* sin *<sup>φ</sup>*/*<sup>c</sup>* − *jT px* sin *θ* cos *φ*/*c* − *iT*

sin *θ*� = sin *θ* − *iλ*/*p*. (13)

= (*i* + *j*) *T*, (7)

*py* sin *<sup>φ</sup>*�/*<sup>c</sup>* . (11)

/*c* = *iT*, (12)

/*c* = *iT*, (8)

/*c* = *jT*. (9)

, (10)

Equation (5) is separated into *x*-dependent and *y*-dependent terms:

 x; *θ*� , *φ*� = *px* 

 x; *θ*� , *φ*� = *py* 

*py* sin *θ*� sin *φ*�

sin *<sup>θ</sup>*� <sup>=</sup> *px* sin *<sup>θ</sup>* cos *<sup>φ</sup>*/*<sup>c</sup>* <sup>−</sup> *iT*

Δ*τ<sup>n</sup>* (*p*; *θ*) − Δ*τn*(*p*; *θ*�

Equation (12) and the relation between cycle *T* and wavelength *λ* : *T* = *λ*/*c* :

*px* sin *θ*� cos *φ*�

Δ*τnm* (x; *θ*, *φ*) − Δ*τnm*

**2.4 Generation of grating lobes**

with phase difference 2(*i* + *j*)*π*.

with the focal length *R*=40 mm.

Δ*τ<sup>n</sup>* (x; *θ*, *φ*) − Δ*τ<sup>n</sup>*

Δ*τ<sup>m</sup>* (x; *θ*, *φ*) − Δ*τ<sup>m</sup>*

tan *<sup>φ</sup>*� <sup>=</sup> *px*

*py*

derived from Equation (11) substituted *p* for *xn*:

Fig. 7. Calculation results of the beam profiles with the delay laws in Fig.6 and calculation region: 100mm square in the *xy*-plane: (a) normal beam with the focal length *R*=20mm, (b) normal beams with *R*=40mm, (c) normal beams with *R*=80mm, (d) 30◦ angled beam with *R*=40mm, and (e) 45◦ angled beam with *R*=40mm.

Equation (13) is consistent with Equation (11) substituted zero for the rotation angles of the main lobe and grating lobes. Grating lobes are generated on the condition that the absolute value of the left side in Equation (13) is less than or equal to one; therefore the grating lobes can be suppressed when an element pitch is less than one-half wave length.

Beam profiles for angled beams generated by two kinds of linear array transducers with different element pitch are compared in Fig.8. The position of the focal point F is the same each other. No grating lobe is observed for the linear array transducer with element pitch equal to one-half wavelength. A grating lobe is generated in the case of the array transducer with one-wavelength element pitch. The incident angle of the grating lobe is obtained as −17.0◦ from Equation (13) for *i*=1 and *θ*=45◦.

#### **3. Modeling of phased array transducer in frequency domain**

Fundamentally, ultrasonic beam can be modeled by superposing spherical waves according to the Huygens' principle. The Rayleigh–Sommerfeld integral (RSI) model is also based on the Huygens' principle and well known as an expression of wave field in liquid from an immersion piston transducer. On the other hand, the wave field in solid from a contact piston transducer can be expressed with the Vezzetti's model (Vezzetti, 1985). This is a little bit complicated because the Vezzetti's model remains the integral form of the fundamental

d*p* is the polarization vector:

and *Kp*(*θ*) is the directivity function:

normal of the transducer surface.

where

 *S*

 *<sup>∂</sup>y*<sup>2</sup> *∂ξ*


*Kp*(*θ*)d*<sup>p</sup>*

where |*Y*| is the Jacobian operator:

*∂y*<sup>3</sup> *∂η* <sup>−</sup> *<sup>∂</sup>y*<sup>3</sup> *∂ξ*

**3.1 Rayleigh–Sommerfeld numerical integral (RSNI)**

*<sup>N</sup>*<sup>1</sup> <sup>=</sup> <sup>1</sup> 2

*<sup>N</sup>*<sup>3</sup> <sup>=</sup> <sup>1</sup> 2

into (14), we can rewrite the integral form as follows.

*∂y*<sup>2</sup> *∂η* 2 + *∂y*<sup>3</sup> *∂ξ*

 1 −1  1 −1

exp(i*kpr*) *r*

(1 − *ξ*) ·

(1 + *ξ*) ·

<sup>d</sup>*<sup>p</sup>* <sup>=</sup> *<sup>x</sup>*<sup>1</sup> <sup>−</sup> *<sup>y</sup>*<sup>1</sup>

*<sup>r</sup>* <sup>e</sup><sup>1</sup> <sup>+</sup>

*Kp*(*θ*) = cos *θκ*2(*κ*2/2 <sup>−</sup> sin2 *<sup>θ</sup>*)

method, but only applicable to circular and rectangular element of the transducer.

y(*ξ*, *η*) =

1 2

1 2

*dS*(y) = <sup>1</sup>

−1

*<sup>G</sup>*(*x*)=(*x*<sup>2</sup> <sup>−</sup> *<sup>κ</sup>*2/2)

2*G*(sin *θ*)

*x*<sup>2</sup> − *y*<sup>2</sup> *<sup>r</sup>* <sup>e</sup><sup>2</sup> <sup>+</sup>

3-D Modelings of an Ultrasonic Phased Array Transducer and Its Radiation Properties in Solid 67

<sup>2</sup> + *x*<sup>2</sup> <sup>1</sup> − *<sup>x</sup>*<sup>2</sup> 

where *cs* is S-wave velocity, *κ* = *cp*/*cs* and *θ* is the angle between the vector (x − y) and the

In the next section, we show two methods to obtain the solution u in Equation (14). One is based on a numerical integral method and the other is an approximation method using superposition of the Gaussian beam. The former method needs some numerical calculation but is effective even if the transducer element has a complicated shape. The latter case is a fast

It is not easy to solve the integral in Equation (14) analytically except for simple element shape such as a rectangle or a circle. Therefore let us consider a numerical integral over *S* in Equation (14). For convenience we use the local coordinate system (*ξ*, *η*) on the transducer surface. Using the four-node two-dimensional element, we can interpolate the coordinate y as

> 4 ∑ *α*=1

(<sup>1</sup> <sup>−</sup> *<sup>η</sup>*), *<sup>N</sup>*<sup>2</sup> <sup>=</sup> <sup>1</sup>

(<sup>1</sup> <sup>+</sup> *<sup>η</sup>*), *<sup>N</sup>*<sup>4</sup> <sup>=</sup> <sup>1</sup>

In Equation (17), y*<sup>α</sup>* is the coordinate of the four element nodes. Substituting Equation (17)

 1 −1

*∂y*<sup>1</sup> *∂η* <sup>−</sup> *<sup>∂</sup>y*<sup>1</sup> *∂ξ*

Equation (19) can be calculated with numerical integration of the Gauss quadrature.

*f*(*ξ*, *η*)*dξdη* ∼=

2

2

*∂y*<sup>3</sup> *∂η* 2 + *∂y*<sup>1</sup> *∂ξ*

*Kp*(*θ*)d*<sup>p</sup>*

*I* ∑ *i*=1

*I* ∑ *j*=1 (1 + *ξ*) ·

(1 − *ξ*) ·

1 2

1 2

exp(i*kp*|x − y(*ξ*, *η*)|)

*∂y*<sup>2</sup> *∂η* <sup>−</sup> *<sup>∂</sup>y*<sup>2</sup> *∂ξ*

*x*<sup>3</sup> − *y*<sup>3</sup>

*<sup>r</sup>* <sup>e</sup><sup>3</sup> (15)

*<sup>κ</sup>*<sup>2</sup> − *<sup>x</sup>*<sup>2</sup> (16)

*Nα*(*ξ*, *η*)y*<sup>α</sup>* (17)

(1 − *η*),

(1 + *η*). (18)

<sup>|</sup><sup>x</sup> <sup>−</sup> <sup>y</sup>(*ξ*, *<sup>η</sup>*)<sup>|</sup> <sup>|</sup>*Y*|*dξd<sup>η</sup>* (19)

*f*(*ξi*, *ηj*)*wiwj* (21)

*∂y*<sup>1</sup> *∂η* 2 1/2 .

(20)

Fig. 8. Calculation results of 45◦ angled beam with focal length *R*=40mm: (a) the element pitch *p*=1.475mm (one-half wavelength); the element number *N*=16, and (b) the element pitch *p*=2.95mm (one wavelength) ; the element number *N*=8.

solution in elastic half space. Schmerr (Schmerr, 1998) has introduced an explicit expression of the Vezzetti's model using the stationary phase method. Here we show numerical modelings of the ultrasonic phased array based on the Schmerr's expression.

Here we assume the harmonic wave of exp(−i*ωt*) time dependency. First of all, consider a single planar piston transducer which generates a P-wave in solid. To model the radiation field from the transducer, we use the geometry shown in Fig.9, where the solid region is the half-space *x*<sup>3</sup> ≥ 0. On the *x*1–*x*<sup>2</sup> plane, we assume that velocity in the *x*<sup>3</sup> direction is zero everywhere except for a finite region *S*. Here the displacement in solid due to the P-wave is written as (Schmerr, 1998)

$$u(x,\omega) = \frac{P\_0}{2\pi\rho c\_p^2} \int\_S K\_p(\theta) d\_p \frac{\exp(\mathrm{i}k\_p r)}{r} dS(y) \tag{14}$$

where, *r* = |x − y|, *cp* and *kp*(= *ω*/*cp*) are the velocity and the wave number of the P-wave, respectively. Also *ρ* is the density and *P*<sup>0</sup> is a constant known pressure in *S*. In Equation (14),

Fig. 9. Coordinate system for a modeling of a contact transducer located on solid.

d*p* is the polarization vector:

8 Ultrasonic Waves

Fig. 8. Calculation results of 45◦ angled beam with focal length *R*=40mm: (a) the element pitch *p*=1.475mm (one-half wavelength); the element number *N*=16, and (b) the element

solution in elastic half space. Schmerr (Schmerr, 1998) has introduced an explicit expression of the Vezzetti's model using the stationary phase method. Here we show numerical modelings

Here we assume the harmonic wave of exp(−i*ωt*) time dependency. First of all, consider a single planar piston transducer which generates a P-wave in solid. To model the radiation field from the transducer, we use the geometry shown in Fig.9, where the solid region is the half-space *x*<sup>3</sup> ≥ 0. On the *x*1–*x*<sup>2</sup> plane, we assume that velocity in the *x*<sup>3</sup> direction is zero everywhere except for a finite region *S*. Here the displacement in solid due to the P-wave is

*Kp*(*θ*)d*<sup>p</sup>*

where, *r* = |x − y|, *cp* and *kp*(= *ω*/*cp*) are the velocity and the wave number of the P-wave, respectively. Also *ρ* is the density and *P*<sup>0</sup> is a constant known pressure in *S*. In Equation (14),

**y**(y1,y2,y3)

**d**p

a

**e**2

T

Fig. 9. Coordinate system for a modeling of a contact transducer located on solid.

**o**

exp(i*kpr*) *r*

**e**1

**x**(x1,x2,x3)

*dS*(y) (14)

pitch *p*=2.95mm (one wavelength) ; the element number *N*=8.

of the ultrasonic phased array based on the Schmerr's expression.

<sup>u</sup>(x, *<sup>ω</sup>*) = *<sup>P</sup>*<sup>0</sup>

S

**e**3

2*πρc*<sup>2</sup> *p S*

written as (Schmerr, 1998)

$$\mathbf{d}\_p = \frac{\mathbf{x}\_1 - y\_1}{r} \mathbf{e}\_1 + \frac{\mathbf{x}\_2 - y\_2}{r} \mathbf{e}\_2 + \frac{\mathbf{x}\_3 - y\_3}{r} \mathbf{e}\_3 \tag{15}$$

and *Kp*(*θ*) is the directivity function:

$$K\_p(\theta) = \frac{\cos \theta \mathbf{x}^2 (\mathbf{x}^2 / 2 - \sin^2 \theta)}{2G(\sin \theta)}$$

$$G(\mathbf{x}) = (\mathbf{x}^2 - \mathbf{x}^2 / 2)^2 + \mathbf{x}^2 \sqrt{1 - \mathbf{x}^2} \sqrt{\mathbf{x}^2 - \mathbf{x}^2} \tag{16}$$

where *cs* is S-wave velocity, *κ* = *cp*/*cs* and *θ* is the angle between the vector (x − y) and the normal of the transducer surface.

In the next section, we show two methods to obtain the solution u in Equation (14). One is based on a numerical integral method and the other is an approximation method using superposition of the Gaussian beam. The former method needs some numerical calculation but is effective even if the transducer element has a complicated shape. The latter case is a fast method, but only applicable to circular and rectangular element of the transducer.

#### **3.1 Rayleigh–Sommerfeld numerical integral (RSNI)**

It is not easy to solve the integral in Equation (14) analytically except for simple element shape such as a rectangle or a circle. Therefore let us consider a numerical integral over *S* in Equation (14). For convenience we use the local coordinate system (*ξ*, *η*) on the transducer surface. Using the four-node two-dimensional element, we can interpolate the coordinate y as

$$y(\xi, \eta) = \sum\_{a=1}^{4} N\_a(\xi, \eta) y^a \tag{17}$$

where

$$\begin{aligned} N\_1 &= \frac{1}{2}(1-\underline{\mathfrak{z}}) \cdot \frac{1}{2}(1-\eta), & N\_2 &= \frac{1}{2}(1+\underline{\mathfrak{z}}) \cdot \frac{1}{2}(1-\eta), \\ N\_3 &= \frac{1}{2}(1+\underline{\mathfrak{z}}) \cdot \frac{1}{2}(1+\eta), & N\_4 &= \frac{1}{2}(1-\underline{\mathfrak{z}}) \cdot \frac{1}{2}(1+\eta). \end{aligned} \tag{18}$$

In Equation (17), y*<sup>α</sup>* is the coordinate of the four element nodes. Substituting Equation (17) into (14), we can rewrite the integral form as follows.

$$\int\_{S} K\_{p}(\theta) d\_{p} \frac{\exp(\mathrm{i}k\_{p}r)}{r} dS(y) = \int\_{-1}^{1} \int\_{-1}^{1} K\_{p}(\theta) d\_{p} \frac{\exp(\mathrm{i}k\_{p}|x - y(\xi, \eta)|)}{|x - y(\xi, \eta)|} |Y| d\xi d\eta \tag{19}$$

where |*Y*| is the Jacobian operator:

$$|Y| = \left[ \left( \frac{\partial y\_2}{\partial \tilde{\xi}} \frac{\partial y\_3}{\partial \eta} - \frac{\partial y\_3}{\partial \tilde{\xi}} \frac{\partial y\_2}{\partial \eta} \right)^2 + \left( \frac{\partial y\_3}{\partial \tilde{\xi}} \frac{\partial y\_1}{\partial \eta} - \frac{\partial y\_1}{\partial \tilde{\xi}} \frac{\partial y\_3}{\partial \eta} \right)^2 + \left( \frac{\partial y\_1}{\partial \tilde{\xi}} \frac{\partial y\_2}{\partial \eta} - \frac{\partial y\_2}{\partial \tilde{\xi}} \frac{\partial y\_1}{\partial \eta} \right)^2 \right]^{1/2} \,\tag{20}$$

Equation (19) can be calculated with numerical integration of the Gauss quadrature.

$$\int\_{-1}^{1} \int\_{-1}^{1} f(\mathfrak{F}, \eta) d\mathfrak{F} d\eta \cong \sum\_{i=1}^{I} \sum\_{j=1}^{I} f(\mathfrak{F}\_{i\prime}, \eta\_{j}) w\_{i} w\_{j} \tag{21}$$

Equations (23) and (24) into (14), we can obtain:

, *<sup>ω</sup>*) = *<sup>P</sup>*0d*pKp*(*θo*) 2*πρc*<sup>2</sup> *p*

> exp − *Bk a*2 1

exp − *Bl a*2 2

<sup>−</sup>*ax*<sup>2</sup> <sup>+</sup> *bx*

10 ∑ *k*

−i*kp* 2

10 ∑ *l*

(*x*� <sup>1</sup>)2/*<sup>R</sup>* 1 + i*BkR*/*D*<sup>1</sup>

*dx* =

× ∞ −∞

× ∞ −∞

exp

, *<sup>ω</sup>*) = <sup>i</sup>*P*0d*pKp ρkpc*<sup>2</sup> *p*

<sup>×</sup> exp

<sup>1</sup>/2 and*D*<sup>2</sup> <sup>=</sup> *kpa*<sup>2</sup>

u(x, *ω*) =

**3.3 Numerical comparison of the RSNI and MGB model**

displacement <sup>|</sup>u<sup>|</sup> which is normalized by multiplying *<sup>ρ</sup>c*<sup>2</sup>

*N*−1 ∑ *n*=0

*M*−1 ∑ *m*=0

the MGB model for a rectangular element can be obtained as

exp(i*kpR*) *R*

3-D Modelings of an Ultrasonic Phased Array Transducer and Its Radiation Properties in Solid 69

<sup>−</sup> <sup>i</sup>*kp* 2*R*

<sup>−</sup> <sup>i</sup>*kp* 2*R*

*π*

<sup>1</sup> <sup>+</sup> <sup>i</sup>*BkR*/*D*<sup>1</sup>

element, we only perform the summation of 10 × 10 in Equation (27). Therefore fast calculation is possible but it is noted that the MGB is not applicable to complicated element

The total beam field of a phased array transducer can be calculated by simple superposition of individual wave fields with the corresponding time delay Δ*τnm*. The P-wave displacement

u(x�

where x is the position vector of the global coordinate. In Equation (27), the origin of x� is always set on the center of each element. The total wave field at the global coordinate can be obtained by adding up the results of Equation (27) in consideration of the delay and location

Here the wave fields from an array transducer in stainless steel (*cp*=5800m/s, *cs*=3200m/s, *ρ* = 7800k*g*/*m*3) are calculated with both the RSNI and the MGB model. The array transducer used in numerical comparisons is a linear phased array with 24 elements and center frequency *f*=3.0MHz. The element pitch *p* is 1.0mm and the gap *g* is 0.1mm. We show the magnitude of

*<sup>p</sup>*/*P*0.

 exp

10 ∑ *k*

 *y*2 <sup>1</sup> <sup>−</sup> <sup>i</sup>*kpx*�

 *y*2 <sup>2</sup> <sup>−</sup> <sup>i</sup>*kpx*�

*<sup>a</sup>* exp *<sup>b</sup>*<sup>2</sup> 4*a* 

*AkAl* exp(i*kpR*)

<sup>1</sup> <sup>+</sup> <sup>i</sup>*BlR*/*D*<sup>2</sup>

<sup>2</sup>/2. To predict the wave field in solid from a rectangular

(*x*� <sup>2</sup>)2/*<sup>R</sup>* 1 + i*BlR*/*D*<sup>2</sup>

, *ω*) exp(i*ω*Δ*τnm*) (28)

−i*kp* 2

10 ∑ *l*

*AkAl*

<sup>1</sup>*y*<sup>1</sup> *R*

<sup>2</sup>*y*<sup>2</sup> *R*

 *dy*<sup>1</sup>

*dy*2. (25)

, �{*a*} > 0 (26)

(27)

u(x�

Using the known integral formula

u(x�

where *D*<sup>1</sup> = *kpa*<sup>2</sup>

of each element.

field can be calculated by

shapes.

 ∞ −∞

where *w* is the integration weight and *I* is the number of the integration points.

In the above procedure, the wave field from a piston transducer, namely the wave field from an element of the array transducer, can be calculated. When the array transducer has elements *N* in the *x*<sup>1</sup> direction and *M* in the *x*<sup>2</sup> direction, we can obtain total wave field by appropriately adding up *N* × *M* wave components. Using the time delay Δ*τnm* for the element positioned at *n*-th in the *x*<sup>1</sup> direction and *m*-th in the *x*<sup>2</sup> direction, the total wave field can be expressed as

$$u(x,\omega) = \sum\_{n=0}^{N-1} \sum\_{m=0}^{M-1} \frac{1}{2\pi\rho c\_p^2} \int\_{S\_{nm}} P\_0 \exp\left(i\omega \Delta \tau\_{nm}\right) \mathbf{d}\_p K\_p(\theta) \frac{\exp(i\mathbf{k}\_p r)}{r} dS(\mathbf{y}) \tag{22}$$

Equation (22) can be applied to not only a flat array element but also a curved one.

#### **3.2 Multi-Gaussian beam (MGB) model**

The multi-Gaussian beam (MGB) model (Schmerr, 2000) is adopted for simulation of beam fields radiated from circular and rectangular transducers. However, the MGB model is based on the assumption of the paraxial approximation, and will lose its accuracy at the high-refracted angle, which is a condition that paraxial approximation is not satisfied (Park et al., 2006). Such problem often happens in simulating the steering beam of phased array transducers with the conventional MGB model. To address such a problem, the MGB without the nonparaxial approximation has been developed by Zhao and Gang (Zhao & Gang, 2009). The paper had some print errors, so here we show the correct formulation of the nonparaxial MGB. As below, we abbreviate "the nonparaxial MGB model" to "the MGB model".

A uniform normal pressure on the transducer surface can be expanded as the superposition of Gaussian beams (Wen & Breazeale, 1988). For a rectangular element with length 2*a*<sup>1</sup> and width 2*a*<sup>2</sup> in the *x*1- and *x*2-directions, respectively, the pressure over the transducer surface can be expressed as

$$P\_0 = \begin{cases} P\_0 & |y\_1| \le a\_{1\prime} |y\_2| \le a\_{2\prime} y\_3 = 0\\ 0 & \text{otherwise} \end{cases}$$

$$= P\_0 \sum\_{k}^{10} \sum\_{l}^{10} A\_k A\_l \exp(-B\_k y\_1^2 / a\_1^2 - B\_l y\_2^2 / a\_2^2) \tag{23}$$

where *A* and *B* are ten complex coefficients obtained with an optimization method. Now the coordinate of wave field to be calculated is defined as x� = (*x*� <sup>1</sup>, *x*� <sup>2</sup>, *x*� <sup>3</sup>). In order to obtain more accurate solution beyond the paraxial approximation region, the distance factor *r* is approximated as

$$r = \sqrt{(\mathbf{x}\_1' - y\_1)^2 + (\mathbf{x}\_2' - y\_2)^2 + (\mathbf{x}\_3')^2} \approx \mathcal{R} + \frac{y\_1^2 + y\_2^2 - 2\mathbf{x}\_1'y\_1 - 2\mathbf{x}\_2'y\_2}{2\mathcal{R}}\tag{24}$$

where *R* = (*x*� 1)<sup>2</sup> + (*x*� 2)<sup>2</sup> + (*x*� <sup>3</sup>)2. The directivity function *Kp*(*θ*) is not sensitive to a small element, so it can be substituted by *Kp*(*θo*) and we move it to outside of the integral. Here *θ<sup>o</sup>* is the angle between the vector (x − **o**) and the surface normal of transducer. Introducing Equations (23) and (24) into (14), we can obtain:

$$\begin{split} u(\mathbf{x}',\omega) &= \frac{\mathrm{P}\_{\rm{0}}\mathrm{d}\_{\rm{p}}\mathrm{K}\_{\rm{p}}(\theta\_{0})}{2\pi\rho c\_{p}^{2}} \frac{\exp(\mathrm{i}\mathbf{k}\_{p}\mathbf{R})}{\mathrm{R}} \sum\_{k}^{10} \frac{10}{l} \,\mathrm{A}\_{k}A\_{l} \\ &\times \int\_{-\infty}^{\infty} \exp\left[-\left(\frac{\mathrm{B}\_{k}}{a\_{1}^{2}} - \frac{\mathrm{i}\mathbf{k}\_{p}}{2\mathrm{R}}\right)y\_{1}^{2} - \frac{\mathrm{i}\mathbf{k}\_{p}\mathbf{x}\_{1}^{\prime}y\_{1}}{\mathrm{R}}\right] dy\_{1} \\ &\times \int\_{-\infty}^{\infty} \exp\left[-\left(\frac{\mathrm{B}\_{l}}{a\_{2}^{2}} - \frac{\mathrm{i}\mathbf{k}\_{p}}{2\mathrm{R}}\right)y\_{2}^{2} - \frac{\mathrm{i}\mathbf{k}\_{p}\mathbf{x}\_{2}^{\prime}y\_{2}}{\mathrm{R}}\right] dy\_{2}. \end{split} \tag{25}$$

Using the known integral formula

10 Ultrasonic Waves

In the above procedure, the wave field from a piston transducer, namely the wave field from an element of the array transducer, can be calculated. When the array transducer has elements *N* in the *x*<sup>1</sup> direction and *M* in the *x*<sup>2</sup> direction, we can obtain total wave field by appropriately adding up *N* × *M* wave components. Using the time delay Δ*τnm* for the element positioned at *n*-th in the *x*<sup>1</sup> direction and *m*-th in the *x*<sup>2</sup> direction, the total wave field can be expressed as

The multi-Gaussian beam (MGB) model (Schmerr, 2000) is adopted for simulation of beam fields radiated from circular and rectangular transducers. However, the MGB model is based on the assumption of the paraxial approximation, and will lose its accuracy at the high-refracted angle, which is a condition that paraxial approximation is not satisfied (Park et al., 2006). Such problem often happens in simulating the steering beam of phased array transducers with the conventional MGB model. To address such a problem, the MGB without the nonparaxial approximation has been developed by Zhao and Gang (Zhao & Gang, 2009). The paper had some print errors, so here we show the correct formulation of the nonparaxial MGB. As below, we abbreviate "the nonparaxial MGB model"

A uniform normal pressure on the transducer surface can be expanded as the superposition of Gaussian beams (Wen & Breazeale, 1988). For a rectangular element with length 2*a*<sup>1</sup> and width 2*a*<sup>2</sup> in the *x*1- and *x*2-directions, respectively, the pressure over the transducer surface

> *<sup>P</sup>*<sup>0</sup> <sup>|</sup>*y*1| ≤ *<sup>a</sup>*1, <sup>|</sup>*y*2| ≤ *<sup>a</sup>*2, *<sup>y</sup>*<sup>3</sup> <sup>=</sup> <sup>0</sup> 0 otherwise

> > *AkAl* exp(−*Bky*<sup>2</sup>

where *A* and *B* are ten complex coefficients obtained with an optimization method. Now the

more accurate solution beyond the paraxial approximation region, the distance factor *r* is

element, so it can be substituted by *Kp*(*θo*) and we move it to outside of the integral. Here *θ<sup>o</sup>* is the angle between the vector (x − **o**) and the surface normal of transducer. Introducing

<sup>2</sup> − *<sup>y</sup>*2)<sup>2</sup> + (*x*�

1/*a*<sup>2</sup>

<sup>3</sup>)<sup>2</sup> <sup>≈</sup> *<sup>R</sup>* <sup>+</sup> *<sup>y</sup>*<sup>2</sup>

<sup>1</sup> <sup>−</sup> *Bly*<sup>2</sup>

<sup>1</sup> <sup>+</sup> *<sup>y</sup>*<sup>2</sup>

<sup>3</sup>)2. The directivity function *Kp*(*θ*) is not sensitive to a small

2/*a*<sup>2</sup>

<sup>1</sup>, *x*� <sup>2</sup>, *x*�

<sup>2</sup> − 2*x*�

<sup>1</sup>*y*<sup>1</sup> − 2*x*�

<sup>2</sup>) (23)

<sup>3</sup>). In order to obtain

<sup>2</sup>*y*<sup>2</sup> <sup>2</sup>*<sup>R</sup>* (24)

*P*<sup>0</sup> exp (i*ω*Δ*τnm*) d*pKp*(*θ*)

exp(i*kpr*) *r*

*dS*(y) (22)

where *w* is the integration weight and *I* is the number of the integration points.

Equation (22) can be applied to not only a flat array element but also a curved one.

u(x, *ω*) =

to "the MGB model".

can be expressed as

approximated as

where *R* =

*r* = (*x*�

 (*x*�

**3.2 Multi-Gaussian beam (MGB) model**

*N*−1 ∑ *n*=0

*M*−1 ∑ *m*=0

*P*<sup>0</sup> =

= *P*<sup>0</sup>

<sup>1</sup> − *<sup>y</sup>*1)<sup>2</sup> + (*x*�

2)<sup>2</sup> + (*x*�

1)<sup>2</sup> + (*x*�

10 ∑ *k*

coordinate of wave field to be calculated is defined as x� = (*x*�

10 ∑ *l*

1 2*πρc*<sup>2</sup> *p Snm*

$$\int\_{-\infty}^{\infty} \exp\left(-a\mathbf{x}^2 + b\mathbf{x}\right) d\mathbf{x} = \sqrt{\frac{\pi}{a}} \exp\left(\frac{b^2}{4a}\right), \text{ \textdegree \text{\textquotedblleft}\{a\} > 0} \tag{26}$$

the MGB model for a rectangular element can be obtained as

$$\begin{split} u(\mathbf{z}',\omega) = \frac{\mathrm{i}P\_0\mathrm{d}\_p\mathrm{K}\_p}{\rho k\_p c\_p^2} \sum\_{k}^{10} \sum\_{l}^{10} \frac{A\_k A\_l \exp(\mathrm{i}k\_p \mathrm{R})}{\sqrt{1 + \mathrm{i}B\_k \mathrm{R}/D\_1} \sqrt{1 + \mathrm{i}B\_l \mathrm{R}/D\_2}} \\ \times \exp\left(-\frac{\mathrm{i}k\_p}{2} \frac{(\mathbf{x}'\_1)^2/\mathrm{R}}{1 + \mathrm{i}B\_k \mathrm{R}/D\_1}\right) \exp\left(-\frac{\mathrm{i}k\_p}{2} \frac{(\mathbf{x}'\_2)^2/\mathrm{R}}{1 + \mathrm{i}B\_l \mathrm{R}/D\_2}\right) \end{split} \tag{27}$$

where *D*<sup>1</sup> = *kpa*<sup>2</sup> <sup>1</sup>/2 and*D*<sup>2</sup> <sup>=</sup> *kpa*<sup>2</sup> <sup>2</sup>/2. To predict the wave field in solid from a rectangular element, we only perform the summation of 10 × 10 in Equation (27). Therefore fast calculation is possible but it is noted that the MGB is not applicable to complicated element shapes.

The total beam field of a phased array transducer can be calculated by simple superposition of individual wave fields with the corresponding time delay Δ*τnm*. The P-wave displacement field can be calculated by

$$u(x,\omega) = \sum\_{n=0}^{N-1} \sum\_{m=0}^{M-1} u(x',\omega) \exp(\mathrm{i}\omega \Delta \tau\_{nm})\tag{28}$$

where x is the position vector of the global coordinate. In Equation (27), the origin of x� is always set on the center of each element. The total wave field at the global coordinate can be obtained by adding up the results of Equation (27) in consideration of the delay and location of each element.

#### **3.3 Numerical comparison of the RSNI and MGB model**

Here the wave fields from an array transducer in stainless steel (*cp*=5800m/s, *cs*=3200m/s, *ρ* = 7800k*g*/*m*3) are calculated with both the RSNI and the MGB model. The array transducer used in numerical comparisons is a linear phased array with 24 elements and center frequency *f*=3.0MHz. The element pitch *p* is 1.0mm and the gap *g* is 0.1mm. We show the magnitude of displacement <sup>|</sup>u<sup>|</sup> which is normalized by multiplying *<sup>ρ</sup>c*<sup>2</sup> *<sup>p</sup>*/*P*0.

**3.5 Properties of linear and matrix array transducers**

the region near the surface of array transducer.

60mm

x1 x2

2.5MHz 5.0MHz

Fig. 11. 3-D wave fields radiated from a matrix array transducer with *px*=*py*=1.0mm and *N* × *M*=16 × 16=256 in the case of 2.5MHz(left side figure) and 5.0MHz(right side figure).

60mm

40mm

x3

focusing and the beam steering.

**3.5.1 Beam focusing**

**3.5.2 Beam steering**

transducer.

Here we investigate properties of both linear and matrix array transducers for the beam

3-D Modelings of an Ultrasonic Phased Array Transducer and Its Radiation Properties in Solid 71

First, the property of beam focusing in the case of a linear array transducer is investigated. As a linear array transducer, the specification of *p*=1.0mm, *g*=0.1mm, *N*=24 and *W*/*A*=0.3 is assumed. The Rayleigh distance *R* of this transducer is 98.48mm in the case of *f*=4.0MHz. In Fig.12(a), the actual focal lengths *FA* are plotted in varying the desired focal length *F*3. The vertical axis in Fig.12(a) shows the parameter *α*(= *FA*/*F*3) which represents the efficiency of the beam focus. When the parameter of *α* comes close to 1, it shows that we can transmit ultrasonic beam to the intended position. The parameter *γ*(= *R*/*F*3) represents the closeness of focal length for the Rayleigh distance. From Fig.12(a), the focal length *FA* in the range of small *γ* is much shorter than the desired length *F*3. The peak of *α* appears at *γ* = 4, and the

Next let us look at the property of beam focusing for a matrix array transducer. Here a matrix array transducer with *px* = *py*=1.0mm, *gx* = *gy*=0.1mm, *N*=*M*=24 and *Ay*/*Ax*=1.0 is modeled. From Fig.12(b), the focal length *FA* comes closer and closer to *F*<sup>3</sup> when *γ* becomes larger and larger. Through the all range of *γ*, the difference between *FA* and *F*<sup>3</sup> is smaller than the case of the linear array transducer. Also ultrasonic beam can be focused efficiently even at

Figure 13(a) shows the deviation between the desired and actual focal points in a linear array transducer. Here the Rayleigh distance is *R*=98.48mm and we keep constant focal length |F |=*R*/4=24.62mm through all steering angles. The actual focus locations are less than the desired focal length and the deviation of the actual focus from desired focal position also increases as the steering angle increases. And Fig.13(b) shows maximal amplitude of normalized displacement and actual focal lengths *FA* versus the steering angle. These simulation results point out that the peak-to-peak amplitude will decrease as the steering angle increases. These changes will reduce the angular sensitivity of the phased array

Main lobe Main lobe

Grating lobes

distance between *FA* and *F*<sup>3</sup> increases when *γ* becomes large more than *γ* = 4.

Fig. 10. Displacements |u| due to radiated ultrasonic wave: (a) on-axis in the case of different desired beam focal lengths (*F*3=*R*, *R*/4 and *R*/8), (b) off-axis in the case of different beam focal lengths (*F*3=*R*, *R*/2 and *R*/4).

In order to investigate the applicability of the MGB model to the UT modeling, the magnitudes of displacement fields are compared in both methods. Here we introduce the Rayleigh distance\*:

$$R = \frac{A^2 f}{4c\_p} (= 73.66 \text{mm}) \tag{29}$$

Fig.10(a) shows the on-axis (*x*1=0, *x*2=0) displacement when we vary *F*3=*R*, *R*/4 and *R*/8 with keeping the steering angle 0◦. From Fig.10(a), the result calculated by the MGB model can conform to the RSNI results. Similar behaviors are given in the off-axis results as shown in Fig.10(b). Fig.10(b) shows displacements on the *x*1-direction on *x*3=*R*, *R*/2 and *R*/4 in the case of *F*3=*R*, *R*/2 and *R*/4, respectively. It is understood that the MGB model is an effective tool in simulating the beam field radiated from phased arrays over a wide range of steering angle.

#### **3.4 Visualization of 3-D wave field from a matrix array transducer**

The following numerical examples are carried out with the MGB model because the MGB model can keep good accuracies in both the on-axis and off-axis regions. Here we show a visualization of 3-D wave field from a matrix array transducer. It is assumed that the array transducer has parameters as *px*=*py*=1.0mm and *N* × *M*=16 × 16=256. The 3-D wave fields in the cases of *f*=2.5 and 5.0MHz when we set the focal point to (*F*1,*F*2,*F*3)=(10mm,10mm,20mm) are illustrated in Fig.11. The main lobe only appears in the case of *f*=2.5MHz, however not only a main lobe but also grating lobes are generated in the case of *f*=5.0MHz. According to Equations (10) and (11), the directions of the grating lobe in *f*=5.0MHz are estimated to (*θ*� , *φ*� )=(58.8◦, −61.5◦) and (58.8◦, 151.5◦), on the other hand the grating lobe in *f*=2.5MHz is nonexistent. These estimation results show good agreements with the visualization results in Fig.11.

<sup>\*</sup> Note that it is well known that the actual focal length is shorter than the desired (designed) one and the focal point is never beyond the Rayleigh distance (Schmerr, 1998).

#### **3.5 Properties of linear and matrix array transducers**

Here we investigate properties of both linear and matrix array transducers for the beam focusing and the beam steering.

#### **3.5.1 Beam focusing**

12 Ultrasonic Waves

Fig. 10. Displacements |u| due to radiated ultrasonic wave: (a) on-axis in the case of different desired beam focal lengths (*F*3=*R*, *R*/4 and *R*/8), (b) off-axis in the case of different beam

In order to investigate the applicability of the MGB model to the UT modeling, the magnitudes of displacement fields are compared in both methods. Here we introduce the Rayleigh

Fig.10(a) shows the on-axis (*x*1=0, *x*2=0) displacement when we vary *F*3=*R*, *R*/4 and *R*/8 with keeping the steering angle 0◦. From Fig.10(a), the result calculated by the MGB model can conform to the RSNI results. Similar behaviors are given in the off-axis results as shown in Fig.10(b). Fig.10(b) shows displacements on the *x*1-direction on *x*3=*R*, *R*/2 and *R*/4 in the case of *F*3=*R*, *R*/2 and *R*/4, respectively. It is understood that the MGB model is an effective tool in simulating the beam field radiated from phased arrays over a wide range of steering

The following numerical examples are carried out with the MGB model because the MGB model can keep good accuracies in both the on-axis and off-axis regions. Here we show a visualization of 3-D wave field from a matrix array transducer. It is assumed that the array transducer has parameters as *px*=*py*=1.0mm and *N* × *M*=16 × 16=256. The 3-D wave fields in the cases of *f*=2.5 and 5.0MHz when we set the focal point to (*F*1,*F*2,*F*3)=(10mm,10mm,20mm) are illustrated in Fig.11. The main lobe only appears in the case of *f*=2.5MHz, however not only a main lobe but also grating lobes are generated in the case of *f*=5.0MHz. According to Equations (10) and (11), the directions of the grating lobe in *f*=5.0MHz are estimated to

)=(58.8◦, −61.5◦) and (58.8◦, 151.5◦), on the other hand the grating lobe in *f*=2.5MHz is nonexistent. These estimation results show good agreements with the visualization results in

\* Note that it is well known that the actual focal length is shorter than the desired (designed) one and the

*<sup>R</sup>* <sup>=</sup> *<sup>A</sup>*<sup>2</sup> *<sup>f</sup>* 4*cp*

**3.4 Visualization of 3-D wave field from a matrix array transducer**

focal point is never beyond the Rayleigh distance (Schmerr, 1998).

RSNI (F3=R)

RSNI (F3=R/4) RSNI (F3=R/8)

MGB (F3=R) MGB (F3=R/4) MGB (F3=R/8)

0 0.2 0.4 0.6 0.8 1 1.2

distance\*:

angle.

(*θ*� , *φ*�

Fig.11.


0 10 20 30 40 50 60 70 80 90 100

(a) (b)

x3(mm)

focal lengths (*F*3=*R*, *R*/2 and *R*/4).

0 0.2 0.4 0.6 0.8 1 1.2



(= 73.66mm) (29)

x1(mm)

RSNI (x3=R, F3=R)

RSNI (x3=R/2, F3=R/2) RSNI (x3=R/4, F3=R/4)

MGB (x3=R, F3=R) MGB (x3=R/2, F3=R/2) MGB (x3=R/4, F3=R/4)

First, the property of beam focusing in the case of a linear array transducer is investigated. As a linear array transducer, the specification of *p*=1.0mm, *g*=0.1mm, *N*=24 and *W*/*A*=0.3 is assumed. The Rayleigh distance *R* of this transducer is 98.48mm in the case of *f*=4.0MHz. In Fig.12(a), the actual focal lengths *FA* are plotted in varying the desired focal length *F*3. The vertical axis in Fig.12(a) shows the parameter *α*(= *FA*/*F*3) which represents the efficiency of the beam focus. When the parameter of *α* comes close to 1, it shows that we can transmit ultrasonic beam to the intended position. The parameter *γ*(= *R*/*F*3) represents the closeness of focal length for the Rayleigh distance. From Fig.12(a), the focal length *FA* in the range of small *γ* is much shorter than the desired length *F*3. The peak of *α* appears at *γ* = 4, and the distance between *FA* and *F*<sup>3</sup> increases when *γ* becomes large more than *γ* = 4.

Next let us look at the property of beam focusing for a matrix array transducer. Here a matrix array transducer with *px* = *py*=1.0mm, *gx* = *gy*=0.1mm, *N*=*M*=24 and *Ay*/*Ax*=1.0 is modeled. From Fig.12(b), the focal length *FA* comes closer and closer to *F*<sup>3</sup> when *γ* becomes larger and larger. Through the all range of *γ*, the difference between *FA* and *F*<sup>3</sup> is smaller than the case of the linear array transducer. Also ultrasonic beam can be focused efficiently even at the region near the surface of array transducer.

#### **3.5.2 Beam steering**

Figure 13(a) shows the deviation between the desired and actual focal points in a linear array transducer. Here the Rayleigh distance is *R*=98.48mm and we keep constant focal length |F |=*R*/4=24.62mm through all steering angles. The actual focus locations are less than the desired focal length and the deviation of the actual focus from desired focal position also increases as the steering angle increases. And Fig.13(b) shows maximal amplitude of normalized displacement and actual focal lengths *FA* versus the steering angle. These simulation results point out that the peak-to-peak amplitude will decrease as the steering angle increases. These changes will reduce the angular sensitivity of the phased array transducer.

Fig. 11. 3-D wave fields radiated from a matrix array transducer with *px*=*py*=1.0mm and *N* × *M*=16 × 16=256 in the case of 2.5MHz(left side figure) and 5.0MHz(right side figure).

heterogeneous material (Schubert & B. Köhler, 2001). In order to perform simulations of the UT accurately, a realistic shape data of the target is required. Here the image-based modeling is adopted as a pre-processing of the FIT. Using the image-base modeling, we can make geometries of targets directly from a digital image such as X-ray photograph, captured curve data of surface, CAD data, etc. Here we describe the 3-D image-based FIT (Nakahata et al.,

3-D Modelings of an Ultrasonic Phased Array Transducer and Its Radiation Properties in Solid 73

We show the formulation of the FIT and the calculation flow as below. We consider the Cartesian coordinates (*x*1, *x*2, *x*3). The governing equations of elastic waves are the Cauchy equation of motion and the equation of deformation rate. These equations are given in integral

*τij*(x, *t*)*njdS* +

<sup>T</sup> x1

Fig. 13. (a)Actual focal points versus the steering angle, (b)variations of maximal

(p=1.0mm, g=0.1mm, W/A=0.3, f=4MHz, N=24, |**F|**=24.62mm)

(mm)

60o 70o

x3

Linear array transducer

where v is the particle velocity vector, τ is the stress second rank tensor, *ρ* is the density, n is the outward normal vector on the surface *S*, and f is the body force vector. In Equation (30), C is the stiffness tensor of rank four. In the case of isotropic materials, C can be written as:

 *V*

*Cklijvi*(x, *t*)*njdS* (30)

*Cijkl* = *λδijδkl* + *μ*(*δikδjl* + *δilδjk*) (32)

**|F|=**24.62mm

Actual focal length FA (mm) Max amplitude of **|u|**

0 10 20 30 40 50 60 70

FA <sup>|</sup>**u**<sup>|</sup>

Steering angle T

*fi*(x, *t*)*dV* (*i* = 1, 2, 3) (31)

0 0.2 0.4 0.6 0.8 1 1.2

2011) and show a numerical example of the phased array UT.

**4.1 3-D finite integration technique (FIT)**

*∂ ∂t V*

*∂ ∂t V*

(mm)

0

5

10

15

x3

20

25

0o 10o 20o 30<sup>o</sup> 40o 50<sup>o</sup>

forms for a finite volume *V* with the surface *S* by

*τkl*(x,*t*)*dV* =

*ρvi*(x,*t*)*dV* =

Desired focal point Actual focal point

(a) (b)

0 5 10 15 20 25

displacement in the case of linear array transducer.

x1

**|F|=**24.62mm

 *S*

 *S*

In contrast, the behavior of the matrix array transducer is different from the linear array transducer. Figure 14(a) shows that the deviation distance between the desired and actual focal points is almost constant as the steering angle increases. However the peak-to-peak amplitude becomes lower as the steering angle increases.

From these results, the effective steering angle for the linear array transducer is up to approx. 40◦. Although the beam steering for the matrix array transducer is possible in higher angle than the linear array transducer, we suggest that the effective steering angle is up to approx. 55◦ from the standpoint of the 6dB down of the peak-to-peak amplitude.

#### **4. Modeling of phased array transducer in time domain**

We here introduce a time domain simulation tool to predict the wave propagation from the array transducer. The method is a combined technique of the finite integration technique (FIT) (Marklein, 1998) and an image-based modeling (Terada et al., 1997). In the FIT, the finite difference equations of the stress and the particle velocity are derived from integral forms of the governing equations. Computational grids of these quantities are arranged in a staggered configuration in space, and the finite difference equations can be solved by marching time steps in the leap-frog manner. The FIT has an advantage that it can treat different boundary conditions without difficulty. This is essential to model the ultrasonic wave propagation in

Linear array transducer (p=1.0mm, g=0.1mm, W/A=0.3, f=4MHz, N=24)

Matrix array transducer (px=1.0mm, gx=0.1mm, py=1.0mm, gy=0.1mm, Ay/Ax=1.0, f=4MHz, N=M=24)

Fig. 12. Ratios of actual focal length *FA* and desired one *F*<sup>3</sup> in the case of the linear array transducer (a) and matrix array transducer (b).

heterogeneous material (Schubert & B. Köhler, 2001). In order to perform simulations of the UT accurately, a realistic shape data of the target is required. Here the image-based modeling is adopted as a pre-processing of the FIT. Using the image-base modeling, we can make geometries of targets directly from a digital image such as X-ray photograph, captured curve data of surface, CAD data, etc. Here we describe the 3-D image-based FIT (Nakahata et al., 2011) and show a numerical example of the phased array UT.

#### **4.1 3-D finite integration technique (FIT)**

14 Ultrasonic Waves

In contrast, the behavior of the matrix array transducer is different from the linear array transducer. Figure 14(a) shows that the deviation distance between the desired and actual focal points is almost constant as the steering angle increases. However the peak-to-peak

From these results, the effective steering angle for the linear array transducer is up to approx. 40◦. Although the beam steering for the matrix array transducer is possible in higher angle than the linear array transducer, we suggest that the effective steering angle is up to approx.

We here introduce a time domain simulation tool to predict the wave propagation from the array transducer. The method is a combined technique of the finite integration technique (FIT) (Marklein, 1998) and an image-based modeling (Terada et al., 1997). In the FIT, the finite difference equations of the stress and the particle velocity are derived from integral forms of the governing equations. Computational grids of these quantities are arranged in a staggered configuration in space, and the finite difference equations can be solved by marching time steps in the leap-frog manner. The FIT has an advantage that it can treat different boundary conditions without difficulty. This is essential to model the ultrasonic wave propagation in

> 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fig. 12. Ratios of actual focal length *FA* and desired one *F*<sup>3</sup> in the case of the linear array

D = FA / F3

012345678 J=R / F3

x1

x2

(px=1.0mm, gx=0.1mm, py=1.0mm, gy=0.1mm,

x3

F3

Matrix array transducer

Ay/Ax=1.0, f=4MHz, N=M=24)

amplitude becomes lower as the steering angle increases.

**4. Modeling of phased array transducer in time domain**

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

D = FA / F3

012345678 J=R / F3

transducer (a) and matrix array transducer (b).

55◦ from the standpoint of the 6dB down of the peak-to-peak amplitude.

x1

x2

(a) (b)

(p=1.0mm, g=0.1mm, W/A=0.3, f=4MHz, N=24)

x3

F3

Linear array transducer

We show the formulation of the FIT and the calculation flow as below. We consider the Cartesian coordinates (*x*1, *x*2, *x*3). The governing equations of elastic waves are the Cauchy equation of motion and the equation of deformation rate. These equations are given in integral forms for a finite volume *V* with the surface *S* by

$$\frac{\partial}{\partial t} \int\_{V} \pi\_{kl}(\mathbf{z}, t)dV = \int\_{S} \mathbb{C}\_{klij} v\_i(\mathbf{z}, t) n\_j dS \tag{30}$$

$$\frac{\partial}{\partial t} \int\_{V} \rho v\_i(\mathbf{z}, t)dV = \int\_{S} \tau\_{i\bar{j}}(\mathbf{z}, t) n\_{\bar{j}} dS + \int\_{V} f\_i(\mathbf{z}, t)dV \quad (i = 1, 2, 3) \tag{31}$$

where v is the particle velocity vector, τ is the stress second rank tensor, *ρ* is the density, n is the outward normal vector on the surface *S*, and f is the body force vector. In Equation (30), C is the stiffness tensor of rank four. In the case of isotropic materials, C can be written as:

$$C\_{ijkl} = \lambda \delta\_{ij} \delta\_{kl} + \mu (\delta\_{ik}\delta\_{jl} + \delta\_{il}\delta\_{jk}) \tag{32}$$

Linear array transducer (p=1.0mm, g=0.1mm, W/A=0.3, f=4MHz, N=24, |**F|**=24.62mm)

Fig. 13. (a)Actual focal points versus the steering angle, (b)variations of maximal displacement in the case of linear array transducer.

v1

(*F*) <sup>2</sup> + *v*

> (*U*) <sup>3</sup> − *v*

<sup>13</sup> <sup>−</sup> *<sup>τ</sup>*(*D*) <sup>13</sup> Δ*x*<sup>2</sup>

<sup>12</sup> <sup>+</sup> *<sup>τ</sup>*(*U*)

(*F*) <sup>2</sup> + *v*

<sup>12</sup> <sup>+</sup> *<sup>τ</sup>*(*U*)

<sup>12</sup> <sup>−</sup> *<sup>τ</sup>*(*F*)

W13 (U)

> W11 (R)

(R) U

v1 integration cell

W12 (B)

(*U*) <sup>3</sup> − *v*

<sup>13</sup> <sup>−</sup> *<sup>τ</sup>*(*D*) <sup>13</sup>

(*D*) 3 

(*D*) 3 Δ*x*<sup>2</sup>

<sup>2</sup> (36)

<sup>2</sup> <sup>+</sup> <sup>Δ</sup>*t*{*τ*˙*ij*}*<sup>z</sup>* (37)

(34)

(35)

W13 (D)

W12 (F)

W11 (L)

(L) U

3-D Modelings of an Ultrasonic Phased Array Transducer and Its Radiation Properties in Solid 75

Fig. 16. *τii*–integration cell (left figure) and *v*1–integration cell (right figure). All material

(*L*) 1 

(*L*) 1 + *λ* Δ*x v* (*B*) <sup>2</sup> − *v*

We consider a discretization form of Equation (30) using a cube as an integral volume *V* in

Δ*x*<sup>2</sup> + *λ*

where *τ*˙11 = *∂τ*11/*∂t* and Δ*x* is the length on a side of *V*. In Equation (34), superscript () expresses positions of physical quantities in the integral volume as shown in Fig.16. Similarly

<sup>12</sup> <sup>−</sup> *<sup>τ</sup>*(*F*)

<sup>11</sup> <sup>+</sup> *<sup>τ</sup>*(*B*)

where we let f = 0. Since the density *ρ* is given in the *τii*-integral volume as shown in Fig.16,

In the time domain, stress components τ are allocated at half-time steps, while the velocities v are at full-time steps. The following time discretization yields an explicit leap-frog scheme:

{*vi*}*<sup>z</sup>* <sup>=</sup> {*vi*}*z*−<sup>1</sup> <sup>+</sup> <sup>Δ</sup>*t*{*v*˙*i*}*z*<sup>−</sup> <sup>1</sup>

where Δ*t* is the time interval and the superscript *z* denotes integer number of the time step. Therefore the FIT repeats the operations of Equations (36) and (37) by means of adequate initial and boundary conditions. A specific stability condition and adequate spatial resolution must be fulfilled to calculate the FIT accurately (Nakahata et al., 2011; Schubert & B. Köhler,

We show a procedure of the 3-D image-based modeling with a 3-D curve measurement system based on the coded pattern projection method (Nakahata et al., 2012). The system consists of a projector that flashes narrow bands of black and white light onto the model's face and a camera that captures the pattern of light. The distortion pattern of light leads to the

<sup>2</sup> <sup>=</sup> {*τij*}*z*<sup>−</sup> <sup>1</sup>

<sup>11</sup> <sup>+</sup> *<sup>τ</sup>*(*B*)

 *v* (*B*) <sup>2</sup> − *v*

v1 (R)

Wii integration cell

v2 (B)

v3 (U)

v3 (D)

*τ*˙11Δ*x*<sup>3</sup> = (*λ* + 2*μ*)

*<sup>τ</sup>*˙11 <sup>=</sup> *<sup>λ</sup>* <sup>+</sup> <sup>2</sup>*<sup>μ</sup>* Δ*x*

a discretization of Equation (31) becomes

2001).

**4.2 Image based modeling**

*ρv*˙1Δ*x*<sup>3</sup> =

parameters are defined in the *τii*–integration cell.

Fig.15. Assuming that *τ*<sup>11</sup> is constant in *V*, we have

 *v* (*R*) <sup>1</sup> − *v*

 *v* (*R*) <sup>1</sup> − *v*

 *<sup>τ</sup>*(*R*) <sup>11</sup> <sup>−</sup> *<sup>τ</sup>*(*L*)

the average value *ρ* = (*ρ*(*L*) + *ρ*(*R*))/2 is used in Equation (35).

{*τij*}*z*+<sup>1</sup>

*<sup>v</sup>*˙1 <sup>=</sup> <sup>1</sup> *ρ*Δ*x <sup>τ</sup>*(*R*) <sup>11</sup> <sup>−</sup> *<sup>τ</sup>*(*L*)

v2 (F)

UPO

v1 (L)

Matrix array transducer (px=py=1.0mm, gx=gy=0.1mm, Ay/Ax=1.0, f=4MHz, N=M=24, |**F**|=24.62mm)

Fig. 14. (a)Actual focal points versus the steering angle, (b)variations of maximal displacement in the case of matrix array transducer.

in terms of the two Lamé constants, *λ* and *μ*. In Equation (32), *δij* is the Kronecker delta tensor. P and S wave velocities are given as:

$$c\_P = \sqrt{\frac{(\lambda + 2\mu)}{\rho}} \quad c\_S = \sqrt{\frac{\mu}{\rho}}.\tag{33}$$

In the case of an acoustic problem, we can use above equations but have to set *τij* = 0 (*i j*) in Equation (31) and *μ* = 0 in Equation (32).

Fig. 15. Finite volume *V* and grid arrangement in the 3-D FIT.

Fig. 16. *τii*–integration cell (left figure) and *v*1–integration cell (right figure). All material parameters are defined in the *τii*–integration cell.

We consider a discretization form of Equation (30) using a cube as an integral volume *V* in Fig.15. Assuming that *τ*<sup>11</sup> is constant in *V*, we have

$$
\dot{\tau}\_{11}\Delta\mathbf{x}^{3} = (\lambda + 2\mu) \left[ v\_{1}^{(R)} - v\_{1}^{(L)} \right] \Delta\mathbf{x}^{2} + \lambda \left[ v\_{2}^{(B)} - v\_{2}^{(F)} + v\_{3}^{(II)} - v\_{3}^{(D)} \right] \Delta\mathbf{x}^{2}
$$

$$
\dot{\tau}\_{11} = \frac{\lambda + 2\mu}{\Delta\mathbf{x}} \left[ v\_{1}^{(R)} - v\_{1}^{(L)} \right] + \frac{\lambda}{\Delta\mathbf{x}} \left[ v\_{2}^{(B)} - v\_{2}^{(F)} + v\_{3}^{(U)} - v\_{3}^{(D)} \right] \tag{34}
$$

where *τ*˙11 = *∂τ*11/*∂t* and Δ*x* is the length on a side of *V*. In Equation (34), superscript () expresses positions of physical quantities in the integral volume as shown in Fig.16. Similarly a discretization of Equation (31) becomes

$$
\overline{\rho}\dot{\nu}\_{1}\Delta x^{3} = \left[\tau\_{11}^{(R)} - \tau\_{11}^{(L)} + \tau\_{12}^{(B)} - \tau\_{12}^{(F)} + \tau\_{13}^{(U)} - \tau\_{13}^{(D)}\right] \Delta x^{2}
$$

$$
\dot{\psi}\_{1} = \frac{1}{\overline{\rho}\Delta x} \left[\tau\_{11}^{(R)} - \tau\_{11}^{(L)} + \tau\_{12}^{(B)} - \tau\_{12}^{(F)} + \tau\_{13}^{(U)} - \tau\_{13}^{(D)}\right] \tag{35}
$$

where we let f = 0. Since the density *ρ* is given in the *τii*-integral volume as shown in Fig.16, the average value *ρ* = (*ρ*(*L*) + *ρ*(*R*))/2 is used in Equation (35).

In the time domain, stress components τ are allocated at half-time steps, while the velocities v are at full-time steps. The following time discretization yields an explicit leap-frog scheme:

$$\{v\_i\}^z = \{v\_i\}^{z-1} + \Delta t \{\dot{v}\_i\}^{z-\frac{1}{2}} \tag{36}$$

$$\{\{\tau\_{l\dot{l}}\}^{z+\frac{1}{2}} = \{\tau\_{l\dot{l}}\}^{z-\frac{1}{2}} + \Delta t \{\dot{\tau}\_{l\dot{l}}\}^z \tag{37}$$

where Δ*t* is the time interval and the superscript *z* denotes integer number of the time step. Therefore the FIT repeats the operations of Equations (36) and (37) by means of adequate initial and boundary conditions. A specific stability condition and adequate spatial resolution must be fulfilled to calculate the FIT accurately (Nakahata et al., 2011; Schubert & B. Köhler, 2001).

#### **4.2 Image based modeling**

16 Ultrasonic Waves

(mm)

<sup>T</sup> x1

in terms of the two Lamé constants, *λ* and *μ*. In Equation (32), *δij* is the Kronecker delta tensor.

In the case of an acoustic problem, we can use above equations but have to set *τij* = 0 (*i j*)

*<sup>ρ</sup>* , *cS* <sup>=</sup>

'x

*μ*

(*λ* + 2*μ*)

(px=py=1.0mm, gx=gy=0.1mm, Ay/Ax=1.0, f=4MHz, N=M=24, |**F**|=24.62mm)

(mm)

Desired focal point Actual focal point

(a) (b)

**|F|**(24.62mm)

x1

displacement in the case of matrix array transducer.

P and S wave velocities are given as:

in Equation (31) and *μ* = 0 in Equation (32).

0 5 10 15 20 25

x3

0

5

10

15

20

25

0o 10<sup>o</sup> 20<sup>o</sup> 30<sup>o</sup> 40<sup>o</sup> 50<sup>o</sup>

(mm) Steering angle T

Fig. 14. (a)Actual focal points versus the steering angle, (b)variations of maximal

x3

Matrix array transducer

*cP* =

x1 x2 x3

Fig. 15. Finite volume *V* and grid arrangement in the 3-D FIT.

'x

V '<sup>x</sup>

60o 70<sup>o</sup> Actual focal length FA (mm) Max amplitude of **|u|**

0 10 20 30 40 50 60 70

FA <sup>|</sup>**u**<sup>|</sup>

0 0.5 1 1.5 2 2.5 3

*<sup>ρ</sup>* . (33)

<sup>W</sup><sup>12</sup> v2

W11, W22, W<sup>33</sup>

W13 W23

v3

v1

**|F|=**24.62mm

We show a procedure of the 3-D image-based modeling with a 3-D curve measurement system based on the coded pattern projection method (Nakahata et al., 2012). The system consists of a projector that flashes narrow bands of black and white light onto the model's face and a camera that captures the pattern of light. The distortion pattern of light leads to the

g=0.1mm

p=0.6mm

PML(1.0mm)

12mm

2mm

x1

x3

30mm

0.85Ps 1.95Ps

3.05Ps 4.15Ps

Lack of penetration (Penny-shaped crack, I=3mm, thickness=0.1mm)

7mm

25mm

3-D Modelings of an Ultrasonic Phased Array Transducer and Its Radiation Properties in Solid 77

Fig. 18. Numerical model of the welded T-joint including an artificial defect (penny-shaped

center frequency 4.0MHz is transmitted from the array transducer toward the defect. Here the ultrasonic wave is generated into solid by giving a time-dependent normal stress at the

Figure 19 shows the isosurface of absolute value of the displacement vector <sup>|</sup>u(= <sup>v</sup>*dt*)<sup>|</sup> and Fig.20 shows |u| at a cross section of the model. These results show that ultrasonic wave propagates toward the artificial defect and scattered waves are generated. After some time steps, the echo from the defect arrives at the array transducer. In the general array UT, the echoes at the receiving process are also synthesized with the appropriate delay time in the same way to the transmitting process. Figure 21 shows the synthesized echo at receiving process, and waveform around 5 *μ*s is the wave component from the defect. The calculation

Scattered P-wave Scattered S-wave

Fig. 19. Snapshots of ultrasonic wave propagation in welded T-joint (isosurface rendering).

12mm

crack).

17mm

x1 x2 x3

surface of the transducer.

Linear array transducer

Fig. 17. Image-based modeling of a welded T-joint using a 3-D curve measurement system.

reconstruction of the surface of model. As shown in Fig.17, a 3-D shape of a welded T-joint is reconstructed from surface profiles which are measured at multiple angles. After unnecessary parts for calculation are trimmed away, the rest part is discretized into a voxel data. Here voxel is "volumetric pixel", representing a value on a regular grid in a 3-D space. The voxel data is directly used as the computational cell in the FIT, therefore the voxel size is set to be equivalent to the cell size of the FIT.

#### **4.3 3-D numerical simulation**

A 3-D simulation of ultrasonic wave propagation is demonstrated using the model in Fig.17. Here an artificial defect which is assumed to be a lack of weld penetration is introduced at the inside of the welding part(see in Fig.18). The diameter of the defect is approx. 3mm and thickness is 0.1mm. Material parameters of the welded T-joint are set as *cP* = 5800m/s, *cS* = 3100m/s and *ρ* = 7800kg/m3. The linear array transducer (total element number *N*=24, the pitch *p*=0.6m, the element width *W*=7mm and the gap *g*=0.1mm) is located on the top of the model. In the simulation, we choose Δ*x* = 0.02mm, Δ*t*=1.0 nano-second (ns) and the total time step is 6500. Total number of the voxel is about 1 billion (1000 millions). The P-wave with the

Fig. 17. Image-based modeling of a welded T-joint using a 3-D curve measurement system.

reconstruction of the surface of model. As shown in Fig.17, a 3-D shape of a welded T-joint is reconstructed from surface profiles which are measured at multiple angles. After unnecessary parts for calculation are trimmed away, the rest part is discretized into a voxel data. Here voxel is "volumetric pixel", representing a value on a regular grid in a 3-D space. The voxel data is directly used as the computational cell in the FIT, therefore the voxel size is set to be

A 3-D simulation of ultrasonic wave propagation is demonstrated using the model in Fig.17. Here an artificial defect which is assumed to be a lack of weld penetration is introduced at the inside of the welding part(see in Fig.18). The diameter of the defect is approx. 3mm and thickness is 0.1mm. Material parameters of the welded T-joint are set as *cP* = 5800m/s, *cS* = 3100m/s and *ρ* = 7800kg/m3. The linear array transducer (total element number *N*=24, the pitch *p*=0.6m, the element width *W*=7mm and the gap *g*=0.1mm) is located on the top of the model. In the simulation, we choose Δ*x* = 0.02mm, Δ*t*=1.0 nano-second (ns) and the total time step is 6500. Total number of the voxel is about 1 billion (1000 millions). The P-wave with the

Projector

PC

3-D reconstruction of welded T-joint

closeup

Voxel mesh

Caputure surface data from four angles

equivalent to the cell size of the FIT.

**4.3 3-D numerical simulation**

Camera

Fig. 18. Numerical model of the welded T-joint including an artificial defect (penny-shaped crack).

center frequency 4.0MHz is transmitted from the array transducer toward the defect. Here the ultrasonic wave is generated into solid by giving a time-dependent normal stress at the surface of the transducer.

Figure 19 shows the isosurface of absolute value of the displacement vector <sup>|</sup>u(= <sup>v</sup>*dt*)<sup>|</sup> and Fig.20 shows |u| at a cross section of the model. These results show that ultrasonic wave propagates toward the artificial defect and scattered waves are generated. After some time steps, the echo from the defect arrives at the array transducer. In the general array UT, the echoes at the receiving process are also synthesized with the appropriate delay time in the same way to the transmitting process. Figure 21 shows the synthesized echo at receiving process, and waveform around 5 *μ*s is the wave component from the defect. The calculation

Fig. 19. Snapshots of ultrasonic wave propagation in welded T-joint (isosurface rendering).

Phased array transducer has a great potential to detect defects in real time. In the future, it is expected that the detectability of defects with the phased array UT can be improved by using

3-D Modelings of an Ultrasonic Phased Array Transducer and Its Radiation Properties in Solid 79

Azar, L.; Shi, Y. & Wooh, S.C. (2000). Beam focusing behavior of linear phased arrays, *NDT &*

Drinkwater, B.W. & Wilcox, P.D. (2006). Ultrasonic arrays for non-destructive evaluation: A

Fellinger, P.; Marklein, R.; Langenberg, K.J. & Klaholz, S. (1995). Numerical modeling of

Kono, N. & Mizota H. (2011). Analysis of characteristics of grating lobes generated with

Kono, N.; Baba, A. & Ehara, K. (2010). Ultrasonic multiple beam technique using single phased

Macovski, A. (1979). Ultrasonic imaging using arrays, *Proceedings of the IEEE*, Vol.67, No.4,

Marklein, R. (1998). *Numerical Methods for the Modeling of Acoustic, Electromagnetic, Elastic, and*

Nakahata, K.; Schubert, F. & Köhler, B. (2011). 3-D image-based simulation for ultrasonic

Nakahata, K.; Ichikawa, S.; Saitoh, T. & Hirose, S. (2012). Acceleration of the 3D image-based

Park, J.S.; Song, S.J. & Kim, H.J. (2006). Calculation of radiation beam field from phased array

Schmerr, L.W. (1998). *Fundamentals of Ultrasonic Nondestructive Evaluation*, Plenum Press, New

Schmerr, L.W. (2000). A multigaussian ultrasonic beam model for high performance

Schubert, F. & Köhler, B. (2001). Three-dimensional time domain modeling of ultrasonic wave

Song, S.J. & Kim, C.H. (2002). Simulation of 3-D radiation beam patterns propagated through

Terada, K.; Miura, T. & Kikuchi, N. (1997). Digital image-based modeling applied to the

Vezzetti, D.J. (1985). Propagation of bounded ultrasonic beams in anisotropic media, *Journal of*

simulation on a personal computer, *Materials Evaluation*, 882–888.

*Technique*, Shaker Verlag, Aachen, ISBN 3826531728. (in German)

elastic wave propagation and scattering with EFIT - elastodynamic finite integration

Gaussian pulse excitation by ultrasonic 2-D array transducer, *NDT & E International*,

array probe for detection and sizing of stress corrosion cracking in austenitic welds,

*Piezoelectric Wave Propagation Problems in the Time Domain Based on the Finite Integration*

wave propagation in heterogeneous and anisotropic materials, *Review of Quantitative*

FIT with an explicit parallelization approach, *Review of Quantitative Nondestructive*

transducers using expanded multi-Gaussian beam model, *Solid State Phenomena*,

propagation in concrete in explicit consideration of aggregates and porosity, *Journal*

a planar interface from ultrasonic phased array transducers, *Ultrasonics*, Vol.40,

homogenization analysis of composite materials, *Computational Mechanics*, Vol.20,

above presented useful numerical tools.

Vol.44, No.6, 477–483.

484–495.

Vol.110, 163–168.

519–524.

No.4, 331–346.

York, ISBN 0306457520.

*E International*, Vol.33, No.3, 189–198.

technique, *Wave motion*, Vol.21, No.1, 47–66.

*Materials Evaluation*, Vol.68, No.10, 1163–1170.

*Nondestructive Evaluation*, Vol.30, pp.51-58.

*Evaluation*, Vol.31, accepted for publication.

*of Computational Acoustics*, Vol.9, No.4, 1543–1560.

*Acoustical Society of America*, Vol.78, No.3, 1103–1108.

review, *NDT & E International*, Vol.39, No.7, 525–541.

**6. References**

Fig. 20. Snapshots of ultrasonic wave propagation in welded T-joint (cross-sectional view)

Fig. 21. Predicted echo which is observed at the array transducer.

time of this 3-D model was about 12 hours with a parallel computer system with 64 CPUs (Flat MPI).

## **5. Summary**

For the reliable application of phased array techniques to NDT, it is essential to have thorough understanding on the properties of the phased array transducer and characteristics of radiation beam in solid. Here we show principle of electronic scanning with phased array transducers and an effective setting of the delay time to avoid grating and side lobes. In order to predict the wave field due to the array transducer, 3-D calculation tools in both frequency and time domains are introduced. From the simulation results with the MGB model in the frequency domain, characteristics about beam steering and focusing are compared between the linear and matrix array transducers. As the time domain calculation tool, the 3-D image-based FIT modeling of the UT for a welded T-joint is demonstrated.

Phased array transducer has a great potential to detect defects in real time. In the future, it is expected that the detectability of defects with the phased array UT can be improved by using above presented useful numerical tools.

#### **6. References**

20 Ultrasonic Waves

Scattered P-wave Scattered S-wave

Fig. 20. Snapshots of ultrasonic wave propagation in welded T-joint (cross-sectional view)

0.0 1.0 2.0 3.0 4.0 5.0

Time (Ps)

time of this 3-D model was about 12 hours with a parallel computer system with 64 CPUs

For the reliable application of phased array techniques to NDT, it is essential to have thorough understanding on the properties of the phased array transducer and characteristics of radiation beam in solid. Here we show principle of electronic scanning with phased array transducers and an effective setting of the delay time to avoid grating and side lobes. In order to predict the wave field due to the array transducer, 3-D calculation tools in both frequency and time domains are introduced. From the simulation results with the MGB model in the frequency domain, characteristics about beam steering and focusing are compared between the linear and matrix array transducers. As the time domain calculation tool, the

3-D image-based FIT modeling of the UT for a welded T-joint is demonstrated.

Echo from defect (predicted)

0.85Ps 1.95Ps

3.05Ps 4.15Ps


Fig. 21. Predicted echo which is observed at the array transducer.

u3

(Flat MPI).

**5. Summary**


**4** 

Hassina Khelladi

 *Algeria* 

**Goldberg's Number Influence on the Validity** 

Nonlinear propagation occurs widely in many acoustic systems, especially in the field of medical ultrasound. Despite the widespread use of ultrasound in diagnosis and therapy, the propagation of ultrasound through biological media was modeled as a linear process for many years. The invalidity of infinitesimal acoustic assumption, at biomedical frequencies and intensities, was demonstrated by Muir and Carstensen (Muir & Carstensen, 1980). It was realized that nonlinear effects are not negligible and must therefore be taken into account in theoretical developments of ultrasound in biomedical research. Indeed, increasing the acoustic frequency or intensity in order to enhance resolution or penetration

Nonlinear effects occur more strongly when ultrasound propagates through slightly dissipative liquids such as water or amniotic fluid. As in medical sonography, the full bladder or the pregnant uterus, which may be filled with amniotic fluid, is used as an acoustic window in many types of diagnoses; a special attention is given to slightly dissipative liquids where the possibility of signal distortions has several implications. However, within soft tissues, the

In absorbing medium, nonlinear effects cannot be examined without considering dissipation. The absorption limits the generation of harmonics by decreasing their amplitudes gradually. In addition, as the absorption coefficient increases with frequency, the energy transformation towards frequencies higher than the fundamental frequency (generation of harmonics) can also lead to significant acoustic losses. Nonlinear effects create all higher harmonics from the energy at the insonation frequency, but, due to the absorption of high frequency components, only the lower harmonic orders and the fundamental

Dissipation can have various origins (Sehgal & Greenleaf, 1982): viscosity (resulting from shear motions between fluid particles), thermal conduction (due to the energy loss resulting from thermal conduction between particles) or molecular relaxation (where the molecular equilibrium state is affected by the pressure variations of the acoustic wave propagation). Nonlinear effects and dissipation are antagonistic phenomena. The nonlinearity mechanism shocks the wave by generating harmonics while dissipation increases with frequency and

remain. So, the tendency for wave distortion to occur is limited by dissipation.

depth may alter the beam shape in a way not predicted by linear theory.

tendency for wave distortion to occur is limited by dissipation.

**1. Introduction** 

**Domain of the Quasi-Linear Approximation** 

 **of Finite Amplitude Acoustic Waves** 

*University of Sciences and Technology Houari Boumediene* 


## **Goldberg's Number Influence on the Validity Domain of the Quasi-Linear Approximation of Finite Amplitude Acoustic Waves**

Hassina Khelladi *University of Sciences and Technology Houari Boumediene Algeria* 

## **1. Introduction**

22 Ultrasonic Waves

80 Ultrasonic Waves

Wen, J.J & Breazeale, M.A. (1988). A diffraction beam field expressed as the superposition of Gaussian beams, *Journal of Acoustical Society of America*, Vol.83, 1752–1756. Zhao, X. & Gang, T.(2009). Nonparaxial multi-Gaussian beam models and measurement

models for phased array transducer, *Ultrasonics*, Vol.49, 126–130.

Nonlinear propagation occurs widely in many acoustic systems, especially in the field of medical ultrasound. Despite the widespread use of ultrasound in diagnosis and therapy, the propagation of ultrasound through biological media was modeled as a linear process for many years. The invalidity of infinitesimal acoustic assumption, at biomedical frequencies and intensities, was demonstrated by Muir and Carstensen (Muir & Carstensen, 1980). It was realized that nonlinear effects are not negligible and must therefore be taken into account in theoretical developments of ultrasound in biomedical research. Indeed, increasing the acoustic frequency or intensity in order to enhance resolution or penetration depth may alter the beam shape in a way not predicted by linear theory.

Nonlinear effects occur more strongly when ultrasound propagates through slightly dissipative liquids such as water or amniotic fluid. As in medical sonography, the full bladder or the pregnant uterus, which may be filled with amniotic fluid, is used as an acoustic window in many types of diagnoses; a special attention is given to slightly dissipative liquids where the possibility of signal distortions has several implications. However, within soft tissues, the tendency for wave distortion to occur is limited by dissipation.

In absorbing medium, nonlinear effects cannot be examined without considering dissipation. The absorption limits the generation of harmonics by decreasing their amplitudes gradually. In addition, as the absorption coefficient increases with frequency, the energy transformation towards frequencies higher than the fundamental frequency (generation of harmonics) can also lead to significant acoustic losses. Nonlinear effects create all higher harmonics from the energy at the insonation frequency, but, due to the absorption of high frequency components, only the lower harmonic orders and the fundamental remain. So, the tendency for wave distortion to occur is limited by dissipation.

Dissipation can have various origins (Sehgal & Greenleaf, 1982): viscosity (resulting from shear motions between fluid particles), thermal conduction (due to the energy loss resulting from thermal conduction between particles) or molecular relaxation (where the molecular equilibrium state is affected by the pressure variations of the acoustic wave propagation).

Nonlinear effects and dissipation are antagonistic phenomena. The nonlinearity mechanism shocks the wave by generating harmonics while dissipation increases with frequency and

Goldberg's Number Influence on the Validity Domain

ranges.

measurements, respectively.

similarities to soft tissues.

**2. Theoretical formulation** 

of the Quasi-Linear Approximation of Finite Amplitude Acoustic Waves 83

Similarly, Haran and Cook (Haran & Cook, 1983) have used the Burgers' equation to elaborate an algorithm for calculating harmonics generation by a finite amplitude plane wave of ultrasound propagating in a lossy and nondispersive medium. Their algorithm accounts for an absorption coefficient of any desired frequency dependence. The variation effect of the absorption coefficient on the second harmonic was demonstrated in a medium similar to carbon tetrachloride. Calculations for several types of tissue and biological fluids were presented. It was shown that for some biological media having a low absorption coefficient, a significant distortion of the plane wave can be observed for large propagation

Recently, D'hooge et al. (D'hooge et al., 1999) have analyzed the nonlinear propagation effects of pulses on broadband attenuation measurements and their implications in ultrasonic tissue characterization by using a simple mathematical model based on the numerical solution, in the time domain, of the Burgers' equation. The developed model has been validated by measuring the absorption coefficient of both a tissue-mimicking phantom in vitro and a liver in vivo at several pressure amplitudes using transmission and reflection

In the present chapter, the intensity effects on the behavior of the fundamental and the generated second harmonic, by using both the numerical solution of the Burgers' equation and the analytical expressions established with the quasi-linear approximation are examined. An analysis on the validity domain of the fundamental and the second harmonic analytical expressions established with the quasi-linear approximation is elaborated. The deviations resulting from the analytical expressions established with the quasi-linear approximation and the numerical solution of the Burgers' equation are estimated. This investigation is based on Krassilnikov et al. (Krassilnikov et al., 1957) experimental results. These experimental data concern water and glycerol that correspond, respectively, to a weakly dissipative liquid approaching the characteristics of urine or amniotic fluid (Bouakaz et al., 2004) and a strongly dissipative liquid with some

It should be noted that in this study all derivations are developed entirely in the frequency domain, thus avoiding both the steep waveform problems and the use of FFT, which alternates between time and frequency domains. The utility of the method resides in the

The description of acoustic waves in a liquid is founded on the theory of motion of a liquid, which is considered to be continuous. In the present investigation, the viscosity and the heat conduction coefficients, although in general are functions of the state variables, are assumed to be constant. The theoretical formulation of the propagation of finite amplitude plane progressive waves in a homogeneous and dissipative liquid is elaborated in section 2.1, and the theoretical model is based on the derivation of a nonlinear partial differential equation in which the longitudinal particle velocity is a function of time and space. In section 2.2, the dimensionless Burgers' equation is presented, which is considered to be among the most

ease with which it can be implemented on a digital computer.

exhaustively studied equations in the theory of nonlinear waves.

attenuates the harmonics resulting from nonlinear effects. The shock length *sl* (Enflo & Hedberg, 2002; Naugolnykh & Ostrovosky, 1998) quantifies the influence of the nonlinear phenomena, and it is necessary to define another parameter, denoted Goldberg's number (Goldberg, 1957), when dissipation is added. represents the ratio of the absorption length *al* (the inverse of the absorption coefficient and corresponds to the beginning of the old age region) to the shock length *<sup>s</sup> l* at which the waveform would shock if absorption phenomena were absent:

$$
\Gamma = \frac{l\_a}{l\_s} = \frac{k\beta\mathcal{M}}{\alpha} \tag{1}
$$

where *k* , *M* , and are, respectively, the wave number, the acoustic Mach number, the acoustic nonlinearity parameter and the absorption coefficient.

It should be noted that higher harmonics may turn the wave into shock state. On the other hand, dissipation attenuates higher harmonics much more than lower harmonics, thus making it more difficult for the waves to go into shock.

The dimensionless parameter measures the relative importance of the nonlinear and dissipative phenomena, which are in perpetual competition. Thus, the Goldberg's number is a reliable indicator for any analysis including these two phenomena. An analysis based on the Goldberg's number is important since it is an essential step for solving general problems involving ultrasound waves of finite amplitude.

Nowadays, Tissue Harmonic Imaging (THI) or second harmonic imaging offers several advantages over conventional pulse-echo imaging. Both harmonic contrast and lateral resolution are improved in harmonic mode. Tissue Harmonic Imaging also provides a better signal to noise ratio which leads to better image quality in many applications. The major benefit of Tissue Harmonic Imaging is artifact reduction resulting in less noisy images, making cysts appear clearer and improving visualization of pathologic conditions and normal structures. Indeed, Tissue Harmonic Imaging is widely used for detecting subtle lesions (e.g., thyroid and breast) and visualizing technically- challenging patients with high body mass index.

In order to create images exclusively from the second harmonic, a theoretical review with some mathematical approximations is elaborated, in this chapter, to derive an analytical expression of the second harmonic. The performance of the simplified model of the second harmonic is interesting, as it can provide a simple, useful model for understanding phenomena in diagnostic imaging.

Despite the significant advantages offered by Tissue Harmonic Imaging, theory has been partially explained. A number of works were elaborated over recent decades. Among these, are Trivett and Van Buren (Trivett & Van Buren, 1981) work which have presented an analysis of the generated harmonics based on the generalized Burgers' equation. Significant differences in the calculated harmonic content were found by Trivett and Van Buren when compared with those obtained by Woodsum (Woodsum, 1981). No explanation was given by Trivett and Van Buren to justify their results. In an author's reply, Woodsum seemed to attribute these differences to the high number of terms retained by Trivett and Van Buren in the Fourier series.

Hedberg, 2002; Naugolnykh & Ostrovosky, 1998) quantifies the influence of the nonlinear phenomena, and it is necessary to define another parameter, denoted Goldberg's number (Goldberg, 1957), when dissipation is added. represents the ratio of the absorption length

*a s*

*l*

*l k M*

It should be noted that higher harmonics may turn the wave into shock state. On the other hand, dissipation attenuates higher harmonics much more than lower harmonics, thus

The dimensionless parameter measures the relative importance of the nonlinear and dissipative phenomena, which are in perpetual competition. Thus, the Goldberg's number is a reliable indicator for any analysis including these two phenomena. An analysis based on the Goldberg's number is important since it is an essential step for solving general problems

Nowadays, Tissue Harmonic Imaging (THI) or second harmonic imaging offers several advantages over conventional pulse-echo imaging. Both harmonic contrast and lateral resolution are improved in harmonic mode. Tissue Harmonic Imaging also provides a better signal to noise ratio which leads to better image quality in many applications. The major benefit of Tissue Harmonic Imaging is artifact reduction resulting in less noisy images, making cysts appear clearer and improving visualization of pathologic conditions and normal structures. Indeed, Tissue Harmonic Imaging is widely used for detecting subtle lesions (e.g., thyroid and breast) and visualizing technically- challenging patients with high

In order to create images exclusively from the second harmonic, a theoretical review with some mathematical approximations is elaborated, in this chapter, to derive an analytical expression of the second harmonic. The performance of the simplified model of the second harmonic is interesting, as it can provide a simple, useful model for understanding

Despite the significant advantages offered by Tissue Harmonic Imaging, theory has been partially explained. A number of works were elaborated over recent decades. Among these, are Trivett and Van Buren (Trivett & Van Buren, 1981) work which have presented an analysis of the generated harmonics based on the generalized Burgers' equation. Significant differences in the calculated harmonic content were found by Trivett and Van Buren when compared with those obtained by Woodsum (Woodsum, 1981). No explanation was given by Trivett and Van Buren to justify their results. In an author's reply, Woodsum seemed to attribute these differences to the high number of terms retained by Trivett and Van Buren in

*l* (Enflo &

and corresponds to the beginning of the old

*l* at which the waveform would shock if absorption

are, respectively, the wave number, the acoustic Mach number, the

(1)

attenuates the harmonics resulting from nonlinear effects. The shock length *<sup>s</sup>*

*al* (the inverse of the absorption coefficient

acoustic nonlinearity parameter and the absorption coefficient.

making it more difficult for the waves to go into shock.

involving ultrasound waves of finite amplitude.

age region) to the shock length *<sup>s</sup>*

 and 

phenomena were absent:

where *k* , *M* ,

body mass index.

the Fourier series.

phenomena in diagnostic imaging.

Similarly, Haran and Cook (Haran & Cook, 1983) have used the Burgers' equation to elaborate an algorithm for calculating harmonics generation by a finite amplitude plane wave of ultrasound propagating in a lossy and nondispersive medium. Their algorithm accounts for an absorption coefficient of any desired frequency dependence. The variation effect of the absorption coefficient on the second harmonic was demonstrated in a medium similar to carbon tetrachloride. Calculations for several types of tissue and biological fluids were presented. It was shown that for some biological media having a low absorption coefficient, a significant distortion of the plane wave can be observed for large propagation ranges.

Recently, D'hooge et al. (D'hooge et al., 1999) have analyzed the nonlinear propagation effects of pulses on broadband attenuation measurements and their implications in ultrasonic tissue characterization by using a simple mathematical model based on the numerical solution, in the time domain, of the Burgers' equation. The developed model has been validated by measuring the absorption coefficient of both a tissue-mimicking phantom in vitro and a liver in vivo at several pressure amplitudes using transmission and reflection measurements, respectively.

In the present chapter, the intensity effects on the behavior of the fundamental and the generated second harmonic, by using both the numerical solution of the Burgers' equation and the analytical expressions established with the quasi-linear approximation are examined. An analysis on the validity domain of the fundamental and the second harmonic analytical expressions established with the quasi-linear approximation is elaborated. The deviations resulting from the analytical expressions established with the quasi-linear approximation and the numerical solution of the Burgers' equation are estimated. This investigation is based on Krassilnikov et al. (Krassilnikov et al., 1957) experimental results. These experimental data concern water and glycerol that correspond, respectively, to a weakly dissipative liquid approaching the characteristics of urine or amniotic fluid (Bouakaz et al., 2004) and a strongly dissipative liquid with some similarities to soft tissues.

It should be noted that in this study all derivations are developed entirely in the frequency domain, thus avoiding both the steep waveform problems and the use of FFT, which alternates between time and frequency domains. The utility of the method resides in the ease with which it can be implemented on a digital computer.

## **2. Theoretical formulation**

The description of acoustic waves in a liquid is founded on the theory of motion of a liquid, which is considered to be continuous. In the present investigation, the viscosity and the heat conduction coefficients, although in general are functions of the state variables, are assumed to be constant. The theoretical formulation of the propagation of finite amplitude plane progressive waves in a homogeneous and dissipative liquid is elaborated in section 2.1, and the theoretical model is based on the derivation of a nonlinear partial differential equation in which the longitudinal particle velocity is a function of time and space. In section 2.2, the dimensionless Burgers' equation is presented, which is considered to be among the most exhaustively studied equations in the theory of nonlinear waves.

Goldberg's Number Influence on the Validity Domain

\* *W W n n* , \* symbolizes the complex conjugate.

Ngoc & Mayer, 1987):

where

1985):

The complex amplitude can be expressed as *<sup>n</sup> in W we n n*

respectively to the amplitude and the phase of the nth harmonic, and <sup>2</sup>

2 0

1

*z c*

for loss due to dissipation relative to the nth harmonic.

*z c*

By using the real notation, knowing that 2

1

1

For a sinusoidal source condition, 0 0 *u u* (0, ) sin( )

2 0 1

2 0 1

Equation (8) is then written more simply as:

*u*

*z c* 

*z c* 

*z c* 

*v*

*u*

2 0 1

 *m*

of the Quasi-Linear Approximation of Finite Amplitude Acoustic Waves 85

For the easiest derivation, equation (4) is substituted into equation (2) (Haran & Cook, 1983;

0 2

0 \* 2

*n n <sup>n</sup> v iu <sup>W</sup>* and

0 2

0 2

0 2

*mu u nu u n u*

*<sup>n</sup> <sup>n</sup> m nm m mn n*

*mu v v u nv u u v nv*

(7)

*mu u v v nu u v v nu*

(8)

0

(10)

 

 

(6)

*<sup>n</sup> <sup>n</sup> m nm m mn n*

*<sup>W</sup> i mW W nW W n W*

yields two coupled partial differential equations governing the behavior of the components *<sup>n</sup> v* and *un* as a function of the spatial coordinate *z* (Aanonsen et al., 1984; Hamilton et al.,

> ( )( ) <sup>2</sup> *<sup>n</sup> <sup>n</sup> mnm m nm m mn m mn <sup>n</sup>*

> ( )( ) <sup>2</sup> *<sup>n</sup> <sup>n</sup> m nm m nm m mn mmn <sup>n</sup>*

> > 1 ( , ) ( )sin( ) *<sup>n</sup>*

*m m n*

*n uz u z n*

*m m n*

*m m n*

al., 1985; Hedberg, 1999; Menounou & Blackstock, 2004), equation (3) becomes:

1

<sup>2</sup> 2 <sup>0</sup> <sup>1</sup>

( ) *<sup>n</sup> m nm n*

*<sup>W</sup> i n mW W nW*

Equation (5) describes the amplitude variation of the nth harmonic in the propagation direction *z* . The summation over *m* expresses nonlinear interactions among various spectral components caused by the energy transfer process, while the other term accounts

Equation (5) is rewritten in another form (Haran & Cook, 1983; Ngoc & Mayer, 1987):

*m m n*

 

where *wn* ,

2 *n n <sup>n</sup> v iu <sup>W</sup>*

(Aanonsen et al., 1984; Hamilton et

(9)

, equation (6)

(5)

*<sup>n</sup>* correspond

*i* 1 . Note that

#### **2.1 Basic equations**

The propagation of finite amplitude plane progressive waves in a homogeneous and dissipative liquid is governed by the Burgers' equation. Here, it is assumed that the ultrasonic wave propagates in the positive *z* direction, and the differential change of the longitudinal particle velocity with respect to *z* is given by (Enflo & Hedberg, 2002; Naugolnykh & Ostrovosky, 1998):

$$\frac{\partial u(z,\tau)}{\partial z} = \frac{\beta}{c\_0^2} u(z,\tau) \frac{\partial u(z,\tau)}{\partial \tau} + \frac{D}{2c\_0^3} \frac{\partial^2 u(z,\tau)}{\partial \tau^2} \tag{2}$$

0 14 11 3 *v p D c c* is the diffusivity of the sound for a thermoviscous fluid. This

parameter is a function of the fluid shear viscosity , the fluid bulk viscosity , the thermal conductivity , the specific heat at constant volume *<sup>v</sup> c* and the specific heat at constant pressure *<sup>p</sup> c* . The acoustic nonlinearity parameter 1 2 *B A* is function of the nonlinearity parameter of the medium *B A*/ , which represents the ratio of quadratic to linear terms in the isentropic pressure-density relation (Hamilton & Blackstock, 1988; Khelladi et al., 2007, 2009). 0 *t zc* is the retarded time, 0*c* is the infinitesimal sound speed and 0 is the undisturbed density of the liquid.

The term on the left hand side of equation (2) is the linear wave propagation. The first term on the right hand side of equation (2) is the nonlinear term that accounts for quadratic nonlinearity producing cumulative effects in progressive plane wave propagation, while the second term represents the loss due to viscosity and heat conduction or any other agencies of dissipation.

Nonlinear propagation in a dissipative liquid is considered using Fourier series expansion. By assuming that the solution of equation (2) is periodic in time with period 0 2 , the solution can be written as the sum of the fundamental and the generated harmonics. Thus *u z*(,) can be developed in Fourier series, with amplitudes that are functions of the spatial coordinate *z* :

$$u(z,\tau) = \sum\_{n=1}^{+\infty} \left[ v\_n(z) \cos(n\phi\_0 \tau) + u\_n(z) \sin(n\phi\_0 \tau) \right] \tag{3}$$

0 is the characteristic angular frequency and , *n n v u* are the Fourier coefficients of the nth harmonic.

When complex notation is used, equation (3) changes to (Haran & Cook, 1983; Ngoc & Mayer, 1987):

$$\ln(z\_{\prime}\mid\tau) = \sum\_{n=-\infty}^{+\infty} W\_n(z) \, e^{in\alpha\_0 \tau} \tag{4}$$

The complex amplitude can be expressed as *<sup>n</sup> in W we n n* where *wn* ,*<sup>n</sup>* correspond respectively to the amplitude and the phase of the nth harmonic, and <sup>2</sup> *i* 1 . Note that \* *W W n n* , \* symbolizes the complex conjugate.

For the easiest derivation, equation (4) is substituted into equation (2) (Haran & Cook, 1983; Ngoc & Mayer, 1987):

$$\frac{\partial \mathcal{W}\_n}{\partial z} = \text{i} \, \frac{\mathcal{B}\alpha\_0}{c\_0^2} \sum\_{m=-\infty}^{+\infty} \text{( $n-m$ )} \, \mathcal{W}\_m \mathcal{W}\_{n-m} - \alpha m^2 \mathcal{W}\_n \tag{5}$$

where 2 0 3 <sup>0</sup> 2 *D c* 

84 Ultrasonic Waves

The propagation of finite amplitude plane progressive waves in a homogeneous and dissipative liquid is governed by the Burgers' equation. Here, it is assumed that the ultrasonic wave propagates in the positive *z* direction, and the differential change of the longitudinal particle velocity with respect to *z* is given by (Enflo & Hedberg, 2002;

> *u z z c c*

2 3 2 0 0 (,) (,) (,) (,) <sup>2</sup> *u z uz D uz*

nonlinearity parameter of the medium *B A*/ , which represents the ratio of quadratic to linear terms in the isentropic pressure-density relation (Hamilton & Blackstock, 1988;

The term on the left hand side of equation (2) is the linear wave propagation. The first term on the right hand side of equation (2) is the nonlinear term that accounts for quadratic nonlinearity producing cumulative effects in progressive plane wave propagation, while the second term represents the loss due to viscosity and heat conduction or any other agencies

Nonlinear propagation in a dissipative liquid is considered using Fourier series expansion. By assuming that the solution of equation (2) is periodic in time with period 0 2

solution can be written as the sum of the fundamental and the generated harmonics. Thus

( , ) [ ( )cos( ) ( )sin( )] *n n*

 

0 is the characteristic angular frequency and , *n n v u* are the Fourier coefficients of the nth

When complex notation is used, equation (3) changes to (Haran & Cook, 1983; Ngoc &

<sup>0</sup> ( , ) ( ) *in n*

*n uz W z e*

*uz v z n u z n*

can be developed in Fourier series, with amplitudes that are functions of the spatial

0 0

   

(4)

(3)

2

, the specific heat at constant volume *<sup>v</sup> c* and the specific heat at

 

is the diffusivity of the sound for a thermoviscous fluid. This

*t zc* is the retarded time, 0*c* is the infinitesimal sound

, the fluid bulk viscosity

1 2 *B A* is function of the

, the

 , the

(2)

**2.1 Basic equations** 

0

speed and

of dissipation.

coordinate *z* :

*u z*(,) 

harmonic.

Mayer, 1987):

*D*

Naugolnykh & Ostrovosky, 1998):

14 11 3 *v p*

 

 

Khelladi et al., 2007, 2009). 0

thermal conductivity

*c c*

parameter is a function of the fluid shear viscosity

constant pressure *<sup>p</sup> c* . The acoustic nonlinearity parameter

0 is the undisturbed density of the liquid.

1

*n*

 Equation (5) describes the amplitude variation of the nth harmonic in the propagation direction *z* . The summation over *m* expresses nonlinear interactions among various spectral components caused by the energy transfer process, while the other term accounts for loss due to dissipation relative to the nth harmonic.

Equation (5) is rewritten in another form (Haran & Cook, 1983; Ngoc & Mayer, 1987):

$$\frac{\partial \mathcal{W}\_n}{\partial z} = i \frac{\beta \alpha\_0}{c\_0^2} \left[ \sum\_{m=1}^{n-1} m \mathcal{W}\_m \mathcal{W}\_{n-m} + \sum\_{m=n}^{+n} n \mathcal{W}\_m \mathcal{W}\_{m-n}^\* \right] - \alpha n^2 \mathcal{W}\_n \tag{6}$$

By using the real notation, knowing that 2 *n n <sup>n</sup> v iu <sup>W</sup>* and 2 *n n <sup>n</sup> v iu <sup>W</sup>* , equation (6) yields two coupled partial differential equations governing the behavior of the components *<sup>n</sup> v* and *un* as a function of the spatial coordinate *z* (Aanonsen et al., 1984; Hamilton et al., 1985):

$$\frac{\partial \upsilon\_n}{\partial z} = \frac{\partial \alpha\_0}{2c\_0^2} \left[ \sum\_{m=1}^{n-1} m(\mu\_m \upsilon\_{n-m} + \upsilon\_m \mu\_{n-m}) - \sum\_{m=n}^{+n} n(\upsilon\_m \mu\_{m-n} - \mu\_m \upsilon\_{m-n}) \right] - \alpha n^2 \upsilon\_n \tag{7}$$

$$\frac{\partial u\_n}{\partial z} = \frac{\partial \alpha\_0}{\mathcal{L}c\_0^2} \left[ \sum\_{m=1}^{n-1} m (u\_m u\_{n-m} - v\_m v\_{n-m}) - \sum\_{m=n}^{+\alpha} n (u\_m u\_{m-n} + v\_m v\_{m-n}) \right] - \alpha n^2 u\_n \tag{8}$$

For a sinusoidal source condition, 0 0 *u u* (0, ) sin( ) (Aanonsen et al., 1984; Hamilton et al., 1985; Hedberg, 1999; Menounou & Blackstock, 2004), equation (3) becomes:

$$
\mu(z,\tau) = \sum\_{n=1}^{+\infty} \mu\_n(z) \sin(n\phi\_0 \tau) \tag{9}
$$

Equation (8) is then written more simply as:

$$\frac{\partial u\_n}{\partial z} = \frac{\beta \alpha \rho\_0}{2c\_0^2} \left[ \sum\_{m=1}^{n-1} m u\_m u\_{n-m} - \sum\_{m=n}^{+\infty} n u\_m u\_{m-n} \right] - \alpha n^2 u\_n \tag{10}$$

Goldberg's Number Influence on the Validity Domain

*f* and <sup>2</sup>

3 1 2

0 0 () () <sup>2</sup> *<sup>p</sup> <sup>z</sup> <sup>p</sup> <sup>z</sup> c* 

fundamental and the second harmonic, respectively.

where <sup>2</sup> 1 0 

0 0 (,) (,) *n n p zt cu zt* 

If <sup>0</sup> 2

1935):

*z* 0 ).

with <sup>0</sup>

*h*

<sup>2</sup> () *<sup>P</sup> p z*

Equation (16) becomes:

3 0 0 2

Moreover, if the term 2 1 ( 2) 1

(BjØrnØ, 2002; Cobb, 1983; Zhang et al.,1991):

 

*c* 

Thuras et al., 1935):

, then <sup>0</sup>

of the Quasi-Linear Approximation of Finite Amplitude Acoustic Waves 87

2 1 22

*uz uz*

3 1 2 11

1

0 2 2 ( ) ( ) *p z <sup>z</sup> hP e p z*

2 1 2

 

1 2

 

1 2 2 ( 2) 2 0 ( ) *<sup>z</sup> p z hP ze* 

The solution of equation (18) is easily obtained. Knowing that for 0 *z* <sup>2</sup> *p* (0) 0 , the acoustic pressure of the second harmonic component can be expressed as (Cobb, 1983;

2

( ) <sup>2</sup> *z z e e p z hP* 

2 0

can be neglected comparatively to 1 1

(18)

*z* , an approximation of equation (19) can be made

(20)

(17)

3 1 22

( ) () () () <sup>2</sup>

*p z <sup>p</sup> zp z p z*

( ) () () <sup>2</sup>

In many situations, the experimental studies are based on pressure measurements. Knowing that the ratio of the nth harmonic pressure to the associated particle velocity is given by

( ) () () () <sup>2</sup>

0

0

 

(Germain et al., 1989); equation (15) is rewritten as:

0 0

*p z pz pz*

( ) () () <sup>2</sup>

acoustic pressure of the fundamental can be written as (Gong et al., 1989; Thuras et al.,

1 0 ( ) *<sup>z</sup> p z Pe*

where *P*0 is the characteristic pressure amplitude (the value of the fundamental pressure at

<sup>1</sup> 2 2 2

*z*

0 0

1 0

*z c*

2 0 2

*z c*

2 1 2 11

*u zu z u z*

*f* denote the absorption coefficients of the

(15)

(16)

(19)

*p* ( ) *z* . The

1 0

*z c*

*z c*

20 1

 4 4 

*u z*

*u z*

2 0 2

The incremental change of the particle velocity can be approximated by the first order truncated power series (Haran & Cook, 1983; Ngoc et al., 1987):

$$
\mu(z + \Delta z, t) = \mu(z, t) + \frac{\partial \mu(z, t)}{\partial z} \Delta z \tag{11}
$$

By combining equations (10) and (11), an iterative description of finite amplitude plane wave propagation in a homogeneous and dissipative liquid, is obtained:

$$\mu\_n(z+\Delta z) = \mu\_n(z) + \frac{\beta \alpha\_0}{2c\_0^2} \left[ \sum\_{m=1}^{n-1} m u\_m(z) \mu\_{n-m}(z) - \sum\_{m=n}^{+n} n u\_m(z) \mu\_{m-n}(z) \right] \Delta z - \alpha n^2 u\_n(z) \Delta z \tag{12}$$

The first summation term on the right hand side of equation (12) represents the contribution of lower order harmonics to the nth harmonic, while the second one is associated with the contribution of higher order harmonics. According to the sign of each contribution the nth harmonic energy can be enhanced or decreased. The last term in this equation represents losses undergone by the nth harmonic.

Generally, the absorption coefficient depends on the propagation medium characteristics and the insonation frequency. For the considered viscous fluids, this frequency dependence is quadratic with frequency and can be represented by (Smith & Beyer, 1948; Willard, 1941):

$$a = a\_0 f^2 \tag{13}$$

where 0 depends upon the nature of the liquid, and 0 *f* 2 is the insonation frequency.

Therefore the Goldberg's number , increases with the amplitude of excitation and decreases with frequency.

Equation (12) becomes:

$$
\mu\_n(z + \Delta z) = \mu\_n(z) + \frac{\beta \alpha\_0}{2c\_0^2} \left[ \sum\_{m=1}^{n-1} m u\_m(z) u\_{n-m}(z) - \sum\_{m=n}^{+\infty} m u\_m(z) u\_{m-n}(z) \right] \Delta z - \alpha\_n u\_n(z) \Delta z \tag{14}
$$

where 2 2 *<sup>n</sup>* <sup>0</sup>*n f*

Equation (14) allows the determination of the nth harmonic amplitude at the location *z z* in terms of all harmonics at the preceding spatial coordinate *z* . This derivation requires an appropriate truncation of the finite series on the right hand side of equation (14) to ensure a negligibly small error in the highest harmonic of interest and to maintain some acceptable accuracy.

In the hypothesis of the quasi-linear approximation, all the harmonics of higher order than two can be neglected in the numerical solution of the Burgers' equation, so equation (14) changes to:

$$\begin{cases} \frac{\partial u\_1(z)}{\partial z} = -\frac{\beta a\_0}{2c\_0^2} u\_1(z)u\_2(z) - a\_1 u\_1(z) \\ \frac{\partial u\_2(z)}{\partial z} = \frac{\beta a\_0}{2c\_0^2} u\_1^2(z) - a\_2 u\_2(z) \end{cases} \tag{15}$$

where <sup>2</sup> 1 0 *f* and <sup>2</sup> 20 1 4 4 *f* denote the absorption coefficients of the fundamental and the second harmonic, respectively.

In many situations, the experimental studies are based on pressure measurements. Knowing that the ratio of the nth harmonic pressure to the associated particle velocity is given by 0 0 (,) (,) *n n p zt cu zt* (Germain et al., 1989); equation (15) is rewritten as:

$$\begin{cases} \frac{\partial p\_1(z)}{\partial z} = -\frac{\beta a\_0}{2\rho\_0 c\_0^3} p\_1(z)p\_2(z) - \alpha\_1 p\_1(z) \\ \frac{\partial p\_2(z)}{\partial z} = \frac{\beta a\_0}{2\rho\_0 c\_0^3} p\_1^2(z) - \alpha\_2 p\_2(z) \end{cases} \tag{16}$$

If <sup>0</sup> 2 <sup>2</sup> () *<sup>P</sup> p z* , then <sup>0</sup> 3 1 2 0 0 () () <sup>2</sup> *<sup>p</sup> <sup>z</sup> <sup>p</sup> <sup>z</sup> c* can be neglected comparatively to 1 1 *p* ( ) *z* . The

acoustic pressure of the fundamental can be written as (Gong et al., 1989; Thuras et al., 1935):

$$p\_1(z) = P\_0 e^{-\alpha\_1 z} \tag{17}$$

where *P*0 is the characteristic pressure amplitude (the value of the fundamental pressure at *z* 0 ).

Equation (16) becomes:

$$\frac{\partial p\_2(z)}{\partial z} = hP\_0^2 e^{-2a\_1z} - a\_2p\_2(z) \tag{18}$$

with <sup>0</sup> 3 0 0 2 *h c* 

86 Ultrasonic Waves

The incremental change of the particle velocity can be approximated by the first order

(,) ( ,) (,) *uzt uz zt uzt z*

By combining equations (10) and (11), an iterative description of finite amplitude plane

( ) () () () () () () <sup>2</sup>

The first summation term on the right hand side of equation (12) represents the contribution of lower order harmonics to the nth harmonic, while the second one is associated with the contribution of higher order harmonics. According to the sign of each contribution the nth harmonic energy can be enhanced or decreased. The last term in this equation represents

Generally, the absorption coefficient depends on the propagation medium characteristics and the insonation frequency. For the considered viscous fluids, this frequency dependence is quadratic with frequency and can be represented by (Smith &

Therefore the Goldberg's number , increases with the amplitude of excitation and

( ) () () () () () () <sup>2</sup>

Equation (14) allows the determination of the nth harmonic amplitude at the location *z z* in terms of all harmonics at the preceding spatial coordinate *z* . This derivation requires an appropriate truncation of the finite series on the right hand side of equation (14) to ensure a negligibly small error in the highest harmonic of interest and to maintain some acceptable

In the hypothesis of the quasi-linear approximation, all the harmonics of higher order than two can be neglected in the numerical solution of the Burgers' equation, so equation (14)

 

(14)

*n n m n m mm n n n m m n u z z u z mu z u z nu z u z z u z z*

 

0 depends upon the nature of the liquid, and 0 *f*

1

*n*

0 2 0 1

*c* 

2 0

 

*n n m n m mm n n m m n u z z u z mu z u z nu z u z z n u z z*

*z* 

0 2

(12)

*f* (13)

is the insonation

 2

(11)

truncated power series (Haran & Cook, 1983; Ngoc et al., 1987):

wave propagation in a homogeneous and dissipative liquid, is obtained:

1

*n*

2 0 1

*c* 

losses undergone by the nth harmonic.

Beyer, 1948; Willard, 1941):

decreases with frequency.

Equation (12) becomes:

where 2 2 *<sup>n</sup>* <sup>0</sup>*n f*

accuracy.

changes to:

where

frequency.

The solution of equation (18) is easily obtained. Knowing that for 0 *z* <sup>2</sup> *p* (0) 0 , the acoustic pressure of the second harmonic component can be expressed as (Cobb, 1983; Thuras et al., 1935):

$$p\_2(z) = hP\_0^2 \left(\frac{e^{-a\_2 z} - e^{-2a\_1 z}}{2a\_1 - a\_2}\right) \tag{19}$$

Moreover, if the term 2 1 ( 2) 1 *z* , an approximation of equation (19) can be made (BjØrnØ, 2002; Cobb, 1983; Zhang et al.,1991):

$$p\_2(z) = l \mathbf{i} P\_0^2 z e^{-(a\_1 + a\_2 f \mathbf{2})z} \tag{20}$$

Goldberg's Number Influence on the Validity Domain

case of water 13 1 2 0 

(Krassilnikov et al., 1957).

selected range 0 1

Parameters Temperature

Table 1. Material properties.

(°C)

<sup>0</sup>*c* ,  of the Quasi-Linear Approximation of Finite Amplitude Acoustic Waves 89

Nonlinear effects occur more strongly when ultrasound propagates through slightly dissipative liquids, so a special attention is given to a propagation medium characterized by a Goldberg number greater than unity. In this case, when the waveform approaches the shock length, nonlinear effects dominate dissipation phenomena. The amplitude of the generated harmonics increases at the expense of the fundamental component. After the shock length, absorption limits the generation of harmonics by decreasing theirs amplitudes gradually with the propagation path. For this reason, all the simulations of the first two

several values of the acoustic intensity. Moreover, all the shock lengths for several intensities are greater than 19.8 cm (Table 2). As in biomedical diagnosis the region of

is amply appropriate for this kind of investigation.

of excitation. In this study, the insonation frequency is fixed at 2 MHz , thus the shock

Among all the configurations presented in this study, including various acoustic intensities and two analyzed mediums, only one case is sensitive in biomedical diagnostic and must be analyzed with extreme caution. Indeed, a more favorable situation where nonlinear effects have sufficient time to be entirely established corresponds to the case of water, for which the acoustic intensity is equal to <sup>2</sup> 4.7 W/cm and as a consequence a shock length equal to 19.8 cm . As the generation of harmonics occurs while moving away from the source and approaching the shock length, the greatest signal distortion may occur in the range of interest. Moreover, the irradiation of living tissue with shock waves in diagnostic processes appears risky since the damage and exposure criteria for these radiations have not been delineated.

It should be noted that all the simulations are made with intensities of <sup>2</sup> 0.2 - 4.7 W/cm

It will be stated by the derivation of the Goldberg number that water surpasses any tissue in its ability to produce extremely distorted waveforms even at relatively low intensity. So, a special attention is given to this liquid where the possibility of distortion occurring has several implications. Indeed, water can generate extreme waveform distortion compared to glycerol, as indicated by the Goldberg's number for water, which is 200 times larger than

> Sound velocity <sup>0</sup> *Cms* ( /)

(Table 2), which correspond to breast lesion diagnosis (Nightingale et al., 1999).

that of glycerol for an acoustic intensity of about <sup>2</sup> 0.2 W/cm (Table 2).

0 

Density

(/ ) *kg m*

3

Water 20 998 1481 3.48 Glycerol 20 1260 1980 5.4

and on the external parameters such as the insonation frequency and the amplitude

It should be pointed out that the shock length *sl* depends on the medium characteristics

interest (ROI) is about 20 cm , it is absolutely useless to explore beyond 1

length for a given medium will depend only upon the amplitude of excitation.

harmonics are plotted as a function of the dimensionless location

0.23 10 . . *Np m Hz* and for glycerol 13 1 2

0 

26 10 . . *Np m Hz*

up to unity and for

and the

0 ,

Acoustic nonlinearity parameter

#### **2.2 Dimensionless equations**

For theoretical analysis as well as for numerical implementation, it is more convenient to define dimensionless variables, by using the characteristic particle velocity *U*<sup>0</sup> , the characteristic time 0 1 and the lossless plane wave shock formation length *<sup>s</sup> l* :

$$
\mathcal{U}\mathcal{U} = \frac{\mathcal{U}}{\mathcal{U}\_0}\;'\;\theta = a\theta\_0\tau\;\text{ and }\;\sigma = \frac{z}{l\_s} \tag{21}
$$

where *U* , and are, respectively, the dimensionless longitudinal particle velocity, the dimensionless time and the dimensionless propagation path.

Insertion of equation (21) into the Burgers' equation (equation (2)), gives the dimensionless equation (BjØrnØ, 2002; Fenlon, 1971; Hedberg, 1994):

$$\frac{\partial \mathcal{U}(\sigma,\theta)}{\partial \sigma} = \mathcal{U}(\sigma,\theta) \frac{\partial \mathcal{U}(\sigma,\theta)}{\partial \theta} + \Gamma^{-1} \frac{\partial^2 \mathcal{U}(\sigma,\theta)}{\partial \theta^2} \tag{22}$$

The dimensionless amplitude of the nth harmonic at the dimensionless location in terms of all harmonics at the preceding dimensionless location can be written as:

$$\mathrm{i}\mathcal{U}\_{n}(\sigma+\Delta\sigma) = \mathrm{i}\mathcal{U}\_{n}(\sigma) + \frac{1}{2} \left[ \sum\_{m=1}^{n-1} m\mathrm{i}\mathcal{U}\_{m}(\sigma)\mathrm{i}\mathcal{U}\_{n-m}(\sigma) - \sum\_{m=n}^{+n} n\mathrm{i}\mathcal{U}\_{m}(\sigma)\mathrm{i}\mathcal{U}\_{m-n}(\sigma) \right] \Delta\sigma - n^{2}\Gamma^{-1}\mathrm{U}\_{n}(\sigma)\Delta\sigma \tag{23}$$

With this dimensionless notation, the acoustic pressure of the fundamental and the second harmonic can be expressed as:

$$p\_1(\sigma) = P\_0 e^{-\alpha\_1 l\_\* \sigma} \tag{24}$$

$$p\_2(\sigma) = \frac{1}{2} P\_0 \left( \frac{e^{-\alpha\_2 l\_s \sigma} - e^{-2\alpha\_1 l\_s \sigma}}{\left(2\alpha\_1 - \alpha\_2\right) l\_s} \right) \tag{25}$$

In the case of 2 1 ( 2) 1 *<sup>s</sup> l* , equation (25) becomes:

$$p\_2(\sigma) = \frac{1}{2} P\_0 \sigma e^{-(a\_1 + a\_2/2)l\_s \sigma} \tag{26}$$

#### **3. Numerical experiments and discussions**

Krassilnikov et al. (Krassilnikov et al., 1957) experimental data for water and for glycerol are used in order to simulate the amplitude of the first two harmonics, by using both the numerical solution of the Burgers' equation (equation (23)) and the analytical expressions established with the quasi-linear approximation (equations (24), (25) and (26)). Table 1 lists material properties.

According to Krassilnikov et al. (Krassilnikov et al., 1957) experimental work, the absorption coefficient is a quadratic function of frequency. The absorption coefficient is that obtained from an infinitesimal acoustic excitation, even though the acoustic intensity increases. In the

For theoretical analysis as well as for numerical implementation, it is more convenient to define dimensionless variables, by using the characteristic particle velocity *U*<sup>0</sup> , the

Insertion of equation (21) into the Burgers' equation (equation (2)), gives the dimensionless

(,) (,) (,) (,) *U UU <sup>U</sup>*

<sup>1</sup> ( ) () () () () () ( ) <sup>2</sup>

 

With this dimensionless notation, the acoustic pressure of the fundamental and the second

1 0 ( ) *<sup>s</sup> <sup>l</sup> p Pe*

<sup>1</sup> ( ) <sup>22</sup>

*sl p Pe* 

Krassilnikov et al. (Krassilnikov et al., 1957) experimental data for water and for glycerol are used in order to simulate the amplitude of the first two harmonics, by using both the numerical solution of the Burgers' equation (equation (23)) and the analytical expressions established with the quasi-linear approximation (equations (24), (25) and (26)). Table 1 lists

According to Krassilnikov et al. (Krassilnikov et al., 1957) experimental work, the absorption coefficient is a quadratic function of frequency. The absorption coefficient is that obtained from an infinitesimal acoustic excitation, even though the acoustic intensity increases. In the

 

*e e p P <sup>l</sup>* 

*n n m n m mm n n m m n*

   

 

0 *<sup>u</sup> <sup>U</sup> U* , 0 and

> 

The dimensionless amplitude of the nth harmonic at the dimensionless location

dimensionless time and the dimensionless propagation path.

 

terms of all harmonics at the preceding dimensionless location

 

2 0

*l* , equation (25) becomes:

2 0 <sup>1</sup> ( ) <sup>2</sup>

1

*n*

**3. Numerical experiments and discussions** 

*UU m*

harmonic can be expressed as:

In the case of 2 1 ( 2) 1 *<sup>s</sup>* 

material properties.

 

1

equation (BjØrnØ, 2002; Fenlon, 1971; Hedberg, 1994):

and the lossless plane wave shock formation length *sl* :

*s z l* 

> 2 1

 

 

1

 2 1 2

 

1 2

1 2 ( 2)

*s s l l*

 

*s*

 

*UU nU U nU*

(23)

(22)

2

 

(24)

(26)

 

are, respectively, the dimensionless longitudinal particle velocity, the

(21)

can be written as:

2 1

(25)

 in

 

**2.2 Dimensionless equations** 

characteristic time 0 1

 and 

where *U* ,

 case of water 13 1 2 0 0.23 10 . . *Np m Hz* and for glycerol 13 1 2 0 26 10 . . *Np m Hz* (Krassilnikov et al., 1957).

Nonlinear effects occur more strongly when ultrasound propagates through slightly dissipative liquids, so a special attention is given to a propagation medium characterized by a Goldberg number greater than unity. In this case, when the waveform approaches the shock length, nonlinear effects dominate dissipation phenomena. The amplitude of the generated harmonics increases at the expense of the fundamental component. After the shock length, absorption limits the generation of harmonics by decreasing theirs amplitudes gradually with the propagation path. For this reason, all the simulations of the first two harmonics are plotted as a function of the dimensionless location up to unity and for several values of the acoustic intensity. Moreover, all the shock lengths for several intensities are greater than 19.8 cm (Table 2). As in biomedical diagnosis the region of interest (ROI) is about 20 cm , it is absolutely useless to explore beyond 1 and the selected range 0 1 is amply appropriate for this kind of investigation.

It should be pointed out that the shock length *sl* depends on the medium characteristics 0 , <sup>0</sup>*c* , and on the external parameters such as the insonation frequency and the amplitude of excitation. In this study, the insonation frequency is fixed at 2 MHz , thus the shock length for a given medium will depend only upon the amplitude of excitation.

Among all the configurations presented in this study, including various acoustic intensities and two analyzed mediums, only one case is sensitive in biomedical diagnostic and must be analyzed with extreme caution. Indeed, a more favorable situation where nonlinear effects have sufficient time to be entirely established corresponds to the case of water, for which the acoustic intensity is equal to <sup>2</sup> 4.7 W/cm and as a consequence a shock length equal to 19.8 cm . As the generation of harmonics occurs while moving away from the source and approaching the shock length, the greatest signal distortion may occur in the range of interest. Moreover, the irradiation of living tissue with shock waves in diagnostic processes appears risky since the damage and exposure criteria for these radiations have not been delineated.

It should be noted that all the simulations are made with intensities of <sup>2</sup> 0.2 - 4.7 W/cm (Table 2), which correspond to breast lesion diagnosis (Nightingale et al., 1999).

It will be stated by the derivation of the Goldberg number that water surpasses any tissue in its ability to produce extremely distorted waveforms even at relatively low intensity. So, a special attention is given to this liquid where the possibility of distortion occurring has several implications. Indeed, water can generate extreme waveform distortion compared to glycerol, as indicated by the Goldberg's number for water, which is 200 times larger than that of glycerol for an acoustic intensity of about <sup>2</sup> 0.2 W/cm (Table 2).


Table 1. Material properties.

Goldberg's Number Influence on the Validity Domain

harmonics.

amplitude of the fundamental and also that of the second harmonic.

of the Quasi-Linear Approximation of Finite Amplitude Acoustic Waves 91

The amplitude p /P 1 0 (figure 1a, figure 1b) and the amplitude p /P 2 0 (figure 2a, figure 2b) increase with (Table 2). So, the effect of the increased acoustic intensity is to enhance the

In the hypothesis of linear acoustics, increasing the absorption coefficient leads systematically to a decrease of the wave amplitude. The finite amplitude waves do not obey to the same principle because nonlinear effects and dissipation are two phenomena in perpetual contest. The interplay between these two phenomena developed along the propagation path is not simply an additive effect as normally assumed in linear acoustics. Therefore, a measure of whether nonlinear effects or absorption will prevail is the Goldberg's number . The larger is, the more nonlinear effects dominate. Whereas for values of 1 , absorption is so strong that no significant nonlinear effects occur. Thus the calculation of the Goldberg's number is required to quantify the amplitude of the generated

By taking water as an example, the most significant amplitude of the generated harmonic, for various values of intensity, corresponds to the highest Goldberg's number (figure 2a). This is in perfect agreement with physical phenomena that take place in the analyzed medium. Indeed, a high Goldberg number corresponds to a predominance of the nonlinearity phenomenon as compared to dissipation, which represents the main factor of

For a slightly dissipative liquid, it can be seen that the second harmonic component grows

(figure 1a, figure 2a). Its growth begins to taper off at the location of the initial shock formation, beyond this location the curves decay as expected. So, the nonlinearity mechanism is a bridge that facilitates the energy exchange among different harmonic modes. An increase of the Goldberg's number enhances the transfer of energy from the fundamental to higher harmonics and between harmonics themselves. Thus, the generated

However, for a strongly dissipative medium, the absorption is so strong that significant nonlinear effects do not occur. Indeed, the old age region begins at a range smaller than the shock length and once nonlinear effects take place, absorption dominates the behavior of the fundamental and the generated harmonic (figure 1b, figure 2b). In absorbing media, the exchange of energy is more complicated, because absorption diminishes amplitude with increasing the propagation path and acts as a low pass filter that reduces the energy of

The evaluation of the relative deviation, for each analytical expression in relation to the

The relative deviation, on the selected range, of the analytical expression of the fundamental component (equation (24)) in relation to the numerical solution of the Burgers' equation is

 analytical xpression- numerical solution(Burgers) (%) <sup>100</sup> numerical solution(Burgers)

*Deviation* (27)

numerical solution of the Burgers' equation, is carried out in the following way:

*e*

harmonics can only follow the evolution of the fundamental which gives them birth.

at the expense of the fundamental

amplitude decrease. This situation is also apparent for glycerol (figure 2b).

cumulatively with increasing the normalized length

higher harmonics (figure 2b).

less than 4% for glycerol (figure 3b).


Table 2. Goldberg's number for water and glycerol with intensities of W/cm4.7 - <sup>2</sup> 0.2 and an insonation frequency of MHz 2 .

Initially, the ultrasonic wave is taken to be purely sinusoidal with a frequency of 2 MHz in the two considered media. Only the fundamental wave exists at the starting location 0 , and the other harmonic modes are generated as the wave propagates from the source. Through an iterative method, the value of the Goldberg number is inserted into the Burgers' equation in order to determine its numerical solution (Table 2). 40 harmonics are retained to simulate the numerical solution of the Burgers' equation which is considered, in the deviation calculus, as an exact solution.

For a better readability and interpretation of the obtained numerical data, a symbol with a defined shape and type is inserted on the graphic layout of the analyzed functions. All the following simulations exploit equations (23), (24), (25) and (26) corresponding respectively to the numerical solution of the Burgers' equation, the quasi-linear approximation of the acoustic pressure of the fundamental, the quasi-linear approximation of the second harmonic and the quasi-linear approximation of the approximated second harmonic.

The simulations relating to water and glycerol are represented in all figures (a) and (b), respectively.

Fig. 1. Pressure amplitude /Pp 01 versus the coordinate.

Water Temperature (°C) Intensity (W/cm2) Shock length (m) Goldberg Number 20 0.34 0.739 14.7 20 2 0.304 35.7 20 4.7 0.198 54.9 Glycerol Temperature (°C) Intensity (W/cm2) Shock length (m) Goldberg Number 22 0.2 1.443 0.07 21 2.5 0.408 0.24 19 4.5 0.304 0.32

Table 2. Goldberg's number for water and glycerol with intensities of W/cm4.7 - <sup>2</sup> 0.2 and

Initially, the ultrasonic wave is taken to be purely sinusoidal with a frequency of 2 MHz in the two considered media. Only the fundamental wave exists at the starting location 0

and the other harmonic modes are generated as the wave propagates from the source. Through an iterative method, the value of the Goldberg number is inserted into the Burgers' equation in order to determine its numerical solution (Table 2). 40 harmonics are retained to simulate the numerical solution of the Burgers' equation which is considered, in the

For a better readability and interpretation of the obtained numerical data, a symbol with a defined shape and type is inserted on the graphic layout of the analyzed functions. All the following simulations exploit equations (23), (24), (25) and (26) corresponding respectively to the numerical solution of the Burgers' equation, the quasi-linear approximation of the acoustic pressure of the fundamental, the quasi-linear approximation of the second

The simulations relating to water and glycerol are represented in all figures (a) and (b),

**0,0**

**0,2**

**0,4**

**Amplitude p1 /P0**

**0,6**

**0,8**

 **I0**

 **I0**

 **I0**

**1,0**

harmonic and the quasi-linear approximation of the approximated second harmonic.

,

**0,0 0,2 0,4 0,6 0,8 1,0**

 **coordinate** 

**(b) : Glycerol Quasi-linear approximation of the fundamental (eqn(24)) Numerical solution of the Burgers equation (eqn(23))** 

**= 0.2 W/cm<sup>2</sup>**

**= 2.5 W/cm<sup>2</sup>**

**= 4.5 W/cm<sup>2</sup>**

an insonation frequency of MHz 2 .

deviation calculus, as an exact solution.

**0,0 0,2 0,4 0,6 0,8 1,0**

 **coordinate**

Fig. 1. Pressure amplitude /Pp 01 versus the coordinate.

 **Numerical solution of the Burgers equation (eqn(23)) Quasi-linear approximation of the fundamental (eqn(24))**

**(a) : Water**

respectively.

**0,5**

**0,6**

**0,7**

**0,8**

**Amplitude p1 /P0**

 **I0 = 0.34 W/cm<sup>2</sup>**

 **I0 = 2 W/cm<sup>2</sup> I0 = 4.7 W/cm<sup>2</sup>**

**0,9**

**1,0**

The amplitude p /P 1 0 (figure 1a, figure 1b) and the amplitude p /P 2 0 (figure 2a, figure 2b) increase with (Table 2). So, the effect of the increased acoustic intensity is to enhance the amplitude of the fundamental and also that of the second harmonic.

In the hypothesis of linear acoustics, increasing the absorption coefficient leads systematically to a decrease of the wave amplitude. The finite amplitude waves do not obey to the same principle because nonlinear effects and dissipation are two phenomena in perpetual contest. The interplay between these two phenomena developed along the propagation path is not simply an additive effect as normally assumed in linear acoustics. Therefore, a measure of whether nonlinear effects or absorption will prevail is the Goldberg's number . The larger is, the more nonlinear effects dominate. Whereas for values of 1 , absorption is so strong that no significant nonlinear effects occur. Thus the calculation of the Goldberg's number is required to quantify the amplitude of the generated harmonics.

By taking water as an example, the most significant amplitude of the generated harmonic, for various values of intensity, corresponds to the highest Goldberg's number (figure 2a). This is in perfect agreement with physical phenomena that take place in the analyzed medium. Indeed, a high Goldberg number corresponds to a predominance of the nonlinearity phenomenon as compared to dissipation, which represents the main factor of amplitude decrease. This situation is also apparent for glycerol (figure 2b).

For a slightly dissipative liquid, it can be seen that the second harmonic component grows cumulatively with increasing the normalized length at the expense of the fundamental (figure 1a, figure 2a). Its growth begins to taper off at the location of the initial shock formation, beyond this location the curves decay as expected. So, the nonlinearity mechanism is a bridge that facilitates the energy exchange among different harmonic modes. An increase of the Goldberg's number enhances the transfer of energy from the fundamental to higher harmonics and between harmonics themselves. Thus, the generated harmonics can only follow the evolution of the fundamental which gives them birth.

However, for a strongly dissipative medium, the absorption is so strong that significant nonlinear effects do not occur. Indeed, the old age region begins at a range smaller than the shock length and once nonlinear effects take place, absorption dominates the behavior of the fundamental and the generated harmonic (figure 1b, figure 2b). In absorbing media, the exchange of energy is more complicated, because absorption diminishes amplitude with increasing the propagation path and acts as a low pass filter that reduces the energy of higher harmonics (figure 2b).

The evaluation of the relative deviation, for each analytical expression in relation to the numerical solution of the Burgers' equation, is carried out in the following way:

$$\text{Deviation(\%)} = \frac{\left| \text{analytic expression-numerical solution(Burgers)} \right|}{\text{numerical solution(Burgers)}} \times 100\tag{27}$$

The relative deviation, on the selected range, of the analytical expression of the fundamental component (equation (24)) in relation to the numerical solution of the Burgers' equation is less than 4% for glycerol (figure 3b).

Goldberg's Number Influence on the Validity Domain

**(a) : Water**

 **Relative deviation of the analytical expression of the approximated 2nd harmonic compared to the numerical solution of the Burgers equation (eqn(27)) Relative deviation of the analytical expression of the second harmonic compared to the numerical solution of the Burgers equation (eqn(27))**

**0,0 0,2 0,4 0,6 0,8 1,0**

**coordinate** 

equation versus the coordinate.

only if 2 1 ( 2) *<sup>s</sup>* 

about 40% at 1

2 1 ( 2) *<sup>s</sup>*  **Relative deviation %**

 **I0 = 0.34 W/cm2**

 **I0 = 2 W/cm<sup>2</sup> I0 = 4.7 W/cm2**

of the Quasi-Linear Approximation of Finite Amplitude Acoustic Waves 93

**0**

**10**

**20**

**30**

**Relative deviation %**

Fig. 4. Relative deviation of the respectively analytical expression of the second harmonic and the approximated second harmonic compared to the numerical solution of the Burgers'

So, for a strongly dissipative liquid, equation (25) is a good approximation of the numerical solution of the Burgers' equation (figure 4b). But, the equivalence of equations (25) and (26) is not checked (figure 4b). Indeed, equation (26) is a good approximation of equation (25)

In the case of water, the relative deviation of the analytical expression of the second harmonic (equation (25)) in relation to the numerical solution of the Burgers' equation is

second harmonic is based on the analytical expression of the fundamental. As in the case of

solution of the Burgers' equation is observed, the deviation of the analytical expression of the second harmonic in relation to the numerical solution of the Burgers' equation becomes more significant. These deviations increase with (figure 4a). Moreover in this case,

Consequently, the preceding comments are also applicable for the analytical expression of

According to this study, all these obtained solutions are valid, since the measurement is made near the source; otherwise some assumptions must be taken into account in the analysis of the propagation of finite amplitude acoustic waves in liquids. In addition, the analytical expressions precision depends essentially on the Goldberg's number value.

Moreover, for a strongly dissipative medium, the analytical expressions of the fundamental and second harmonic (equations (24) and (25)) can constitute a good approximation of the

For a slightly dissipative medium, the analytical expressions established show discrepancies when compared to the numerical solution of the Burgers' equation. Indeed, equation (24) assumes that the differential variation of the fundamental component with respect to the spatial coordinate is only proportional to the product of the absorption coefficient and the

(figure 4a). In fact, the determination of the analytical expression of the

*l* is weak comparatively to unity and equations (25) and (26) are equivalent.

*l* is weak comparatively to unity.

a slightly dissipative medium a noticeable deviation between 1 *p* ( )

the approximated second harmonic (equation (26)) (figure 4a).

numerical solution of the Burgers' equation.

**40**

**50**

**0,0 0,2 0,4 0,6 0,8 1,0**

 **coordinate** 

and the numerical

**(b) : Glycerol I0**

 **I0**

 **I0**

 **Relative deviation of the analytical expression of the second harmonic compared to the numerical solution of the Burgers equation (eqn(27)) Relative deviation of the analytical expression of the approximated 2nd harmonic compared to the numerical solution of the Burgers equation (eqn(27))**

**= 0.2 W/cm<sup>2</sup>**

**= 2.5 W/cm<sup>2</sup>**

**= 4.5 W/cm<sup>2</sup>**

Fig. 2. Pressure amplitude /Pp 02 versus the coordinate.

Thus, for a strongly dissipative liquid, equation (24) can be considered as a good approximation of equation (23). In fact, in this case the Goldberg's number is lower than unity (Table 2); then dissipation becomes important and dominates nonlinear effects.

As for water, the relative deviation of the analytical expression of the fundamental component (equation (24)) in relation to the numerical solution of the Burgers' equation is about 12% at 1 (figure 3a). It should be noted that for water, the deviations increase with (figure 3a). Indeed, in this case nonlinear effects become important ( <sup>2</sup> *p* ( ) *z* much greater than 0 2 *P* ) and the analytical expression of the fundamental established with the quasi-linear approximation is not valid.

For glycerol, the relative deviation of the analytical expression of the second harmonic (equation (25)) in relation to the numerical solution of the Burgers' equation is much weaker than that resulting from equation (26) (figure 4b). As an example, for 0.1 the deviation obtained from equation (25) is lower than 1% , and that produced by equation (26) can reach 40% .

Fig. 3. Relative deviation of the analytical expression of the fundamental compared to the numerical solution of the Burgers' equation versus the coordinate.

**0,000**

Thus, for a strongly dissipative liquid, equation (24) can be considered as a good approximation of equation (23). In fact, in this case the Goldberg's number is lower than

As for water, the relative deviation of the analytical expression of the fundamental component (equation (24)) in relation to the numerical solution of the Burgers' equation is

with (figure 3a). Indeed, in this case nonlinear effects become important ( <sup>2</sup> *p* ( ) *z* much greater than 0 2 *P* ) and the analytical expression of the fundamental established with the

For glycerol, the relative deviation of the analytical expression of the second harmonic (equation (25)) in relation to the numerical solution of the Burgers' equation is much weaker

obtained from equation (25) is lower than 1% , and that produced by equation (26) can reach

**0**

Fig. 3. Relative deviation of the analytical expression of the fundamental compared to the

numerical solution of the Burgers' equation versus the coordinate.

**1**

**2**

**Relative deviation %**

**3**

 **I0 = 0.2 W/cm<sup>2</sup>**

 **I0 = 2.5 W/cm<sup>2</sup>**

 **I0 = 4.5 W /cm<sup>2</sup>**

**4**

**5**

than that resulting from equation (26) (figure 4b). As an example, for 0.1

(figure 3a). It should be noted that for water, the deviations increase

unity (Table 2); then dissipation becomes important and dominates nonlinear effects.

**0,004**

**0,008**

**0,012**

**Amplitude p2 /P0**

**0,016**

**0,020**

**0,0 0,2 0,4 0,6 0,8 1,0**

 **coordinate**  

**(b) : Glycerol**

 **Relative deviation of the analytical expression of the fundamental compared to the numerical solution of the Burgers equation (eqn(27))**

**0,0 0,2 0,4 0,6 0,8 1,0**

**coordinate** 

the deviation

**(b) : Glycerol Quasi-linear approximation of the approximated 2nd harmonic (eqn(26)) Quasi-linear approximation of the second harmonic (eqn(25)) Numerical solution of the Burgers equation (eqn(23))**

**= 0.2 W/cm2**

**= 2.5 W/cm2**

**= 4.5 W/cm2**

 **I0**

 **I0**

 **I0**

**0,0 0,2 0,4 0,6 0,8 1,0**

 **coordinate**  

Fig. 2. Pressure amplitude /Pp 02 versus the coordinate.

**(a) : Water Quasi-linear approximation of the approximated 2nd harmonic (eqn(26)) Quasi-linear approximation of the second harmonic (eqn(25)) Numerical solution of the Burgers equation (eqn(23))**

**= 0.34 W/cm<sup>2</sup>**

**= 2 W/cm2**

**= 4.7 W/cm2**

**0,0**

**0,1**

**0,2**

**Amplitude p2 /P0**

**0,3**

**0,4**

 **I0**

 **I0**

 **I0**

about 12% at 1

 **I0**

 **I0**

 **I0**

**= 0.34 W/cm<sup>2</sup>**

**= 2 W/cm<sup>2</sup>**

**= 4.7 W /cm<sup>2</sup>**

40% .

**Relative deviation %**

quasi-linear approximation is not valid.

**0,0 0,2 0,4 0,6 0,8 1,0**

**coordinate**

**(a) : Water Relative deviation of the analytical expression of the fundamental compared to the numerical solution of the Burgers equation (eqn(27))**

**0,5**

Fig. 4. Relative deviation of the respectively analytical expression of the second harmonic and the approximated second harmonic compared to the numerical solution of the Burgers' equation versus the coordinate.

So, for a strongly dissipative liquid, equation (25) is a good approximation of the numerical solution of the Burgers' equation (figure 4b). But, the equivalence of equations (25) and (26) is not checked (figure 4b). Indeed, equation (26) is a good approximation of equation (25) only if 2 1 ( 2) *<sup>s</sup> l* is weak comparatively to unity.

In the case of water, the relative deviation of the analytical expression of the second harmonic (equation (25)) in relation to the numerical solution of the Burgers' equation is about 40% at 1 (figure 4a). In fact, the determination of the analytical expression of the second harmonic is based on the analytical expression of the fundamental. As in the case of a slightly dissipative medium a noticeable deviation between 1 *p* ( ) and the numerical solution of the Burgers' equation is observed, the deviation of the analytical expression of the second harmonic in relation to the numerical solution of the Burgers' equation becomes more significant. These deviations increase with (figure 4a). Moreover in this case, 2 1 ( 2) *<sup>s</sup> l* is weak comparatively to unity and equations (25) and (26) are equivalent. Consequently, the preceding comments are also applicable for the analytical expression of the approximated second harmonic (equation (26)) (figure 4a).

According to this study, all these obtained solutions are valid, since the measurement is made near the source; otherwise some assumptions must be taken into account in the analysis of the propagation of finite amplitude acoustic waves in liquids. In addition, the analytical expressions precision depends essentially on the Goldberg's number value.

Moreover, for a strongly dissipative medium, the analytical expressions of the fundamental and second harmonic (equations (24) and (25)) can constitute a good approximation of the numerical solution of the Burgers' equation.

For a slightly dissipative medium, the analytical expressions established show discrepancies when compared to the numerical solution of the Burgers' equation. Indeed, equation (24) assumes that the differential variation of the fundamental component with respect to the spatial coordinate is only proportional to the product of the absorption coefficient and the

Goldberg's Number Influence on the Validity Domain

*Biology*, Vol. 30, pp. 469-476

*America*, Vol. 86, pp. 1560-1565

*Acoustics*, Vol. 3, pp. 340-347

1525-1531

1821-1828

*Acoustical Society of America*, Vol. 75, pp. 749-768

*the Acoustical Society of America*, Vol. 106, pp. 1126-1133

*Acoustical Society of America*, Vol. 50, pp. 1299-1312

*the Acoustical Society of America*, Vol. 86, pp. 1-5

Hamilton, M. F., & Blackstock, D. T. (1988). On the coefficient of nonlinearity

*Journal of the Acoustical Society of America*, Vol. 73, pp. 774-779

nonlinearity parameter B/A. *Ultrasonics*, Vol. 38, pp. 292-296

*International Congress on Ultrasonics*, Vienna, April 9 – 13, 2007

Academic Publishers, ISBN 1-4020-0572-5, the Netherlands

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Aanonsen, S. I., Barkve, T., TjØtta, J. N., & TjØtta, S. (1984). Distortion and harmonic

Bouakaz, A., Merks, E., Lancée, C., & Bom, N. (2004). Noninvasive bladder volume

Cobb, W. N. (1983). Finite amplitude method for the determination of the acoustic

D'hooge, J., Bijnens, B., Nuyts, J., Gorce, J. M., Friboulet, D., Thoen, J., Van de Werf, F., &

Enflo, B. O., & Hedberg, C. M. (2002). *Theory of Nonlinear Acoustics in Fluids*, Kluwer

Fenlon, F. H. (1971). A recursive procedure for computing the nonlinear spectral interactions

Germain, L., Jacques, R., & Cheeke, J. D. N. (1989). Acoustics microscopy applied to

Goldberg, Z. A. (1957). On the propagation of plane waves of finite amplitude. *Soviet Physics* 

Gong, X. F., Zhu, Z. M., Shi, T., & Huang, J. H. (1989). Determination of the acoustic

Hamilton, M. F., TjØtta, J. N., & TjØtta, S. (1985). Nonlinear effects in the farfield of a directive sound source. *Journal of the Acoustical Society of America*, Vol. 78, pp. 202-216

Hedberg, C. (1994). Nonlinear propagation through a fluid of waves originating from a

Hedberg, C. (1999). Multifrequency plane, nonlinear, and dissipative waves at arbitrary distances. *Journal of the Acoustical Society of America*, Vol. 106, pp. 3150-3155 Labat, V., Remenieras, J. P., Bou Matar, O., Ouahabi, A., & Patat, F. (2000). Harmonic

Khelladi H., Plantier F., Daridon J. L., & Djelouah H. (2007). An experimental Study about

biharmonic sound source. *Journal of the Acoustical Society of America*, Vol. 96. pp.

propagation of finite amplitude sound beams: experimental determination of the

the Combined Effects of the Temperature and the Static Pressure on the Nonlinearity Parameter B/A in a Weakly Dissipative Liquid, *Proceedings* 

acoustics. *Journal of the Acoustical Society of America*, Vol. 83, pp. 74-77 Haran, M. E., & Cook, B. D. (1983). Distortion of finite amplitude ultrasound in lossy media.

BjØrnØ, L. (2002). Forty years of nonlinear ultrasound. *Ultrasonics*, Vol. 40, pp. 11-17

generation in the nearfield of a finite amplitude sound beam. *Journal of the* 

measurements based on nonlinear wave distortion. *Ultrasound in Medicine and* 

nonlinearity parameter B/A. *Journal of the Acoustical Society of America*, Vol. 73, pp.

Suetens, P. (1999). Nonlinear Propagation effects on broadband attenuation measurements and its implications for ultrasonic tissue characterization. *Journal of* 

of progressive finite-amplitude waves in nondispersive fluids. *Journal of the* 

nonlinear characterization of biological media. *Journal of the Acoustical Society of* 

nonlinearity parameter in biological media using FAIS and ITD methods. *Journal of* 

in nonlinear

acoustic pressure of the fundamental ( <sup>1</sup> 1 1 ( ) ( ) *p z <sup>p</sup> <sup>z</sup> z* ). This hypothesis is not always checked (equation (16)).

As mentioned at the beginning of this chapter, the performance of the simplified model (equation (26)) is interesting, as it can provide a simple, useful model for understanding phenomena in diagnostic imaging. In fact, tissue harmonic imaging offers several unique advantages over conventional imaging. The greater clarity, contrast and details of the harmonic images are evident and have been quantitatively verified, like the ability to identify suspected cysts... Despite the significant advantages offered by harmonic imaging, theory has been only partially explained. According to the theoretical development established in this chapter, equation (26) is valid only if 2 0 *p P* () 2 and 2 1 ( 2) 1 *<sup>s</sup> l* . Not taking into account these assumptions can generate erroneous numerical results.

On the other hand, as the finite amplitude method is based on pressure measurements of the finite amplitude wave distortion during its propagation, the analytical expressions of the fundamental (equation (24)), the second harmonic (equation (25)) and the approximated second harmonic (equation (26)) lead also to the measurement of the acoustic nonlinearity parameter . However, this method necessitates an accurate model taking into account diffraction effects (Labat et al., 2000; Gong et al., 1989; Zhang et al., 1991). The omission of this phenomenon can explain the discrepancies observed of the nonlinearity parameter values measured by the finite amplitude method compared to those achieved by the thermodynamic method (Law et al. 1983; Plantier et al., 2002; Sehgal et al., 1984; Zhang & Dunn, 1991). The latter is potentially very accurate. The major advantage of the thermodynamic method is that it does not depend on the characteristics of the acoustic field (Khelladi et al., 2007, 2009).

## **4. Conclusion**

The validity domain of the fundamental and the second harmonic analytical expressions established with the quasi-linear approximation can be preset only on the derivation of the Goldberg's number, which can be considered as a reliable indicator for any analysis incorporating nonlinear effects and dissipation.

The obtained numerical results illustrate that the analytical expressions of the fundamental and the second harmonic established with the quasi-linear approximation provide a good approximation of the numerical solution of the Burgers' equation for a propagation medium characterized by a Goldberg number that is small compared to unity.

In the other hand, for a propagation medium characterized by a Goldberg number greater than unity, the analytical expressions of the fundamental and the second harmonic already established with the quasi-linear approximation are not checked and must be redefined.

For that purpose, future studies will concentrate on a new mathematical formulation of the fundamental and second harmonic for a propagation medium characterized by a Goldberg number that is large compared to unity.

## **5. References**

94 Ultrasonic Waves

*z*

established in this chapter, equation (26) is valid only if 2 0 *p P* () 2

On the other hand, as the finite amplitude method is based on pressure measurements of the finite amplitude wave distortion during its propagation, the analytical expressions of the fundamental (equation (24)), the second harmonic (equation (25)) and the approximated second harmonic (equation (26)) lead also to the measurement of the acoustic nonlinearity parameter . However, this method necessitates an accurate model taking into account diffraction effects (Labat et al., 2000; Gong et al., 1989; Zhang et al., 1991). The omission of this phenomenon can explain the discrepancies observed of the nonlinearity parameter values measured by the finite amplitude method compared to those achieved by the thermodynamic method (Law et al. 1983; Plantier et al., 2002; Sehgal et al., 1984; Zhang & Dunn, 1991). The latter is potentially very accurate. The major advantage of the thermodynamic method is that it does not depend on the characteristics of the acoustic field

The validity domain of the fundamental and the second harmonic analytical expressions established with the quasi-linear approximation can be preset only on the derivation of the Goldberg's number, which can be considered as a reliable indicator for any analysis

The obtained numerical results illustrate that the analytical expressions of the fundamental and the second harmonic established with the quasi-linear approximation provide a good approximation of the numerical solution of the Burgers' equation for a propagation medium

In the other hand, for a propagation medium characterized by a Goldberg number greater than unity, the analytical expressions of the fundamental and the second harmonic already established with the quasi-linear approximation are not checked and must be redefined.

For that purpose, future studies will concentrate on a new mathematical formulation of the fundamental and second harmonic for a propagation medium characterized by a Goldberg

As mentioned at the beginning of this chapter, the performance of the simplified model (equation (26)) is interesting, as it can provide a simple, useful model for understanding phenomena in diagnostic imaging. In fact, tissue harmonic imaging offers several unique advantages over conventional imaging. The greater clarity, contrast and details of the harmonic images are evident and have been quantitatively verified, like the ability to identify suspected cysts... Despite the significant advantages offered by harmonic imaging, theory has been only partially explained. According to the theoretical development

1 1 ( ) ( ) *p z <sup>p</sup> <sup>z</sup>*

). This hypothesis is not always

and

*l* . Not taking into account these assumptions can generate erroneous

acoustic pressure of the fundamental ( <sup>1</sup>

checked (equation (16)).

2 1 ( 2) 1 *<sup>s</sup>*

numerical results.

(Khelladi et al., 2007, 2009).

incorporating nonlinear effects and dissipation.

number that is large compared to unity.

characterized by a Goldberg number that is small compared to unity.

**4. Conclusion** 

 


**5** 

*China* 

Gaowei Hu and Yuguang Ye *Qingdao Institute of Marine Geology* 

**Ultrasonic Waves on Gas Hydrates Experiments** 

In this chapter, the acoustic properties of gas hydrate-bearing sediments are investigated experimentally. The flat-plate transducers and a new kind of bender elements are developed to measure both compressional wave velocity (Vp) and shear wave velocity (Vs) of hydrated consolidated sediments and hydrated unconsolidated sediments, respectively. The main purpose is to construct a relation between gas hydrate saturation and acoustic velocities of the hydrate-bearing sediments, with which we can give suggestions on the usage of various

Gas hydrates, or clathrates, are ice-like crystalline solids composed of water molecules surrounding gas molecules (usually methane) under certain pressure and temperature conditions [Sloan, 1998]. In recent years, gas hydrates have been widely studied because of their potential as a future energy resource [Kvenvolden, 1998; Milkov and Sassen, 2003], their important role in the global carbon cycle and global warming [Dickens, 2003; Dickens, 2004], and their potential as a geotechnical hazard [Brown et al., 2006; Pecher et al., 2008]. To assess the impact of gas hydrates within these areas of interest, an understanding of their distribution within the seabed and their relationship with the host sediment is essential and helpful.

Seismic techniques have been widely used for mapping and quantifying gas hydrates in oceanic sediments [Shipley et al., 1979; Holbrook et al., 1996; Carcione and Gei, 2004]. In general, gas hydrates exhibit relatively high elastic velocities (both Vp and Vs), compared to the pore-filling fluids; therefore, the velocity of gas hydrate-bearing sediments is usually elevated [Stoll, 1974; Tucholke et al., 1977]. To quantify the amount of gas hydrate or to infer the physical properties of gas hydrate-bearing sediments, an understanding of the relationship between the amount of gas hydrate in the pore space of sediments and the

Two different approaches were used to relate the hydrate saturation and velocity in oceanic sediments: (1) empirical methods including Wyllie's time average [Wyllie et al., 1958; Pearson et al., 1983], Wood's equation [Wood, 1941] and weighted combinations of the Wyllie's time average and Wood's equation [Lee et al., 1996], and (2) physics-based models, such as the effective medium theory (EMT) [Helgerud et al., 1999; Dvorkin and Prasad, 1999] and the Biot-Gassmann theory modified by Lee (BGTL) [Lee, 2002a, 2002b, 2003]. However, the gas hydrate volumes within sediments estimated with these approaches are quite different. Chand et al. [2004] made a comparison of four current models, i.e., the WE (Weighted Equation) [Lee et al., 1996], the EMT, the three-phase Biot theory (TPB) [Carcione

**1. Introduction** 

elastic velocities is needed.

velocity-models in field gas hydrate explorations.


## **Ultrasonic Waves on Gas Hydrates Experiments**

Gaowei Hu and Yuguang Ye

*Qingdao Institute of Marine Geology China* 

## **1. Introduction**

96 Ultrasonic Waves

Khelladi H., Plantier F., Daridon J. L., & Djelouah H. (2009). Measurement under High

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Ngoc, D. K., King, K. R., & Mayer W. G. (1987). A numerical model for nonlinear and

Nightingale, K. R., Kornguth, P. J., & Trahey, G. E. (1999). The use of acoustic streaming in

Plantier, F., Daridon, J. L., Lagourette, B. ( 2002). Measurement of the B/A nonlinearity

Sehgal, C. M., & Greenleaf, J. F. (1982). Ultrasonic absorption and dispersion in biological

Sehgal, C. M., Bahn, R. C., Greenleaf, J. F. (1984). Measurement of the acoustic nonlinearity

Smith, M. C., & Beyer, R. T. (1948). Ultrasonic absorption in water in the temperature range 0°-80°C. *Journal of the Acoustical Society of America*, Vol. 20, pp. 608-610 Thuras, A. L., Jenkins, R. T., & O'Neil, H. T. (1935). Extraneous frequency generated in air

Trivett, D. H., & Van Buren, A. L. (1981). Propagation of plane, cylindrical and spherical finite amplitude waves. *Journal of the Acoustical Society of America*, Vol. 69, pp. 943-949 Willard, G. W. (1941). Ultrasonic absorption and velocity measurement in numerous liquids.

Woodsum, H. C. (1981). Author's reply. *Journal of Sound and Vibration*, Vol. 76, pp. 297-298 Zhang, J., Dunn, F. (1991). A small volume thermodynamic system for B/A measurement.

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In this chapter, the acoustic properties of gas hydrate-bearing sediments are investigated experimentally. The flat-plate transducers and a new kind of bender elements are developed to measure both compressional wave velocity (Vp) and shear wave velocity (Vs) of hydrated consolidated sediments and hydrated unconsolidated sediments, respectively. The main purpose is to construct a relation between gas hydrate saturation and acoustic velocities of the hydrate-bearing sediments, with which we can give suggestions on the usage of various velocity-models in field gas hydrate explorations.

Gas hydrates, or clathrates, are ice-like crystalline solids composed of water molecules surrounding gas molecules (usually methane) under certain pressure and temperature conditions [Sloan, 1998]. In recent years, gas hydrates have been widely studied because of their potential as a future energy resource [Kvenvolden, 1998; Milkov and Sassen, 2003], their important role in the global carbon cycle and global warming [Dickens, 2003; Dickens, 2004], and their potential as a geotechnical hazard [Brown et al., 2006; Pecher et al., 2008]. To assess the impact of gas hydrates within these areas of interest, an understanding of their distribution within the seabed and their relationship with the host sediment is essential and helpful.

Seismic techniques have been widely used for mapping and quantifying gas hydrates in oceanic sediments [Shipley et al., 1979; Holbrook et al., 1996; Carcione and Gei, 2004]. In general, gas hydrates exhibit relatively high elastic velocities (both Vp and Vs), compared to the pore-filling fluids; therefore, the velocity of gas hydrate-bearing sediments is usually elevated [Stoll, 1974; Tucholke et al., 1977]. To quantify the amount of gas hydrate or to infer the physical properties of gas hydrate-bearing sediments, an understanding of the relationship between the amount of gas hydrate in the pore space of sediments and the elastic velocities is needed.

Two different approaches were used to relate the hydrate saturation and velocity in oceanic sediments: (1) empirical methods including Wyllie's time average [Wyllie et al., 1958; Pearson et al., 1983], Wood's equation [Wood, 1941] and weighted combinations of the Wyllie's time average and Wood's equation [Lee et al., 1996], and (2) physics-based models, such as the effective medium theory (EMT) [Helgerud et al., 1999; Dvorkin and Prasad, 1999] and the Biot-Gassmann theory modified by Lee (BGTL) [Lee, 2002a, 2002b, 2003]. However, the gas hydrate volumes within sediments estimated with these approaches are quite different. Chand et al. [2004] made a comparison of four current models, i.e., the WE (Weighted Equation) [Lee et al., 1996], the EMT, the three-phase Biot theory (TPB) [Carcione

Ultrasonic Waves on Gas Hydrates Experiments 89

L/(t1 - t0) and Vs = L/(t2 - t0), where L is the sample length and t0 is the inherent travel time of the transducers. Two different lengths of standardized cylindrical aluminum rods were used to calibrate the t0 of the transducers. The final errors in estimating compressional wave velocity and shear wave velocity were ±1.2% (±50 m s-1) and ±1.6% (±40 m s-1), respectively. The

Fig. 1. Cross section through the high-pressure vessel. A single probe and a stainless steel

Fig. 2. Ultrasonic waveform measured by flat-plate tranducers. Arrival times of the first arrival wave of compressional and shear waves are t1 and t2, respectively. Amplitude 1 and amplitude 2 represent the amplitudes of compressional and shear waves, respectively.

TDR was initially used for detecting the position of breaks in transmission line cables. The technique was introduced to measure water contents of soil samples in 1980s, and then it developed rapidly [Topp et al., 1980; Dalton et al., 1986]. Topp et al. [1980] was probably the first one who measured water contents of soil samples with TDR technique. They found a practical relation between dielectric constants and water contents of the soil samples based

**2.1.2 TDR technique** 

circle around the cylindrical core are two poles of the TDR coaxial probe.

amplitudes of compressional and shear waves can be read directly (Fig. 2).

and Tinivella, 2000; Gei and Carcione, 2003] and the differential effective medium theory (DEM) [Jakobsen et al., 2000], in predicting hydrate saturation with field data sets obtained from ODP Leg 164 on Blake Ridge, and from the Mallik 2L-38 well, Mackenzie Delta, Canada. The results show that three of the models predict consistent hydrate saturation of 60-80% for the Mallik 2L-38 well, but the EMT model predicts 20 per cent higher. For the clay-rich sediments of Blake Ridge, the DEM, EMT and WE models predict 10-20% hydrate saturation, which is similar to the result inferred from resistivity data, but lower than the result predicted by the TPB model. Ojha and Sain [2008] estimated the saturation of gas hydrate at Makran accretionary prism using the BGTL and EMT models. The BGTL model shows hydrate saturation of 7-9%, but the EMT model predicts the saturation of gas hydrate as 14-33%. Apparently, it is important to validate these elastic velocity models with data obtained from synthesized hydrate-bearing sediment in the laboratory.

The relationship between acoustic velocities and hydrate saturation has been studied in laboratory by several researchers. Although the experimental results may not automatically be applied to the seismic frequencies of field data, however, as the general variability of acoustic velocities with the variable pore fluid is similar with that of seismic velocities [Sothcott et al., 2000], the experiments can provide some basic geophysical parameters for the exploration of gas hydrate reservoirs.

## **2. Ultrasonic waves in hydrate-bearing consolidated sediments**

In this section, acoustic properties of gas hydrate-bearing consolidated sediments are investigated experimentally. Gas hydrate was formed and subsequently dissociated in consolidated sediments. In the whole process, ultrasonic methods and Time Domain Reflectometry (TDR) are simultaneously used to measure the acoustic properties and hydrate saturations of the host sediments, respectively. The whole experimental processes and results are presented here.

#### **2.1 Methods**

It's needed to keep a suitable circumstance, e.g. a certain high pressure and low temperature, for measuring parameters of hydrate-bearing sediments. Thus, ultrasonic method and TDR technique used in this study are a little different from their conventional ways because they must sustain pressure during the measurements. A characteristic of our methods is that real-time measurements of both hydrate saturation and acoustic velocities were conducted in one system.

#### **2.1.1 Ultrasonic method**

P wave and S wave velocities were measured by transmission using two transducers (0.5 MHz frequency) placed at each end of the cylindrical core (Fig. 1). Signals were digitized by a CompuScope card from GaGe Applied Technologies. Because the CompuScope 14100 is a 14 bit, 50 million samples per second dual-channel waveform digitizer card and data transfer rates from the Compu-Scope memory to PC memory run as high as 80 Mb s-1, it is thought that the CompuScope card caused few errors in velocity estimation. However, the errors in velocity estimation resulted mainly from picking t1 and t2, which are the travel times of the compressional and shear waves, respectively (Fig. 2). The velocities were calculated by Vp =

and Tinivella, 2000; Gei and Carcione, 2003] and the differential effective medium theory (DEM) [Jakobsen et al., 2000], in predicting hydrate saturation with field data sets obtained from ODP Leg 164 on Blake Ridge, and from the Mallik 2L-38 well, Mackenzie Delta, Canada. The results show that three of the models predict consistent hydrate saturation of 60-80% for the Mallik 2L-38 well, but the EMT model predicts 20 per cent higher. For the clay-rich sediments of Blake Ridge, the DEM, EMT and WE models predict 10-20% hydrate saturation, which is similar to the result inferred from resistivity data, but lower than the result predicted by the TPB model. Ojha and Sain [2008] estimated the saturation of gas hydrate at Makran accretionary prism using the BGTL and EMT models. The BGTL model shows hydrate saturation of 7-9%, but the EMT model predicts the saturation of gas hydrate as 14-33%. Apparently, it is important to validate these elastic velocity models with data

The relationship between acoustic velocities and hydrate saturation has been studied in laboratory by several researchers. Although the experimental results may not automatically be applied to the seismic frequencies of field data, however, as the general variability of acoustic velocities with the variable pore fluid is similar with that of seismic velocities [Sothcott et al., 2000], the experiments can provide some basic geophysical parameters for

In this section, acoustic properties of gas hydrate-bearing consolidated sediments are investigated experimentally. Gas hydrate was formed and subsequently dissociated in consolidated sediments. In the whole process, ultrasonic methods and Time Domain Reflectometry (TDR) are simultaneously used to measure the acoustic properties and hydrate saturations of the host sediments, respectively. The whole experimental processes

It's needed to keep a suitable circumstance, e.g. a certain high pressure and low temperature, for measuring parameters of hydrate-bearing sediments. Thus, ultrasonic method and TDR technique used in this study are a little different from their conventional ways because they must sustain pressure during the measurements. A characteristic of our methods is that real-time measurements of both hydrate saturation and acoustic velocities

P wave and S wave velocities were measured by transmission using two transducers (0.5 MHz frequency) placed at each end of the cylindrical core (Fig. 1). Signals were digitized by a CompuScope card from GaGe Applied Technologies. Because the CompuScope 14100 is a 14 bit, 50 million samples per second dual-channel waveform digitizer card and data transfer rates from the Compu-Scope memory to PC memory run as high as 80 Mb s-1, it is thought that the CompuScope card caused few errors in velocity estimation. However, the errors in velocity estimation resulted mainly from picking t1 and t2, which are the travel times of the compressional and shear waves, respectively (Fig. 2). The velocities were calculated by Vp =

obtained from synthesized hydrate-bearing sediment in the laboratory.

**2. Ultrasonic waves in hydrate-bearing consolidated sediments** 

the exploration of gas hydrate reservoirs.

and results are presented here.

were conducted in one system.

**2.1.1 Ultrasonic method** 

**2.1 Methods** 

L/(t1 - t0) and Vs = L/(t2 - t0), where L is the sample length and t0 is the inherent travel time of the transducers. Two different lengths of standardized cylindrical aluminum rods were used to calibrate the t0 of the transducers. The final errors in estimating compressional wave velocity and shear wave velocity were ±1.2% (±50 m s-1) and ±1.6% (±40 m s-1), respectively. The amplitudes of compressional and shear waves can be read directly (Fig. 2).

Fig. 1. Cross section through the high-pressure vessel. A single probe and a stainless steel circle around the cylindrical core are two poles of the TDR coaxial probe.

Fig. 2. Ultrasonic waveform measured by flat-plate tranducers. Arrival times of the first arrival wave of compressional and shear waves are t1 and t2, respectively. Amplitude 1 and amplitude 2 represent the amplitudes of compressional and shear waves, respectively.

### **2.1.2 TDR technique**

TDR was initially used for detecting the position of breaks in transmission line cables. The technique was introduced to measure water contents of soil samples in 1980s, and then it developed rapidly [Topp et al., 1980; Dalton et al., 1986]. Topp et al. [1980] was probably the first one who measured water contents of soil samples with TDR technique. They found a practical relation between dielectric constants and water contents of the soil samples based

Ultrasonic Waves on Gas Hydrates Experiments 91

As a result, the hydrate pore saturation of the hydrated sediments can be calculated with:

Sh=(-θv)/ 100% (6)

Gas hydrates could be formed in sediments by at least four different methods in many types of apparatus. Gas can be introduced to specimens containing (1) partially water-saturated sediment [Waite et al., 2004], (2) water-saturated sediment [Winters et al., 2007], (3) seed icesediment [Priest et al., 2005] and (4) continually feeding gas-saturated water into the specimen, which is rarely used because of much difficult and time-consuming. The "gas + watersaturated sediment" system was used to synthesize hydrate-bearing sediments in this study. The geophysical experimental apparatus in the Gas Hydrate Laboratory of Qingdao Institute of Marine Geology (GHL-QIMG) [Ye et al., 2005, 2008; Hu et al., 2010] can simulate in situ pressure and temperature conditions conducive to hydrate formation (Fig. 4). The apparatus is composed of five functioning units: (1) A high-pressure vessel with a plastic inner barrel for simulating in situ pressure and temperature, in which there are two platinum (Pt100) resistance thermometers with precision of ±0.1°C used for measuring the temperature of inner and surface of the sample. (2) a vessel used for making gas-saturated water, (3) a gas compressor and a pressure transducer (precision, ±0.1MPa) responsible for gas pressure control, (4) a cooling system and a bathing through for temperature control ,

v 11.9677 4.506072566Ka 0.14615Ka 0.0021399Ka (5)

Where is the porosity of the sample.

**2.1.3 Synthesized method of hydrated sediments** 

and (5) a computer system for measuring and logging data.

Magnetic stirrer

T P

TDR

T Acoustic

P

Computer System

Bathing trough Sediment specimen Pressure vessel

hydrate-bearing sediments.

Pressure vessel for saturated water

2 3

Pt100

Transducer 1

(a) (b)

Metallic Mandril

TDR Coaxial Probe

**Water-saturated core** 

Free gas

Transducer 2 Pressure vessel

Cooling system

Gas compressor

Valve

Fig. 4. Schematic diagram of experimental apparatus for geophysical research on gas

The experimental processes are as follows: (1) the artificial core was immersed by pure water or 300ppm SDS solution to gravimetric water content of about 40%, and then loaded

Gas

Inner barrel

Stainless steel circle

on various types of experimental results. With regard to some particular substance, special calibration is needed before measurements. For example, Regalado et al. [2003] proposed a empirical equation for calculating water contents of volcanic soils; Wright et al. [2002] found the relationship between dielectric constants and water contents of hydrate-bearing sediments. Thereafter, TDR was effectively used to measure hydrate pore saturations of the hydrated sediments [Ye et al., 2008; Hu et al., 2010].

Fig. 3. TDR waveform of low-salty sediments with traditional TDR probe

TDR waveform of a soil or sediment sample is shown in Fig 3. The electromagnetic wave is generated by TDR instruments, and transmitted along the coaxial cable and the TDR probe (Fig 1 & Fig 3). Because there is loss current during electromagnetic wave transmitting in the samples, the characteristics of the entry point and the end point are obviously. The velocity of electromagnetic wave transmitting in the samples can be calculated with:

$$\mathbf{V} = \mathbf{1}/\mathbf{t} \tag{1}$$

Where l is the length of the TDR probe, t is the time-interval between entry point and end point (Fig 3). At the same time, the velocity of electromagnetic wave in the samples can be also related to dielectric constants:

$$\mathbf{V} = \mathbf{c} / \,\mathrm{K}\mathbf{a}^{1/2} \tag{2}$$

Where c is the velocity of propagation in free space (approximately 3×108 m/s). From equation 1 and 2 it solves:

$$\mathbf{Ka} = \begin{bmatrix} \mathbf{ct} \end{bmatrix}^2 \tag{3}$$

When the Ka is calculated, with the relationship between Ka and water contents we can obtain the water contents of the sample. For soils [Topp et al.,1980]:

$$
\theta \text{v} = -5.3 \times 10^{-2} + 2.92 \times 10^{-2} \text{K} \text{a} - 5.5 \times 10^{-4} \text{K} \text{a}^2 + 4.3 \times 10^{-6} \text{K} \text{a}^3 \tag{4}
$$

For hydrate-bearing sediments, it's effective to use Wright et al. [2002] 's empirical equation:

on various types of experimental results. With regard to some particular substance, special calibration is needed before measurements. For example, Regalado et al. [2003] proposed a empirical equation for calculating water contents of volcanic soils; Wright et al. [2002] found the relationship between dielectric constants and water contents of hydrate-bearing sediments. Thereafter, TDR was effectively used to measure hydrate pore saturations of the

hydrated sediments [Ye et al., 2008; Hu et al., 2010].

Fig. 3. TDR waveform of low-salty sediments with traditional TDR probe

of electromagnetic wave transmitting in the samples can be calculated with:

obtain the water contents of the sample. For soils [Topp et al.,1980]:

also related to dielectric constants:

equation 1 and 2 it solves:

TDR waveform of a soil or sediment sample is shown in Fig 3. The electromagnetic wave is generated by TDR instruments, and transmitted along the coaxial cable and the TDR probe (Fig 1 & Fig 3). Because there is loss current during electromagnetic wave transmitting in the samples, the characteristics of the entry point and the end point are obviously. The velocity

Where l is the length of the TDR probe, t is the time-interval between entry point and end point (Fig 3). At the same time, the velocity of electromagnetic wave in the samples can be

Where c is the velocity of propagation in free space (approximately 3×108 m/s). From

When the Ka is calculated, with the relationship between Ka and water contents we can

For hydrate-bearing sediments, it's effective to use Wright et al. [2002] 's empirical equation:

22 4 2 6 3

v 5.3 10 2.92 10 Ka 5.5 10 Ka 4.3 10 Ka (4)

V l/t (1)

1/2 V c /Ka (2)

<sup>2</sup> Ka ct /l (3)

$$1.\theta\text{v} = -11.9677 + 4.506072566\text{Ka} - 0.14615\text{Ka}^2 + 0.0021399\text{Ka}^3\tag{5}$$

As a result, the hydrate pore saturation of the hydrated sediments can be calculated with:

$$\text{Sh} \equiv (\text{q-}\theta\text{v}) / \text{ q} \times 100\text{°} \tag{6}$$

Where is the porosity of the sample.

## **2.1.3 Synthesized method of hydrated sediments**

Gas hydrates could be formed in sediments by at least four different methods in many types of apparatus. Gas can be introduced to specimens containing (1) partially water-saturated sediment [Waite et al., 2004], (2) water-saturated sediment [Winters et al., 2007], (3) seed icesediment [Priest et al., 2005] and (4) continually feeding gas-saturated water into the specimen, which is rarely used because of much difficult and time-consuming. The "gas + watersaturated sediment" system was used to synthesize hydrate-bearing sediments in this study.

The geophysical experimental apparatus in the Gas Hydrate Laboratory of Qingdao Institute of Marine Geology (GHL-QIMG) [Ye et al., 2005, 2008; Hu et al., 2010] can simulate in situ pressure and temperature conditions conducive to hydrate formation (Fig. 4). The apparatus is composed of five functioning units: (1) A high-pressure vessel with a plastic inner barrel for simulating in situ pressure and temperature, in which there are two platinum (Pt100) resistance thermometers with precision of ±0.1°C used for measuring the temperature of inner and surface of the sample. (2) a vessel used for making gas-saturated water, (3) a gas compressor and a pressure transducer (precision, ±0.1MPa) responsible for gas pressure control, (4) a cooling system and a bathing through for temperature control , and (5) a computer system for measuring and logging data.

Fig. 4. Schematic diagram of experimental apparatus for geophysical research on gas hydrate-bearing sediments.

The experimental processes are as follows: (1) the artificial core was immersed by pure water or 300ppm SDS solution to gravimetric water content of about 40%, and then loaded

Ultrasonic Waves on Gas Hydrates Experiments 93

Figure 6 shows the changes of acoustic velocities (Vp, Vs) and hydrate saturations in hydrate-formation process. The Vp and Vs of the water-saturated sediment are 4242m/s and 2530m/s, respectively. During the first stage of hydrate-formation process, Vp changes hardly (time: 7.7h-~9.7h;Sh: 0-~20%). Later, it begins to increase and gets to 4643m/s when the hydrate saturation is up to ~65.5% (time: 12.75h). The Vs of the sediment decreases slightly to 2470m/s at the beginning of hydrate-formation process (time: 7.7h-8.95h; Sh: 0- ~10%). After that it begins to increase and gets to 2725m/s when the hydrate saturation is up to ~65.5%. Although the pressure-temperature condition was maintained 1~2d after hydrate saturation up to 65.5%, the hydrate saturation didn't increase. However, both Vp and Vs increase slightly in hydrate-maintained process. Vp and Vs increase to 4770m/s and 2770m/s, respectively. During the hydrate-dissociation process, Vp and Vs of the sediments decrease with the increasing water contents (decreasing hydrate saturation). And they decrease to 4250m/s and 2550m/s respectively when gas hydrate was completely

Fig. 6. Variation in hydrate saturation (Sh) and acoustic velocities (Vp & Vs) during gas hydrate formation. Vp changes very little during hydrate saturation 0~20%. Vs decrease

The compressional (or shear) wave velocity measured in the hydrate-dissociation process is much higher than that measured in the hydrate-formation process at the same saturation degree. It may be caused by the hydrate morphology. As indicated in Yoslim and Englezos [2008], when surfactant (SDS) is present in the system, porous hydrate is believed to form at the gas-water interface. In addition, hydrate formation may occur in two stages: first the formation of a water-hydrate slurry, and then a very slow solidification stage [Beltrán and Servio, 2008]. Thus, in the hydrate formation process, the hydrates are porous and soft; as time lapses, the hydrates become rigid solids and consequently lead to an increase of the

from 2530m/s to 2470m/s during hydrate saturation 0~10%.

**2.3 Relate hydrate saturation to acoustic velocities** 

**2.2.2 Acoustic properties** 

dissociated (Fig. 5).

into the high-pressure vessel; (2) Methane gas was introduced into the vessel to a scheduled pressure after vacuum, and more than 24 hours are allowed for methane dissolving into the fluid; (3) Temperature in the high pressure-vessel was controlled to ~2°C for hydrate formation; and (4) Temperature was increased naturally to room temperature for hydrate dissociation.

During the whole process of hydrate formation and dissociation, real-time measurements on temperature, pressure, the ultrasonic waveform, and the TDR waveform were recorded by the computer system.

### **2.2 Acoustic properties of hydrate-bearing consolidated sediments**

During gas hydrate formation and subsequent dissociation in the consolidated sediments, the acoustic properties of the sample were measured and the phenomenon is described below. Also, with the results we obtain the relationship between hydrate saturations and acoustic properties of the hydrate-bearing consolidated sediments.

#### **2.2.1 Hydrate formation and dissociation processes**

Methane gas was charged into the specimen until the pressure reached to ~5MPa. Then the temperature was decreased gradually and finally at 5°C hydrates began to form (Fig. 5). A temperature-anomaly could be detected in the sample due to the exothermic reaction when hydrate forms. At the same time, the decrease of water content and pressure also indicated that hydrates began to form. In order to get more hydrates, we kept the temperature-pressure condition of the high-pressure vessel for 1~2d. The hydrate saturations range from 0% to 65.5% (water content 40.18% to 13.85%). Gas hydrate dissociation was induced by increasing temperature and the hydrate-dissociation process typically lasts about 8~10 hours.

Fig. 5. Variation in temperature (T), pressure (P), water content and acoustic velocities (Vp&Vs) during gas hydrate formation and subsequent dissociation in the sediment core. Ta and Tb are temperatures of the inner and surface of the sediment core, respectively.

#### **2.2.2 Acoustic properties**

92 Ultrasonic Waves

into the high-pressure vessel; (2) Methane gas was introduced into the vessel to a scheduled pressure after vacuum, and more than 24 hours are allowed for methane dissolving into the fluid; (3) Temperature in the high pressure-vessel was controlled to ~2°C for hydrate formation; and (4) Temperature was increased naturally to room temperature for hydrate

During the whole process of hydrate formation and dissociation, real-time measurements on temperature, pressure, the ultrasonic waveform, and the TDR waveform were recorded by

During gas hydrate formation and subsequent dissociation in the consolidated sediments, the acoustic properties of the sample were measured and the phenomenon is described below. Also, with the results we obtain the relationship between hydrate saturations and

Methane gas was charged into the specimen until the pressure reached to ~5MPa. Then the temperature was decreased gradually and finally at 5°C hydrates began to form (Fig. 5). A temperature-anomaly could be detected in the sample due to the exothermic reaction when hydrate forms. At the same time, the decrease of water content and pressure also indicated that hydrates began to form. In order to get more hydrates, we kept the temperature-pressure condition of the high-pressure vessel for 1~2d. The hydrate saturations range from 0% to 65.5% (water content 40.18% to 13.85%). Gas hydrate dissociation was induced by increasing

temperature and the hydrate-dissociation process typically lasts about 8~10 hours.

Fig. 5. Variation in temperature (T), pressure (P), water content and acoustic velocities (Vp&Vs) during gas hydrate formation and subsequent dissociation in the sediment core. Ta

and Tb are temperatures of the inner and surface of the sediment core, respectively.

**2.2 Acoustic properties of hydrate-bearing consolidated sediments** 

acoustic properties of the hydrate-bearing consolidated sediments.

**2.2.1 Hydrate formation and dissociation processes** 

dissociation.

the computer system.

Figure 6 shows the changes of acoustic velocities (Vp, Vs) and hydrate saturations in hydrate-formation process. The Vp and Vs of the water-saturated sediment are 4242m/s and 2530m/s, respectively. During the first stage of hydrate-formation process, Vp changes hardly (time: 7.7h-~9.7h;Sh: 0-~20%). Later, it begins to increase and gets to 4643m/s when the hydrate saturation is up to ~65.5% (time: 12.75h). The Vs of the sediment decreases slightly to 2470m/s at the beginning of hydrate-formation process (time: 7.7h-8.95h; Sh: 0- ~10%). After that it begins to increase and gets to 2725m/s when the hydrate saturation is up to ~65.5%. Although the pressure-temperature condition was maintained 1~2d after hydrate saturation up to 65.5%, the hydrate saturation didn't increase. However, both Vp and Vs increase slightly in hydrate-maintained process. Vp and Vs increase to 4770m/s and 2770m/s, respectively. During the hydrate-dissociation process, Vp and Vs of the sediments decrease with the increasing water contents (decreasing hydrate saturation). And they decrease to 4250m/s and 2550m/s respectively when gas hydrate was completely dissociated (Fig. 5).

Fig. 6. Variation in hydrate saturation (Sh) and acoustic velocities (Vp & Vs) during gas hydrate formation. Vp changes very little during hydrate saturation 0~20%. Vs decrease from 2530m/s to 2470m/s during hydrate saturation 0~10%.

#### **2.3 Relate hydrate saturation to acoustic velocities**

The compressional (or shear) wave velocity measured in the hydrate-dissociation process is much higher than that measured in the hydrate-formation process at the same saturation degree. It may be caused by the hydrate morphology. As indicated in Yoslim and Englezos [2008], when surfactant (SDS) is present in the system, porous hydrate is believed to form at the gas-water interface. In addition, hydrate formation may occur in two stages: first the formation of a water-hydrate slurry, and then a very slow solidification stage [Beltrán and Servio, 2008]. Thus, in the hydrate formation process, the hydrates are porous and soft; as time lapses, the hydrates become rigid solids and consequently lead to an increase of the

Ultrasonic Waves on Gas Hydrates Experiments 95

A waveform of the hydrated sediments measured by bender elements is shown in Fig. 8. From the waveform, it's easy to read the arrival time of shear wave. However, the arrival time of compressional wave is hard to get because of the noise. Thus, we combine the FFT transformation and wavelet-transformation (we called FFT-WT method hereafter) to interpret the compressional wave and obtain the Vp data. Calibration has been made and

Bender elements are commonly used to measure shear wave velocity of unconsolidated sediments. In order to obtain both Vp and Vs of the hydrate-bearing unconsolidated sediments, a new kind of bender element transducers are developed. With the FFT-WT

Bender elements consist of two sheets of piezoceramic plates rigidly bonded to a center shim of brass or stainless steel plate (Fig. 9a). When the "cantilever beam" of the transducer is excited by an input voltage, it changes its shape and generates a mechanical excitation (Fig. 9b), and then the signal transmits to the receiver bender element. The in-plane directivity of bender elements was explored by Lee and Santamarina (2005). The results show that amplitude of the signal is more pronounced when the installations of bender elements are parallel (Fig. 10). The amplitude in the transverse configuration is about 75% of the amplitude at 0° in the parallel axes configuration, which suggest the potential use of bender elements in a wide range of in-plane configurations besides the standard tip-to-tip alignment. In order to

Fig. 9. (a) Schematic representation of bender element; (b) Mechanical excitation of bender

method, the new transducers are used successfully in this study.

obtain good signal, we use the parallel installations in our experiments.

**3.1.1 Preparation of bender element transducers** 

the method is considered to be correct.

**3.1 Bender elements technique** 

element.

velocity. Because it's difficult to judge whether in situ gas hydrates are in the process of formation or dissociation during gas hydrate exploration, we use the average Vp (or Vs) of the compressional (or shear) wave velocities obtained in the two processes as the measured velocity to relate with gas hydrate saturations in this paper (Fig. 7). The result shows that acoustic velocities are insensitive to low hydrate saturations (0-~10%). However, the velocities increase rapidly with hydrate saturation when saturation is higher than 10%, especially in the range of 10-30%.

Fig. 7. Variation in Vp during hydrate formation (Vp(form)) and hydrate dissociation (Vp(dis)), Vs during hydrate formation (Vs(form)) and hydrate dissociation (Vs(dis)), the average Vp of Vp(form) and Vp(dis), and the average Vs of Vs(form) and Vs(dis).

## **3. Ultrasonic waves in hydrate-bearing unconsolidated sediments**

The attenuation of ultrasonic wave in unconsolidated sediments is usually much higher than that in consolidated sediments. In order to obtain both Vp and Vs of the hydratebearing unconsolidated sediments, various techniques including bender elements, resonant column, etc, are developed to measure acoustic properties of the hydrated samples. In this section, the bender elements are successfully used in measuring both Vp and Vs of the hydrate-bearing unconsolidated sediments.

Fig. 8. Waveform of the hydrated unconsolidated sediments measured by bender elements

velocity. Because it's difficult to judge whether in situ gas hydrates are in the process of formation or dissociation during gas hydrate exploration, we use the average Vp (or Vs) of the compressional (or shear) wave velocities obtained in the two processes as the measured velocity to relate with gas hydrate saturations in this paper (Fig. 7). The result shows that acoustic velocities are insensitive to low hydrate saturations (0-~10%). However, the velocities increase rapidly with hydrate saturation when saturation is higher than 10%,

Fig. 7. Variation in Vp during hydrate formation (Vp(form)) and hydrate dissociation (Vp(dis)), Vs during hydrate formation (Vs(form)) and hydrate dissociation (Vs(dis)), the average Vp of Vp(form) and Vp(dis), and the average Vs of Vs(form) and Vs(dis).

The attenuation of ultrasonic wave in unconsolidated sediments is usually much higher than that in consolidated sediments. In order to obtain both Vp and Vs of the hydratebearing unconsolidated sediments, various techniques including bender elements, resonant column, etc, are developed to measure acoustic properties of the hydrated samples. In this section, the bender elements are successfully used in measuring both Vp and Vs of the

Fig. 8. Waveform of the hydrated unconsolidated sediments measured by bender elements

**3. Ultrasonic waves in hydrate-bearing unconsolidated sediments** 

hydrate-bearing unconsolidated sediments.

especially in the range of 10-30%.

A waveform of the hydrated sediments measured by bender elements is shown in Fig. 8. From the waveform, it's easy to read the arrival time of shear wave. However, the arrival time of compressional wave is hard to get because of the noise. Thus, we combine the FFT transformation and wavelet-transformation (we called FFT-WT method hereafter) to interpret the compressional wave and obtain the Vp data. Calibration has been made and the method is considered to be correct.

#### **3.1 Bender elements technique**

Bender elements are commonly used to measure shear wave velocity of unconsolidated sediments. In order to obtain both Vp and Vs of the hydrate-bearing unconsolidated sediments, a new kind of bender element transducers are developed. With the FFT-WT method, the new transducers are used successfully in this study.

#### **3.1.1 Preparation of bender element transducers**

Bender elements consist of two sheets of piezoceramic plates rigidly bonded to a center shim of brass or stainless steel plate (Fig. 9a). When the "cantilever beam" of the transducer is excited by an input voltage, it changes its shape and generates a mechanical excitation (Fig. 9b), and then the signal transmits to the receiver bender element. The in-plane directivity of bender elements was explored by Lee and Santamarina (2005). The results show that amplitude of the signal is more pronounced when the installations of bender elements are parallel (Fig. 10). The amplitude in the transverse configuration is about 75% of the amplitude at 0° in the parallel axes configuration, which suggest the potential use of bender elements in a wide range of in-plane configurations besides the standard tip-to-tip alignment. In order to obtain good signal, we use the parallel installations in our experiments.

Fig. 9. (a) Schematic representation of bender element; (b) Mechanical excitation of bender element.

Ultrasonic Waves on Gas Hydrates Experiments 97

vibration is occurred. At the same time, a small longitudinal movement is also occurred on the cantilever beam. In order to magnify the longitudinal movement, we add a longitudinal piezoelectric slice clinging to the bender elements. Therefore, there is also compressional wave in the integrative waveform (Fig. 8). From the waveform, it's easy to read the first arrival of shear wave. However, as the noise is largely, we develop the FFT-WT method to

Fig. 12. (a) Photography of the bender element transducers; (b) mechanical excitation of the

Compressional and shear wave velocities are calculated with: Vp=Ltt/(tp-t0p), Vs=Ltt/(ts-t0s), where Ltt is the tip-to-tip distance of two bender elements, tp and ts are travel times of compressional wave and shear wave in the sediments respectively, t0p and t0s are the measured travel times of compressional wave and shear wave in the bender element transducers respectively. Four different lengths of cylindrical Polyoxy-methylene (POM) columns were used to calibrate t0p and t0s of the bender elements (Table 1). The diameter of the POM columns are about 6cm, which is close to the diameter of samples (6.8cm). The waveform of the POM column is shown in Fig. 13. It shows that it's easy to read both arrival times of the P-wave and S-wave using the new type of bender elements. According to lengths and wave-arrival times of the four POM columns, we obtained the t0p and t0s, which are 7.017us and 18.63us respectively. And the P-wave and S-wave velocities of the POM material are 2294.5m/s and 933.9m/s respectively [Fig. 14], which is very close to the reported values [Choy et al., 1983]. The P-wave velocity is also close to the measured results

by our flat-plate transducers, which is 2280m/s for POM-I and 2319m/s for POM-II.

analysis the first arrival time of compressional wave.

new bender element.

**3.1.2 Calibration** 

Fig. 10. Source-receiver directivity: (a) test setup, and (b) polar plot of peak amplitudes [Lee and Santamarina, 2005]

Fig. 11. (a) Schematic map of the high pressure cell for hydrate formation and acoustic measurements; (b) Measuring system of bender elements

Simultaneous measurements of compressional and shear wave velocity of methane hydrate bearing sediments using bender elements are explored. The apparatus is shown in Fig. 11. In the acoustic measuring system (Fig. 11b), signal is generated, amplified and then transmitted by the source bender element. Because the mechanical excitation of bender element is transverse, the waveform received by the receiver bender element is mainly shear wave. In order to obtain both shear wave and compressional wave, a new kind of bender element is developed (Fig. 12).

To overcome a high pressure environment, the bender elements are filled with phenolic resin and protected by stainless steel shell. The mechanical excitation of the new bender element is shown in Fig 12b. The cantilever beam is driven by two piezoelectric circles. When the excitation is generated, the cantilever beam will be distorted and the torsional

(a) (b) Fig. 10. Source-receiver directivity: (a) test setup, and (b) polar plot of peak amplitudes [Lee

Fig. 11. (a) Schematic map of the high pressure cell for hydrate formation and acoustic

Simultaneous measurements of compressional and shear wave velocity of methane hydrate bearing sediments using bender elements are explored. The apparatus is shown in Fig. 11. In the acoustic measuring system (Fig. 11b), signal is generated, amplified and then transmitted by the source bender element. Because the mechanical excitation of bender element is transverse, the waveform received by the receiver bender element is mainly shear wave. In order to obtain both shear wave and compressional wave, a new kind of bender

To overcome a high pressure environment, the bender elements are filled with phenolic resin and protected by stainless steel shell. The mechanical excitation of the new bender element is shown in Fig 12b. The cantilever beam is driven by two piezoelectric circles. When the excitation is generated, the cantilever beam will be distorted and the torsional

measurements; (b) Measuring system of bender elements

element is developed (Fig. 12).

and Santamarina, 2005]

vibration is occurred. At the same time, a small longitudinal movement is also occurred on the cantilever beam. In order to magnify the longitudinal movement, we add a longitudinal piezoelectric slice clinging to the bender elements. Therefore, there is also compressional wave in the integrative waveform (Fig. 8). From the waveform, it's easy to read the first arrival of shear wave. However, as the noise is largely, we develop the FFT-WT method to analysis the first arrival time of compressional wave.

Fig. 12. (a) Photography of the bender element transducers; (b) mechanical excitation of the new bender element.

#### **3.1.2 Calibration**

Compressional and shear wave velocities are calculated with: Vp=Ltt/(tp-t0p), Vs=Ltt/(ts-t0s), where Ltt is the tip-to-tip distance of two bender elements, tp and ts are travel times of compressional wave and shear wave in the sediments respectively, t0p and t0s are the measured travel times of compressional wave and shear wave in the bender element transducers respectively. Four different lengths of cylindrical Polyoxy-methylene (POM) columns were used to calibrate t0p and t0s of the bender elements (Table 1). The diameter of the POM columns are about 6cm, which is close to the diameter of samples (6.8cm). The waveform of the POM column is shown in Fig. 13. It shows that it's easy to read both arrival times of the P-wave and S-wave using the new type of bender elements. According to lengths and wave-arrival times of the four POM columns, we obtained the t0p and t0s, which are 7.017us and 18.63us respectively. And the P-wave and S-wave velocities of the POM material are 2294.5m/s and 933.9m/s respectively [Fig. 14], which is very close to the reported values [Choy et al., 1983]. The P-wave velocity is also close to the measured results by our flat-plate transducers, which is 2280m/s for POM-I and 2319m/s for POM-II.

Ultrasonic Waves on Gas Hydrates Experiments 99

approach to determine the travel time of shear wave. However, as compressional wave is significantly influenced by the acoustic noise of the samples, the FFT-WT method is

The analysis process is as follows: (1) measuring the main frequencies of the bender element transducers; (2) choosing the compressional waveform, make a Fast Fourier Transform (FFT) on the waveform to obtain the main frequency of compressional wave; (3) making Wavelet Transform (WT) on the chosen compressional waveform to obtain frequencies versus arrival time, from which the arrival time of compressional wave can be obtained. An example of the

Firstly, the frequencies of the bender element transducers are determined by admittance curves. The results indicate that the main shear frequency is 30kHz, while the main compressional frequencies are 75kHz, 125kHz, and 140kHz. Secondly, the frequency of compressional wave is analyzed by FFT (Fig. 15). It shows that the frequencies of compressional wave mainly consist of 122kHz and 73kHz. Thirdly, the chosen compressional waveform is analyzed by WT (Fig. 16). With the frequency versus arrival time by WT, it shows that at about 96.1μs the frequency characteristics are the same with that analyzed by FFT. Thus, the travel time of compressional wave is 96.1μs with the above

Using above FFT-WT method, the travel time of shear wave can be also obtained. The results of Vs obtained using FFT-WT method are comparable with that measured by the first approach (in which the travel time of shear wave is read directly) (Fig. 17), which indicates

Fig. 15. (a) chosen compressional waveform for FFT analysis; (b) main frequencies of

developed to determine the travel time of compressional wave.

analysis process is given below.

that the FFT-WT method is credible.

FFT-WT analysis.

compressional wave


Table 1. Parameters of the POM columns

Fig. 13. Waveform of POM-I by bender element transducers

Fig. 14. Calibrating results of the new bender elements

#### **3.1.3 FFT-WT method**

Usually, there are two approaches to obtain the travel time of shear wave when using bender elements. In the first approach, the travel time can be directly read from the waveform of the receiver bender element. The characteristic point of wave's arrival must be very markedly when using this approach. The second approach is based on detailed analysis of the waveform, such as the Dynamic Finite Element Analysis, Cross-Correction Analysis, Phase Velocity Analysis, Phase Sensitive Detection, etc. Because it's easy to read the characteristic point of shear wave's arrival in our experiments, we used the first

POM-I 60.3 120 3.9~4.0 POM-II 60.3 150 3.9~4.0 POM-III 60.4 204 3.94 POM-IV 60.4 250 3.84

Length (mm)

Trough depth (mm)

Number Diameter

Fig. 13. Waveform of POM-I by bender element transducers

Fig. 14. Calibrating results of the new bender elements

Usually, there are two approaches to obtain the travel time of shear wave when using bender elements. In the first approach, the travel time can be directly read from the waveform of the receiver bender element. The characteristic point of wave's arrival must be very markedly when using this approach. The second approach is based on detailed analysis of the waveform, such as the Dynamic Finite Element Analysis, Cross-Correction Analysis, Phase Velocity Analysis, Phase Sensitive Detection, etc. Because it's easy to read the characteristic point of shear wave's arrival in our experiments, we used the first

**3.1.3 FFT-WT method** 

Table 1. Parameters of the POM columns

(mm)

approach to determine the travel time of shear wave. However, as compressional wave is significantly influenced by the acoustic noise of the samples, the FFT-WT method is developed to determine the travel time of compressional wave.

The analysis process is as follows: (1) measuring the main frequencies of the bender element transducers; (2) choosing the compressional waveform, make a Fast Fourier Transform (FFT) on the waveform to obtain the main frequency of compressional wave; (3) making Wavelet Transform (WT) on the chosen compressional waveform to obtain frequencies versus arrival time, from which the arrival time of compressional wave can be obtained. An example of the analysis process is given below.

Firstly, the frequencies of the bender element transducers are determined by admittance curves. The results indicate that the main shear frequency is 30kHz, while the main compressional frequencies are 75kHz, 125kHz, and 140kHz. Secondly, the frequency of compressional wave is analyzed by FFT (Fig. 15). It shows that the frequencies of compressional wave mainly consist of 122kHz and 73kHz. Thirdly, the chosen compressional waveform is analyzed by WT (Fig. 16). With the frequency versus arrival time by WT, it shows that at about 96.1μs the frequency characteristics are the same with that analyzed by FFT. Thus, the travel time of compressional wave is 96.1μs with the above FFT-WT analysis.

Using above FFT-WT method, the travel time of shear wave can be also obtained. The results of Vs obtained using FFT-WT method are comparable with that measured by the first approach (in which the travel time of shear wave is read directly) (Fig. 17), which indicates that the FFT-WT method is credible.

Fig. 15. (a) chosen compressional waveform for FFT analysis; (b) main frequencies of compressional wave

Ultrasonic Waves on Gas Hydrates Experiments 101

measured simultaneously with the new type of bender element transducers and analyzed with the FFT-WT method. Also, the water content, temperature, pressure of the porous media are measured (Fig. 18). The results show that the time point of gas hydrates begin to form (or dissociate) detected by the acoustic velocities is the same with that detected by the temperature-pressure method, which indicates that the bender element technique is very sensitive with gas hydrate formation and dissociation. Thus, it is effective for using the new type of bender elements in measuring both Vp and Vs of hydrate-bearing unconsolidated

Fig. 18. Changes of parameters during gas hydrate formation and subsequent dissociation

The experimental results also show that the compressional (or shear) wave velocity measured in the hydrate-dissociation process is much lower than that measured in the hydrate-formation process at the same saturation degree (Fig. 19). This may be caused by the influence of gas hydrates on the sediment frame. In the unconsolidated sediments, gas hydrates may act as a kind of cement. A small amount of gas hydrates may dramatically affects the acoustic velocities in this condition (Priest et al., 2005). During gas hydrate formation and dissociation in the unconsolidated sediments, the influences of gas hydrates on the sediment frame became smaller as time lapse. As a result, compressional (or shear) wave velocity of the hydrated unconsolidated sediments in the hydrate-dissociation process is lower than that in the hydrate-formation process. With the average Vp (or Vs) of the compressional (or shear) wave velocities obtained in the two processes, we obtained the relationship between gas hydrate saturation and acoustic velocities of hydrate-bearing unconsolidated sediments. The result shows that Vp and Vs increase rapidly with hydrate saturations, although they increase relatively slow in the range of saturation 25%~60%. It indicates that gas hydrate may first cement grain particles of the unconsolidated sediments,

sediments under high pressure conditions.

Fig. 16. WT analysis of waveform by bender element measurement

Fig. 17. Comparison of Vs determined by the direct method and the FFT-WT method

#### **3.2 Acoustic properties of hydrate-bearing unconsolidated sediments**

Methane hydrate was formed and then dissociated in the 0.09~0.125mm sands (with saturated water), during the process the acoustic velocities (Vp and Vs) of the samples are

Fig. 16. WT analysis of waveform by bender element measurement

**0.00 2.00 4.00 6.00 8.00 10.00 12.00 Time (hours)**

Fig. 17. Comparison of Vs determined by the direct method and the FFT-WT method

Methane hydrate was formed and then dissociated in the 0.09~0.125mm sands (with saturated water), during the process the acoustic velocities (Vp and Vs) of the samples are

**3.2 Acoustic properties of hydrate-bearing unconsolidated sediments** 

**Vs(direct method) Vs(FFT-WT method)**

2D frequency @96.1us

**Vs**(**m/s**)

measured simultaneously with the new type of bender element transducers and analyzed with the FFT-WT method. Also, the water content, temperature, pressure of the porous media are measured (Fig. 18). The results show that the time point of gas hydrates begin to form (or dissociate) detected by the acoustic velocities is the same with that detected by the temperature-pressure method, which indicates that the bender element technique is very sensitive with gas hydrate formation and dissociation. Thus, it is effective for using the new type of bender elements in measuring both Vp and Vs of hydrate-bearing unconsolidated sediments under high pressure conditions.

Fig. 18. Changes of parameters during gas hydrate formation and subsequent dissociation

The experimental results also show that the compressional (or shear) wave velocity measured in the hydrate-dissociation process is much lower than that measured in the hydrate-formation process at the same saturation degree (Fig. 19). This may be caused by the influence of gas hydrates on the sediment frame. In the unconsolidated sediments, gas hydrates may act as a kind of cement. A small amount of gas hydrates may dramatically affects the acoustic velocities in this condition (Priest et al., 2005). During gas hydrate formation and dissociation in the unconsolidated sediments, the influences of gas hydrates on the sediment frame became smaller as time lapse. As a result, compressional (or shear) wave velocity of the hydrated unconsolidated sediments in the hydrate-dissociation process is lower than that in the hydrate-formation process. With the average Vp (or Vs) of the compressional (or shear) wave velocities obtained in the two processes, we obtained the relationship between gas hydrate saturation and acoustic velocities of hydrate-bearing unconsolidated sediments. The result shows that Vp and Vs increase rapidly with hydrate saturations, although they increase relatively slow in the range of saturation 25%~60%. It indicates that gas hydrate may first cement grain particles of the unconsolidated sediments,

Ultrasonic Waves on Gas Hydrates Experiments 103

that in the hydrate-formation process. The experimental results may provide basic

In this chapter, two kinds of ultrasonic methods, namely, the flat-plate transducers and a new kind of bender elements have been successfully used in measuring the acoustic properties of gas hydrate bearing sediments. The results show that it's an effective way to use classic flat-plate transducers to measure both Vp and Vs of the consolidated sediments. However, in unconsolidated sediments the bender element technique is much appropriate because the bender elements can sustain larger attenuation. Thus, although significant attenuation was occurred during the unconsolidated experimental process, a developed FFT-WT method is capable of reading the time of the first arrival of P-wave, while the S-

Both methods have shown sensitivity in detecting the formation and dissociation of gas hydrate in sediments. With the flat-plate transducers, the acoustic properties of the hydrate bearing consolidated sediments were obtained, which shows that acoustic velocities are insensitive to low hydrate saturations but they increase rapidly with hydrate saturation when saturation is higher than 10%, especially in the range of 10-30%. The measurements by bender element technique figure out that Vp and Vs of the unconsolidated sediments increase rapidly with hydrate saturations, although they increase relatively slow in the range of saturation 25%~60%. It indicates that gas hydrate may first cement grain particles of the unconsolidated sediments, and then contact with

This work was financially supported by National Natural Science Foundation of China (40576028, 41104086), Natural Gas Hydrate in China Sea Exploration and Evaluation Project (GZH200200202), Gas Hydrate Reservoir Mechanism Research Project (GZH201100306) and Key Laboratory of Marine Hydrocarbon Resources and Environmental Geology, Ministry of

Sloan, E. D., Jr. (1998), *Clathrate Hydrates of Natural Gases*, CRC Press, Boca Raton, Fla. Kvenvolden, K. A. (1998), *A primer on the geological occurrence of gas hydrate*, Geological

continental slope, *Marine and Petroleum Geology*, 20, 111–128. Dickens, G. R. (2003), A methane trigger for rapid warming?, *Science*, 299, 1017. Dickens, G. R. (2004), Hydrocarbon-driven warming, *Nature*, 429, 513-515.

Milkov, A. V., and R. Sassen (2003), Preliminary assessment of resources and economic

Brown, H. E., W. S. Holbrook, M. J. Hornbach and J. Nealon (2006), Slide structure and

potential of individual gas hydrate accumulations in the Gulf of Mexico

role of gas hydrate at the northern boundary of the Storegga Slide, offshore

Society, London, Special publications, 137, 9-30.

Norway, *Marine Geology*, 229, 179-186.

knowledge for field geophysical interpretations.

wave can be read directly.

the sediment frame.

**6. References** 

**5. Acknowledgment** 

Land and Resources (MRE201113).

when hydrate saturation is higher, gas hydrate may contact with the sediment frame, or continue cementing sediment particles.

Fig. 19. For unconsolidated sediments: variation in Vp during hydrate formation (Vp(form)) and hydrate dissociation (Vp(dis)), Vs during hydrate formation (Vs(form)) and hydrate dissociation (Vs(dis)), the average Vp of Vp(form) and Vp(dis), and the average Vs of Vs(form) and Vs(dis).

## **4. Discussions and conclusions**

Some interesting acoustic phenomena have been observed in the above experiments. For example, we noticed that compressional (or shear) wave velocities are different at the same saturation degree during hydrate-formation process and hydrate-dissociation process. The morphology of gas hydrates may be a possible factor. As discussed above, hydrate formation may occur in two stages: first the formation of a water-hydrate slurry and then a very slow solidification stage. In the consolidated sediments, gas hydrates became more and more rigid as time lapse during hydrate formation and dissociation. Thus, although the amounts of gas hydrates is the same, compressional (or shear) wave velocity in the hydratedissociation process is much higher than that in the hydrate-formation process. However, in the unconsolidated sediments, gas hydrates may act as a kind of cement which mainly affects the sediment frame. A small amount of gas hydrates may dramatically affects the acoustic velocities in this condition (Priest et al., 2005). During gas hydrate formation and dissociation in the unconsolidated sediments, the influences of gas hydrates on the sediment frame became smaller as time lapse. As a result, compressional (or shear) wave velocity of the hydrated unconsolidated sediments in the hydrate-dissociation process is lower than

when hydrate saturation is higher, gas hydrate may contact with the sediment frame, or

Fig. 19. For unconsolidated sediments: variation in Vp during hydrate formation (Vp(form)) and hydrate dissociation (Vp(dis)), Vs during hydrate formation (Vs(form)) and hydrate dissociation (Vs(dis)), the average Vp of Vp(form) and Vp(dis), and the average Vs of

Some interesting acoustic phenomena have been observed in the above experiments. For example, we noticed that compressional (or shear) wave velocities are different at the same saturation degree during hydrate-formation process and hydrate-dissociation process. The morphology of gas hydrates may be a possible factor. As discussed above, hydrate formation may occur in two stages: first the formation of a water-hydrate slurry and then a very slow solidification stage. In the consolidated sediments, gas hydrates became more and more rigid as time lapse during hydrate formation and dissociation. Thus, although the amounts of gas hydrates is the same, compressional (or shear) wave velocity in the hydratedissociation process is much higher than that in the hydrate-formation process. However, in the unconsolidated sediments, gas hydrates may act as a kind of cement which mainly affects the sediment frame. A small amount of gas hydrates may dramatically affects the acoustic velocities in this condition (Priest et al., 2005). During gas hydrate formation and dissociation in the unconsolidated sediments, the influences of gas hydrates on the sediment frame became smaller as time lapse. As a result, compressional (or shear) wave velocity of the hydrated unconsolidated sediments in the hydrate-dissociation process is lower than

continue cementing sediment particles.

Vs(form) and Vs(dis).

**4. Discussions and conclusions** 

that in the hydrate-formation process. The experimental results may provide basic knowledge for field geophysical interpretations.

In this chapter, two kinds of ultrasonic methods, namely, the flat-plate transducers and a new kind of bender elements have been successfully used in measuring the acoustic properties of gas hydrate bearing sediments. The results show that it's an effective way to use classic flat-plate transducers to measure both Vp and Vs of the consolidated sediments. However, in unconsolidated sediments the bender element technique is much appropriate because the bender elements can sustain larger attenuation. Thus, although significant attenuation was occurred during the unconsolidated experimental process, a developed FFT-WT method is capable of reading the time of the first arrival of P-wave, while the Swave can be read directly.

Both methods have shown sensitivity in detecting the formation and dissociation of gas hydrate in sediments. With the flat-plate transducers, the acoustic properties of the hydrate bearing consolidated sediments were obtained, which shows that acoustic velocities are insensitive to low hydrate saturations but they increase rapidly with hydrate saturation when saturation is higher than 10%, especially in the range of 10-30%. The measurements by bender element technique figure out that Vp and Vs of the unconsolidated sediments increase rapidly with hydrate saturations, although they increase relatively slow in the range of saturation 25%~60%. It indicates that gas hydrate may first cement grain particles of the unconsolidated sediments, and then contact with the sediment frame.

## **5. Acknowledgment**

This work was financially supported by National Natural Science Foundation of China (40576028, 41104086), Natural Gas Hydrate in China Sea Exploration and Evaluation Project (GZH200200202), Gas Hydrate Reservoir Mechanism Research Project (GZH201100306) and Key Laboratory of Marine Hydrocarbon Resources and Environmental Geology, Ministry of Land and Resources (MRE201113).

## **6. References**

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edited by I. Kaplan, pp. 235-248, Springer, New York.

Wood, A. B. (1941), *A text book of sound*, Macmillan, New York, 578 pp

gas hydrate dissociation around boreholes – could it be feasible? A conceptual 2D study linking geomechanical and seismic FD models, *paper presented at Sixth* 

L. Worzel (1979), Seismic evidence for widespread occurrence of possible gas hydrate horizons on continental slopes and rises, *AAPG Bull.*, 63(12), 2204-2213. Holbrook, W. S., H. Hoskins, W. T. Wood, R. A. Stephen, and D. Lizarralde (1996),

Methane hydrate and free gas on the Blake Ridge from Vertical Seismic Profiling,

wave velocities at the Mallik 2L-38 research well, Mackenzie Delta, Canada,

seismic-profile data from the western North Atlantic, *Am. Assoc. Petrol. Geol.* 

of factors affecting elastic wave velocities in porous media, *Geophysics*, 23, 459-

hydrate-bearing sediments using weighted equation, *J. Geophys. Res.*, 101, 20347-

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Stoll, R. D.(1974), Effects of gas hydrate in sediments, in *Natural Gases in Marine Sediment*,

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**6** 

Hikaru Miura *Nihon university* 

*Japan* 

**Intense Aerial Ultrasonic Source and** 

 **Removal of Unnecessary Gas by the Source** 

Ultrasonic energy is widely used with liquids and solids, for example, for cleaning substances in liquids, atomization of liquids (Miura, 2007b), (Ueha et al., 1985), sedimentation of dispersed fine particles (Miura, 2004), deflection of water (Ito, 2005) , removal of liquid in a pore (Ito & Takamura, 2010), and cutting (Asami & Miura, 2010, 2011) and welding solids (Miura, 2003, 2008). In addition, it is also used with gases, for example, to enhance the removal of unnecessary gases (Miura, 2007a), (Kobayashi et al., 1997), the aggregation and removal of airborne substances such as smoke, antifoaming, drying of wet substances containing moisture, thawing of frozen materials, and the decomposition of methane hydrates (Miura et al., 2006). However, extremely intense sound waves (with a sound pressure of at least 160 dB) are required to perform the above processes in air. Therefore, intense aerial ultrasonic sources are required so that ultrasonic waves can be used in air. Unfortunately, such ultrasonic sources have seldom been developed because it is

In this chapter, I describe the structure of an ultrasonic source with a flexurally vibrating plate that can radiate extremely intense ultrasonic waves in air as well as the vibration distribution and directivity of the radiated ultrasonic waves. Next, I examine the effect of using the above-mentioned intense aerial ultrasonic source on the enhancement of the removal rate of an unnecessary gas in air. In general, some gas components in air may have adverse effects on humans and the environment, causing environmental problems. Currently, gas absorption, in which a gas is dissolved in a liquid for transport, is widely adopted as a means of collecting gas components in a mixed gas and removing unnecessary gas components. General absorption systems are roughly classified into gas-dispersion-type systems, in which a gas is dispersed in a liquid in the form of microbubbles, and liquiddispersion-type systems, in which a liquid is sprayed into containers filled with the gas. Both systems allow the gas and liquid to come into contact without applying an external

I attempted to enhance the effect of gas absorption by applying ultrasonic waves to such systems. In this chapter, I describe a method of allowing liquid mist obtained using aerial

**1. Introduction** 

force.

ultrasonic waves to absorb a gas.

difficult to generate intense ultrasonic waves in air.


## **Intense Aerial Ultrasonic Source and Removal of Unnecessary Gas by the Source**

Hikaru Miura *Nihon university Japan* 

## **1. Introduction**

106 Ultrasonic Waves

Lee J S, Santamarina J C (2005). Bender elements: Performance and signal interpretation. *Journal of Geotechnical and Geoenvironmental Engineering*, 1063-1070. Choy C L, Leung W P, and Huang C W (1983). Elastic moduli of highly oriented polyoxymethylene. *Polymer Engineering and Science*, 23(16): 910-922. Priest, J., C. R. I. Best, and R. I. Clayton (2005), A laboratory investigation into the seismic

doi:10.1029/2004JB003259.

velocities of methane gas hydrate-bearing sand, *J. Geophys. Res.*, 110, B04102,

Ultrasonic energy is widely used with liquids and solids, for example, for cleaning substances in liquids, atomization of liquids (Miura, 2007b), (Ueha et al., 1985), sedimentation of dispersed fine particles (Miura, 2004), deflection of water (Ito, 2005) , removal of liquid in a pore (Ito & Takamura, 2010), and cutting (Asami & Miura, 2010, 2011) and welding solids (Miura, 2003, 2008). In addition, it is also used with gases, for example, to enhance the removal of unnecessary gases (Miura, 2007a), (Kobayashi et al., 1997), the aggregation and removal of airborne substances such as smoke, antifoaming, drying of wet substances containing moisture, thawing of frozen materials, and the decomposition of methane hydrates (Miura et al., 2006). However, extremely intense sound waves (with a sound pressure of at least 160 dB) are required to perform the above processes in air. Therefore, intense aerial ultrasonic sources are required so that ultrasonic waves can be used in air. Unfortunately, such ultrasonic sources have seldom been developed because it is difficult to generate intense ultrasonic waves in air.

In this chapter, I describe the structure of an ultrasonic source with a flexurally vibrating plate that can radiate extremely intense ultrasonic waves in air as well as the vibration distribution and directivity of the radiated ultrasonic waves. Next, I examine the effect of using the above-mentioned intense aerial ultrasonic source on the enhancement of the removal rate of an unnecessary gas in air. In general, some gas components in air may have adverse effects on humans and the environment, causing environmental problems. Currently, gas absorption, in which a gas is dissolved in a liquid for transport, is widely adopted as a means of collecting gas components in a mixed gas and removing unnecessary gas components. General absorption systems are roughly classified into gas-dispersion-type systems, in which a gas is dispersed in a liquid in the form of microbubbles, and liquiddispersion-type systems, in which a liquid is sprayed into containers filled with the gas. Both systems allow the gas and liquid to come into contact without applying an external force.

I attempted to enhance the effect of gas absorption by applying ultrasonic waves to such systems. In this chapter, I describe a method of allowing liquid mist obtained using aerial ultrasonic waves to absorb a gas.

Intense Aerial Ultrasonic Source and Removal of Unnecessary Gas by the Source 109

These parts are joined with screws. In general, transducers are required to generate longitudinal vibration with a frequency of approximately 20-100 kHz; I used a 20 kHz boltclamped Langevin-type piezoelectric (PZT) transducer (BLT transducer) to obtain large amplitudes. To increase the amplitude of the vibration of the transducer, an exponential horn (duralumin; diameter of thin end face, 10 mm; amplification rate, approximately was used. A rod (duralumin; diameter, 10 mm; length, 93 mm) was used to tune the resonant frequency of the 20 kHz longitudinal vibration of the transducer to that of the flexural vibration of the plate. The length of this rod should be an integral multiple of the halfwavelength of its longitudinal vibration; however, the rod can be omitted if not required. The flexurally vibrating plate attached at the end of the rod is allowed to vibrate by applying an amplified longitudinal vibration to the plate center as the driving point. Thus, lattice-mode vibration, in which the nodes of the flexural vibration are distributed in a

**2.3 Design and fabrication of plate vibrating flexurally in the lattice mode** 

stripe mode between opposite sides of the plate is given as

of side *L* given by eq. (2), lattice-mode vibration can be obtained.

2

To distribute the nodes of a high-order flexural vibration in a lattice pattern by vibrating the center of a square plate, all sides of which undergo free vibration, high-order stripe-mode flexural vibration is induced between each pair of parallel sides of the plate, and the sum of the two flexural vibrations, which orthogonally intersect, is considered to be the latticemode flexural vibration. The equation used to design a plate vibrating flexurally in the

> <sup>2</sup> *<sup>P</sup> <sup>t</sup> C h f*

where *λt* is the wavelength of the flexural vibration of the plate, *CP* is a constant specific to the plate material, *h* is the thickness of the plate, and *f* is the frequency (Yamane et al., 1983). The length of the plate in the direction perpendicular to the striped pattern, *L*, is given by

0.5

where *N* is the number of nodal lines and is even. When the plate is assumed to be a square

Figure 2 shows a schematic of a square flexurally vibrating plate viewed from above. When the center of the plate is assumed to be the origin and the *x*- and *y*-axes are set as shown in the figure, the vibration displacement at an arbitrary point (*x,y*) on the plate, *ξ*, is

> <sup>0</sup> 0.5 0.5 cos cos

*N xN y L L*

*<sup>t</sup> L N*

1/2

2

(1)

(2)

 

(3)

lattice pattern, can be realized.

**2.3.1 Design of vibrating plate** 

**2.3.2 Vibration mode of plate** 

approximated using eq. (3) for *N* ≥ 10.

## **2. Intense aerial ultrasonic source**

## **2.1 Outline**

Ultrasonic sources using a thin metal plate that vibrates flexurally can be used to radiate extremely intense ultrasonic waves in air. Such ultrasonic sources consist of a longitudinal vibration transducer, a horn for increasing the amplitude, and a flexurally vibrating plate attached at the end of the horn. The length and width of the flexurally vibrating plate are sufficiently greater than the wavelength of flexural vibration, and the plate shape may be square, rectangular, or circular. Flexural vibration is usually generated by vibrating the center of the plate (Miura & Honda, 2002). Such ultrasonic sources can efficiently radiate sound waves in air and achieve a high electroacoustic conversion efficiency by generating an appropriate mode of vibration, i.e., a vibration mode in which the vibration nodes are distributed in a lattice pattern for square plates, a striped pattern for rectangular plates, and a concentric circular pattern for circular plates (Onishi & Miura, 2005), (Miura & Ishikawa, 2009).

In this section, I describe a method for designing a square plate that vibrates flexurally in the lattice mode and can be used as an ultrasonic source, and I discuss the vibration mode of the fabricated plate (Miura, 1994). Next, the measured distributions of vibration displacement and sound pressure near the plate surface are described. Moreover, the directivity of the sound waves radiated from the vibrating plate into remote acoustic fields is theoretically and experimentally examined.

## **2.2 Intense aerial ultrasonic source**

Figure 1 shows a schematic diagram of an ultrasonic source that can radiate intense ultrasonic waves in air. As shown in the figure, the ultrasonic source consists of a longitudinal vibration transducer, a horn for increasing the amplitude, a rod for tuning the resonance of the longitudinal vibration, and a flexurally vibrating plate attached at the end of the horn.

Fig. 1. Schematic diagram of an aerial ultrasonic source.

These parts are joined with screws. In general, transducers are required to generate longitudinal vibration with a frequency of approximately 20-100 kHz; I used a 20 kHz boltclamped Langevin-type piezoelectric (PZT) transducer (BLT transducer) to obtain large amplitudes. To increase the amplitude of the vibration of the transducer, an exponential horn (duralumin; diameter of thin end face, 10 mm; amplification rate, approximately was used. A rod (duralumin; diameter, 10 mm; length, 93 mm) was used to tune the resonant frequency of the 20 kHz longitudinal vibration of the transducer to that of the flexural vibration of the plate. The length of this rod should be an integral multiple of the halfwavelength of its longitudinal vibration; however, the rod can be omitted if not required. The flexurally vibrating plate attached at the end of the rod is allowed to vibrate by applying an amplified longitudinal vibration to the plate center as the driving point. Thus, lattice-mode vibration, in which the nodes of the flexural vibration are distributed in a lattice pattern, can be realized.

#### **2.3 Design and fabrication of plate vibrating flexurally in the lattice mode**

#### **2.3.1 Design of vibrating plate**

108 Ultrasonic Waves

Ultrasonic sources using a thin metal plate that vibrates flexurally can be used to radiate extremely intense ultrasonic waves in air. Such ultrasonic sources consist of a longitudinal vibration transducer, a horn for increasing the amplitude, and a flexurally vibrating plate attached at the end of the horn. The length and width of the flexurally vibrating plate are sufficiently greater than the wavelength of flexural vibration, and the plate shape may be square, rectangular, or circular. Flexural vibration is usually generated by vibrating the center of the plate (Miura & Honda, 2002). Such ultrasonic sources can efficiently radiate sound waves in air and achieve a high electroacoustic conversion efficiency by generating an appropriate mode of vibration, i.e., a vibration mode in which the vibration nodes are distributed in a lattice pattern for square plates, a striped pattern for rectangular plates, and a concentric circular pattern for circular plates (Onishi & Miura, 2005), (Miura & Ishikawa, 2009). In this section, I describe a method for designing a square plate that vibrates flexurally in the lattice mode and can be used as an ultrasonic source, and I discuss the vibration mode of the fabricated plate (Miura, 1994). Next, the measured distributions of vibration displacement and sound pressure near the plate surface are described. Moreover, the directivity of the sound waves radiated from the vibrating plate into remote acoustic fields is theoretically

Figure 1 shows a schematic diagram of an ultrasonic source that can radiate intense ultrasonic waves in air. As shown in the figure, the ultrasonic source consists of a longitudinal vibration transducer, a horn for increasing the amplitude, a rod for tuning the resonance of the

longitudinal vibration, and a flexurally vibrating plate attached at the end of the horn.

**2. Intense aerial ultrasonic source** 

and experimentally examined.

**2.2 Intense aerial ultrasonic source** 

Fig. 1. Schematic diagram of an aerial ultrasonic source.

**2.1 Outline** 

To distribute the nodes of a high-order flexural vibration in a lattice pattern by vibrating the center of a square plate, all sides of which undergo free vibration, high-order stripe-mode flexural vibration is induced between each pair of parallel sides of the plate, and the sum of the two flexural vibrations, which orthogonally intersect, is considered to be the latticemode flexural vibration. The equation used to design a plate vibrating flexurally in the stripe mode between opposite sides of the plate is given as

$$\mathcal{A}\_t = \left\{ \frac{2\pi \mathcal{C}\_p h}{f} \right\}^{1/2} \tag{1}$$

where *λt* is the wavelength of the flexural vibration of the plate, *CP* is a constant specific to the plate material, *h* is the thickness of the plate, and *f* is the frequency (Yamane et al., 1983). The length of the plate in the direction perpendicular to the striped pattern, *L*, is given by

$$L = (N - 0.5)\frac{\mathcal{A}\_t}{2} \tag{2}$$

where *N* is the number of nodal lines and is even. When the plate is assumed to be a square of side *L* given by eq. (2), lattice-mode vibration can be obtained.

#### **2.3.2 Vibration mode of plate**

Figure 2 shows a schematic of a square flexurally vibrating plate viewed from above. When the center of the plate is assumed to be the origin and the *x*- and *y*-axes are set as shown in the figure, the vibration displacement at an arbitrary point (*x,y*) on the plate, *ξ*, is approximated using eq. (3) for *N* ≥ 10.

$$\zeta = \frac{\xi\_0}{2} \left\{ \cos \frac{(N - 0.5)\pi x}{L} + \cos \frac{(N - 0.5)\pi y}{L} \right\} \tag{3}$$

Intense Aerial Ultrasonic Source and Removal of Unnecessary Gas by the Source 111

Fig. 3. Chladni sand figure showing nodal pattern of the square vibrating plate.

plates vibrating in a lattice mode when *N* = 10 and *h* = 1, 2, 3, or 5 mm.

**2.4 Amplitude distribution of vibration displacement of plate** 

region (A) shown in Fig. 4, where the nodes are indicated by dashed lines.

Duralumin (JIS A2017P-T3; thickness *h*, 1-5 mm) was used to fabricate vibrating plates. The resonant frequency of the lattice mode vibration was approximately 20 kHz. The vibrating plates were designed using eqs. (1) and (2). Table 1 summarizes the details of the fabricated

To determine the flexural vibration mode of each plate, Chladni sand figures were observed. Figure 3 shows the result for the plate with *h* = 3 mm (see Table 1). In the figure, the nodal lines on which sand particles (silicon carbide, #100) are concentrated are distributed in a lattice pattern similar to the nodal pattern (dotted lines) shown in Fig. 2. Vibrating plates with *N* not equal to 10 were also fabricated to observe the Chladni sand figures, and the vibration mode was confirmed to be the lattice mode with a different number of nodal lines.

The vibration displacement at each position on the plate surface was measured using a noncontact microdisplacement meter to determine its distribution. The vibrating plate with *h* = 3 mm, which was used to obtain Fig. 3 (see Table 1), was used also for this measurement. The electric power input to the transducer was maintained at 1 W, and the square region indicated as (A) in Fig. 4 was targeted. Figure 5 shows the distribution of the normalized vibration displacement at various positions on the plate surface (using the center of the plate as the reference). The nodal lines of vibration displacement were distributed in a lattice pattern and the antinodes were positioned at the centers of the regions surrounded by the nodal lines, in good agreement with the schematic of the distribution of the vibration displacement within

**2.3.3 Fabrication of vibrating plate** 

**2.3.4 Chladni sand figures** 

Here, *ξ*0 is the maximum vibration displacement.

The displacements of the plate flexurally vibrating in the stripe mode along the *x*- and *y*directions are shown at the bottom and to the left of Fig. 2, respectively, and were calculated using eq. (3) assuming *N* = 10. Because the vibration mode is obtained by superimposing the vibration displacement in the *x*-direction (the first term in eq. (3)) onto that in the *y*-direction (the second term in eq. (3)), the nodal lines of the flexural vibration are represented by the broken lines in the figure. These nodal lines form a lattice pattern having an angle of 45° to each side of the plate. The number of nodal lines in each direction of the lattice mode is equal to *N* in the stripe mode. The interval between the lattice-mode nodal lines, *ds*, is given by

$$d\_s = \frac{\lambda\_t}{\sqrt{2}}\tag{4}$$

The antinodes of vibration displacement are positioned at the center of the regions surrounded by the nodal lines, and the displacements at adjacent antinodes have opposite phases (Miura, 1994).

Fig. 2. Outline of a square plate vibrating in a lattice mode. Broken lines are nodal lines. *ds* is the interval of the lattice mode.


Table 1. Details of the square plate vibrating in the lattice mode.

Fig. 3. Chladni sand figure showing nodal pattern of the square vibrating plate.

## **2.3.3 Fabrication of vibrating plate**

Duralumin (JIS A2017P-T3; thickness *h*, 1-5 mm) was used to fabricate vibrating plates. The resonant frequency of the lattice mode vibration was approximately 20 kHz. The vibrating plates were designed using eqs. (1) and (2). Table 1 summarizes the details of the fabricated plates vibrating in a lattice mode when *N* = 10 and *h* = 1, 2, 3, or 5 mm.

## **2.3.4 Chladni sand figures**

110 Ultrasonic Waves

The displacements of the plate flexurally vibrating in the stripe mode along the *x*- and *y*directions are shown at the bottom and to the left of Fig. 2, respectively, and were calculated using eq. (3) assuming *N* = 10. Because the vibration mode is obtained by superimposing the vibration displacement in the *x*-direction (the first term in eq. (3)) onto that in the *y*-direction (the second term in eq. (3)), the nodal lines of the flexural vibration are represented by the broken lines in the figure. These nodal lines form a lattice pattern having an angle of 45° to each side of the plate. The number of nodal lines in each direction of the lattice mode is equal to *N* in the stripe mode. The interval between the lattice-mode nodal lines, *ds*, is given by

> 2 *<sup>t</sup> <sup>s</sup> d*

The antinodes of vibration displacement are positioned at the center of the regions surrounded by the nodal lines, and the displacements at adjacent antinodes have opposite

Fig. 2. Outline of a square plate vibrating in a lattice mode. Broken lines are nodal lines. *ds* is

Table 1. Details of the square plate vibrating in the lattice mode.

(4)

Here, *ξ*0 is the maximum vibration displacement.

phases (Miura, 1994).

the interval of the lattice mode.

To determine the flexural vibration mode of each plate, Chladni sand figures were observed. Figure 3 shows the result for the plate with *h* = 3 mm (see Table 1). In the figure, the nodal lines on which sand particles (silicon carbide, #100) are concentrated are distributed in a lattice pattern similar to the nodal pattern (dotted lines) shown in Fig. 2. Vibrating plates with *N* not equal to 10 were also fabricated to observe the Chladni sand figures, and the vibration mode was confirmed to be the lattice mode with a different number of nodal lines.

## **2.4 Amplitude distribution of vibration displacement of plate**

The vibration displacement at each position on the plate surface was measured using a noncontact microdisplacement meter to determine its distribution. The vibrating plate with *h* = 3 mm, which was used to obtain Fig. 3 (see Table 1), was used also for this measurement. The electric power input to the transducer was maintained at 1 W, and the square region indicated as (A) in Fig. 4 was targeted. Figure 5 shows the distribution of the normalized vibration displacement at various positions on the plate surface (using the center of the plate as the reference). The nodal lines of vibration displacement were distributed in a lattice pattern and the antinodes were positioned at the centers of the regions surrounded by the nodal lines, in good agreement with the schematic of the distribution of the vibration displacement within region (A) shown in Fig. 4, where the nodes are indicated by dashed lines.

Intense Aerial Ultrasonic Source and Removal of Unnecessary Gas by the Source 113

In this section, I theoretically and experimentally examine the directivity of the sound waves

The directivity of the sound waves radiated from the square plate vibrating in the lattice

As shown in Fig. 7, it is assumed that the distance from the plate center to a sufficiently remote observation point P is *R0* and that the distance from an arbitrary point (*x,y*) on the

Fig. 6. Distributions of the sound pressure near the vibrating plate. Lines (I) - (IV) are

Fig. 7. Coordinate system for the evaluating directivity of radiated sound waves.

radiated from the plate vibrating in the lattice mode into remote acoustic fields.

**2.6 Directivity of sound waves in remote acoustic fields** 

mode into remote acoustic fields is calculated as follows.

Square

vibrating plate

**2.6.1 Method of calculating directivity** 

indicated in Fig.4.

Fig. 4. Measurement positions of vibration displacement and sound pressure.

Fig. 5. Distribution of the vibration displacement.

#### **2.5 Distribution of sound pressure near vibrating plate surface**

To determine the distribution of sound pressure near the vibrating plate surface, the sound pressure at various positions approximately 1 mm above the plate along measurement lines (I)-(IV), indicated by bold lines in Fig. 4, was measured using condenser microphones (Bruel & Kjaer, 4138) while maintaining the input electric power at 1 W. Figure 6 shows the results, where the ordinate represents the normalized sound pressure and the abscissa represents the distance along each measurement line (using the foot of the perpendicular from the plate center to each measurement line as the reference). The distributions of sound pressure along measurement lines (I) and (III), which passed through antinodes of sound pressure, were similar to a sine wave with a specific amplitude and period. In this case, *ds*, the interval between adjacent nodes of sound pressure, was approximately 26 mm, in good agreement with the value of 26.4 mm calculated using eq. (4). In contrast, sound pressures along measurement lines (II) and (IV), which were located on the nodal lines, were low.

#### **2.6 Directivity of sound waves in remote acoustic fields**

In this section, I theoretically and experimentally examine the directivity of the sound waves radiated from the plate vibrating in the lattice mode into remote acoustic fields.

#### **2.6.1 Method of calculating directivity**

112 Ultrasonic Waves

Fig. 4. Measurement positions of vibration displacement and sound pressure.

Fig. 5. Distribution of the vibration displacement.

Normalized displacement

(Mutual value)

**2.5 Distribution of sound pressure near vibrating plate surface** 

To determine the distribution of sound pressure near the vibrating plate surface, the sound pressure at various positions approximately 1 mm above the plate along measurement lines (I)-(IV), indicated by bold lines in Fig. 4, was measured using condenser microphones (Bruel & Kjaer, 4138) while maintaining the input electric power at 1 W. Figure 6 shows the results, where the ordinate represents the normalized sound pressure and the abscissa represents the distance along each measurement line (using the foot of the perpendicular from the plate center to each measurement line as the reference). The distributions of sound pressure along measurement lines (I) and (III), which passed through antinodes of sound pressure, were similar to a sine wave with a specific amplitude and period. In this case, *ds*, the interval between adjacent nodes of sound pressure, was approximately 26 mm, in good agreement with the value of 26.4 mm calculated using eq. (4). In contrast, sound pressures along

Position along x-direction [mm]

measurement lines (II) and (IV), which were located on the nodal lines, were low.

The directivity of the sound waves radiated from the square plate vibrating in the lattice mode into remote acoustic fields is calculated as follows.

As shown in Fig. 7, it is assumed that the distance from the plate center to a sufficiently remote observation point P is *R0* and that the distance from an arbitrary point (*x,y*) on the

Fig. 6. Distributions of the sound pressure near the vibrating plate. Lines (I) - (IV) are indicated in Fig.4.

Fig. 7. Coordinate system for the evaluating directivity of radiated sound waves.

Intense Aerial Ultrasonic Source and Removal of Unnecessary Gas by the Source 115

The sound pressure at a distance of 1.8 m from the plate center was experimentally measured using a 6.4-mm-diameter condenser microphone (B&K 4136) while maintaining

The characteristics of the vibrating plates used in the experiment are summarized in Table 1. No baffles were used because the dimensions of the vibrating plates were sufficiently greater than the wavelength of the sound waves and baffles were considered to have little effect. The back of the vibrating plates was covered with glass wool with a thickness of 50 mm to absorb the sound waves radiated from the back. The measurement distance was set at values greater than the Fresnel last maximum (at which the phase difference due to differences in the distance from the vibrating plate surface is negligible and a remote acoustic field is assumed to be formed) on the centerline of a disc piston with a diameter

First, sound pressures in various directions were measured using the vibrating plate with *h*  = 3 mm and *N* = 10 (see Table 1) for different angles between OP and the *z*-axis, *γ*, from -90 (*x*-*y* plane), 0 (*z*-axis), to 90° (*x*-*y* plane) to examine the radiation direction of the sound waves. Figure 8 shows the measurement positions: (a) *γ* = -90 — 90° in the *y*-*z* plane (*α* = 90°), (b) *γ* = -90 — 90° in the *x*-*z* plane (*β* = 90°), (c) *γ* = -90 — 90° in the *y* = *x* and *z*–axis plane (*α* = *β*), and (d) *γ* = -90 — 90° in the *y* = -*x* and *z*–axis plane. The measurement results are shown by the solid lines in Figs. 9(a)—9(d). The abscissa represents *γ* and the ordinate represents the normalized sound pressure. In Figs. 9(a) and 9(b), sharp main lobes of the radiated sound waves are observed in two directions; they have specific *z*–axis-symmetric angles in the *y*-*z* and *x*-*z* planes, respectively. However, no sharp main lobes are observed in the *y* = *x* and *z*–axis plane and the *y* = -*x* and *z*–axis plane, as shown in Figs. 9(c) and 9(d), respectively. The dashed lines in Figs. 9(a)—9(d) represent the calculation results obtained

using eq. (9), and are in good agreement with the experimental results in all cases.

The directions of the main lobe are symmetric about the *z*-axis and the angle between the *z*-

equal to the length of the diagonal of the vibrating plates (David & Cheeke, 2002).

**2.6.2 Method of measuring directivity** 

Fig. 8. Measurement positions of directivity.

**2.6.3 Directivity in various directions** 

axis and the main lobe, *γ*m, is given by

the electric power input to the acoustic source at 1 W.

plate surface to P is *R*. When the angles between OP and the *x*-, *y*-, and *z*-axes are assumed to be *α*, *β*, and *γ*, respectively, *R* is given by

$$R = R\_0 - (\arccos \alpha + y \cos \beta) \tag{5}$$

Therefore, the component of the volume velocity over a small area *dxdy* in the OP direction is *ξdxdy*cos*γ*, and the velocity potential *dΦ* at P is given by

$$d\Phi = \frac{\xi d\mathbf{x} dy}{2\pi R} \cos \varphi \ e^{j(\alpha t - k\mathbb{R})} \tag{6}$$

where *ω* is the angular frequency and *k* is the wavelength constant of sound waves. When P is sufficiently remote from the plate center, ܴ ب ݔߙ ݕߚ. When assuming that *R* only affects the phase difference of the velocity potential *Φ*, *Φ* at P is obtained by integrating eq. (6) over the entire plate area.

$$\Phi = \frac{\cos \gamma}{2\pi R\_0} \int\_{-\frac{L}{2} - \frac{L}{2}}^{\frac{L}{2}} \int\_{\frac{\omega}{2}}^{\frac{L}{2}} \xi e^{j(\cot - k\theta)} dx dy \tag{7}$$

The sound pressure *p* at P is expressed by

$$p = \rho \frac{\partial \Phi}{\partial t} \tag{8}$$

where *ρ* is the density of air. By substituting eq. (7) into eq. (8),

$$p = \frac{\alpha \rho \varepsilon\_0^x \cos \gamma}{2 \pi R\_0} \left( \frac{A \sin \frac{k\_\beta L}{2}}{k\_\beta} + \frac{B \sin \frac{k\_\alpha L}{2}}{k\_\alpha} \right) \cdot e^{j(\alpha t - k R\_0 + \frac{\pi}{2})} \tag{9}$$

is obtained. Here, *A*, *B*, *kα*, *kβ*, and *kN* are given as follows.

$$A = \frac{1}{k\_{\alpha} + k\_{N}} \sin \frac{\left(k\_{\alpha} + k\_{N}\right)L}{2} + \frac{1}{k\_{\alpha} - k\_{N}} \sin \frac{\left(k\_{\alpha} - k\_{N}\right)L}{2}$$

$$B = \frac{1}{k\_{\beta} + k\_{N}} \sin \frac{\left(k\_{\beta} + k\_{N}\right)L}{2} + \frac{1}{k\_{\beta} - k\_{N}} \sin \frac{\left(k\_{\beta} - k\_{N}\right)L}{2}$$

$$k\_{\alpha} = k \cos \alpha \quad \therefore \quad k\_{\beta} = k \cos \beta$$

$$k\_{N} = \frac{\left(N - 0.5\right)\pi}{L}$$

*L*

## **2.6.2 Method of measuring directivity**

114 Ultrasonic Waves

plate surface to P is *R*. When the angles between OP and the *x*-, *y*-, and *z*-axes are assumed

<sup>0</sup> *RR x y* ( cos cos ) 

Therefore, the component of the volume velocity over a small area *dxdy* in the OP direction

*dxdy <sup>j</sup> t kR d e R*

where *ω* is the angular frequency and *k* is the wavelength constant of sound waves. When P is sufficiently remote from the plate center, ܴ ب ݔߙ ݕߚ. When assuming that *R* only affects the phase difference of the velocity potential *Φ*, *Φ* at P is obtained by integrating eq.

2 2

*e dxdy <sup>R</sup>*

<sup>0</sup> ( ) <sup>0</sup> <sup>2</sup>

 

2 2 *N N*

2 2 *N N*

*k kL k kL*

*k kL k kL*

*j t kR*

 

2

0 2 2

 *L L*

*L L*

*<sup>p</sup> <sup>t</sup>* 

sin sin os 2 2

*N N*

*N N*

 

 

*k k k k* 

*k k k k* 

> *N <sup>N</sup> <sup>k</sup>*

*k k*cos 

*p e Rk k*

*k L k L A B*

1 1 sin sin

1 1 sin sin

 

 

 , *k k*cos 

0.5

*L* 

cos 2

where *ρ* is the density of air. By substituting eq. (7) into eq. (8),

*c*

0

 

2

is obtained. Here, *A*, *B*, *kα*, *kβ*, and *kN* are given as follows.

*A*

*B*

( ) cos

 

(6)

(7)

*j t kR*

 

 

(8)

(9)

(5)

to be *α*, *β*, and *γ*, respectively, *R* is given by

(6) over the entire plate area.

The sound pressure *p* at P is expressed by

is *ξdxdy*cos*γ*, and the velocity potential *dΦ* at P is given by

The sound pressure at a distance of 1.8 m from the plate center was experimentally measured using a 6.4-mm-diameter condenser microphone (B&K 4136) while maintaining the electric power input to the acoustic source at 1 W.

The characteristics of the vibrating plates used in the experiment are summarized in Table 1. No baffles were used because the dimensions of the vibrating plates were sufficiently greater than the wavelength of the sound waves and baffles were considered to have little effect. The back of the vibrating plates was covered with glass wool with a thickness of 50 mm to absorb the sound waves radiated from the back. The measurement distance was set at values greater than the Fresnel last maximum (at which the phase difference due to differences in the distance from the vibrating plate surface is negligible and a remote acoustic field is assumed to be formed) on the centerline of a disc piston with a diameter equal to the length of the diagonal of the vibrating plates (David & Cheeke, 2002).

Fig. 8. Measurement positions of directivity.

#### **2.6.3 Directivity in various directions**

First, sound pressures in various directions were measured using the vibrating plate with *h*  = 3 mm and *N* = 10 (see Table 1) for different angles between OP and the *z*-axis, *γ*, from -90 (*x*-*y* plane), 0 (*z*-axis), to 90° (*x*-*y* plane) to examine the radiation direction of the sound waves. Figure 8 shows the measurement positions: (a) *γ* = -90 — 90° in the *y*-*z* plane (*α* = 90°), (b) *γ* = -90 — 90° in the *x*-*z* plane (*β* = 90°), (c) *γ* = -90 — 90° in the *y* = *x* and *z*–axis plane (*α* = *β*), and (d) *γ* = -90 — 90° in the *y* = -*x* and *z*–axis plane. The measurement results are shown by the solid lines in Figs. 9(a)—9(d). The abscissa represents *γ* and the ordinate represents the normalized sound pressure. In Figs. 9(a) and 9(b), sharp main lobes of the radiated sound waves are observed in two directions; they have specific *z*–axis-symmetric angles in the *y*-*z* and *x*-*z* planes, respectively. However, no sharp main lobes are observed in the *y* = *x* and *z*–axis plane and the *y* = -*x* and *z*–axis plane, as shown in Figs. 9(c) and 9(d), respectively. The dashed lines in Figs. 9(a)—9(d) represent the calculation results obtained using eq. (9), and are in good agreement with the experimental results in all cases.

The directions of the main lobe are symmetric about the *z*-axis and the angle between the *z*axis and the main lobe, *γ*m, is given by

Intense Aerial Ultrasonic Source and Removal of Unnecessary Gas by the Source 117

using eq. (10). When similar experiments were carried out for planes other than the abovementioned planes, the sound pressures were low, similarly to the cases in Figs. 9(c) and 9(d). Therefore, the sound waves radiated from the plate vibrating in the lattice mode have main lobes only in four symmetric directions with a specific angle from the *z*-axis in the *y*-*z* and *x*-

Fig. 11. Directivity patterns for various *λ*t/*λ*a. ----- : Experimental, - - - : calculated. (a) changing *λ*t/*λ*a = 1.29 (*h*=1 mm), (b) *λ*t/*λ*a = 1.83 (*h*=2 mm), (c) *λ*t/*λ*a = 2.20 (*h*=3 mm), (d)

Table 2. Directivity of the square plate vibrating in the lattice mode.

*z* planes, as shown in Fig. 10.

*λ*t/*λ*a = 2.74 (*h*=5 mm).

Fig. 9. Directivity patterns in various directions. ------ : Experimental, - - - : calculated. Position (a) is in the *y-z* axis plane, (b) is in the *x-z* axis plane, (c) is in the *y*=*x* and the *z-*axis plane, (d) is in the *y=* - *x* and the *z-*axis plane.

Fig. 10. Outline of the directivity.

where *λ*a is the wavelength of the sound waves in air. The measurement results in Figs. 9(a) and 9(b) reveal that *γ*m = 27.2°, which is in good agreement with the value of 27.0° calculated

<sup>1</sup> <sup>a</sup> <sup>m</sup>

γ sin

Fig. 9. Directivity patterns in various directions. ------ : Experimental, - - - : calculated. Position (a) is in the *y-z* axis plane, (b) is in the *x-z* axis plane, (c) is in the *y*=*x* and the *z-*axis

where *λ*a is the wavelength of the sound waves in air. The measurement results in Figs. 9(a) and 9(b) reveal that *γ*m = 27.2°, which is in good agreement with the value of 27.0° calculated

plane, (d) is in the *y=* - *x* and the *z-*axis plane.

Fig. 10. Outline of the directivity.

t

(10)

using eq. (10). When similar experiments were carried out for planes other than the abovementioned planes, the sound pressures were low, similarly to the cases in Figs. 9(c) and 9(d). Therefore, the sound waves radiated from the plate vibrating in the lattice mode have main lobes only in four symmetric directions with a specific angle from the *z*-axis in the *y*-*z* and *xz* planes, as shown in Fig. 10.

Fig. 11. Directivity patterns for various *λ*t/*λ*a. ----- : Experimental, - - - : calculated. (a) changing *λ*t/*λ*a = 1.29 (*h*=1 mm), (b) *λ*t/*λ*a = 1.83 (*h*=2 mm), (c) *λ*t/*λ*a = 2.20 (*h*=3 mm), (d) *λ*t/*λ*a = 2.74 (*h*=5 mm).


Table 2. Directivity of the square plate vibrating in the lattice mode.

Intense Aerial Ultrasonic Source and Removal of Unnecessary Gas by the Source 119

examined the effect of the combined use of aerial ultrasonic waves with water mist on the removal of a gas, with the aim of applying it to the removal of an unnecessary gas. As a result, we found that both aerial ultrasonic waves and water mist are necessary to remove a low-hydrophilicity lemon-odor gas and that the maximum removal percentage reached

In this method, intense ultrasonic waves are propagated into air from the acoustic source using the square plate flexurally vibrating in the lattice mode, and water is added dropwise onto the positions of the antinodes of flexural vibration on the plate to generate a mist of water microparticles. Ultrasonic waves are irradiated onto the water microparticles and a gas to enhance the absorption of the gas by the water microparticles. The flexurally vibrating plate has two roles: the radiation of sound waves and the formation of water microparticles. The frequency of collisions between microparticles and gas molecules increases because of differences in the velocity of particles in the sound waves and in their momentum owing to differences in their size, enhancing the aggregation and settlement of

To further increase the removal efficiency for a low-hydrophilicity gas, water mist of small particles (average particle diameter, approximately 3 μm) generated using a water spray system was used, as well as water mist of large particles (average particle diameter, approximately 60 μm) formed by the flexural vibration of the plate. The gas removal efficiency achieved when aerial ultrasonic waves are irradiated onto these water particles is discussed. Here, the particle diameter was calculated using Lang's equation (Lang, 1962).

In this section, I describe the process of the removal of a gas by applying water mist of small particles when intense standing-wave acoustic fields were formed within a gas removal chamber. Also, I discuss the gas removal efficiency achieved upon changing the amounts of the water mist of large and small particles, the electric power input to the transducer, and

Although a toxic gas such as dioxin could be used as the gas to be removed, a gas harmless to humans should be used because the experiments were carried out in a simple laboratory. The gas also was required to have low hydrophilicity so as not to easily aggregate in water and be usable at normal temperatures and pressures, and its gas concentration was required to be easily measurable. To satisfy these conditions, a gas evaporated from lemon oil (Sigma-Aldrich Corporation; Product No., W262528; chemical formula, C10H16O; specific

In this section, I give an outline of the gas removal apparatus used in the experiment.

Figure 12 shows a schematic of the experimental setup. The gas removal apparatus used in this experiment consists of a supply fan with an activated carbon filter (amount, 92 cm3/s), a unit for generating the lemon-odor gas, a digital constant-rate pump for supplying water to the vibrating plate, a water spray system for generating water mist of small particles, an

approximately 40%, (Miura, 2007a).

water microparticles and gas molecules.

the initial gas concentration.

gravity, 0.855) was used.

**3.2 Gas removal apparatus** 

**3.2.1 Outline of apparatus** 

## **2.6.4 Directivities for different wavelengths of flexural vibration of plate**

Next, to examine the directivity for different wavelengths (*λ*t) of the flexural vibration of the plate, the sound pressure on the *y*-*z* plane (*α* = 90°, the case of (a) in Fig. 8), on which main lobes were observed, was measured by changing γ from -90 to 90° for the vibrating plates with a constant frequency with *N* = 10 and *h* in the range of 1-5 mm (see Table 1). The measurement results are shown by the solid lines in Fig. 11: (a) *λ*t/*λ*a = 1.29 (*h* = 1 mm), (b) *λ*t/*λ*a = 1.83 (*h* = 2 mm), (c) *λ*t/*λ*a = 2.20 (*h* = 3 mm), and (d) *λ*t/*λ*a = 2.74 (*h* = 5 mm). The ordinate and abscissa are the same as those in Fig. 9(a). From Figs. 11(a)—11(d), it is found that the radiated sound waves have main lobes in two directions and that their directions are symmetric about the *z*-axis with an angle depending on *λ*t/*λ*a, similarly to the cases in Figs. 9(a) and 9(b). The dashed lines in Fig. 11 represent the calculation results obtained using eq. (9), and are in good agreement with the experimental results.

As shown in Table 2, *γ*m decreased when *λ*t/*λ*a increased, i.e., when *h*, and thereby the wavelength of flexural vibration, increased. The experimental and calculated values of *γ*<sup>m</sup> were in good agreement.

Moreover, the angle range of full-width half-maximum of the sound pressure was compared among the main lobes, as shown in the rightmost column in Table 2. This angle range decreased with increasing *λ*t/*λ*a, indicating that the directivity became sharp.

### **2.7 Summary**

The details of the aerial ultrasonic source with a square plate vibrating in the lattice mode described in this section are summarized as follows.


## **3. Removal of unnecessary gas by intense aerial ultrasonic source**

#### **3.1 Outline**

In this section, I discuss the effect of the intense aerial ultrasonic source described in section 2 on enhancing the removal of an unnecessary gas from air. In previous studies, We

Next, to examine the directivity for different wavelengths (*λ*t) of the flexural vibration of the plate, the sound pressure on the *y*-*z* plane (*α* = 90°, the case of (a) in Fig. 8), on which main lobes were observed, was measured by changing γ from -90 to 90° for the vibrating plates with a constant frequency with *N* = 10 and *h* in the range of 1-5 mm (see Table 1). The measurement results are shown by the solid lines in Fig. 11: (a) *λ*t/*λ*a = 1.29 (*h* = 1 mm), (b) *λ*t/*λ*a = 1.83 (*h* = 2 mm), (c) *λ*t/*λ*a = 2.20 (*h* = 3 mm), and (d) *λ*t/*λ*a = 2.74 (*h* = 5 mm). The ordinate and abscissa are the same as those in Fig. 9(a). From Figs. 11(a)—11(d), it is found that the radiated sound waves have main lobes in two directions and that their directions are symmetric about the *z*-axis with an angle depending on *λ*t/*λ*a, similarly to the cases in Figs. 9(a) and 9(b). The dashed lines in Fig. 11 represent the calculation results obtained

As shown in Table 2, *γ*m decreased when *λ*t/*λ*a increased, i.e., when *h*, and thereby the wavelength of flexural vibration, increased. The experimental and calculated values of *γ*<sup>m</sup>

Moreover, the angle range of full-width half-maximum of the sound pressure was compared among the main lobes, as shown in the rightmost column in Table 2. This angle range

The details of the aerial ultrasonic source with a square plate vibrating in the lattice mode

2. The angle between the nodal lines of the lattice-mode vibration and the sides of the

3. The calculation results for the distributions of the vibration displacement of the plate were confirmed by measuring the distributions of vibration displacement and sound

4. Regarding the directivity of the plate vibrating in a lattice mode, the theoretical values calculated using eq. (9) and the experimental values for the fabricated plates were in

5. The sound waves that were radiated from the vibrating plate had sharp main lobes in four symmetric directions on the *y*-*z* and *x*-*z* planes, with a specific angle from the *z*-

6. The angle between the *z*-axis and the main lobe, *γ*m, decreased when *λ*t/*λ*a increased,

7. The angle range of the main lobe in which the sound pressure was half the maximum value decreased with increasing *λ*t/*λ*a, indicating that the directivity became sharp.

In this section, I discuss the effect of the intense aerial ultrasonic source described in section 2 on enhancing the removal of an unnecessary gas from air. In previous studies, We

i.e., when *h*, and thereby the wavelength of flexural vibration, increased.

**3. Removal of unnecessary gas by intense aerial ultrasonic source** 

**2.6.4 Directivities for different wavelengths of flexural vibration of plate** 

using eq. (9), and are in good agreement with the experimental results.

decreased with increasing *λ*t/*λ*a, indicating that the directivity became sharp.

1. A practical method for designing the flexurally vibrating plate was proposed.

described in this section are summarized as follows.

pressure near the plate surface.

were in good agreement.

**2.7 Summary** 

plate was 45°.

good agreement.

axis.

**3.1 Outline** 

examined the effect of the combined use of aerial ultrasonic waves with water mist on the removal of a gas, with the aim of applying it to the removal of an unnecessary gas. As a result, we found that both aerial ultrasonic waves and water mist are necessary to remove a low-hydrophilicity lemon-odor gas and that the maximum removal percentage reached approximately 40%, (Miura, 2007a).

In this method, intense ultrasonic waves are propagated into air from the acoustic source using the square plate flexurally vibrating in the lattice mode, and water is added dropwise onto the positions of the antinodes of flexural vibration on the plate to generate a mist of water microparticles. Ultrasonic waves are irradiated onto the water microparticles and a gas to enhance the absorption of the gas by the water microparticles. The flexurally vibrating plate has two roles: the radiation of sound waves and the formation of water microparticles. The frequency of collisions between microparticles and gas molecules increases because of differences in the velocity of particles in the sound waves and in their momentum owing to differences in their size, enhancing the aggregation and settlement of water microparticles and gas molecules.

To further increase the removal efficiency for a low-hydrophilicity gas, water mist of small particles (average particle diameter, approximately 3 μm) generated using a water spray system was used, as well as water mist of large particles (average particle diameter, approximately 60 μm) formed by the flexural vibration of the plate. The gas removal efficiency achieved when aerial ultrasonic waves are irradiated onto these water particles is discussed. Here, the particle diameter was calculated using Lang's equation (Lang, 1962).

In this section, I describe the process of the removal of a gas by applying water mist of small particles when intense standing-wave acoustic fields were formed within a gas removal chamber. Also, I discuss the gas removal efficiency achieved upon changing the amounts of the water mist of large and small particles, the electric power input to the transducer, and the initial gas concentration.

Although a toxic gas such as dioxin could be used as the gas to be removed, a gas harmless to humans should be used because the experiments were carried out in a simple laboratory. The gas also was required to have low hydrophilicity so as not to easily aggregate in water and be usable at normal temperatures and pressures, and its gas concentration was required to be easily measurable. To satisfy these conditions, a gas evaporated from lemon oil (Sigma-Aldrich Corporation; Product No., W262528; chemical formula, C10H16O; specific gravity, 0.855) was used.

## **3.2 Gas removal apparatus**

In this section, I give an outline of the gas removal apparatus used in the experiment.

## **3.2.1 Outline of apparatus**

Figure 12 shows a schematic of the experimental setup. The gas removal apparatus used in this experiment consists of a supply fan with an activated carbon filter (amount, 92 cm3/s), a unit for generating the lemon-odor gas, a digital constant-rate pump for supplying water to the vibrating plate, a water spray system for generating water mist of small particles, an

Intense Aerial Ultrasonic Source and Removal of Unnecessary Gas by the Source 121

Exponential horn Thick end face 70 mm Thin end face 9 mm

vibration rod Diameter 10 mm Length 116 mm

1/2-wavelength longitudinal

1/2-wavelength longitudinal

42 mm

14 mm

Exponential horn

vibration rod

Fig. 13. Schematic of ultrasonic transducer.

Bolt-clamped Langevin-type ultrasonic transducer

Upper reflective board

Transverse vibrating plate Square of side 217 mm Thickness 3 mm

Resonance frequency 19.8 kHz

Bolt-clamped Langevin-type ultrasonic transducer

Fig. 14. Positions of reflective boards.

Transverse vibrating plate

Lower reflective

board

the vibrating plate into free space is as follows.

**3.2.3 Distances between flexurally vibrating plate and reflective boards** 

Planar reflective boards were installed above and below the vibrating plate to form an

The relationship between the velocity *c* and wavelength *λ*a of the sound waves radiated from

intense standing-wave acoustic field within the acrylic chamber, as shown in Fig. 14.

acoustic source mainly comprising a 20 kHz BLT transducer and a square plate vibrating flexurally in the lattice mode, an acrylic chamber as the main body, and an exhaust fan for removing air passing through the chamber. The flexurally vibrating plate of the acoustic source, 36 narrow hypodermic needles that supply water to the vibrating plate to form water mist of large particles, and reflective boards used to form standing-wave acoustic fields are included in the acrylic chamber. The digital constant-rate pump is used to supply water at a constant rate to the vibrating plate within the acrylic chamber. In addition, a 2.4 MHz ultrasonic humidifier equipped with a water spray system is used to generate water mist of small particles.

Fig. 12. Schematic of gas removal apparatus.

To measure the gas concentration, a hot-wire semiconductor-type gas sensor is attached to the pipe that connects the acrylic chamber to the exhaust fan.

## **3.2.2 Aerial ultrasonic source used in experiment**

Figure 13 shows a schematic of the ultrasonic source used in the experiment. As shown in the figure, an exponential horn (diameter of thick end face, 70 mm; diameter of thin end face, 10 mm; length, 150 mm; amplification rate, 7.0) and a half-wavelength longitudinal vibration rod (diameter, 10 mm; length, 116 mm) were connected to a 20 kHz BLT transducer (D45520). A duralumin square plate vibrating flexurally in the lattice mode (plate constant, *CP* = 1.51 × 106 Hz・mm; *N* = 12; *h* = 3 mm; *L* = 217 mm; resonant frequency, 19.8 kHz) was screwed onto the end of the rod.

Fig. 13. Schematic of ultrasonic transducer.

acoustic source mainly comprising a 20 kHz BLT transducer and a square plate vibrating flexurally in the lattice mode, an acrylic chamber as the main body, and an exhaust fan for removing air passing through the chamber. The flexurally vibrating plate of the acoustic source, 36 narrow hypodermic needles that supply water to the vibrating plate to form water mist of large particles, and reflective boards used to form standing-wave acoustic fields are included in the acrylic chamber. The digital constant-rate pump is used to supply water at a constant rate to the vibrating plate within the acrylic chamber. In addition, a 2.4 MHz ultrasonic humidifier equipped with a water spray system is used to generate water

To measure the gas concentration, a hot-wire semiconductor-type gas sensor is attached to

**Drainage Air flow direction**

**Lemon oil**

**Hypodermic needles 1/2-wavelength longitudinal vibration rod**

**Gas generating Supply fan Gas removal chamber area**

**Unit for generating water mist of small particles**

**Exponential horn**

**Bolt-clamped Langevin-type ultrasonic transducer** 

Figure 13 shows a schematic of the ultrasonic source used in the experiment. As shown in the figure, an exponential horn (diameter of thick end face, 70 mm; diameter of thin end face, 10 mm; length, 150 mm; amplification rate, 7.0) and a half-wavelength longitudinal vibration rod (diameter, 10 mm; length, 116 mm) were connected to a 20 kHz BLT transducer (D45520). A duralumin square plate vibrating flexurally in the lattice mode (plate constant, *CP* = 1.51 × 106 Hz・mm; *N* = 12; *h* = 3 mm; *L* = 217 mm; resonant frequency, 19.8

mist of small particles.

**Digital constantrate pump**

**Water**

Fig. 12. Schematic of gas removal apparatus.

kHz) was screwed onto the end of the rod.

the pipe that connects the acrylic chamber to the exhaust fan.

**Transverse vibrating plate (Unit for generating water mist of large particles)**

**3.2.2 Aerial ultrasonic source used in experiment** 

**Exhaust fan**

**Gas concentration measurement section** 

**Gas sensor**

Fig. 14. Positions of reflective boards.

## **3.2.3 Distances between flexurally vibrating plate and reflective boards**

Planar reflective boards were installed above and below the vibrating plate to form an intense standing-wave acoustic field within the acrylic chamber, as shown in Fig. 14.

The relationship between the velocity *c* and wavelength *λ*a of the sound waves radiated from the vibrating plate into free space is as follows.

Intense Aerial Ultrasonic Source and Removal of Unnecessary Gas by the Source 123

In the experiment, water mist to absorb the gas was generated by supplying water to the vibrating plate. Figure 15 shows the positions of the narrow hypodermic needles installed to supply water to the vibrating plate. The dashed lines in the figure represent nodal lines of vibration. Thirty-six narrow needles are positioned exactly above the antinodes at the

**3.3 Effect of combined use of aerial ultrasonic waves and water mist on removal of** 

In this section, on the basis of experimental results, We examine the effects of using intense aerial ultrasonic waves and two types of water mist of particles with different diameters on

The process for removing the lemon-odor gas by irradiating aerial ultrasonic waves was

Fig. 16. Gas removal process. The input electric power is 50 W and the driving frequency is

Because the rate of vaporization of the generated gas was unstable when the fan started operating, the apparatus was left to stand for approximately 180 s to stabilize the rate of generation of the gas. The time at which the vaporization rate was stabilized was assigned a time of -180 s, and the apparatus was left to stand for another 180 s. The electric power was

examined in the presence and absence of water mist of large and small particles.

**3.2.4 Position of narrow needles** 

the removal of an unnecessary gas.

**3.3.1 Gas removal process** 

**unnecessary gas** 

19.8 kHz.

centers of the regions surrounded by the nodal lines.

$$
\mathcal{A}\_a = \frac{c}{f} \tag{11}
$$

Here, *c* = 331.5 + 0.6*t* (temperature (t) = 25 °C) and *f* is the frequency. The wavelength of the sound waves transmitted in the vertical direction, *λy*, is given by

$$
\mathcal{A}\_y = \frac{\mathcal{A}\_u}{\sin \theta} \tag{12}
$$

where *θ* is the radiation angle. Assuming the wavelength of the flexural vibration of the plate to be *λt*, *θ* is expressed by

$$\theta = \cos^{-1}\left(\frac{\mathcal{A}\_u}{\mathcal{A}\_t}\right) \tag{13}$$

To form standing waves between the vibrating plate and the upper reflective board, the distance between them, *W*, must be set at an integral multiple of the half-wavelength of the sound waves transmitted in the vertical direction and is given as follows.

$$\mathcal{W} = \frac{\lambda\_y}{2} n \text{ ( $n$  is an arbitrary integer)}\tag{14}$$

Here, *W* is set to 42 mm, which is double the wavelength of the sound waves transmitted in the vertical direction. In this case, the electric impedance of the transducer is maximized. The distance between the vibrating plate and the lower reflective board is set to 14 mm, at which the electric impedance of the transducer is minimized.

Fig. 15. Positions of narrow needles.

## **3.2.4 Position of narrow needles**

122 Ultrasonic Waves

(11)

(12)

(*n* is an arbitrary integer) (14)

(13)

*a c f*

Here, *c* = 331.5 + 0.6*t* (temperature (t) = 25 °C) and *f* is the frequency. The wavelength of the

sin *<sup>a</sup> <sup>y</sup>* 

where *θ* is the radiation angle. Assuming the wavelength of the flexural vibration of the

<sup>1</sup> cos *<sup>a</sup>*

 

To form standing waves between the vibrating plate and the upper reflective board, the distance between them, *W*, must be set at an integral multiple of the half-wavelength of the

Here, *W* is set to 42 mm, which is double the wavelength of the sound waves transmitted in the vertical direction. In this case, the electric impedance of the transducer is maximized. The distance between the vibrating plate and the lower reflective board is set to 14 mm, at

*t* 

sound waves transmitted in the vertical direction and is given as follows.

2 *<sup>y</sup> W n* 

which the electric impedance of the transducer is minimized.

sound waves transmitted in the vertical direction, *λy*, is given by

plate to be *λt*, *θ* is expressed by

Fig. 15. Positions of narrow needles.

In the experiment, water mist to absorb the gas was generated by supplying water to the vibrating plate. Figure 15 shows the positions of the narrow hypodermic needles installed to supply water to the vibrating plate. The dashed lines in the figure represent nodal lines of vibration. Thirty-six narrow needles are positioned exactly above the antinodes at the centers of the regions surrounded by the nodal lines.

#### **3.3 Effect of combined use of aerial ultrasonic waves and water mist on removal of unnecessary gas**

In this section, on the basis of experimental results, We examine the effects of using intense aerial ultrasonic waves and two types of water mist of particles with different diameters on the removal of an unnecessary gas.

### **3.3.1 Gas removal process**

The process for removing the lemon-odor gas by irradiating aerial ultrasonic waves was examined in the presence and absence of water mist of large and small particles.

Fig. 16. Gas removal process. The input electric power is 50 W and the driving frequency is 19.8 kHz.

Because the rate of vaporization of the generated gas was unstable when the fan started operating, the apparatus was left to stand for approximately 180 s to stabilize the rate of generation of the gas. The time at which the vaporization rate was stabilized was assigned a time of -180 s, and the apparatus was left to stand for another 180 s. The electric power was

Intense Aerial Ultrasonic Source and Removal of Unnecessary Gas by the Source 125

Figure 17 shows the experimental results, where the ordinate represents the concentration of lemon-odor gas and the abscissa represents the elapsed time, with the amount of water mist of small particles as a parameter. In all cases, the gas concentration changed negligibly between -180 and 0 s, during which no ultrasonic waves were irradiated. When ultrasonic waves and the water mist were generated at 0 s, however, the gas concentration sharply

Fig. 18. Relationship between amount of water mist of small particles and removal rate. The amount of water mist of large particles is 2.78 cm3/s, and the driving frequency is 19.8 kHz.

Similar experiments were also performed for input electric powers of 20, 30, and 40 W as well as above 50 W. The gas removal rate *Px* was defined as the index describing the

> *G G <sup>P</sup> G*

*a b a*

*<sup>x</sup>* (15)

**3.3.3 Gas removal rate with different amounts of water mist of small particles** 

**3.3.2 Gas removal efficiency with different amounts of water mist of small particles**  To examine the gas removal efficiency with different amounts of the water mist of small particles, a gas removal experiment was carried out with the electric power input to the transducer fixed at 50 W and the amount of water mist of large particles maintained at 2.78 cm3/s while the amount of water mist of small particles was set at 0, 0.07, 0.13, 0.20, or 0.27

cm3/s. The experimental procedure followed was that described in sec. 3.3.1.

decreased.

removal rate of gas.

input to the ultrasonic source at a time of 0 s to generate ultrasonic waves in air. Simultaneously, water was supplied to the vibrating plate through the narrow needles attached to the upper reflective board, and water mist was formed by vibrating the plate. Moreover, water mist of small particles was also generated by the water spray system. This condition was maintained for 600 s. Then, the generation of the ultrasonic waves and water mist was stopped, and the state within the acrylic chamber was observed.

The experimental conditions were as follows: the input electric power was maintained at 50 W, the driving frequency was 19.8 kHz, the amount of water mist of large particles was 0 or 2.78 cm3/s, and the amount of water mist of small particles was 0, 0.13, or 0.27 cm3/s. Figure 16 shows the experimental results, where the ordinate represents the concentration of lemon-odor gas and the abscissa represents the elapsed time. When no mist was generated, case of (a), the gas concentration negligibly decreased during the measurement time from 0 to 600 s (with ultrasonic irradiation). When the amounts of water mist of large and small particles were 2.78 and 0 cm3/s, respectively, case of (b), the gas concentration decreased slightly. However, when the amount of water mist of small particles was increased to 0.13, case of (c), and 0.27 cm3/s, case of (d), the gas concentration sharply decreased after the start of ultrasonic irradiation, maintained an almost constant value, then increased to the initial value shortly after the end of ultrasonic irradiation (600 s). The results for water mist of small particles in the absence of water mist of large particles are not show, since aggregation and drainage did not occurred in this case. Therefore, the gas was effectively removed by the combined use of ultrasonic waves and the water mist of both large and small particles.

Fig. 17. Relationship between elapsed time and gas concentration. The input electric power is 50 W, the amount of water mist of large particles is 2.78 cm3/s, and the driving frequency is 19.8 kHz.

input to the ultrasonic source at a time of 0 s to generate ultrasonic waves in air. Simultaneously, water was supplied to the vibrating plate through the narrow needles attached to the upper reflective board, and water mist was formed by vibrating the plate. Moreover, water mist of small particles was also generated by the water spray system. This condition was maintained for 600 s. Then, the generation of the ultrasonic waves and water

The experimental conditions were as follows: the input electric power was maintained at 50 W, the driving frequency was 19.8 kHz, the amount of water mist of large particles was 0 or 2.78 cm3/s, and the amount of water mist of small particles was 0, 0.13, or 0.27 cm3/s. Figure 16 shows the experimental results, where the ordinate represents the concentration of lemon-odor gas and the abscissa represents the elapsed time. When no mist was generated, case of (a), the gas concentration negligibly decreased during the measurement time from 0 to 600 s (with ultrasonic irradiation). When the amounts of water mist of large and small particles were 2.78 and 0 cm3/s, respectively, case of (b), the gas concentration decreased slightly. However, when the amount of water mist of small particles was increased to 0.13, case of (c), and 0.27 cm3/s, case of (d), the gas concentration sharply decreased after the start of ultrasonic irradiation, maintained an almost constant value, then increased to the initial value shortly after the end of ultrasonic irradiation (600 s). The results for water mist of small particles in the absence of water mist of large particles are not show, since aggregation and drainage did not occurred in this case. Therefore, the gas was effectively removed by the combined use of ultrasonic waves and the water mist of both large and

Fig. 17. Relationship between elapsed time and gas concentration. The input electric power is 50 W, the amount of water mist of large particles is 2.78 cm3/s, and the driving frequency

mist was stopped, and the state within the acrylic chamber was observed.

small particles.

is 19.8 kHz.

#### **3.3.2 Gas removal efficiency with different amounts of water mist of small particles**

To examine the gas removal efficiency with different amounts of the water mist of small particles, a gas removal experiment was carried out with the electric power input to the transducer fixed at 50 W and the amount of water mist of large particles maintained at 2.78 cm3/s while the amount of water mist of small particles was set at 0, 0.07, 0.13, 0.20, or 0.27 cm3/s. The experimental procedure followed was that described in sec. 3.3.1.

Figure 17 shows the experimental results, where the ordinate represents the concentration of lemon-odor gas and the abscissa represents the elapsed time, with the amount of water mist of small particles as a parameter. In all cases, the gas concentration changed negligibly between -180 and 0 s, during which no ultrasonic waves were irradiated. When ultrasonic waves and the water mist were generated at 0 s, however, the gas concentration sharply decreased.

Fig. 18. Relationship between amount of water mist of small particles and removal rate. The amount of water mist of large particles is 2.78 cm3/s, and the driving frequency is 19.8 kHz.

#### **3.3.3 Gas removal rate with different amounts of water mist of small particles**

Similar experiments were also performed for input electric powers of 20, 30, and 40 W as well as above 50 W. The gas removal rate *Px* was defined as the index describing the removal rate of gas.

$$P\mathfrak{X} = \frac{G\_a - G\_b}{G\_a} \tag{15}$$

Intense Aerial Ultrasonic Source and Removal of Unnecessary Gas by the Source 127

transducer fixed at 50 W and the amount of water mist of small particles maintained at 0.27 cm3/s while the amount of water mist of large particles was set at 0.83, 1.39, 1.95, or 2.78

Figure 19 shows the experimental results, where the ordinate represents the concentration of lemon-odor gas and the abscissa represents the elapsed time, with the amount of water mist of large particles as a parameter. In all cases, the gas concentration changed negligibly between -180 and 0 s, during which no ultrasonic waves were irradiated. When ultrasonic waves and the water mist were generated at 0 s, however, the gas concentration sharply

Next, similar experiments were performed for input electric powers of 30 and 40 as well as 50 W, to obtain the gas removal rate. The gas removal rate was then calculated using

Figure 20 shows the experimental results, where the ordinate represents the gas removal rate and the abscissa represents the amount of water mist of large particles, with the input electric power as a parameter. When the input electric power was in the range examined in this experiment and the amount of water mist of small particles was constant, the gas removal rate tended to increase as the amount of water mist of large particles and the input

Fig. 20. Relationship between amount of water mist of large particles and removal rate. The Amount of water mist of small particles is 0.27 cm3/s, and the driving frequency is 19.8 kHz.

cm3/s. The experimental procedure followed was that described in sec. 3.3.1.

decreased and reached a constant value after approximately 200 s.

eq. (15).

electric power increased.

Here, *Ga* and *Gb* are the average concentrations of lemon-odor gas [g/m3] between -180 and 0 s and between 500 and 600 s, respectively. Equation (15) indicates that the greater the value of *Px*, the higher the gas removal rate.

Figure 18 shows the experimental results, where the ordinate represents the gas removal rate and the abscissa represents the amount of water mist of small particles, with the electric power input to the transducer as a parameter. When the amount of water mist of large particles was constant, the gas removal rate increased with the amount of water mist of small particles; however, no marked difference in the gas removal rate was observed within the range of input electric power examined in this experiment. The gas removal rate reached approximately 90% when the amount of water mist of small particles was 0.27 cm3/s and the input electric power was 50 W. Moreover, the gas was removed when the amount of water mist of small particles was 0 cm3/s, i.e., when only water mist of large particles was used; in this case, the gas removal rate reached approximately 60% when the input electric power was 50 W. These results indicate that most of the gas can be removed by using water mist of both large and small particles.

Fig. 19. Relationship between elapsed time and gas concentration. The input electric power is 50 W, the amount of water mist of small particles is 0.27 cm3/s, and the driving frequency is 19.8 kHz.

#### **3.3.4 Gas removal rate with different amounts of water mist of large particles**

To examine the gas removal efficiency with different amounts of water mist of large particles, a gas removal experiment was carried out with the electric power input to the

Here, *Ga* and *Gb* are the average concentrations of lemon-odor gas [g/m3] between -180 and 0 s and between 500 and 600 s, respectively. Equation (15) indicates that the greater the

Figure 18 shows the experimental results, where the ordinate represents the gas removal rate and the abscissa represents the amount of water mist of small particles, with the electric power input to the transducer as a parameter. When the amount of water mist of large particles was constant, the gas removal rate increased with the amount of water mist of small particles; however, no marked difference in the gas removal rate was observed within the range of input electric power examined in this experiment. The gas removal rate reached approximately 90% when the amount of water mist of small particles was 0.27 cm3/s and the input electric power was 50 W. Moreover, the gas was removed when the amount of water mist of small particles was 0 cm3/s, i.e., when only water mist of large particles was used; in this case, the gas removal rate reached approximately 60% when the input electric power was 50 W. These results indicate that most of the gas can be removed by using water

Fig. 19. Relationship between elapsed time and gas concentration. The input electric power is 50 W, the amount of water mist of small particles is 0.27 cm3/s, and the driving frequency

To examine the gas removal efficiency with different amounts of water mist of large particles, a gas removal experiment was carried out with the electric power input to the

**3.3.4 Gas removal rate with different amounts of water mist of large particles** 

value of *Px*, the higher the gas removal rate.

mist of both large and small particles.

is 19.8 kHz.

transducer fixed at 50 W and the amount of water mist of small particles maintained at 0.27 cm3/s while the amount of water mist of large particles was set at 0.83, 1.39, 1.95, or 2.78 cm3/s. The experimental procedure followed was that described in sec. 3.3.1.

Figure 19 shows the experimental results, where the ordinate represents the concentration of lemon-odor gas and the abscissa represents the elapsed time, with the amount of water mist of large particles as a parameter. In all cases, the gas concentration changed negligibly between -180 and 0 s, during which no ultrasonic waves were irradiated. When ultrasonic waves and the water mist were generated at 0 s, however, the gas concentration sharply decreased and reached a constant value after approximately 200 s.

Next, similar experiments were performed for input electric powers of 30 and 40 as well as 50 W, to obtain the gas removal rate. The gas removal rate was then calculated using eq. (15).

Figure 20 shows the experimental results, where the ordinate represents the gas removal rate and the abscissa represents the amount of water mist of large particles, with the input electric power as a parameter. When the input electric power was in the range examined in this experiment and the amount of water mist of small particles was constant, the gas removal rate tended to increase as the amount of water mist of large particles and the input electric power increased.

Fig. 20. Relationship between amount of water mist of large particles and removal rate. The Amount of water mist of small particles is 0.27 cm3/s, and the driving frequency is 19.8 kHz.

Intense Aerial Ultrasonic Source and Removal of Unnecessary Gas by the Source 129

Fig. 22. Relationship between initial gas concentration and removal rate. The input electric power is 50 W, the amount of water mist of large particles is 2.78 cm3/s, the amount of

0 0.1 0.2 0.3 0.4 Initial gas concentration [g/cm3]

In this section, I examined the effect on the increase in gas removal rate obtained when a low-hydrophilicity gas was absorbed and aggregated by irradiating aerial ultrasonic waves onto water mist of large particles (average particle diameter, approximately 60 μm) formed by the vibration of the plate and water mist of small particles (average particle diameter,

1. Even a low-hydrophilicity gas was effectively removed by the combined use of

2. The gas removal rate increased when two kinds of water mist of large and small

3. When the amount of water mist of large particles was constant, the gas removal rate increased with the electric power input to the transducer and the amount of water mist of small particles for the ranges of input power and amount examined in this study.

water mist of small particles is 0.27 cm3/s, and driving frequency is 19.8 kHz.

approximately 3 μm) generated using a water spray system.

The following findings were obtained.

0

50

Removal rate [%]

100

ultrasonic waves and water mist.

particles were simultaneously used.

**3.4 Summary** 

#### **3.3.5 Examination of gas removal efficiency with different initial gas concentrations**

To examine the gas removal efficiency with different initial concentrations of lemon-odor gas, a gas removal experiment was performed. In this experiment, the area of the base of the petri dish into which the lemon oil was placed was changed to vary the amount of generated gas. The experimental procedure was the same as before.

Figure 21 shows the experimental results, where the ordinate represents the concentration of lemon-odor gas and the abscissa represents the elapsed time, with the initial gas concentration as a parameter. In all cases, the gas concentration remained almost constant between -180 and 0 s, during which no ultrasonic waves were irradiated. When ultrasonic waves and the water mist were generated at 0 s, however, the gas concentration sharply decreased.

Next, a similar experiment was carried out to obtain the gas removal rate with different initial gas concentrations. The gas removal rate was calculated from the experimental results using eq. (15). Figure 22 shows the results, where the ordinate represents the gas removal rate and the abscissa represents the initial gas concentration. When the amounts of water mist of small and large particles and the input electric power were constant, the gas removal rate tended to increase with the initial gas concentration in the range examined in this experiment. This was considered to be because the higher the gas concentration, the greater the number of opportunities for the water mist to absorb gas particles.

Fig. 21. Relationship between elapsed time and gas concentration. The input electric power is 50 W, the amount of water mist of large particles is 2.78 cm3/s, the amount of water mist of small particles is 0.27 cm3/s, and the driving frequency is 19.8 kHz.

Fig. 22. Relationship between initial gas concentration and removal rate. The input electric power is 50 W, the amount of water mist of large particles is 2.78 cm3/s, the amount of water mist of small particles is 0.27 cm3/s, and driving frequency is 19.8 kHz.

## **3.4 Summary**

128 Ultrasonic Waves

Figure 21 shows the experimental results, where the ordinate represents the concentration of lemon-odor gas and the abscissa represents the elapsed time, with the initial gas concentration as a parameter. In all cases, the gas concentration remained almost constant between -180 and 0 s, during which no ultrasonic waves were irradiated. When ultrasonic waves and the water

Next, a similar experiment was carried out to obtain the gas removal rate with different initial gas concentrations. The gas removal rate was calculated from the experimental results using eq. (15). Figure 22 shows the results, where the ordinate represents the gas removal rate and the abscissa represents the initial gas concentration. When the amounts of water mist of small and large particles and the input electric power were constant, the gas removal rate tended to increase with the initial gas concentration in the range examined in this experiment. This was considered to be because the higher the gas concentration, the greater

Fig. 21. Relationship between elapsed time and gas concentration. The input electric power is 50 W, the amount of water mist of large particles is 2.78 cm3/s, the amount of water mist

of small particles is 0.27 cm3/s, and the driving frequency is 19.8 kHz.

**3.3.5 Examination of gas removal efficiency with different initial gas concentrations**  To examine the gas removal efficiency with different initial concentrations of lemon-odor gas, a gas removal experiment was performed. In this experiment, the area of the base of the petri dish into which the lemon oil was placed was changed to vary the amount of

generated gas. The experimental procedure was the same as before.

mist were generated at 0 s, however, the gas concentration sharply decreased.

the number of opportunities for the water mist to absorb gas particles.

In this section, I examined the effect on the increase in gas removal rate obtained when a low-hydrophilicity gas was absorbed and aggregated by irradiating aerial ultrasonic waves onto water mist of large particles (average particle diameter, approximately 60 μm) formed by the vibration of the plate and water mist of small particles (average particle diameter, approximately 3 μm) generated using a water spray system.

The following findings were obtained.


Intense Aerial Ultrasonic Source and Removal of Unnecessary Gas by the Source 131

Ito, Y. & Takamura, E. (2010). Removal of Liquid in a Long Pore Opened at Both Ends Using

Kobayashi, M., Kamata, C. & Ito, K. (1997). Cold Model Experiments of Gas Removal from

Lang, R. J. (1962). Ultrasonic Atomization of Liquids. *The Journal of the Acoustical Society of* 

Miura, H. (1994). Aerial Ultrasonic Vibration Source using a Square Plate Vibrating in a

Miura, H., & Honda, Y. (2002). Aerial Ultrasonic Source Using a Striped Mode Transverse

Miura, H. (2003). Eggshell Cutter Using Ultrasonic Vibration. *Japanese Journal of Applied* 

Miura, H. (2004). Promotion of Sedimentation of Dispersed Fine Particles Using Underwater

Miura, H., Takata, M., Tajima, D. & Tsuyuki, K. (2006). Promotion of Methane Hydrate

Miura, H. (2007a). Removal of Unnecessary Gas by Spraying Water Particles Formed by

Miura, H. (2007b). Atomization of High-Viscosity Materials by One Point Convergence of

Miura, H. (2008). Vibration Characteristics of Stepped Horn Joined Cutting Tip Employed in

Miura, H. & Ishikawa, H. (2009). Aerial Ultrasonic Source Using Stripes-Mode Transverse

Onishi, Y. & Miura, H. (2005). Convergence of Sound Waves Radiated from Aerial

Ueha, S., Maehara, N. & Mori,E. (1985). Mechanism of Ultrasonic Atomization using a

*Japan (ISIJ) International,* Vol. 37, No. 1, (January 1997), pp. 9-1 5.

07HE22, (July 2010), pp. 07HE22-1-6.

*America*, Vol. 34, No. 1, (January 1962), pp. 6-8.

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47, No. 5, (May 2008), pp. 4282-4286.

(January 1985), pp. 21-26.

48, 07GM10, (July 2009), pp. 07GM10-1-4.

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4688.

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High-Intensity Aerial Ultrasonic Waves. *Japanese Journal of Applied Physics*, Vol. 49,

Molten Metal by an Irradiation of Ultrasonic Waves. *The Iron and Steel Institute of* 

Transverse Lattice-Mode. *The Journal of the Acoustical Society of Japan*, Vol. 50, No. 9,

Vibrating Plate with Two Driving Frequencies. *Japanese Journal of Applied Physics,* 

Ultrasonic Wave. *Japanese Journal of Applied Physics,* Vol. 43, Part 1, No. 5B, (May

Dissociation by Underwater Ultrasonic Wave. *Japanese Journal of Applied Physics,*

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Circular Cutting Using Ultrasonic Vibration. *Japanese Journal of Applied Physics,* Vol.

Vibrating Plate with Jutting Driving Point. *Japanese Journal of Applied Physics,* Vol.

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Multipinhole Plate. *Journal of the Acoustical Society of Japan (E)*, Vol. 6, No. 1,


From these results, it was found that a gas can be effectively removed by irradiating ultrasonic waves onto water mist of both large and small particles.

## **4. Conclusion**

In this chapter, I described a method of designing square plates vibrating flexurally in the lattice mode that can be used as an intense aerial ultrasonic source, as well as the vibration mode of the fabricated plates. The distributions of the vibration displacement of the fabricated plates and the sound pressure near the plate surface were also described. In addition, the validity of the ultrasonic source was verified by theoretically and experimentally determining the directivity of the sound waves radiated from the vibrating plates into remote acoustic fields.

As an application of intense aerial ultrasonic waves, the enhancement of the removal of an unnecessary gas was examined. It was demonstrated that up to 90% of the gas was removed when aerial ultrasonic waves were irradiated onto water mist of large particles (average particle diameter, approximately 60 μm) formed by the flexural vibration of the plate and water mist of small particles (average particle diameter, approximately 3 μm) separately generated using a water spray system.

Thus, the use of aerial ultrasonic energy to remove unwanted gases appears to be promising; however, many unclear points still remain. Their clarification is expected to lead to an expansion in the range of applications of ultrasonic energy.

## **5. References**


4. When the amount of water mist of small particles was constant, the gas removal rate tended to increase with the electric power input to the transducer and the amount of

5. When the two kinds of water mist of large and small particles were used and subjected to ultrasonic irradiation, the gas removal rate increased with the initial gas concentration. 6. The gas removal rate reached approximately 90% when the amounts of water mist of small and large particles were 0.27 and 2.78 cm3/s, respectively, and the electric power

From these results, it was found that a gas can be effectively removed by irradiating

In this chapter, I described a method of designing square plates vibrating flexurally in the lattice mode that can be used as an intense aerial ultrasonic source, as well as the vibration mode of the fabricated plates. The distributions of the vibration displacement of the fabricated plates and the sound pressure near the plate surface were also described. In addition, the validity of the ultrasonic source was verified by theoretically and experimentally determining the directivity of the sound waves radiated from the vibrating

As an application of intense aerial ultrasonic waves, the enhancement of the removal of an unnecessary gas was examined. It was demonstrated that up to 90% of the gas was removed when aerial ultrasonic waves were irradiated onto water mist of large particles (average particle diameter, approximately 60 μm) formed by the flexural vibration of the plate and water mist of small particles (average particle diameter, approximately 3 μm) separately

Thus, the use of aerial ultrasonic energy to remove unwanted gases appears to be promising; however, many unclear points still remain. Their clarification is expected to lead

Asami, T. & Miura, H. (2010). Longitudinal Vibration Characteristics Required to Cut a

Asami, T. & Miura, H. (2011). Vibrator Development for Hole Machining by Ultrasonic

David, J. & Cheeke, N. (2002). *Fundamentals and applications of ultrasonic waves,* CRC Press,

Ito, Y. (2005). Experimental Investigation of Deflection of High-Speed Water Current with

Circle by Ultrasonic Vibration. *Japanese Journal of Applied Physics,* Vol. 49, 07HE23,

Longitudinal and Torsional Vibration. *Japanese Journal of Applied Physics,* Vol. 50,

Aerial Ultrasonic Waves. *Japanese Journal of Applied Physics*, Vol. 44, Part 1, No. 6B,

water mist of large particles.

input to the transducer was 50 W.

plates into remote acoustic fields.

generated using a water spray system.

(July 2010), pp. 07HE23-1-7

(June 2005), pp. 4669-4673.

07HE31, (July 2010), pp. 07HE31-1-9

ISBN 0-8493-0130-0, Florida, USA.

**4. Conclusion** 

**5. References** 

ultrasonic waves onto water mist of both large and small particles.

to an expansion in the range of applications of ultrasonic energy.


**7** 

*Algeria* 

**Application of Pulsed** 

*University Ibn Khaldoun of Tiaret* 

N. Sad Chemloul, K. Chaib and K. Mostefa

 **Ultrasonic Doppler Velocimetry to the** 

**Simultaneous Measurement of Velocity** 

 **and Concentration Profiles in Two Phase Flow** 

The improvement of knowledge which governs the transport of the solid-liquid suspensions is the subject of many works having generally led to empirical or semi-empirical models which are valid only for specific conditions. Research carried out on solid-liquid suspensions has investigated the continuous phase and particularly the influence of turbulence modulation. In the works by Elghobashi and Truesdell (1993), Michaelides and Stock (1989), Owen (1969), and Parthasarathy and Faeth (1990a, 1990b), dissipation or production of turbulent kinetic energy in the continuous phase were reported. The effect of particles on the carrier flow turbulence was investigated numerically by Varaksin and Zaichik (2000) and, Lei (2000). Recently PUDV (pulsed ultrasonic Doppler velocimetry) was applied to study fluid flow alone or with solid particles by Takeda (1995), Aritomi et al. (1996), Nakamura (1996), Rolland and Lemmin (1996, 1997), Cellino and Graf (2000), Brito et al (2001), Eckert and Gerbeth (2002), Kikura et al. (1999), Kikura et al. (2004), Xu (2003), Alfonsi et al. (2003). Note that the measurement techniques cited

The aim of this experimental chapter is the simultaneous measurement of the local parameters which are the velocity and the concentration fields of the solid particles in flows in a horizontal pipe. The difference between our application of PUDV and those cited in the above references resides in the use of a new measurement approach of the local concentration of the solid particles. This approach consists of the representation of the local concentration profile by the ratio of the number of solid particles crossing the measurement volume, to the number of solid particles crossing the control volume. PUDV technique was selected rather than hot wire or film, or Laser, or the PIV technique, because the first would be destroyed by the particles, and with the second the ultrasonic signal is more attenuated when the volumetric concentration CV of particles increases. The third technique as reported

The working principle of pulsed ultrasound Doppler velocimetry is to detect and process many ultrasonic echoes issued from pulses reflected by micro particles contained in a

in the above references are limited by the nature of the suspensions.

by Jensen (2004) requires many conditions for its application.

**1. Introduction** 

**2. Experiment** 

Yamane, H., Ito, Y., & Kawamura, M. (1983). Sound Radiation from Rectangular Plate Vibrating in Stripes Mode. *The Journal of the Acoustical Society of Japan*, Vol. 39, No. 6, (June 1983), pp. 380-387.[in Japanese]

## **Application of Pulsed Ultrasonic Doppler Velocimetry to the Simultaneous Measurement of Velocity and Concentration Profiles in Two Phase Flow**

N. Sad Chemloul, K. Chaib and K. Mostefa *University Ibn Khaldoun of Tiaret Algeria* 

## **1. Introduction**

132 Ultrasonic Waves

Yamane, H., Ito, Y., & Kawamura, M. (1983). Sound Radiation from Rectangular Plate

(June 1983), pp. 380-387.[in Japanese]

Vibrating in Stripes Mode. *The Journal of the Acoustical Society of Japan*, Vol. 39, No. 6,

The improvement of knowledge which governs the transport of the solid-liquid suspensions is the subject of many works having generally led to empirical or semi-empirical models which are valid only for specific conditions. Research carried out on solid-liquid suspensions has investigated the continuous phase and particularly the influence of turbulence modulation. In the works by Elghobashi and Truesdell (1993), Michaelides and Stock (1989), Owen (1969), and Parthasarathy and Faeth (1990a, 1990b), dissipation or production of turbulent kinetic energy in the continuous phase were reported. The effect of particles on the carrier flow turbulence was investigated numerically by Varaksin and Zaichik (2000) and, Lei (2000). Recently PUDV (pulsed ultrasonic Doppler velocimetry) was applied to study fluid flow alone or with solid particles by Takeda (1995), Aritomi et al. (1996), Nakamura (1996), Rolland and Lemmin (1996, 1997), Cellino and Graf (2000), Brito et al (2001), Eckert and Gerbeth (2002), Kikura et al. (1999), Kikura et al. (2004), Xu (2003), Alfonsi et al. (2003). Note that the measurement techniques cited in the above references are limited by the nature of the suspensions.

The aim of this experimental chapter is the simultaneous measurement of the local parameters which are the velocity and the concentration fields of the solid particles in flows in a horizontal pipe. The difference between our application of PUDV and those cited in the above references resides in the use of a new measurement approach of the local concentration of the solid particles. This approach consists of the representation of the local concentration profile by the ratio of the number of solid particles crossing the measurement volume, to the number of solid particles crossing the control volume. PUDV technique was selected rather than hot wire or film, or Laser, or the PIV technique, because the first would be destroyed by the particles, and with the second the ultrasonic signal is more attenuated when the volumetric concentration CV of particles increases. The third technique as reported by Jensen (2004) requires many conditions for its application.

## **2. Experiment**

The working principle of pulsed ultrasound Doppler velocimetry is to detect and process many ultrasonic echoes issued from pulses reflected by micro particles contained in a

Application of Pulsed Ultrasonic Doppler Velocimetry to

**2.2 Ultrasonic measurement system of the velocity profiles** 

m/s to 2.5 m/s.

the Simultaneous Measurement of Velocity and Concentration Profiles in Two Phase Flow 135

This study was performed using water as the continuous phase and glass beads for solid particles. The particles are spherical with a 5% sphericity defect and density of 2640 kg/m3. Four samples of different particle size distributions were tested. The volume-averaged mean particle diameters dp corresponding to these samples are 0.27, 0.3, 0.4, and 0.7 mm. These particles were chosen to be larger than the Kolmogorov length scale , estimated to 206.6 m near the wall. The particle diameter is ranging between 1.30 and 3.38. The volumetric concentrations (Cv) of the glass beads used in the suspension are 0.5%, 1%, 1.5% and 2%, and were determined from the volume of the flow circuit. Performed measurements within this work consider a two way coupling, i.e., taking into account the effects of the particles on the carrier fluid and vice versa. The classification of the suspension flow used is made according to Sato (1996), Elghobashi (1994) and Crowe, et al. (1996) who proposed the particle volume fractions as the criterion of classification. The particle volume fractions are ranging between 1.25 x10-3 and 5x10-3. Fine starch particles with 6 m of diameter and density of 1530 kg/m3 were used as a tracer. The maximum concentration of the starch was fixed at 3% in order to reduce the attenuation of the Doppler signal. The flow mean velocity Umoy is ranging from 1

The method used for measurement local velocity of the large particles (larger than the wavelength of the ultrasonic wave) is based on a combination of the measurement technique PUDV (7) (type ECHOVAR CF8) used in medical physics, and a data processor (9). The technical specifications of the velocimeter are: emission frequency 8 MHz, pulse durations 0.5 s, 1 s, 2 s, and pulse repetition frequency 64 s and 32 s. The maximum measurable

The position adjustment of the measurement volume is done by time step of 0.5 s between the emission and reception of the ultrasonic wave. This time step corresponds to the penetration depth of the measurement volume of 0.37 mm for an angle of 67° ( angle between the internal pipe wall and the direction of the propagation of the ultrasonic waves). The value of is nearly equal to that determined by the calibration of the ultrasonic transducer (67.4°). The dimensions of the cylindrical measurement volume are the same as

In this experimental study, the signal is emitting and receiving by the same ultrasonic transducer, in this case the velocity U (axial velocity component) of the solid particles

with fD the Doppler shift frequency, fe the emitted frequency by the transducer, c speed of the sound in the water, and the angle between the ultrasonic beam and the pipe axis.

The use of the PUDV technique for the solid-liquid suspensions remains however, limited by the concentration of the solid particles. Indeed, the preliminary study results show that the concentration of the solid particles and depth of the measurement volume affect the ultrasonic signal. Figure 2 shows that for a concentration higher than 2.5%, the attenuation

D e cf U =

2f cosθ (1)

distance of these two pulse repetition frequency are respectively 48 mm and 24 mm.

those of the ultrasonic transducer, diameter of 2 mm and length of 0.8 mm.

determined from the Doppler frequency is:

flowing liquid. A single transducer emits the ultrasonic pulses and receives the echoes. By sampling the incoming echoes at the same time relative to the emission of the pulses, the variation of the positions of scatterers are measured and therefore their velocities. The measurement of the time lapse between the emission of ultrasonic bursts and the reception of the pulse (echo generated by particles flowing in the liquid) gives the position of the particles. By measuring the Doppler frequency in the echo as a function of time shifts of these particles, a velocity profile after few ultrasonic emissions is obtained.

In this study, PUDV technique originally applied in the medical field is used only for the emission and the reception of the ultrasonic signal. This technique was combined with a data processor for the flow measurement of solid-liquid suspension. This combination allows the determination of a local velocity and a local concentration of the solid particles larger than the wavelength of the ultrasonic wave.

#### **2.1 Flow circuit**

The flow circuit (Fig. 1a.) consists of a closed loop made of glass pipes with an internal diameter D of 20 mm. The flow is driven by a variable speed centrifugal pump (1). The suspension is kept at a constant temperature by the heat exchanger (2) during measurement. The test section (4) (for detail see Fig 1b), realized in a Plexiglas box which is 150 mm long, 100 mm wide, and 50 mm high, was located at 75D downstream of the pump where the flow was fully-developed. Plexiglas was chosen in order to reduce the reflection of the ultrasonic beam when it crosses the wall. The pressure differential along the test pipe given by two differential pressure transducer (3) allows the determination of the wall shear stress.

Fig. 1. a) Flow circuit: 1 pump, 2 heat exchanger, 3 differential pressure transducer, 4 test section, 5 tank of suspension; b) Detail of the test section 4: ( ) glass pipe, ( ) Plexiglas box, ( ) Ultrasonic transducer; c) Ultrasonic measurement with acquisition and treatment system: 6 displacement system of the measurement volume, 7 ultrasonic Doppler velocimeter (ECHOVAR CF8, ALVAR), 8 digital storage oscilloscope, 9 data processor (plurimat S), 10 computer

flowing liquid. A single transducer emits the ultrasonic pulses and receives the echoes. By sampling the incoming echoes at the same time relative to the emission of the pulses, the variation of the positions of scatterers are measured and therefore their velocities. The measurement of the time lapse between the emission of ultrasonic bursts and the reception of the pulse (echo generated by particles flowing in the liquid) gives the position of the particles. By measuring the Doppler frequency in the echo as a function of time shifts of

In this study, PUDV technique originally applied in the medical field is used only for the emission and the reception of the ultrasonic signal. This technique was combined with a data processor for the flow measurement of solid-liquid suspension. This combination allows the determination of a local velocity and a local concentration of the solid particles

The flow circuit (Fig. 1a.) consists of a closed loop made of glass pipes with an internal diameter D of 20 mm. The flow is driven by a variable speed centrifugal pump (1). The suspension is kept at a constant temperature by the heat exchanger (2) during measurement. The test section (4) (for detail see Fig 1b), realized in a Plexiglas box which is 150 mm long, 100 mm wide, and 50 mm high, was located at 75D downstream of the pump where the flow was fully-developed. Plexiglas was chosen in order to reduce the reflection of the ultrasonic beam when it crosses the wall. The pressure differential along the test pipe given by two differential pressure transducer (3) allows the determination of

Fig. 1. a) Flow circuit: 1 pump, 2 heat exchanger, 3 differential pressure transducer, 4 test section, 5 tank of suspension; b) Detail of the test section 4: ( ) glass pipe, ( ) Plexiglas box, ( ) Ultrasonic transducer; c) Ultrasonic measurement with acquisition and treatment

system: 6 displacement system of the measurement volume, 7 ultrasonic Doppler velocimeter (ECHOVAR CF8, ALVAR), 8 digital storage oscilloscope, 9 data processor

these particles, a velocity profile after few ultrasonic emissions is obtained.

larger than the wavelength of the ultrasonic wave.

**2.1 Flow circuit** 

the wall shear stress.

(plurimat S), 10 computer

This study was performed using water as the continuous phase and glass beads for solid particles. The particles are spherical with a 5% sphericity defect and density of 2640 kg/m3. Four samples of different particle size distributions were tested. The volume-averaged mean particle diameters dp corresponding to these samples are 0.27, 0.3, 0.4, and 0.7 mm. These particles were chosen to be larger than the Kolmogorov length scale , estimated to 206.6 m near the wall. The particle diameter is ranging between 1.30 and 3.38. The volumetric concentrations (Cv) of the glass beads used in the suspension are 0.5%, 1%, 1.5% and 2%, and were determined from the volume of the flow circuit. Performed measurements within this work consider a two way coupling, i.e., taking into account the effects of the particles on the carrier fluid and vice versa. The classification of the suspension flow used is made according to Sato (1996), Elghobashi (1994) and Crowe, et al. (1996) who proposed the particle volume fractions as the criterion of classification. The particle volume fractions are ranging between 1.25 x10-3 and 5x10-3. Fine starch particles with 6 m of diameter and density of 1530 kg/m3 were used as a tracer. The maximum concentration of the starch was fixed at 3% in order to reduce the attenuation of the Doppler signal. The flow mean velocity Umoy is ranging from 1 m/s to 2.5 m/s.

## **2.2 Ultrasonic measurement system of the velocity profiles**

The method used for measurement local velocity of the large particles (larger than the wavelength of the ultrasonic wave) is based on a combination of the measurement technique PUDV (7) (type ECHOVAR CF8) used in medical physics, and a data processor (9). The technical specifications of the velocimeter are: emission frequency 8 MHz, pulse durations 0.5 s, 1 s, 2 s, and pulse repetition frequency 64 s and 32 s. The maximum measurable distance of these two pulse repetition frequency are respectively 48 mm and 24 mm.

The position adjustment of the measurement volume is done by time step of 0.5 s between the emission and reception of the ultrasonic wave. This time step corresponds to the penetration depth of the measurement volume of 0.37 mm for an angle of 67° ( angle between the internal pipe wall and the direction of the propagation of the ultrasonic waves). The value of is nearly equal to that determined by the calibration of the ultrasonic transducer (67.4°). The dimensions of the cylindrical measurement volume are the same as those of the ultrasonic transducer, diameter of 2 mm and length of 0.8 mm.

In this experimental study, the signal is emitting and receiving by the same ultrasonic transducer, in this case the velocity U (axial velocity component) of the solid particles determined from the Doppler frequency is:

$$\mathbf{U} = \frac{\mathbf{c}\mathbf{f}\_{\rm D}}{2\mathbf{f}\_{\rm e}\cos\theta} \tag{1}$$

with fD the Doppler shift frequency, fe the emitted frequency by the transducer, c speed of the sound in the water, and the angle between the ultrasonic beam and the pipe axis.

The use of the PUDV technique for the solid-liquid suspensions remains however, limited by the concentration of the solid particles. Indeed, the preliminary study results show that the concentration of the solid particles and depth of the measurement volume affect the ultrasonic signal. Figure 2 shows that for a concentration higher than 2.5%, the attenuation

Application of Pulsed Ultrasonic Doppler Velocimetry to

the power spectral density.

versus the wall distance

have:

spectrum.

the Simultaneous Measurement of Velocity and Concentration Profiles in Two Phase Flow 137

maximum power spectrum were taken into account. The peak amplitude of the power spectra increases with increasing particle diameter; this confirms the difference in energy between the signals coming from the bead glass and those from the starch particles. The separation between the Doppler signals of the continuous phase and large particle is made using two fixed thresholds on the integral of the power spectral density. The higher threshold Ssup and the lower threshold Sinf are respectively given by the following relations:

sup max

  E -E S =E -

E -E S =E +

where Emax and Emin are respectively, the maximal and the minimal value of the integral of

Fig. 3. Doppler power spectrum in the case of glass beads (dp= 0.7 mm, Cv= 1%) and tracer

where K is a constant and the Dirac's impulse. According to the formula of Poisson, we

where X( ) is the Fourier transform of the Doppler signal x(t) . The values treated using the Fourier transform allow to the calculation of the power spectral density P(f) of the Doppler signal from the product of the frequency spectrum and its conjugate. The average frequency Doppler fD is given by the normalized moment of order 1 of the ensemble average

e

<sup>K</sup> x(t) = x(t) (t - ) <sup>f</sup> (3)

<sup>e</sup> x( ) X( ) ( - nf ) (4)

The sampled function x(t) ˆ of the Doppler signal x(t) is given by:

inf min

max min

(2)

max min

3

3

of the signal is about 80% (or 20% of coherent signal), and thus the maximum volumetric concentration of solid particles used in this study was Cv= 2%. Because of the non-uniform distribution of the concentration in the test section, the flow is divided in two regions having the horizontal line passing through the position of the maximum concentration as a boundary. To obtain this boundary line, which corresponds to the great number of the Doppler signal visualised by a digital storage oscilloscope (Fig. 1c), we have scanned the test section by the displacement of the ultrasonic transducer along the vertical diameter. Before each measurement, the boundary line is located to be taken as the first measurement point. The first measurements with a two same ultrasonic transducers (one fixed on the pipe top wall and the other on the pipe bottom wall) shown that at the same position of the measurement volume, the velocity measured presents a difference about 3 - 4 %. For the high concentration, this difference is more significant.

Fig. 2. Signal attenuation function of the distance from the ultrasonic transducer and the volumetric concentration of particle, dp = 0.7 mm

#### **2.2.1 Signal processing**

The treatment of the Doppler signal is done using a data processor associated with the ultrasonic velocimeter. The successive Doppler signals received by the ultrasonic transducer are result from either the same particle reached by successive impulses, or various particles crossing the measurement volume. These signals depend on the particles size, the dimensions of the measurement volume, and the velocity of the particles. To avoid the spectrum overlap phenomenon and thus the loss of information, the output signal of the velocimeter is sampled with a sampling frequency about 5 to 10 times the greatest frequency of the signal spectrum. The numerical data of the sampling process are stored in the memory of the data processor to be treated. The signal processing is made by an elaborate software where frequency domain and Fourier transform were used. Figure 3 shows that the power spectral density obtained from different wall distance have the same trend as that of Gauss with a value correlation coefficient close to 0.97. To satisfy the symmetry condition of the power spectrum, a threshold was fixed, only the values greater than 2/3 of the

of the signal is about 80% (or 20% of coherent signal), and thus the maximum volumetric concentration of solid particles used in this study was Cv= 2%. Because of the non-uniform distribution of the concentration in the test section, the flow is divided in two regions having the horizontal line passing through the position of the maximum concentration as a boundary. To obtain this boundary line, which corresponds to the great number of the Doppler signal visualised by a digital storage oscilloscope (Fig. 1c), we have scanned the test section by the displacement of the ultrasonic transducer along the vertical diameter. Before each measurement, the boundary line is located to be taken as the first measurement point. The first measurements with a two same ultrasonic transducers (one fixed on the pipe top wall and the other on the pipe bottom wall) shown that at the same position of the measurement volume, the velocity measured presents a difference about 3 - 4 %. For the

Fig. 2. Signal attenuation function of the distance from the ultrasonic transducer and the

The treatment of the Doppler signal is done using a data processor associated with the ultrasonic velocimeter. The successive Doppler signals received by the ultrasonic transducer are result from either the same particle reached by successive impulses, or various particles crossing the measurement volume. These signals depend on the particles size, the dimensions of the measurement volume, and the velocity of the particles. To avoid the spectrum overlap phenomenon and thus the loss of information, the output signal of the velocimeter is sampled with a sampling frequency about 5 to 10 times the greatest frequency of the signal spectrum. The numerical data of the sampling process are stored in the memory of the data processor to be treated. The signal processing is made by an elaborate software where frequency domain and Fourier transform were used. Figure 3 shows that the power spectral density obtained from different wall distance have the same trend as that of Gauss with a value correlation coefficient close to 0.97. To satisfy the symmetry condition of the power spectrum, a threshold was fixed, only the values greater than 2/3 of the

high concentration, this difference is more significant.

volumetric concentration of particle, dp = 0.7 mm

**2.2.1 Signal processing** 

maximum power spectrum were taken into account. The peak amplitude of the power spectra increases with increasing particle diameter; this confirms the difference in energy between the signals coming from the bead glass and those from the starch particles. The separation between the Doppler signals of the continuous phase and large particle is made using two fixed thresholds on the integral of the power spectral density. The higher threshold Ssup and the lower threshold Sinf are respectively given by the following relations:

$$\begin{cases} \mathbf{S\_{sup}} = \mathbf{E\_{max}} \cdot \frac{\mathbf{E\_{max}} \cdot \mathbf{E\_{min}}}{\mathcal{B}} \\ \mathbf{S\_{inf}} = \mathbf{E\_{min}} + \frac{\mathbf{E\_{max}} \cdot \mathbf{E\_{min}}}{\mathcal{B}} \end{cases} \tag{2}$$

where Emax and Emin are respectively, the maximal and the minimal value of the integral of the power spectral density.

Fig. 3. Doppler power spectrum in the case of glass beads (dp= 0.7 mm, Cv= 1%) and tracer versus the wall distance

The sampled function x(t) ˆ of the Doppler signal x(t) is given by:

$$\hat{\mathbf{x}}(\mathbf{t}) = \mathbf{x}(\mathbf{t}) \sum \mathbf{\hat{s}}(\mathbf{t} \cdot \frac{\mathbf{K}}{\mathbf{f}\_{\mathbf{e}}}) \tag{3}$$

where K is a constant and the Dirac's impulse. According to the formula of Poisson, we have:

$$\hat{\mathbf{x}}(\mathbf{v}) \boxdot \ \boxdot \ \overleftarrow{\boxdot} \ \mathbf{X}(\mathbf{v}) \sum \delta(\mathbf{v} - \mathbf{n} \mathbf{f}\_{\mathbf{e}}) \tag{4}$$

where X( ) is the Fourier transform of the Doppler signal x(t) . The values treated using the Fourier transform allow to the calculation of the power spectral density P(f) of the Doppler signal from the product of the frequency spectrum and its conjugate. The average frequency Doppler fD is given by the normalized moment of order 1 of the ensemble average spectrum.

Application of Pulsed Ultrasonic Doppler Velocimetry to

along the vertical diameter of the test section (Fig. 4).

transducer

**transducer** 

the Simultaneous Measurement of Velocity and Concentration Profiles in Two Phase Flow 139

n pt p i i=1

with n the number of the measurement point. The local concentration profile is obtained by the plot of the variation of the ratio Np / Npt versus the depth of the measurement volume

Fig. 5. a) Validation of the method and calibration of the ultrasonic transducer: 1 glass tank of calibration, 2 lateral wall of the tank, 3 ultrasonic transducer, 4 micrometric displacement system of the transducer, 5 plastic disc, 6 Abrasive cloth band or cylindrical metal rods, 7 ultrasonic Doppler velocimeter, 8 digital storage oscilloscope, 9 Data processor (Plurimat S); b) Validation curve of the signal processing method; c) Calibration curve of the ultrasonic

**2.2.3 Validation of the processing signal method and calibration of the ultrasonic** 

The validation of the method applied for the signal processing consists of a comparison between the results obtained by the data processor (9) and those of the processing signal system integrated into the medical apparatus (7). For this validation we developed a device illustrated in Fig. 5a, it consists of the plastic disc (5) of 160 mm diameter turning with a variable speed motor, and on which a abrasive cloth band (6) is stuck (in Figure 5a, only the

N = (N ) (7)

$$\overline{\mathbf{f\_D}} = \frac{\sum\_{n=1}^{N} \mathbf{f\_nG(f\_n)}}{\sum\_{n=1}^{N} \mathbf{G(f\_n)}} \tag{5}$$

where N k n n=1 G(f ) = P (f ) <sup>k</sup> and k = 1, 2, 3,...........,N. Considering that the enlargement of the spectrum is due only to the turbulent velocity fluctuations, we can calculate the normalized moment of second order of the energy spectrum integral associated with the Doppler signals. This moment with a Gaussian trend represents the turbulent intensity given by the relation:

$$\sqrt{\mathbf{f\_D^2}} = \left[\sum\_{n=1}^{N} (\mathbf{f\_n} \cdot \overline{\mathbf{f\_d}})^2 \mathbf{G(f\_n)} / \sum\_{n=1}^{N} \mathbf{G(f\_n)}\right]^{1/2} \tag{6}$$

#### **2.2.2 Measurement method of the local concentration profile**

This method consists of determining the ratio of the number of particles Np crossing the measurement volume to the total number of particles Npt crossing the control volume. This control volume is obtained by the displacement of the measurement volume along the vertical diameter of the test section (Fig. 4).

Fig. 4. Determination of the concentration profile: (-) measurement volume Np, (---) control volume Npt , (-..-) test section.

The numbers of particles Np and Npt are obtained by counting the number of the Doppler signal respectively in the volume measurement and in the control volume. For this counting made for each measurement point of the particle velocity, two thresholds are fixed, one on the amplitude of the Doppler signal and the other on the integral of the power spectral density. The total number of particles Npt is given by:

n=1 N

D

where

relation:

N k n n=1

f =

N n

n=1

spectrum is due only to the turbulent velocity fluctuations, we can calculate the normalized moment of second order of the energy spectrum integral associated with the Doppler signals. This moment with a Gaussian trend represents the turbulent intensity given by the

N N

D nd n n n=1 n=1

This method consists of determining the ratio of the number of particles Np crossing the measurement volume to the total number of particles Npt crossing the control volume. This control volume is obtained by the displacement of the measurement volume along the

Fig. 4. Determination of the concentration profile: (-) measurement volume Np, (---) control

The numbers of particles Np and Npt are obtained by counting the number of the Doppler signal respectively in the volume measurement and in the control volume. For this counting made for each measurement point of the particle velocity, two thresholds are fixed, one on the amplitude of the Doppler signal and the other on the integral of the power spectral

'2 2

**2.2.2 Measurement method of the local concentration profile** 

vertical diameter of the test section (Fig. 4).

volume Npt , (-..-) test section.

density. The total number of particles Npt is given by:

n

(5)

f G(f )

n

G(f )

G(f ) = P (f ) <sup>k</sup> and k = 1, 2, 3,...........,N. Considering that the enlargement of the

1 2 f = [ (f - f ) G(f ) / G(f )] (6)

$$\mathbf{N}\_{\mathbf{P}^\mathrm{t}} = \sum\_{\mathbf{i}=1}^{n} (\mathbf{N}\_{\mathbf{P}})\_{\mathbf{i}} \tag{7}$$

with n the number of the measurement point. The local concentration profile is obtained by the plot of the variation of the ratio Np / Npt versus the depth of the measurement volume along the vertical diameter of the test section (Fig. 4).

Fig. 5. a) Validation of the method and calibration of the ultrasonic transducer: 1 glass tank of calibration, 2 lateral wall of the tank, 3 ultrasonic transducer, 4 micrometric displacement system of the transducer, 5 plastic disc, 6 Abrasive cloth band or cylindrical metal rods, 7 ultrasonic Doppler velocimeter, 8 digital storage oscilloscope, 9 Data processor (Plurimat S); b) Validation curve of the signal processing method; c) Calibration curve of the ultrasonic transducer

### **2.2.3 Validation of the processing signal method and calibration of the ultrasonic transducer**

The validation of the method applied for the signal processing consists of a comparison between the results obtained by the data processor (9) and those of the processing signal system integrated into the medical apparatus (7). For this validation we developed a device illustrated in Fig. 5a, it consists of the plastic disc (5) of 160 mm diameter turning with a variable speed motor, and on which a abrasive cloth band (6) is stuck (in Figure 5a, only the

Application of Pulsed Ultrasonic Doppler Velocimetry to

volumetric concentration of the starch is CV = 0.3%.

**3.2.1 Effect of the mean flow velocity** 

= 2 m/s.

heterogeneous regime.

**3.2 Velocity and concentration profiles of large particles** 

represented by the model of Pai.

the Simultaneous Measurement of Velocity and Concentration Profiles in Two Phase Flow 141

suspension is negligible. In the following results the velocity profile of water alone will be

Fig. 6. Velocity and concentration profiles of water-starch suspension, Re= 42000, the

the results corresponding to the particles of diameter 0.13 and 0.4 mm are presented.

Large particles of high density compared to the carrier fluid, i.e., those where the diameter exceeds the wavelength of the ultrasonic wave, do not follow the flow. In this paper, only

A mean flow velocity range of 1 m/s to 2.5 m/s was used. This range corresponds to the heterogeneous and saltation flow regimes. Figure 7 show the influence of the mean flow velocity on the velocity and the concentration profiles of the solid particles of diameter 0.13 and 0.4 mm respectively. The velocity profiles are parabolas with a top around the pipe axis, very near for dp= 0.13 mm and below for dp= 0.4 mm. For the other particle diameter of 0.27 mm and 0.7 mm (which are not presented in this paper), the tops are below the pipe axis. The presence of the particles near the top wall was observed only for dp = 0.13 mm and Umoy

The concentration profiles are in a better agreement with the observations of the flow made during the experimental tests. Indeed, for a constant volumetric concentration and a mean flow velocity ranging between 1 and 2 m/s, the flow of the particles suspension of diameter 0.27 mm, 0.4 mm and 0.7 mm is a saltation regime. It becomes heterogeneous when the flow mean velocity exceeds 2 m/s. For the particles of diameter 0.13 mm, the flow suspension is

cylindrical metal rods used for the calibration of the ultrasonic transducer are represented). The fine emery grains act as the centers of diffraction moving with a known tangential velocity in the volume measurement. This intersection between the emery grains and the volume measurement is obtained by a micrometric displacement system of the ultrasonic transducer (4) fixed on one of the lateral walls of the tank calibration (2). The Doppler frequency is determined by the acquisition and processing system (9). Figure 5b shows that, in a large tension range, the results obtained from the signal processing of the data processor have a better linearity than those from the signal processing system integrated into the medical apparatus.

For the calibration of the ultrasonic transducer (3), which consists of determining the exact value of the inclination angle of the ultrasonic transducer, the same device as that of the validation of the signal processing method was used, except for the plastic disc where the abrasive cloth band is replaced by cylindrical metal rods (6) of 1 mm diameter and 15 mm length. Figure 5c shows a better linearity between a known tangential velocity Ut of the rods crossing the measurement volume and the value of Ucos determined from the signal processing. The inclination angle of the ultrasonic transducer (*=* 67.4°) determined from the calibration curve is very close to that used in the medical apparatus (*= 67°*).

## **3. Experimental results**

#### **3.1 Velocity and concentration profiles of fine particles**

The rheological study of the starch suspension in water made in a rotating viscometer of coaxial cylinders (Haake RV12), showed that the suspension have a shear thickening behaviour described by the power law rheological model.

$$
\pi = \mathbb{k} \text{ } \varepsilon^n \text{ } \tag{8}
$$

where is the shear stress, k is the flow consistency index and n is the flow behaviour index. These two rheological parameters determined from the flow curves, k = 0.33x10-3 kg/ms and n = 1.1, are necessary to determine the velocity profile using the model of Pai (Brodkey, Lee, Chase (1961) and Brodkey (1963)) valid for the Newtonian and non-Newtonian fluids when the Reynolds number Re is lower than 105. This model is given by:

$$\frac{\mathbf{U(r)}}{\mathbf{U\_{max}}} = \mathbf{1} + \mathbf{a}\_1 \left(\frac{\mathbf{r}}{\mathbf{R}}\right)^{\frac{\mathbf{n}+1}{\mathbf{n}}} + \mathbf{a}\_2 \left(\frac{\mathbf{r}}{\mathbf{R}}\right)^{2\mathbf{m}} \tag{9}$$

where Umax is the maximum velocity generally taken on the pipe axis, R the pipe radius, and r = R - y the variable pipe radius (y is the wall normal distance). The constants a1, a2 and m depend on the nature of the fluid, the boundary conditions and the flow mean velocity. Figure 6 shows the velocity and the concentration profiles of the water-starch suspension obtained along the test section diameter. The velocity profile coincides well with the theoretical profile of Pai, and the concentration profile has a uniform distribution.

According to Furuta et al. (1977) we can thus assume that fine particle suspension which represents a tracer behaves as a homogeneous fluid and the slip velocity of the solid - liquid suspension is negligible. In the following results the velocity profile of water alone will be represented by the model of Pai.

Fig. 6. Velocity and concentration profiles of water-starch suspension, Re= 42000, the volumetric concentration of the starch is CV = 0.3%.

## **3.2 Velocity and concentration profiles of large particles**

Large particles of high density compared to the carrier fluid, i.e., those where the diameter exceeds the wavelength of the ultrasonic wave, do not follow the flow. In this paper, only the results corresponding to the particles of diameter 0.13 and 0.4 mm are presented.

## **3.2.1 Effect of the mean flow velocity**

140 Ultrasonic Waves

cylindrical metal rods used for the calibration of the ultrasonic transducer are represented). The fine emery grains act as the centers of diffraction moving with a known tangential velocity in the volume measurement. This intersection between the emery grains and the volume measurement is obtained by a micrometric displacement system of the ultrasonic transducer (4) fixed on one of the lateral walls of the tank calibration (2). The Doppler frequency is determined by the acquisition and processing system (9). Figure 5b shows that, in a large tension range, the results obtained from the signal processing of the data processor have a better linearity than those from the signal processing system integrated into the

For the calibration of the ultrasonic transducer (3), which consists of determining the exact value of the inclination angle of the ultrasonic transducer, the same device as that of the validation of the signal processing method was used, except for the plastic disc where the abrasive cloth band is replaced by cylindrical metal rods (6) of 1 mm diameter and 15 mm length. Figure 5c shows a better linearity between a known tangential velocity Ut of the rods crossing the measurement volume and the value of Ucos determined from the signal processing. The inclination angle of the ultrasonic transducer (*=* 67.4°) determined from the

The rheological study of the starch suspension in water made in a rotating viscometer of coaxial cylinders (Haake RV12), showed that the suspension have a shear thickening

where is the shear stress, k is the flow consistency index and n is the flow behaviour index. These two rheological parameters determined from the flow curves, k = 0.33x10-3 kg/ms and n = 1.1, are necessary to determine the velocity profile using the model of Pai (Brodkey, Lee, Chase (1961) and Brodkey (1963)) valid for the Newtonian and non-Newtonian fluids

1 2

where Umax is the maximum velocity generally taken on the pipe axis, R the pipe radius, and r = R - y the variable pipe radius (y is the wall normal distance). The constants a1, a2 and m depend on the nature of the fluid, the boundary conditions and the flow mean velocity. Figure 6 shows the velocity and the concentration profiles of the water-starch suspension obtained along the test section diameter. The velocity profile coincides well with the

According to Furuta et al. (1977) we can thus assume that fine particle suspension which represents a tracer behaves as a homogeneous fluid and the slip velocity of the solid - liquid

 

U(r) r <sup>r</sup> =1+a +a U RR

theoretical profile of Pai, and the concentration profile has a uniform distribution.

n+1 2m <sup>n</sup>

<sup>n</sup> = k (8)

(9)

calibration curve is very close to that used in the medical apparatus (*= 67°*).

when the Reynolds number Re is lower than 105. This model is given by:

max

**3.1 Velocity and concentration profiles of fine particles** 

behaviour described by the power law rheological model.

medical apparatus.

**3. Experimental results** 

A mean flow velocity range of 1 m/s to 2.5 m/s was used. This range corresponds to the heterogeneous and saltation flow regimes. Figure 7 show the influence of the mean flow velocity on the velocity and the concentration profiles of the solid particles of diameter 0.13 and 0.4 mm respectively. The velocity profiles are parabolas with a top around the pipe axis, very near for dp= 0.13 mm and below for dp= 0.4 mm. For the other particle diameter of 0.27 mm and 0.7 mm (which are not presented in this paper), the tops are below the pipe axis. The presence of the particles near the top wall was observed only for dp = 0.13 mm and Umoy = 2 m/s.

The concentration profiles are in a better agreement with the observations of the flow made during the experimental tests. Indeed, for a constant volumetric concentration and a mean flow velocity ranging between 1 and 2 m/s, the flow of the particles suspension of diameter 0.27 mm, 0.4 mm and 0.7 mm is a saltation regime. It becomes heterogeneous when the flow mean velocity exceeds 2 m/s. For the particles of diameter 0.13 mm, the flow suspension is heterogeneous regime.

Application of Pulsed Ultrasonic Doppler Velocimetry to

profiles tend to that of the carrying fluid.

profiles for Cv= 1%, Re = 44600.

**4. Conclusion** 

**3.2.3 Effect of diameter** 

the Simultaneous Measurement of Velocity and Concentration Profiles in Two Phase Flow 143

The figure 9a shows that for all the particle diameters used, the velocity profiles of the solid particles have the same trend as that of the carrying fluid. The difference between the velocity profile of the continuous phase and that of the solid phase confirms the existence of the solid-liquid slip velocity which, increases with increasing particle diameter. Figure. 9b shows that for the solid particles of diameter 0.13 mm and 0.27 mm, the local concentration

(a) (b)

Fig. 9. Influence of the particle diameter on: a) the velocity profiles; b) the concentration

The maximum of local concentration profile appears for dp= 0.4 mm and dp = 0.7 mm. The solid particles are present near the pipe top wall only for the small diameter (0.13 mm).

In this experimental chapter, we have tested a new approach measurement in order to determine simultaneously the velocity profiles and the concentration profiles of the solid

Fig. 7. Velocity and concentration profiles of glass bead for Cv= 1%: a) dp= 0.13 mm; b) dp= 0.40 mm

#### **3.2.2 Effect of volumetric concentration**

Figure 8 shows the influence of the volumetric concentration of the solid particles on the velocity and local concentration profiles. Note that the difference between the velocity profile of the solid particles and that of the carrying fluid (Fig. 8a), which represents the solid-liquid slip velocity, increases with the increase in the volumetric concentration. The local concentration profiles (Fig. 8b) present a maximum value, whose position moves away from the internal pipe bottom wall when the volumetric concentration decreases. For the lower volumetric concentration, the local concentration profile tends to that of the carrying fluid (or the fine particles) which is almost uniform. The presence of the particles near the pipe top wall was observed for the volumetric concentration lower than 0.5 %.

Fig. 8. Effect of the particle volumetric concentration on: a) the velocity profiles; b) the local concentration profiles for dp*=* 0.27 mm and Re= 44600.

## **3.2.3 Effect of diameter**

142 Ultrasonic Waves

(a) (b) Fig. 7. Velocity and concentration profiles of glass bead for Cv= 1%: a) dp= 0.13 mm; b) dp=

Figure 8 shows the influence of the volumetric concentration of the solid particles on the velocity and local concentration profiles. Note that the difference between the velocity profile of the solid particles and that of the carrying fluid (Fig. 8a), which represents the solid-liquid slip velocity, increases with the increase in the volumetric concentration. The local concentration profiles (Fig. 8b) present a maximum value, whose position moves away from the internal pipe bottom wall when the volumetric concentration decreases. For the lower volumetric concentration, the local concentration profile tends to that of the carrying fluid (or the fine particles) which is almost uniform. The presence of the particles near the

(a) (b) Fig. 8. Effect of the particle volumetric concentration on: a) the velocity profiles; b) the local

concentration profiles for dp*=* 0.27 mm and Re= 44600.

pipe top wall was observed for the volumetric concentration lower than 0.5 %.

0.40 mm

**3.2.2 Effect of volumetric concentration** 

The figure 9a shows that for all the particle diameters used, the velocity profiles of the solid particles have the same trend as that of the carrying fluid. The difference between the velocity profile of the continuous phase and that of the solid phase confirms the existence of the solid-liquid slip velocity which, increases with increasing particle diameter. Figure. 9b shows that for the solid particles of diameter 0.13 mm and 0.27 mm, the local concentration profiles tend to that of the carrying fluid.

Fig. 9. Influence of the particle diameter on: a) the velocity profiles; b) the concentration profiles for Cv= 1%, Re = 44600.

The maximum of local concentration profile appears for dp= 0.4 mm and dp = 0.7 mm. The solid particles are present near the pipe top wall only for the small diameter (0.13 mm).

## **4. Conclusion**

In this experimental chapter, we have tested a new approach measurement in order to determine simultaneously the velocity profiles and the concentration profiles of the solid

Application of Pulsed Ultrasonic Doppler Velocimetry to

*Ohio*.

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Kikura, H., Yamanaka, G., Aritomi, M. (2004). Effect of Measurement Volume Size on

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particles (glass bead) and the continuous phase (water) of two phase flow in horizontal pipe. The distinction between the Doppler signals coming from the solid phase and the continuous phase was obtained by imposing a threshold on the integral of the power spectral density. The use of this approach measurement is limited to small concentrations lower than 2%. Indeed when a concentration of 2% is exceeded, the ultrasonic signal is attenuated. This approach measurement shows the effects of the particle diameter and volumetric concentration on the local mean velocity and local mean concentration profiles of the suspensions.

The results obtained show that for the fine particles, the suspension behaves like a homogeneous fluid; the velocity profile is in a better agreement with the Pai's model and the concentration profile is almost uniform. For the large particles, saltation and heterogeneous flow regimes were obtained. These two regimes depend on the diameter and volumetric concentration of the particle, and on the flow mean velocity. The slip velocity which is responsible for the fluid-particle interaction depends on the flow regime.

In our next chapter concerning the solid liquid suspension, PUDV and Particle–Tracking Velocimetry (PTV) will be applied together. The most advantage of PTV is the possibility of measurements of large particles, by which the fluid–particle interactions and the particle– response property will be able to be examined. This allows us perhaps the best understanding of the phenomenon caused by the fluid–particle interactions, and also probably by the particle–particle interactions because of high concentration near the wall. It is also necessary to investigate interactions between these phenomenon and particle motion due to two-way and four-way couplings in a wide range of particle diameter, specific density, and sediment concentration, and then to develop reasonable computer simulation models of two phase horizontal pipe flow.

## **5. Acknowledgment**

This work has been supported by Research Laboratory of Industrial Technologies, University Ibn Khaldoun of Tiaret. The first author would like to thank Professor S. Hadj Ziane, Institute of Physics, University Ibn Khaldoun of Tiaret, for useful discussions as well as for providing support to perform this work.

## **6. References**


particles (glass bead) and the continuous phase (water) of two phase flow in horizontal pipe. The distinction between the Doppler signals coming from the solid phase and the continuous phase was obtained by imposing a threshold on the integral of the power spectral density. The use of this approach measurement is limited to small concentrations lower than 2%. Indeed when a concentration of 2% is exceeded, the ultrasonic signal is attenuated. This approach measurement shows the effects of the particle diameter and volumetric concentration on the local mean velocity and local mean concentration profiles of

The results obtained show that for the fine particles, the suspension behaves like a homogeneous fluid; the velocity profile is in a better agreement with the Pai's model and the concentration profile is almost uniform. For the large particles, saltation and heterogeneous flow regimes were obtained. These two regimes depend on the diameter and volumetric concentration of the particle, and on the flow mean velocity. The slip velocity which is

In our next chapter concerning the solid liquid suspension, PUDV and Particle–Tracking Velocimetry (PTV) will be applied together. The most advantage of PTV is the possibility of measurements of large particles, by which the fluid–particle interactions and the particle– response property will be able to be examined. This allows us perhaps the best understanding of the phenomenon caused by the fluid–particle interactions, and also probably by the particle–particle interactions because of high concentration near the wall. It is also necessary to investigate interactions between these phenomenon and particle motion due to two-way and four-way couplings in a wide range of particle diameter, specific density, and sediment concentration, and then to develop reasonable computer simulation

This work has been supported by Research Laboratory of Industrial Technologies, University Ibn Khaldoun of Tiaret. The first author would like to thank Professor S. Hadj Ziane, Institute of Physics, University Ibn Khaldoun of Tiaret, for useful discussions as well

Alfonsi, G.,Brambilla, S., Chiuch, D. (2003). The Use of an Ultrasonic Doppler Velocimeter in

Aritomi, M., Zhou, S., Nakajima, M., Takeda, Y., Yoshioka, Y. (1996). Measurement System

Brito, D., Nataf, H.C., Cardin, P., Aubert, J., Masson, J.P. (2001). Ultrasonic Doppler Velocimtry in Liquid Gallium, *Experiments in Fluids*, Vol.31, No. 6, pp. 653-663. Brodkey, R.S., Lee, J., Chasse, R.C. (1961). A Generalized Velocity Distribution for Non-

of Bubbly Flow Using Ultrasonic Velocity Profile Monitor and Video Data

Turbulent Pipe Flow, *Experiments in Fluids*, Vol.35, pp. 553-559.

Processing Unit, *J. Nucl. Sci. Technol*, Vol.33, pp. 915-923.

Newtonian Fluids, *A. I. Ch. E. Journal, Ohio*.

responsible for the fluid-particle interaction depends on the flow regime.

models of two phase horizontal pipe flow.

as for providing support to perform this work.

**5. Acknowledgment** 

**6. References** 

the suspensions.


**1. Introduction** 

132dB (re 1

between various propulsion technologies in Table 1.

**8** 

**Ultrasonic Thruster** 

*United States of America* 

Alfred C. H. Tan and Franz S. Hover *Massachusetts Institute of Technology* 

Acoustic streaming refers to the bulk net flow of fluid generated as a result of intense, freefield ultrasound. This phenomenon, also known as 'quartz wind' or 'sonic wind', is induced by a loss in mean momentum flux due to sound absorption in the fluid medium, leading to a net flow along the transducer axial direction. The transmission of intense sound energy into the fluid is also associated with a resultant force acting on the transducer surface. This resultant "backthrust" can be exploited in underwater vehicles for propulsion or maneuvering purposes. We refer to the device as ultrasonic thruster (UST), shown in Fig. 1, and define it as an ultrasonic transducer made from piezoelectric material, excited by an alternating high-voltage source in the megahertz. We provide a general comparison

Fig. 1. An ultrasonic thrusters (UST) directing a jet through the free water surface.

Jet

The UST we describe is being considered as an alternative small-scale propulsor for underwater robotic devices and systems, because it has no moving parts beyond its membrane and can be mounted flushed with the body of the watercraft. At a sound level of

biofouling and could perform self-cleaning under prolonged water submersion. These novelties in low maintenance and design robustness, coupled with low cost commercially off-the-shelf (COTS) transducers, are attributes which are not found in rotary or biomimetic propulsors of today. For a small transducer diameter of 1cm, thrust generated is in the order of tens of milli-newtons (mN), suitable for systems operating at low Reynolds numbers.

Pa) (Panchal, Takahashi, & Avery, 1995; Tan & Tanaka, 2006), it is destructive to

UltraSonic Thruster (UST)


## **Ultrasonic Thruster**

Alfred C. H. Tan and Franz S. Hover *Massachusetts Institute of Technology United States of America* 

## **1. Introduction**

146 Ultrasonic Waves

Varaksin, A;Y., Zaichik, L.I. (2000). Effect of Particles on the Carrier Flow Turbulence,

Xu, H. (2003). *Measurement of Fiber Suspension Flow and Forming Jet Velocity Profile by Pulsed* 

*Ultrasonic Doppler Velocimetry*, Ph.D. Thesis, Atlanta, Institute of Paper Science and

*Thermophys. Aeromech*., Vol.7, pp. 237-248.

Technology.

Acoustic streaming refers to the bulk net flow of fluid generated as a result of intense, freefield ultrasound. This phenomenon, also known as 'quartz wind' or 'sonic wind', is induced by a loss in mean momentum flux due to sound absorption in the fluid medium, leading to a net flow along the transducer axial direction. The transmission of intense sound energy into the fluid is also associated with a resultant force acting on the transducer surface. This resultant "backthrust" can be exploited in underwater vehicles for propulsion or maneuvering purposes. We refer to the device as ultrasonic thruster (UST), shown in Fig. 1, and define it as an ultrasonic transducer made from piezoelectric material, excited by an alternating high-voltage source in the megahertz. We provide a general comparison between various propulsion technologies in Table 1.

Fig. 1. An ultrasonic thrusters (UST) directing a jet through the free water surface.

The UST we describe is being considered as an alternative small-scale propulsor for underwater robotic devices and systems, because it has no moving parts beyond its membrane and can be mounted flushed with the body of the watercraft. At a sound level of 132dB (re 1Pa) (Panchal, Takahashi, & Avery, 1995; Tan & Tanaka, 2006), it is destructive to biofouling and could perform self-cleaning under prolonged water submersion. These novelties in low maintenance and design robustness, coupled with low cost commercially off-the-shelf (COTS) transducers, are attributes which are not found in rotary or biomimetic propulsors of today. For a small transducer diameter of 1cm, thrust generated is in the order of tens of milli-newtons (mN), suitable for systems operating at low Reynolds numbers.

Ultrasonic Thruster 149

works on thrust by ultrasonic means can be found in (Allison, Springer, & Van Dam, 2008; Nobunaga, 2004; Wang et al., 2011; Yu & Kim, 2004); we review and expand upon these. We have also made a comparison among these UST technologies, and provide some insights

The UST is made from a membrane actuator mounted in a specially designed waterproof housing, excited by an electrical source at ultrasonic frequency; the UST is generally applied underwater to generate thrust. For this work, the actuator is made of a thin, circular piezoelectric plate. We establish some fundamental concepts of the UST physics leading to thrust as experienced on the transducer surface, and its resulting jet of acoustic streaming into the farfield. Several nomenclatures are defined and used to describe the experimental

As described in the introduction, there are some prior works on the UST found in the literature. The experimental model used in those examples varies in shapes and sizes, and in the following, we provide a basis of comparison among them in terms of thrust density and electrical power density between each UST design. As an introduction, we refer to the mathematical treatment of thrust and acoustic power generation in Eqs. (1) to (4) as originally proposed by (Allison, et al., 2008), and relates thrust to the transducer voltage

Fig. 2. An elemental control volume in the far-field; the elevation angle, azimuth angle and

In Fig. 2, several variables for the transducer and the acoustic field are introduced. The acoustic energy transmits to the right semi-hemisphere, propagating perpendicularly

, and *s* respectively.

*ds*

sin *s d*

> *s d*

. According to (Allison, et al., 2008), thrust experienced on the surface of

 

*x* 

*x'*

of a control volume

, 

*S* 

through an elemental cross-sectional surface area <sup>2</sup> *dS s d d* sin

into some of the design parameters of ultrasonic propulsion.

supplied, *E*, a parameter reported in most UST-related studies.

2*a*

**2. Underwater Ultrasonic Thruster (UST)** 

results in the preceding sections.

distance from the center of transducer are

<sup>2</sup> *dS s d d ds* sin 

the transducer is expressed as

**2.1 Thrust generation** 


Table 1. Broad comparison of propulsor technologies (Carlton, 2007; Sfakiotakis, Lane, & Davies, 1999; Tan & Hover, 2009).

At the same time, large-scale collaborative swarm of small "microrobots" or "pods" systems are also gaining more interest in terms of low cost and practical operation. Small clusters of exploratory underwater vehicles could also acquire true flexibility in formation morphing, and wide spatial/temporal coverage in search-survey work (Trimmer & Jebens, 1989). More importantly, small water submersible is valuable in cluttered or confined environments such as inside a piping network or complex underwater structures (Egeskov, Bech, Bowley, & Aage, 1995). Naval reconnaissance missions could involve the deployment of clusters of expendable (even biodegradable) small underwater robots for hazardous/security missions such as mine-hunting or surveillance mapping (Doty et al., 1998).

In the following sections, we examine the theoretical background of the UST thrust generation, introduce scaled parameters for comparison between various UST devices found in the literature, examine its transit efficiency, identify some thermal anomalies, and discuss some of the UST design considerations. A detailed construction of the UST will be outlined with underlying insights to materials selection and design principles. For practical demonstration, we have also built a small underwater vehicle named *Huygens*, to establish a miniaturized platform for supporting multi-objectives subsea tasks. The main focus will be exclusively on the transducer-only testing (without nozzle appendages) for thrust and wake characterization as these are fundamentals toward understanding the UST technology. Prior

Maneuverability; reduced impact on vessel internal layout; noise isolation

High efficiency; directional stability; robustness against

advance speeds and loadings

Efficient and maneuverable at

High maneuverability at low

Very small size; no moving parts; short-range acoustic

Table 1. Broad comparison of propulsor technologies (Carlton, 2007; Sfakiotakis, Lane, &

At the same time, large-scale collaborative swarm of small "microrobots" or "pods" systems are also gaining more interest in terms of low cost and practical operation. Small clusters of exploratory underwater vehicles could also acquire true flexibility in formation morphing, and wide spatial/temporal coverage in search-survey work (Trimmer & Jebens, 1989). More importantly, small water submersible is valuable in cluttered or confined environments such as inside a piping network or complex underwater structures (Egeskov, Bech, Bowley, & Aage, 1995). Naval reconnaissance missions could involve the deployment of clusters of expendable (even biodegradable) small underwater robots for hazardous/security missions

In the following sections, we examine the theoretical background of the UST thrust generation, introduce scaled parameters for comparison between various UST devices found in the literature, examine its transit efficiency, identify some thermal anomalies, and discuss some of the UST design considerations. A detailed construction of the UST will be outlined with underlying insights to materials selection and design principles. For practical demonstration, we have also built a small underwater vehicle named *Huygens*, to establish a miniaturized platform for supporting multi-objectives subsea tasks. The main focus will be exclusively on the transducer-only testing (without nozzle appendages) for thrust and wake characterization as these are fundamentals toward understanding the UST technology. Prior

communication

such as mine-hunting or surveillance mapping (Doty et al., 1998).

many speeds; quiet Complex physical design

speeds; quiet Complex physical design

supply

suitable for shallow water

Cycloidal propeller Low-speed maneuverability Complex mechanical

Load- and speed-dependent

High bearing loads; costly

May be inefficient at off design conditions

Complex actuation system

Poor performance at low speeds; vulnerable to ingested debris

structure and maintenance

Poor propulsive efficiency; requires a high-voltage

and maintenance

performance

and complex

Propulsors Advantages Disadvantages

Fixed-pitch propeller Very mature technology; cost

Controllable-pitch propeller High efficiency at different

Waterjet Efficient at high vessel speeds;

Podded drive

Ducted propeller

Biomimetic – body/caudal fin (BCF) locomotion

Davies, 1999; Tan & Hover, 2009).

Biomimetic – Median and/or paired fin (MPF)

Ultrasonic thruster

locomotion

(this work)

effective

line fouling

works on thrust by ultrasonic means can be found in (Allison, Springer, & Van Dam, 2008; Nobunaga, 2004; Wang et al., 2011; Yu & Kim, 2004); we review and expand upon these. We have also made a comparison among these UST technologies, and provide some insights into some of the design parameters of ultrasonic propulsion.

## **2. Underwater Ultrasonic Thruster (UST)**

The UST is made from a membrane actuator mounted in a specially designed waterproof housing, excited by an electrical source at ultrasonic frequency; the UST is generally applied underwater to generate thrust. For this work, the actuator is made of a thin, circular piezoelectric plate. We establish some fundamental concepts of the UST physics leading to thrust as experienced on the transducer surface, and its resulting jet of acoustic streaming into the farfield. Several nomenclatures are defined and used to describe the experimental results in the preceding sections.

#### **2.1 Thrust generation**

As described in the introduction, there are some prior works on the UST found in the literature. The experimental model used in those examples varies in shapes and sizes, and in the following, we provide a basis of comparison among them in terms of thrust density and electrical power density between each UST design. As an introduction, we refer to the mathematical treatment of thrust and acoustic power generation in Eqs. (1) to (4) as originally proposed by (Allison, et al., 2008), and relates thrust to the transducer voltage supplied, *E*, a parameter reported in most UST-related studies.

Fig. 2. An elemental control volume in the far-field; the elevation angle, azimuth angle and distance from the center of transducer are , , and *s* respectively.

In Fig. 2, several variables for the transducer and the acoustic field are introduced. The acoustic energy transmits to the right semi-hemisphere, propagating perpendicularly through an elemental cross-sectional surface area <sup>2</sup> *dS s d d* sin of a control volume <sup>2</sup> *dS s d d ds* sin . According to (Allison, et al., 2008), thrust experienced on the surface of the transducer is expressed as

Ultrasonic Thruster 151

fluid particle is less than it entered due to sound absorption. The distribution of energy gradient then moves the particle away from the source resulting in an overall streaming effect. In another words, the loss in momentum flux across a control volume (Fig. 2) leads to a net force in the direction of the acoustic path. This net force in turn generates hydrodynamic flow

0

where *ux* denotes the time-averaged streaming velocity along the *x*-axis, *Fx* is the radiation

 *c*, and

the fluid. Since mass flow is conserved within the control volume, that is, 0 *ux*

2 2

 

(m2/s),

 

where *A* and *B* are characteristic coefficients of the fully developed axial velocity profile (*t*

(dB/m) is the absorption coefficient at a particular sound transmission frequency (Rudenko & Soluian, 1977). Accordingly, Lighthill demonstrated that the radiation force can also be

> <sup>2</sup> *ux <sup>F</sup> x*

In this case, it follows from Eq. (7) that *F* is proportional to <sup>2</sup> *ux* and knowing that *F* is also

Next, the total fluid discharge across the lateral section of the flow in Fig. 3 is given by

From (Tan & Hover, 2009) and later in Section 2.3.4, it can be seen that the streaming field across a lateral section can be approximated by a Gaussian distribution of the following

distance and *C* is a constant associated with the standard deviation of the Gaussian distribution. *ux* reveals a vital difference from conventional propeller design and biomimetic actuation – that at *x* = 0, the streaming velocity at the surface of the transducer is zero. This observation is further verified in the experimental results in Section 2.3.4. Hence the total kinetic energy of the velocity field in unit time from *x* = 0 to *x* = *xf*, where *xf* (m) is the focal

(Fig. 4), and the mass flow rate through the elemental disc is 2

, where *ux* (m/s) is the axial velocity in unit time, *r* (m) is the radial

*A u ee B*

*x*

represented by the Reynolds stress along the axial direction:

proportional to <sup>2</sup> *E* , *ux* is seen to scale directly with *E*.

distance from the flat transducer surface, is given by

*<sup>x</sup> <sup>x</sup> x x u p <sup>u</sup> u F x x x*

  2

2

*x Bx*

 = 2

(5)

and using boundary conditions *u u x x* 0 0, 0 ,

(6)

(7)

(kg/m·s) is the dynamic viscosity of

for low intensity sound, and

 *u r dr <sup>r</sup>* .

*x* 

, Eq. (5)

in a steady state incompressible medium, as governed by the Navier-Stokes equation,

force along the *x*-axis, *p*0 is defined as 0 0 *p*

is the kinematic viscosity

can be solved by assuming *<sup>x</sup> F Ae <sup>x</sup>*

to obtain

), and

2 *u A u rr r r*

 

*<sup>C</sup> u ue r x* 

form,

$$T = \rho \upsilon\_o^2 a^2 \,\pi \int\_0^{\theta\_1} \frac{J\_1(ka\sin\theta)^2}{\sin\theta} d\theta \tag{1}$$

where (kg/m3), *o* (m/s), *a* (m), *k*, , and *J*1() denote the fluid density, transducer surface velocity, radius of the transducer, wavenumber, sound absorption coefficient, and Bessel function of the first kind, respectively, and we consider <sup>1</sup> 1 3.832 sin *ka* (Blackstock, 2000) as the upper limit of the dominant ultrasonic beamwidth.

The relationship between the acoustic power along the transducer axial direction x and the ultrasonic thrust is given by a simple relationship (Allison, et al., 2008)

$$P\_{\mathfrak{x}} = {}^{c}\!\!\!\!\!\!\!\!/ \mathfrak{y}\tag{2}$$

where *c* (m/s) denotes the sound speed, and relates to the efficiency which will be elaborated in the next paragraph. As we verify below, this also means the thrust is considerably lower than would a rotary propulsor operating at the same power level.

At *s* = 0, this acoustic power radiation is associated with the electrical power consumption across the transducer. The electrical power would provide an approximation to the acoustic power loading, which relates to the thrust force, and is reflected in the following considerations through an efficiency constant, ; (i) electrical power lost across the transducer is not wholly transferred into the medium; (ii) acoustic loading at the sharp dominant resonant frequency of the transducer may not be precisely tuned; (iii) it is difficult to consider the equivalent acoustic load in the lumped circuit impedance; (iv) although the acoustic power is mainly generated within the narrow ultrasonic beam, some losses also occur outside the beamwidth.

By relating the thrust production to the root mean square of the transducer voltage supplied, *Erms*, it can be equated as

$$T = \frac{\eta E\_{rms}^2}{c \operatorname{Re}(\Re)}\tag{3}$$

where Re() is the real component of the transducer impedance, and the electrical power,

$$P = \frac{E\_{rms}^2}{\text{Re}(\Re)}\tag{4}$$

corresponds to *Px* at *s* = 0. This thrust scaling with squared voltage in (3) will be used in Section 2.3.4.

#### **2.2 Acoustic streaming**

Following Lighthill (Lighthill, 1978), acoustic streaming arises because of acoustic energy absorption along the path of propagation in a viscous, dissipative fluid medium. A simple explanation of the mechanism can be thought of as exit momentum flux from each exposed

2 2 1 0

 

where 

(kg/m3),

*o* (m/s), *a* (m), *k*,

the upper limit of the dominant ultrasonic beamwidth.

where *c* (m/s) denotes the sound speed, and

considerations through an efficiency constant,

occur outside the beamwidth.

Section 2.3.4.

**2.2 Acoustic streaming** 

supplied, *Erms*, it can be equated as

function of the first kind, respectively, and we consider <sup>1</sup>

ultrasonic thrust is given by a simple relationship (Allison, et al., 2008)

sin *<sup>o</sup> J ka T a <sup>d</sup>* 

velocity, radius of the transducer, wavenumber, sound absorption coefficient, and Bessel

The relationship between the acoustic power along the transducer axial direction x and the

*xP cT* 

elaborated in the next paragraph. As we verify below, this also means the thrust is

At *s* = 0, this acoustic power radiation is associated with the electrical power consumption across the transducer. The electrical power would provide an approximation to the acoustic power loading, which relates to the thrust force, and is reflected in the following

transducer is not wholly transferred into the medium; (ii) acoustic loading at the sharp dominant resonant frequency of the transducer may not be precisely tuned; (iii) it is difficult to consider the equivalent acoustic load in the lumped circuit impedance; (iv) although the acoustic power is mainly generated within the narrow ultrasonic beam, some losses also

By relating the thrust production to the root mean square of the transducer voltage

where Re() is the real component of the transducer impedance, and the electrical power,

 2 Re *Erms <sup>T</sup> c* 

> 2 Re

corresponds to *Px* at *s* = 0. This thrust scaling with squared voltage in (3) will be used in

Following Lighthill (Lighthill, 1978), acoustic streaming arises because of acoustic energy absorption along the path of propagation in a viscous, dissipative fluid medium. A simple explanation of the mechanism can be thought of as exit momentum flux from each exposed

considerably lower than would a rotary propulsor operating at the same power level.

<sup>1</sup> <sup>2</sup>

sin

1

(1)

, and *J*1() denote the fluid density, transducer surface

3.832 sin *ka*

(2)

relates to the efficiency which will be

; (i) electrical power lost across the

(3)

*Erms <sup>P</sup>* (4)

(Blackstock, 2000) as

fluid particle is less than it entered due to sound absorption. The distribution of energy gradient then moves the particle away from the source resulting in an overall streaming effect. In another words, the loss in momentum flux across a control volume (Fig. 2) leads to a net force in the direction of the acoustic path. This net force in turn generates hydrodynamic flow in a steady state incompressible medium, as governed by the Navier-Stokes equation,

$$
\rho \,\overline{u}\_{\mathbf{x}} \frac{\partial \overline{u}\_{\mathbf{x}}}{\partial \mathbf{x}} = -\frac{\partial \overline{p}\_0}{\partial \mathbf{x}} + \mu \frac{\partial^2 \overline{u}\_{\mathbf{x}}}{\partial \mathbf{x}^2} + F\_{\mathbf{x}} \tag{5}
$$

where *ux* denotes the time-averaged streaming velocity along the *x*-axis, *Fx* is the radiation force along the *x*-axis, *p*0 is defined as 0 0 *p c* , and (kg/m·s) is the dynamic viscosity of the fluid. Since mass flow is conserved within the control volume, that is, 0 *ux x* , Eq. (5) can be solved by assuming *<sup>x</sup> F Ae <sup>x</sup>* and using boundary conditions *u u x x* 0 0, 0 , to obtain

$$\overline{\mu}\_x = \frac{A}{\nu \left(\mathcal{B}^2 - \mathcal{B}^2\right)} \left[e^{-\mathcal{B}^\chi x} - e^{-\mathcal{B}^\chi x}\right] \tag{6}$$

where *A* and *B* are characteristic coefficients of the fully developed axial velocity profile (*t* ), and is the kinematic viscosity (m2/s), = 2 for low intensity sound, and (dB/m) is the absorption coefficient at a particular sound transmission frequency (Rudenko & Soluian, 1977). Accordingly, Lighthill demonstrated that the radiation force can also be represented by the Reynolds stress along the axial direction:

$$\overline{F}' = -\frac{\partial \,\rho \,\overline{u}\_x^2}{\partial x} \tag{7}$$

In this case, it follows from Eq. (7) that *F* is proportional to <sup>2</sup> *ux* and knowing that *F* is also proportional to <sup>2</sup> *E* , *ux* is seen to scale directly with *E*.

Next, the total fluid discharge across the lateral section of the flow in Fig. 3 is given by 2 *u A u rr r r* (Fig. 4), and the mass flow rate through the elemental disc is 2 *u r dr <sup>r</sup>* . From (Tan & Hover, 2009) and later in Section 2.3.4, it can be seen that the streaming field across a lateral section can be approximated by a Gaussian distribution of the following form, 2 <sup>2</sup> 2 *r <sup>C</sup> u ue r x* , where *ux* (m/s) is the axial velocity in unit time, *r* (m) is the radial distance and *C* is a constant associated with the standard deviation of the Gaussian distribution. *ux* reveals a vital difference from conventional propeller design and biomimetic actuation – that at *x* = 0, the streaming velocity at the surface of the transducer is zero. This observation is further verified in the experimental results in Section 2.3.4. Hence the total kinetic energy of the velocity field in unit time from *x* = 0 to *x* = *xf*, where *xf* (m) is the focal distance from the flat transducer surface, is given by

Ultrasonic Thruster 153

1 111 <sup>1</sup>

While Eq. (9) sums up the total kinetic energy within the streaming field up to the focal distance, streaming at distances above *xf* will evidently slow down, and free turbulence occurs due to the boundary between the stationary ambient water and the insonified flow – a process called entrainment. We consider regime *x* > *xf* to be no longer reliable or valid for *K.E.* calculation. Eq. (9) will be used in Section 2.3.4. Finally, it follows earlier that *ux* scales directly with *E* and knowing that *K.E.* is also proportional to <sup>2</sup> *ux* from Eq. (8), *K.E.* is seen to scale directly with *E*2. This scaling of *K.E.* with square voltage will be used in Section 2.3.4 to

*B B B*

 

determine the streaming energy and a comparison is made across other UST devices.

In this section, we characterize thrust and wake energy for a specific UST design, and then investigate and how these properties can be modified using various source voltages. An underwater vehicle prototype was also constructed to demonstrate the ultrasonic

We use standard piezoelectric transducer technology for the conversion of electrical to acoustical energy. The transducer (Murata Manufacturing Co. Ltd) is made from a circular PZT plate measuring 7mm in diameter, housed in a 10mm diameter waterproof metallic

> PZT Metal casing UST Thin wire

Plastic base Potting material Lead terminal

> Connected to power amplifier

Fig. 5. Construction of the UST (adapted from Murata Manufacturing Co. Ltd).

In our prototype vehicle described below, three USTs are connected to switches for actuation control, and 50 coaxial cables connect the three switches to a single power amplifier (ENI 3100L). The ENI unit accepts an oscillatory input up to a maximum of 1Vrms, and amplifies the output voltage by a gain of 50dB for a 50 output impedance. It is a Class A amplifier which means it will be unconditionally stable, and maintains linearity even with a combination of mismatched source and load impedance. When the ultrasonic transducer

Coaxial cable

*B B*

2

 

<sup>2</sup> <sup>2</sup>

*e e*

*Bx <sup>C</sup>*

 

*f f*

*x B x*

 

2 2 (9)

*R*

 

<sup>2</sup> 22 2

**2.3 Thrust and wake experimental setup** 

**2.3.1 UST hardware and methods** 

propulsion capability.

casing as shown in Fig. 5.

2 2

1 1 . .

*A C K E <sup>e</sup> <sup>e</sup>*

*f*

$$\begin{aligned} K.E. &= \int\_0^{\chi\_f} \int\_0^R \rho \, u\_x u\_r \, 2\pi \, r \, dr d\mathbf{x} \\ &= 2\pi \rho \int\_0^{\chi\_f} \int\_0^R u\_x^2 \, r \, e^{-\frac{r^2}{2C^2}} dr d\mathbf{x} \end{aligned} \tag{8}$$

Fig. 4. An elemental disc sectioned laterally from the ultrasonic field. Axis of the transducer passes through the center of the concentric circles. The shaded annulus area, 2 *r r* , is an elemental area through which flow discharges.

where *R* (m) is the radius of *ur* (m/s) profile subtended by 1 at *xf*, and the overbar is omitted for simplicity. Substituting Eq. (6) into Eq. (8), and solving the double integral, the total kinetic energy in unit time becomes

*x r*

*<sup>r</sup> x R <sup>C</sup> <sup>x</sup>*

0 0

*x R*

*f*

2

*r* 

gradual decline.

. . 2

*K E u u r drdx*

Fig. 3. Velocity distribution of the acoustic streaming. Lateral velocity profile approximates a Gaussian distribution while axial velocity follows a rapid increase in velocity before a

Lateral velocity profile, *ur*

Fig. 4. An elemental disc sectioned laterally from the ultrasonic field. Axis of the transducer

omitted for simplicity. Substituting Eq. (6) into Eq. (8), and solving the double integral, the

passes through the center of the concentric circles. The shaded annulus area, 2

*r* 

 *r* 

where *R* (m) is the radius of *ur* (m/s) profile subtended by

*R* 

elemental area through which flow discharges.

2a

*u* 

UST

total kinetic energy in unit time becomes

0 0

*f*

2 2

Axial velocity profile, *ux*

(8)

*x* 

*ur|*

*u* 

*xf*

 *r r* , is an

*ur*

1 at *xf*, and the overbar is

*u r e drdx*

*xf*

*u* 

*r* 

2 2

 

$$\begin{aligned} \text{K.E.} &= \frac{\pi \rho \, A^2 \mathbb{C}^2}{\nu^2 \left(\mathcal{B}^2 - \beta^2\right)^2} \Big( -\frac{1}{\beta} e^{-2\beta x\_f} + \frac{1}{\beta + B} e^{-(\beta + B)x\_f} - \\ &\frac{1}{B} e^{-2\beta x\_f} + \frac{1}{\beta} - \frac{1}{\beta + B} + \frac{1}{B} \right) \times \left( 1 - e^{-\frac{R^2}{2\mathcal{C}^2}} \right) \end{aligned} \tag{9}$$

While Eq. (9) sums up the total kinetic energy within the streaming field up to the focal distance, streaming at distances above *xf* will evidently slow down, and free turbulence occurs due to the boundary between the stationary ambient water and the insonified flow – a process called entrainment. We consider regime *x* > *xf* to be no longer reliable or valid for *K.E.* calculation. Eq. (9) will be used in Section 2.3.4. Finally, it follows earlier that *ux* scales directly with *E* and knowing that *K.E.* is also proportional to <sup>2</sup> *ux* from Eq. (8), *K.E.* is seen to scale directly with *E*2. This scaling of *K.E.* with square voltage will be used in Section 2.3.4 to determine the streaming energy and a comparison is made across other UST devices.

#### **2.3 Thrust and wake experimental setup**

In this section, we characterize thrust and wake energy for a specific UST design, and then investigate and how these properties can be modified using various source voltages. An underwater vehicle prototype was also constructed to demonstrate the ultrasonic propulsion capability.

#### **2.3.1 UST hardware and methods**

We use standard piezoelectric transducer technology for the conversion of electrical to acoustical energy. The transducer (Murata Manufacturing Co. Ltd) is made from a circular PZT plate measuring 7mm in diameter, housed in a 10mm diameter waterproof metallic casing as shown in Fig. 5.

Fig. 5. Construction of the UST (adapted from Murata Manufacturing Co. Ltd).

In our prototype vehicle described below, three USTs are connected to switches for actuation control, and 50 coaxial cables connect the three switches to a single power amplifier (ENI 3100L). The ENI unit accepts an oscillatory input up to a maximum of 1Vrms, and amplifies the output voltage by a gain of 50dB for a 50 output impedance. It is a Class A amplifier which means it will be unconditionally stable, and maintains linearity even with a combination of mismatched source and load impedance. When the ultrasonic transducer

Ultrasonic Thruster 155

20

15

10

Torque (mNm)

5

0

Water

amp. Nd-YLF

Digital PIV camera

Laser sheet

UST

Power

Fig. 7. Calibration of the test rig. Each of the four points shown is an average of four separate

0 9.8 19.6 29.4 39.2 49 Dial gauge force (mN)

A 2-dimensional Digital Particle Image Velocimetry (DPIV) provides an accurate depiction of the UST wake, and beyond the entrainment boundary in the far-field. In similarly-scaled conditions, it has been reported that a 2% standard deviation at the point of maximum velocity can be expected with DPIV (Myers, Hariharan, & Banerjee, 2008). The DPIV system is set up in a water tank measuring 2.4m0.7m0.7m, as shown in Fig. 8. Calculated particle velocities are subject to noise depending predominantly on the interrogation window size, the number of seeded particles, and the sampling rate. We made efforts to tune these for the

(a) (b)

Sync

hronize

image acquisitio

n

d

Camera

UST

Laser sheet

Fig. 8. (a) Schematic diagram of the DPIV tank setup. Acoustic streaming is illuminated by a laser sheet, and images are captured by a digital camera for post-processing. (b) A top view

Sheet optics

A pulsed laser sheet, produced by a Quantronix diode pumped Q-switched frequency laser (Darwin-527-30M) and spreads horizontally through the water tank from the outside, coincident with the UST axis. The tank is seeded with 50m polyamide particles, and a camera viewing the laser sheet perpendicularly traces the streaming particles when the UST is operating. The camera samples 300 timed, paired images at 400Hz, in a 0.2m0.2m field of

force applications using a sensitive dial gauge.

**2.3.3 Acoustic streaming measurement** 

of the UST wake measurement setup.

lowest noise level.

Signal gen. (B&K 4011A)

emits intense acoustic energy into the fluid, the thin metal housing provides excellent heat dissipation. Natural convection and acoustic streaming also aid in carrying away heat from the transducer surface (Tan & Hover, 2010a).

## **2.3.2 Thrust force measurement**

It is important to develop a reliable underwater thrust measurement method, especially for small thrust magnitudes as is the case of the UST. Most load cells are either non-submersible or could not provide sufficient sensitivity/resolution required at small driving forces. While thrust can also be inferred from acoustic intensity measurement using a hydrophone, it poses some challenges unique to the UST setup, such as membrane cavitation, heating effect, and reading errors averaged from a finite-size hydrophone. Indeed, (Hariharan et al., 2008) reported that acoustic intensity measurements do not perform well, having an error in excess of 20% with experimental data.

In order to accurately measure the thrust produced by the UST, we developed an approach which allows sensitivity control by sliding the UST along an L-shaped arm. Moment is measured with a high precision torque sensor. Other methods have been proposed for measuring these very fine-scale forces, for example, to attach and submerge a UST on one end of a vertical pendulum, hinged off-center, with the other end flexing a strain gauge (Allison, et al., 2008). Another approach is to attach the UST to a free-hanging wire and take photographs of the displacement as the UST is being actuated (Wang, et al., 2011; Yu & Kim, 2004). We found that the setup in Fig. 6 provides very good accuracy and repeatability, as indicated in the calibration plot of Fig. 7. The torque meter accuracy provided by the manufacturer is 0.09mNm, and the absolute error in the calibration thrust force averages about 0.6mN, when our dial gauge is positioned 0.5m from the torque axis on the L-arm.

Fig. 6. (a) The semi-anechoic water tank and torque sensor used to measure UST thrust; thrust is generated in the direction perpendicular to the page. (b) A plan view of the UST thrust measurement setup.

Hence the total force uncertainty is around 0.8mN. Torque calibration is performed prior to all test sets. Thrust stabilizes and is recorded about ten seconds after turning on the power; then the power is turned off. We maintain a minimum of ten-second rest period between all tests, to allow for cooling and for the water to settle. The UST together with the L-shaped arm is submerged in an ultrasound semi-anechoic tank measuring 1.2m0.6m0.6m, filled with distilled water and covered with an acrylic sheet.

emits intense acoustic energy into the fluid, the thin metal housing provides excellent heat dissipation. Natural convection and acoustic streaming also aid in carrying away heat from

It is important to develop a reliable underwater thrust measurement method, especially for small thrust magnitudes as is the case of the UST. Most load cells are either non-submersible or could not provide sufficient sensitivity/resolution required at small driving forces. While thrust can also be inferred from acoustic intensity measurement using a hydrophone, it poses some challenges unique to the UST setup, such as membrane cavitation, heating effect, and reading errors averaged from a finite-size hydrophone. Indeed, (Hariharan et al., 2008) reported that acoustic intensity measurements do not perform well, having an error in

In order to accurately measure the thrust produced by the UST, we developed an approach which allows sensitivity control by sliding the UST along an L-shaped arm. Moment is measured with a high precision torque sensor. Other methods have been proposed for measuring these very fine-scale forces, for example, to attach and submerge a UST on one end of a vertical pendulum, hinged off-center, with the other end flexing a strain gauge (Allison, et al., 2008). Another approach is to attach the UST to a free-hanging wire and take photographs of the displacement as the UST is being actuated (Wang, et al., 2011; Yu & Kim, 2004). We found that the setup in Fig. 6 provides very good accuracy and repeatability, as indicated in the calibration plot of Fig. 7. The torque meter accuracy provided by the manufacturer is 0.09mNm, and the absolute error in the calibration thrust force averages about 0.6mN, when our dial gauge is positioned 0.5m from the torque axis on the L-arm.

(a) (b)

Torque

meter UST

Fig. 6. (a) The semi-anechoic water tank and torque sensor used to measure UST thrust; thrust is generated in the direction perpendicular to the page. (b) A plan view of the UST

Power amp.

Signal generator (B&K 4011A)

Hence the total force uncertainty is around 0.8mN. Torque calibration is performed prior to all test sets. Thrust stabilizes and is recorded about ten seconds after turning on the power; then the power is turned off. We maintain a minimum of ten-second rest period between all tests, to allow for cooling and for the water to settle. The UST together with the L-shaped arm is submerged in an ultrasound semi-anechoic tank measuring 1.2m0.6m0.6m, filled

the transducer surface (Tan & Hover, 2010a).

**2.3.2 Thrust force measurement** 

excess of 20% with experimental data.

thrust measurement setup.

Torque display Torque meter

> balance weight

with distilled water and covered with an acrylic sheet.

Acrylic sheet

UST Counter-

90

50 coaxial cable

Water

Ultrasound absorber

Fig. 7. Calibration of the test rig. Each of the four points shown is an average of four separate force applications using a sensitive dial gauge.

#### **2.3.3 Acoustic streaming measurement**

A 2-dimensional Digital Particle Image Velocimetry (DPIV) provides an accurate depiction of the UST wake, and beyond the entrainment boundary in the far-field. In similarly-scaled conditions, it has been reported that a 2% standard deviation at the point of maximum velocity can be expected with DPIV (Myers, Hariharan, & Banerjee, 2008). The DPIV system is set up in a water tank measuring 2.4m0.7m0.7m, as shown in Fig. 8. Calculated particle velocities are subject to noise depending predominantly on the interrogation window size, the number of seeded particles, and the sampling rate. We made efforts to tune these for the lowest noise level.

Fig. 8. (a) Schematic diagram of the DPIV tank setup. Acoustic streaming is illuminated by a laser sheet, and images are captured by a digital camera for post-processing. (b) A top view of the UST wake measurement setup.

A pulsed laser sheet, produced by a Quantronix diode pumped Q-switched frequency laser (Darwin-527-30M) and spreads horizontally through the water tank from the outside, coincident with the UST axis. The tank is seeded with 50m polyamide particles, and a camera viewing the laser sheet perpendicularly traces the streaming particles when the UST is operating. The camera samples 300 timed, paired images at 400Hz, in a 0.2m0.2m field of

Ultrasonic Thruster 157

reported in another publication (Tan & Hover, 2010a). The kinetic energy data in Fig. 9(b) show similar trend as well. Using the *K.E.*-*E*2 proportionality relationship from Section 2.2, the scaled kinetic energy at 54Vrms is calculated to be 1.510-7W/V2 respectively. Both scaling of thrust and *K.E.* is important when we compare various UST devices later in Section 2.5.

(a) (b)

50

0

UST

100

150

Axis

distance (mm)

200

20 40 Output voltage (Vrms)

<sup>0</sup> <sup>60</sup>


0 50 Transverse distance (mm)

Scaled

0

80

1

2

3

4 5

6 10- <sup>7</sup>

kinetic energy

(a) (b)

0

3

Kinetic energy

4

5

10- <sup>4</sup>

K.E. Scaled K.E.

2

1

Fig. 11. (a) Measured thrust versus input voltage to the power amplifier using a 7MHz sinusoidal signal. Data appear to saturate above 60Vrms, which implies a disproportionate relationship. (b) Total kinetic energy versus output voltage of the power amplifier using a 7MHz sinusoidal signal. Data appear to saturate above 55Vrms, which also implies a

Although increasing the output voltage always increases thrust, the wake velocity of the sinusoidal source appears to saturate near 55Vrms. The phenomenon of saturation can be

Fig. 10. (a) Digital Particle Image Velocimetry (DPIV) image of a jet at 7MHz, 54Vrms. (b) Processed DPIV velocity field with each velocity contour step at 0.05m/s interval. A focal

0.01

0.008

0.006

0.004

0.002

Scaled t

hrust (mN

/

V2)

0

Streaming velocity (m

/s)

0.2

0.25

0.15

0.1

0.05

point can be clearly seen at (0mm, 120mm).

Thrust Scaled thrust

0 20 40 60 80 Output voltage (Vrms)


Axis

20

15

10

5

T

hrust (mN)

0

distance (mm)

50

0

100

150

200

0 50 Transverse distance (mm)

disproportionate relationship.

view. Post-processing is carried out using DaVis 7.1 software. The UST transducer is positioned at 1.2m from the sheet optics, 0.4m from each adjacent tank wall, and 0.3m below the water surface. The time taken for the stream to become established has been reported variously at about 0.5s (Loh & Lee, 2004) and 20s (Kamakura, Sudo, Matsuda, & Kumamoto, 1996). We allow at least one minute of flow before the camera starts recording. Specific kinetic energy of the flow up to the focal distance *xf* is shown in the next section, which averages 150 measurements; recording the frames takes less than one second. Then the power is turned off, and the tank water is allowed to settle for at least one minute. This schedule is not the same for thrust measurements, as the transducer is powered and cooled for a considerably longer time during DPIV tests. A more detailed analysis of the transducer heating is presented in Section 3.

#### **2.3.4 Thrust and streaming results**

Fig. 9(a) shows thrust force as a function of frequency, for a sinusoidal waveform. The thrust has obvious peaks near 11mN when operated at 7MHz, which can also be computed using the thickness mode frequency constant, *N fh <sup>t</sup>* <sup>0</sup> , where *h* (m) is the thickness of the PZT, and 1970 *Nt* as specified by the manufacturer. We will focus on this frequency in most of the discussion to follow. Fig. 9(b) illustrates the DPIV velocity field for a sinusoidal waveform at 7MHz, with amplified output voltage 54Vrms. From Fig. 10(a) and 10(b), the maximum axial streaming velocity is observed at the point (0mm, 120mm) in the DPIV image, illustrating a fundamental feature of the UST – *that net fluid flow is zero at the transducer face*. Considering the same waveform configurations and frequencies as in Fig. 9(a), a similar peak at 7MHz in the total kinetic energy can be seen in Fig. 9(b).

Fig. 9. (a) Measured thrust versus source frequency for sinusoidal waveforms supplied at 59Vrms. (b) Total kinetic energy versus source frequency, from DPIV. The sinusoidal signal is supplied at 54Vrms.

Figs. 11(a) and 11(b) summarize our findings specifically at the 7MHz resonant point. Thrust generally increases with output voltage, but then starts to flatten out above the amplified output voltage of 60Vrms. From Fig. 12 and from Eq. (3), the corresponding scaled thrust level is 3.810-3mN/V2 at 59Vrms. Scaled thrust decreases gradually as the output voltage increases, and the absolute thrust appears to saturate above 60Vrms; see Fig. 12 – a result also

view. Post-processing is carried out using DaVis 7.1 software. The UST transducer is positioned at 1.2m from the sheet optics, 0.4m from each adjacent tank wall, and 0.3m below the water surface. The time taken for the stream to become established has been reported variously at about 0.5s (Loh & Lee, 2004) and 20s (Kamakura, Sudo, Matsuda, & Kumamoto, 1996). We allow at least one minute of flow before the camera starts recording. Specific kinetic energy of the flow up to the focal distance *xf* is shown in the next section, which averages 150 measurements; recording the frames takes less than one second. Then the power is turned off, and the tank water is allowed to settle for at least one minute. This schedule is not the same for thrust measurements, as the transducer is powered and cooled for a considerably longer time during DPIV tests. A more detailed analysis of the transducer heating is presented in Section 3.

Fig. 9(a) shows thrust force as a function of frequency, for a sinusoidal waveform. The thrust has obvious peaks near 11mN when operated at 7MHz, which can also be computed using the thickness mode frequency constant, *N fh <sup>t</sup>* <sup>0</sup> , where *h* (m) is the thickness of the PZT, and 1970 *Nt* as specified by the manufacturer. We will focus on this frequency in most of the discussion to follow. Fig. 9(b) illustrates the DPIV velocity field for a sinusoidal waveform at 7MHz, with amplified output voltage 54Vrms. From Fig. 10(a) and 10(b), the maximum axial streaming velocity is observed at the point (0mm, 120mm) in the DPIV image, illustrating a fundamental feature of the UST – *that net fluid flow is zero at the transducer face*. Considering the same waveform configurations and frequencies as in Fig.

(a) (b)

4

3

4

10- <sup>4</sup>

2

Kinetic energy

1

0

6 8 10 Frequency (MHz)

Fig. 9. (a) Measured thrust versus source frequency for sinusoidal waveforms supplied at 59Vrms. (b) Total kinetic energy versus source frequency, from DPIV. The sinusoidal signal is

Figs. 11(a) and 11(b) summarize our findings specifically at the 7MHz resonant point. Thrust generally increases with output voltage, but then starts to flatten out above the amplified output voltage of 60Vrms. From Fig. 12 and from Eq. (3), the corresponding scaled thrust level is 3.810-3mN/V2 at 59Vrms. Scaled thrust decreases gradually as the output voltage increases, and the absolute thrust appears to saturate above 60Vrms; see Fig. 12 – a result also

9(a), a similar peak at 7MHz in the total kinetic energy can be seen in Fig. 9(b).

<sup>0</sup> 6 8 <sup>10</sup> Frequency (MHz)

**2.3.4 Thrust and streaming results** 

12

6 4 2

8 10

4

T

hrust (mN)

supplied at 54Vrms.

reported in another publication (Tan & Hover, 2010a). The kinetic energy data in Fig. 9(b) show similar trend as well. Using the *K.E.*-*E*2 proportionality relationship from Section 2.2, the scaled kinetic energy at 54Vrms is calculated to be 1.510-7W/V2 respectively. Both scaling of thrust and *K.E.* is important when we compare various UST devices later in Section 2.5.

Fig. 10. (a) Digital Particle Image Velocimetry (DPIV) image of a jet at 7MHz, 54Vrms. (b) Processed DPIV velocity field with each velocity contour step at 0.05m/s interval. A focal point can be clearly seen at (0mm, 120mm).

Fig. 11. (a) Measured thrust versus input voltage to the power amplifier using a 7MHz sinusoidal signal. Data appear to saturate above 60Vrms, which implies a disproportionate relationship. (b) Total kinetic energy versus output voltage of the power amplifier using a 7MHz sinusoidal signal. Data appear to saturate above 55Vrms, which also implies a disproportionate relationship.

Although increasing the output voltage always increases thrust, the wake velocity of the sinusoidal source appears to saturate near 55Vrms. The phenomenon of saturation can be

Ultrasonic Thruster 159

To our knowledge, there are only three experimental works on ultrasonic propulsors reported from (Allison, et al., 2008), (Yu & Kim, 2004), and (Wang, et al., 2011); we compare them with our UST in terms of thrust density, scaled thrust density and power density, in Table 2. As our UST system was operating at a resonance frequency and high voltage, it is difficult to make a fair and direct comparison of performance. We have employed a much higher power level, resulting in a very high thrust level and higher scaled thrust density.

> Allison et al*.* (Allison, et al., 2008)

Frequency (MHz) 5.5 10.8 17.8 7.0 Voltage (V) 24.5 46 140 59 Electrical power (W) 5 -- -- 69.5 Transducer surface area (mm2) /4102 55 /21.282 /472 Thrust (mN) 2.25 5.6 2.3 13.5 Thrust density (N/m2) 28.6 224 893.7 350 Scaled thrust density (N/m2V2) 0.05 0.11 0.04 0.10 Electrical power density (kW/m2) 64 -- -- 1806

This latter property is important in applications because it indicates a very compact force source operating with reasonable voltage levels, exploitable to benefit from many of its

Fig. 13. Thrust force produced by the UST versus driving electrical power. The *l* and *h* subscripts indicate evident low- and high-power regimes. The output voltage to the

50 100 Electrical power (W)

7MHz Sinusoid 7MHz Squarewave

*h,square* =3.9%

<sup>0</sup> <sup>150</sup>

Yu et al*.* (Yu & Kim, 2004)

Z. Wang et al*.* (Wang, et al., 2011)

This work

**2.5 Ultrasonic propulsion design considerations** 

Table 2. Performance of different UST devices.

sinusoid is 59Vrms, and for the square wave is 54Vrms.

0

5

T

hrust (mN)

10

*l,sinusoid* =33.6%

15

unique ultrasonic attributes we discussed earlier in the introduction.

*h,sinusoid* =8.9%

*l,square* =17.5%

Properties

explained by distortion in finite-amplitude traveling waves, according to weak shock theory. On the other hand, the fact that thrust in this case increases with input power despite the saturation of velocity highlights an unusual observation – that thrust production mechanism involves the wake only indirectly, and in a manner that is distinct from other propulsors. This fact may offer some interesting avenues for UST design, where the wake and the thrust force could be manipulated independently. This is especially useful in a scenario where larger thrust is desired but the wake has to be weak at the same time, for example, to minimize stirring up particulates near the seabed.

In summary, increasing the output voltage of the power amplifier will no doubt increase the thrust and kinetic energy production of the transducer, but at the same time, introduces an undesirable disproportionate relationship (Tan & Hover, 2010a). This will be further discussed in Section 3.

#### **2.4 Small underwater vehicle,** *Huygens*

We designed and constructed a small, streamlined shell with three embedded UST devices for propulsion and steering in the horizontal plane. The shell measures 215mm160mm80mm, profiled by a truncated NACA 0054 airfoil in the side view, and a truncated NACA 0025 airfoil in the plan view, as shown in Fig. 12(a). The shell is made from high-strength urethane foam for buoyancy, and coated with polyurethane. A UST is positioned at the rear end of the craft to provide forward thrust, and two USTs subtending 120° are symmetrically located on each side of the front end – at the "fish eyes" position. Together, these provide the right/left steering and backing thrust. Inside the shell, two rectangular cavities are machined, measuring 100mm100mm35mm, and 45mm60mm30mm. The shell can be opened into two halves via a stepped mid-section opening, lined with double O-rings for a watertight seal. The cylindrical UST seats are also lined with O-rings. The vent shown in the top left corner of Fig. 12(b) allows for a tether or an antenna for shallow-water wireless control.

Fig. 12. (a) 3-D wire mesh view of *Huygens*. (b) A side view of *Huygens* prototype. Three transducers are installed on the shell with two at the frontal "eyes" positioned for steering and another in the rear for forward thrust. Tethered signal generates a multiplexed 7MHz sinusoidal input to the three USTs.

explained by distortion in finite-amplitude traveling waves, according to weak shock theory. On the other hand, the fact that thrust in this case increases with input power despite the saturation of velocity highlights an unusual observation – that thrust production mechanism involves the wake only indirectly, and in a manner that is distinct from other propulsors. This fact may offer some interesting avenues for UST design, where the wake and the thrust force could be manipulated independently. This is especially useful in a scenario where larger thrust is desired but the wake has to be weak at the same time, for

In summary, increasing the output voltage of the power amplifier will no doubt increase the thrust and kinetic energy production of the transducer, but at the same time, introduces an undesirable disproportionate relationship (Tan & Hover, 2010a). This will be further

We designed and constructed a small, streamlined shell with three embedded UST devices for propulsion and steering in the horizontal plane. The shell measures 215mm160mm80mm, profiled by a truncated NACA 0054 airfoil in the side view, and a truncated NACA 0025 airfoil in the plan view, as shown in Fig. 12(a). The shell is made from high-strength urethane foam for buoyancy, and coated with polyurethane. A UST is positioned at the rear end of the craft to provide forward thrust, and two USTs subtending 120° are symmetrically located on each side of the front end – at the "fish eyes" position. Together, these provide the right/left steering and backing thrust. Inside the shell, two rectangular cavities are machined, measuring 100mm100mm35mm, and 45mm60mm30mm. The shell can be opened into two halves via a stepped mid-section opening, lined with double O-rings for a watertight seal. The cylindrical UST seats are also lined with O-rings. The vent shown in the top left corner of Fig. 12(b) allows for a tether or

(a) (b)

Fig. 12. (a) 3-D wire mesh view of *Huygens*. (b) A side view of *Huygens* prototype. Three transducers are installed on the shell with two at the frontal "eyes" positioned for steering and another in the rear for forward thrust. Tethered signal generates a multiplexed 7MHz

NACA 0025

example, to minimize stirring up particulates near the seabed.

discussed in Section 3.

Tether/ antenna

UST seat

**2.4 Small underwater vehicle,** *Huygens*

an antenna for shallow-water wireless control.

NACA 0054 UST seats

sinusoidal input to the three USTs.

#### **2.5 Ultrasonic propulsion design considerations**

To our knowledge, there are only three experimental works on ultrasonic propulsors reported from (Allison, et al., 2008), (Yu & Kim, 2004), and (Wang, et al., 2011); we compare them with our UST in terms of thrust density, scaled thrust density and power density, in Table 2. As our UST system was operating at a resonance frequency and high voltage, it is difficult to make a fair and direct comparison of performance. We have employed a much higher power level, resulting in a very high thrust level and higher scaled thrust density.


Table 2. Performance of different UST devices.

This latter property is important in applications because it indicates a very compact force source operating with reasonable voltage levels, exploitable to benefit from many of its unique ultrasonic attributes we discussed earlier in the introduction.

Fig. 13. Thrust force produced by the UST versus driving electrical power. The *l* and *h* subscripts indicate evident low- and high-power regimes. The output voltage to the sinusoid is 59Vrms, and for the square wave is 54Vrms.

Ultrasonic Thruster 161

and 1.00210-3kg/ms respectively. The parameter <sup>2</sup> Re *Cd* , solved using Eq. (10), is 23.4106, and for *Huygens*, a unique point can be identified on the Re *Cd* Moody diagram; Re = 1.17104 and 0.17 *Cd* . It is thus estimated that *Huygens* will advance at a velocity of

Regarding mission duration and length, a small 11V, 0.75Ah lithium-ion battery would occupy about 10% of the *Huygens* vehicle volume. We assume an average power capacity reduction of 90% in a single discharge cycle, providing about 7.4Wh of energy. The UST consumes 69W of electrical power with an acoustic efficiency of 33.6% (Fig. 13), to produce 13.5mN of thrust with a constant vehicle advance velocity of about 0.05m/s. If we assume the instrumentation and other loads are small compared to the propulsive load, a simple straight-path mission will last around six minutes and travel a distance of about twenty meters. A somewhat larger battery could power the vehicle for perhaps thirty minutes, with a mission length of one hundred meters. While the UST is clearly not competitive with rotary or some biomimetic propulsors in terms of transit efficiency, nonetheless these estimates show that maneuvering a very small-scale vehicle utilizing USTs offers a

(a) (b)

Battery

(c)

We have also demonstrated a wireless version of *Huygens* (Fig. 14(c)) powered by a small on board battery (Fig. 14(b)). However, the electronics board could only supply limited power to each UST, and was able to slowly move the vehicle but inadequate to overcome the

Fig. 14. (a) A specialized UST constructed from a one centimeter diameter piezoelectric transducer for underwater operation. (b) An internal view of the components of the wireless

underwater vehicle, *Huygens*. (c) Wireless testing of *Huygens* in a laboratory tank.

0.054m/s – very close to the observed value.

propulsive force with interesting thrust and wake characteristics.

Fig. 13 details the acoustic efficiency of the UST transmitting at a sinusoidal 7MHz. From Eq. (2), *T* = *P*/*c*, where *c* is known (1480m/s), and *T* and *P* are measured. It can be made out that the sinusoidal waveform has two regimes relating thrust to electrical power – one of lower power with proportionally increasing thrust, and another somewhat saturated thrust at higher power. Below 70W electrical power, the power-thrust curve for sinusoidal input shows an acoustic efficiency of about 34%. Above 70W, the UST efficiency falls to less than 10%. Efficiency is calculated incrementally for each of the lines, that is, using the change in thrust versus the change in power. A square waveform is added in Fig. 13 for the sake of comparison. In general, the sinusoidal excitation is much more efficient than the square waveform. We note that the efficiency numbers given in (Allison, et al., 2008) are somewhat higher than what we show here, in part because of their custom transducer design, but also because they operated at much lower power levels. Below 2mN, we also achieve high efficiency around 70%. In respect to improving acoustic efficiency, other features such as nozzle appendages, shaping of UST, stacked piezoelectric layers could be considered.

#### **2.5.1 Vehicle mission**

In this subsection, we infer the expected speed and mission length that could reasonably be achieved from a UST-propelled underwater vehicle similar to *Huygens*. For our vehicle speed measurements, we used a high resolution Vision Research digital camera (Phantom V10) mounted with a wide-angle 20mm lens from Sigma. We sampled the advance speed of *Huygens* at 40samples/second over a straight course of 0.6m; the craft was allowed to accelerate for ten seconds before beginning the velocity measurement. Although a tether was attached to the vehicle for these tests, we maintained a large loop hanging below the vehicle, and moved the top of the tether along with the vehicle, using a sliding car and guiding post. With the recorded images of *Huygens*, the advance speed measured is a constant 0.049m/s.

The vehicle is quite streamlined, with only small holes around the frontal USTs, and a flat trim at the rear. From the top sectional view of *Huygens*, we can approximate the overall profile as an airfoil with a thickness-to-chord ratio of 0.37, and a span of 0.16m. The Reynolds number is Re *U lc* , where (kg/m3), *U* (m/s), *lc* (m) and (kg/ms) denote the fluid density, advance speed of the vehicle, chord length of the vehicle, and dynamic viscosity of the fluid respectively. The drag coefficient is 2 2 *d w <sup>T</sup> <sup>C</sup> A U* , where *T* (N) and *Aw*

(m3) denote the thrust force, and wetted surface area respectively. Expressing *U* on the left hand side of Re and *Cd* separately, the Reynolds number and drag coefficient is related by

$$\text{Re}^2 \, \text{C}\_d = \frac{2 \, \rho \, l\_c^2 \, T}{A\_w \, \mu^2} \,. \tag{10}$$

Using the Moody chart for a streamlined strut (Hoerner, 1965), *U* can then be estimated. To obtain the thrust, we recall that a sinusoidal input at 7MHz and 59Vrms, creates a thrust force of *T* = 13.5mN (Fig. 11(a)). The constants , *lc*, *Aw*, and are 1000kg/m3, 0.215m, 0.053m2,

Fig. 13 details the acoustic efficiency of the UST transmitting at a sinusoidal 7MHz. From Eq.

that the sinusoidal waveform has two regimes relating thrust to electrical power – one of lower power with proportionally increasing thrust, and another somewhat saturated thrust at higher power. Below 70W electrical power, the power-thrust curve for sinusoidal input shows an acoustic efficiency of about 34%. Above 70W, the UST efficiency falls to less than 10%. Efficiency is calculated incrementally for each of the lines, that is, using the change in thrust versus the change in power. A square waveform is added in Fig. 13 for the sake of comparison. In general, the sinusoidal excitation is much more efficient than the square waveform. We note that the efficiency numbers given in (Allison, et al., 2008) are somewhat higher than what we show here, in part because of their custom transducer design, but also because they operated at much lower power levels. Below 2mN, we also achieve high efficiency around 70%. In respect to improving acoustic efficiency, other features such as nozzle appendages, shaping of UST, stacked piezoelectric layers could be considered.

In this subsection, we infer the expected speed and mission length that could reasonably be achieved from a UST-propelled underwater vehicle similar to *Huygens*. For our vehicle speed measurements, we used a high resolution Vision Research digital camera (Phantom V10) mounted with a wide-angle 20mm lens from Sigma. We sampled the advance speed of *Huygens* at 40samples/second over a straight course of 0.6m; the craft was allowed to accelerate for ten seconds before beginning the velocity measurement. Although a tether was attached to the vehicle for these tests, we maintained a large loop hanging below the vehicle, and moved the top of the tether along with the vehicle, using a sliding car and guiding post. With the recorded images of *Huygens*, the advance speed measured is a

The vehicle is quite streamlined, with only small holes around the frontal USTs, and a flat trim at the rear. From the top sectional view of *Huygens*, we can approximate the overall profile as an airfoil with a thickness-to-chord ratio of 0.37, and a span of 0.16m. The

the fluid density, advance speed of the vehicle, chord length of the vehicle, and dynamic

(m3) denote the thrust force, and wetted surface area respectively. Expressing *U* on the left hand side of Re and *Cd* separately, the Reynolds number and drag coefficient is related by

2

*w l T <sup>C</sup> A* 

, *lc*, *Aw*, and

2

(kg/m3), *U* (m/s), *lc* (m) and

*d*

2

*<sup>T</sup> <sup>C</sup>* 

*w*

. (10)

*A U* , where *T* (N) and *Aw*

are 1000kg/m3, 0.215m, 0.053m2,

(kg/ms) denote

2

<sup>2</sup> Re *<sup>c</sup> <sup>d</sup>*

Using the Moody chart for a streamlined strut (Hoerner, 1965), *U* can then be estimated. To obtain the thrust, we recall that a sinusoidal input at 7MHz and 59Vrms, creates a thrust force

viscosity of the fluid respectively. The drag coefficient is 2

*P*/*c*, where *c* is known (1480m/s), and *T* and *P* are measured. It can be made out

(2), *T* =

**2.5.1 Vehicle mission** 

constant 0.049m/s.

Reynolds number is Re

*U lc* , where

of *T* = 13.5mN (Fig. 11(a)). The constants

and 1.00210-3kg/ms respectively. The parameter <sup>2</sup> Re *Cd* , solved using Eq. (10), is 23.4106, and for *Huygens*, a unique point can be identified on the Re *Cd* Moody diagram; Re = 1.17104 and 0.17 *Cd* . It is thus estimated that *Huygens* will advance at a velocity of 0.054m/s – very close to the observed value.

Regarding mission duration and length, a small 11V, 0.75Ah lithium-ion battery would occupy about 10% of the *Huygens* vehicle volume. We assume an average power capacity reduction of 90% in a single discharge cycle, providing about 7.4Wh of energy. The UST consumes 69W of electrical power with an acoustic efficiency of 33.6% (Fig. 13), to produce 13.5mN of thrust with a constant vehicle advance velocity of about 0.05m/s. If we assume the instrumentation and other loads are small compared to the propulsive load, a simple straight-path mission will last around six minutes and travel a distance of about twenty meters. A somewhat larger battery could power the vehicle for perhaps thirty minutes, with a mission length of one hundred meters. While the UST is clearly not competitive with rotary or some biomimetic propulsors in terms of transit efficiency, nonetheless these estimates show that maneuvering a very small-scale vehicle utilizing USTs offers a propulsive force with interesting thrust and wake characteristics.

(c)

Fig. 14. (a) A specialized UST constructed from a one centimeter diameter piezoelectric transducer for underwater operation. (b) An internal view of the components of the wireless underwater vehicle, *Huygens*. (c) Wireless testing of *Huygens* in a laboratory tank.

We have also demonstrated a wireless version of *Huygens* (Fig. 14(c)) powered by a small on board battery (Fig. 14(b)). However, the electronics board could only supply limited power to each UST, and was able to slowly move the vehicle but inadequate to overcome the

Ultrasonic Thruster 163

the interface between the transducer and epoxy layer (Sherrit, et al., 2001). *Qp* (W) is the average heat transfer rate of the piezoelectric material, which is also the average power

> <sup>2</sup> *Q fC E <sup>p</sup>* 2 tan

is the dielectric dissipation factor, and *Erms* (Vrms) is the root-mean-square of the applied voltage. It is important to understand that under high power and temperature conditions, the transducer's dielectric dissipation factor will change with the voltage applied, temperature and fluidic load. Consequently, the capacitance and dielectricity also vary nonlinearly as the voltage and temperature increase, incurring significant errors if these are not carefully characterized under elevated settings. The temperature profile of the piezoelectric material takes on a parabolic distribution, peaking at *Tmid* (C), and either surface of the transducer has the same temperature, *Tp*; see Fig. 16. In addition, as we verify in Section 3.4 for the Biot number, we consider convection to be more important than the internal conduction which exhibits a nearly uniform temperature gradient within the

(a) (b) Fig. 15. (a) Sectioned view of the modular UST. A female Teflon cap is screwed onto the base holder which holds the transducer. O-rings provide the water-tight seals, and silicone potting provides flexibility and waterproofing to the coaxial cable connection. (b) An

Matching layer

Piezo Air Seal

The 1-dimensional Fourier's law is used to describe the thermal conduction of heat through the layer of epoxy cast on the UST water-side surface, and is governed by the heat flux, *q*

where *ke* (W/mC) is the thermal conductivity, *Te* (C) is the surface temperature of the epoxy cast facing the water, and *xe* (m) is the thickness of the epoxy cast. *q* is positive if heat

*e e ee*

*Q qA k A*

*<sup>p</sup> <sup>e</sup>*

(13)

*T T*

*x*

*e*

where *f* (Hz) is the resonance frequency, *C* (F) is the capacitance of the transducer, tan

 

*rms* (12)

(%)

dissipation, given by

homogenous solid body (PZT).

shrink Coaxial cable

Silicone potting

Silicone potting Adhesive heat

exploded view of the components of the modular UST.

Seal

(W/m2), and the heat transfer rate (W) is given by

flows along the positive *x*-direction, and vice versa.

vehicle drag very well. One solution could be to improve the UST power output through a small size ultrasound amplifier such as reported in (Lewis & Olbricht, 2008). We continue to make improvements to optimize the output thrust density with considerations to on board space budget, using specially constructed underwater UST (Fig. 14(a)), and higher energy density batteries as well.

## **3. Thermal dissipation of UST**

As we observed in our previous work (Tan & Hover, 2009, 2010b), a disproportionate loss in thrust exist under elevated voltage applied across the transducer. As in all piezoelectric transducers, most of the electrical energy is converted into acoustical energy with some lost as superfluous heat through the transducer. In the presence of a large potential voltage, heat dissipation increases significantly, and dependent variables include the dielectric dissipation factor, transducer capacitance, and the presence of heat retardant materials next to the transducer. In instances of high power, localized heating at the soldered points may result in a failure or other undesirable outcome (Zhou & Rogers, 1995).

While temperature studies on PZT have been adequately described and investigated in the literature (Duck, Starritt, ter Haar, & Lunt, 1989; Sherrit et al., 2001), we are not aware of any work that makes a direct connection between transducer temperature rise and the propulsive thrust generated. As the UST is an underwater propulsor, knowledge of the conditions leading it to become a thermal source is important in many applications. In the following, we experimentally quantify the thermal distribution on the surface of the transducer under ultrasonic thrusting conditions, and introduce a dimensionless parameter to relate the thermal loss. In certain strategic applications, knowledge of this heat signature could aid in critical UST and system propulsion designs.

#### **3.1 Heat transfer equations**

We designed and constructed a larger UST unit based on (Allison, et al., 2008) but with several new features as shown in Fig. 15(a) and 15(b). Two o-rings are designed to seal the PZT against the water pressure and the air-backed layer provides maximum acoustic power transfer into the water. The screwed-on base holding the transducer is made of polytetrafluoroethylene (PTFE), so is the capping component holding the o-rings. Adhesive heat shrink and silicone potting seal the water from entering the air backing via the coaxial cable. The transducer material and medium will determine the heat transfer profile, rate of heat transfer and its dominant mode of heat transfer. For effective propulsion purpose, we add a layer of epoxy cast to match the acoustic impedance between the PZT and water.

From Fig. 16, in order to calculate the temperature profile through the transducer, *x* = 0 is taken at the centerline through the thickness of the PZT, and the average temperature at the centerline is given by

$$T\_{mid} = \frac{Q\_p \mathbf{x}\_p}{2k\_p A\_p} \left(\frac{\mathbf{1}}{\pi^2} + \frac{\mathbf{1}}{\mathbf{4}}\right) + T\_p \tag{11}$$

where *xp* (m) is the thickness of the transducer, *kp* (W/mC) is the thermal conductivity of the transducer, *Ap* (m2) is the surface area of the transducer, and *Tp* (C) is the temperature at

vehicle drag very well. One solution could be to improve the UST power output through a small size ultrasound amplifier such as reported in (Lewis & Olbricht, 2008). We continue to make improvements to optimize the output thrust density with considerations to on board space budget, using specially constructed underwater UST (Fig. 14(a)), and higher energy

As we observed in our previous work (Tan & Hover, 2009, 2010b), a disproportionate loss in thrust exist under elevated voltage applied across the transducer. As in all piezoelectric transducers, most of the electrical energy is converted into acoustical energy with some lost as superfluous heat through the transducer. In the presence of a large potential voltage, heat dissipation increases significantly, and dependent variables include the dielectric dissipation factor, transducer capacitance, and the presence of heat retardant materials next to the transducer. In instances of high power, localized heating at the soldered points may result in

While temperature studies on PZT have been adequately described and investigated in the literature (Duck, Starritt, ter Haar, & Lunt, 1989; Sherrit et al., 2001), we are not aware of any work that makes a direct connection between transducer temperature rise and the propulsive thrust generated. As the UST is an underwater propulsor, knowledge of the conditions leading it to become a thermal source is important in many applications. In the following, we experimentally quantify the thermal distribution on the surface of the transducer under ultrasonic thrusting conditions, and introduce a dimensionless parameter to relate the thermal loss. In certain strategic applications, knowledge of this heat signature

We designed and constructed a larger UST unit based on (Allison, et al., 2008) but with several new features as shown in Fig. 15(a) and 15(b). Two o-rings are designed to seal the PZT against the water pressure and the air-backed layer provides maximum acoustic power transfer into the water. The screwed-on base holding the transducer is made of polytetrafluoroethylene (PTFE), so is the capping component holding the o-rings. Adhesive heat shrink and silicone potting seal the water from entering the air backing via the coaxial cable. The transducer material and medium will determine the heat transfer profile, rate of heat transfer and its dominant mode of heat transfer. For effective propulsion purpose, we add a layer of epoxy cast to match the acoustic impedance between the PZT and water.

From Fig. 16, in order to calculate the temperature profile through the transducer, *x* = 0 is taken at the centerline through the thickness of the PZT, and the average temperature at the

> 2 1 1

(11)

2 4 *p p mid p p p Q x T T k A*

where *xp* (m) is the thickness of the transducer, *kp* (W/mC) is the thermal conductivity of the transducer, *Ap* (m2) is the surface area of the transducer, and *Tp* (C) is the temperature at

density batteries as well.

**3. Thermal dissipation of UST** 

**3.1 Heat transfer equations** 

centerline is given by

a failure or other undesirable outcome (Zhou & Rogers, 1995).

could aid in critical UST and system propulsion designs.

the interface between the transducer and epoxy layer (Sherrit, et al., 2001). *Qp* (W) is the average heat transfer rate of the piezoelectric material, which is also the average power dissipation, given by

$$Q\_p = 2\pi \, f \, \text{C} \tan \delta \, E\_{ms}^2 \tag{12}$$

where *f* (Hz) is the resonance frequency, *C* (F) is the capacitance of the transducer, tan (%) is the dielectric dissipation factor, and *Erms* (Vrms) is the root-mean-square of the applied voltage. It is important to understand that under high power and temperature conditions, the transducer's dielectric dissipation factor will change with the voltage applied, temperature and fluidic load. Consequently, the capacitance and dielectricity also vary nonlinearly as the voltage and temperature increase, incurring significant errors if these are not carefully characterized under elevated settings. The temperature profile of the piezoelectric material takes on a parabolic distribution, peaking at *Tmid* (C), and either surface of the transducer has the same temperature, *Tp*; see Fig. 16. In addition, as we verify in Section 3.4 for the Biot number, we consider convection to be more important than the internal conduction which exhibits a nearly uniform temperature gradient within the homogenous solid body (PZT).

Fig. 15. (a) Sectioned view of the modular UST. A female Teflon cap is screwed onto the base holder which holds the transducer. O-rings provide the water-tight seals, and silicone potting provides flexibility and waterproofing to the coaxial cable connection. (b) An exploded view of the components of the modular UST.

The 1-dimensional Fourier's law is used to describe the thermal conduction of heat through the layer of epoxy cast on the UST water-side surface, and is governed by the heat flux, *q* (W/m2), and the heat transfer rate (W) is given by

$$Q\_e = qA\_e = k\_eA\_e \frac{\left(T\_p - T\_e\right)}{\chi\_e} \tag{13}$$

where *ke* (W/mC) is the thermal conductivity, *Te* (C) is the surface temperature of the epoxy cast facing the water, and *xe* (m) is the thickness of the epoxy cast. *q* is positive if heat flows along the positive *x*-direction, and vice versa.

Ultrasonic Thruster 165

(a) (b)

PC

Power amp.

Torque meter

UST

Temp. logger

Fig. 17. (a) The experimental setup used to measure the UST temperature and thrust at the same time. Thrust is generated in the normal direction and out of the page. (b) A top view of

Signal generator (B&K 4011A)

Similar torque calibration is performed prior to each set of test as discussed in Section 2.3.2. The sensing tip of five AWG 36 T-type (copper-constantan) thermocouples (SW-TTC2-F36- CL1) is attached to the transducer surface at five locations; T1 (north), T2 (east), T3 (west), T4 (south), and T5 (center) when viewed face-on as shown in Fig. 18. Thermal grease is applied to the junction of the thermocouples for improved thermal contact and each junction is affixed to the surface with a small adhesive tape. All thermocouples measure 2m in length and each terminates at a type-T miniature plug (CN001-T) is plugged into an Eight Channel Thermocouple Logger (OctTemp, MadgeTech), which has corrected cold-junction compensation internally for improved accuracy and response time. Each of the thermocouple is wrapped with an electromagnetic interference (EMI) tape and grounded to remove any RF interference. All temperature sensors have been calibrated by the manufacturer and are prescribed with an accuracy of 0.5C. Temperature data is sampled at 4Hz for all channels. The data logger has a background temperature noise of less than 0.1C. Overall, temperature measurement uncertainty is estimated to be 1C in all cases. We consider a quasi-steady state of heat transfer to be defined by a temperature change of 0.2C in a minute for 3 minutes.

Fig. 18. Locations of thermocouples on the UST surface for average temperature

measurement. Thermocouple cables are wrapped with RFI tape to ground RF interference.

**T1**

**T5**

**T2 T3**

**T4**

Each temperature profile at a particular applied power is repeated four times independently under ambient water conditions. The experiment begins by recording 5 seconds of ambient

the UST temperature measurement setup.

90

Acrylic sheet

Ultrasound absorber

Water

Thermocouple

UST

50 coaxial cable

Temp. data logger

Torque display

> Torque meter

Counterbalance weight

**3.3 Thermal characteristics results** 

As heat transfers into the water, convection will be dominant (heat radiation is negligible in our case) and using the steady-state Newton's Law of cooling, the convection governing equation is described as

$$Q\_{\rm c} = \overline{h}A\_{\rm c} \left( T\_{\rm c} - T\_{\rm c} \right) \tag{14}$$

where *h* is the averaged convective heat transfer (W/m2C), *Ac* is the area of transducer exposed to the water (m2), and *T*is the ambient water tank temperature (C).

Fig. 16. A schematic diagram of the heat transfer profile. Acoustic transmission is generated by a transducer made from a PZT, through an epoxy cast layer into the water. Temperature gradient in each material outlines their relative heat transmission.

#### **3.2 Temperature experimental setup**

In this section, we characterize temperature for a 50mm UST transducer under increasing power, and then investigate in detail how various parameters could provide insights to the overall UST design consideration.

#### **3.2.1 Hardware and methods**

Fig. 17(a) shows the experimental setup similar to Fig. 6(a) except with an addition temperature data logger. Voltage to the amplifier is increased incrementally in steps of 0.2Vpp up to a maximum of 1Vpp. The output power to the transducer depends on the transducer load capacitance and electrical impedance, and is not explicitly controlled.

The PZT (*kp* = 1.25W/mC) measures 50mm in diameter and 2.1mm thick, and resonates at 1MHz. The epoxy cast layer (*ke* = 0.22W/mC) is designed to be 0.6mm thick, with the same diameter as the transducer but the o-ring seals part of it, and only 42.2mm of the diameter is exposed to the water. The UST is vertically positioned with the surface facing the side wall of the 1.2m0.6m0.6m water tank, submerged at 0.2m from the bottom of the tank. Tap water is filled and stood for at least one day before the experiment begins, and ambient water temperature is regularly maintained at 21.5C with sufficient cooling. Water is filled up to a depth of 0.45m.

As heat transfers into the water, convection will be dominant (heat radiation is negligible in our case) and using the steady-state Newton's Law of cooling, the convection governing

where *h* is the averaged convective heat transfer (W/m2C), *Ac* is the area of transducer

Fig. 16. A schematic diagram of the heat transfer profile. Acoustic transmission is generated by a transducer made from a PZT, through an epoxy cast layer into the water. Temperature

Conduction Convection

*Te*

cast

*Tp*

In this section, we characterize temperature for a 50mm UST transducer under increasing power, and then investigate in detail how various parameters could provide insights to the

Fig. 17(a) shows the experimental setup similar to Fig. 6(a) except with an addition temperature data logger. Voltage to the amplifier is increased incrementally in steps of 0.2Vpp up to a maximum of 1Vpp. The output power to the transducer depends on the transducer load capacitance and electrical impedance, and is not explicitly controlled.

The PZT (*kp* = 1.25W/mC) measures 50mm in diameter and 2.1mm thick, and resonates at 1MHz. The epoxy cast layer (*ke* = 0.22W/mC) is designed to be 0.6mm thick, with the same diameter as the transducer but the o-ring seals part of it, and only 42.2mm of the diameter is exposed to the water. The UST is vertically positioned with the surface facing the side wall of the 1.2m0.6m0.6m water tank, submerged at 0.2m from the bottom of the tank. Tap water is filled and stood for at least one day before the experiment begins, and ambient water temperature is regularly maintained at 21.5C with sufficient cooling. Water is filled

is the ambient water tank temperature (C).

Water

*x* 

*T*

*T*

*xe xp /2 xp /2*

 *Tmid*

*Tp*

gradient in each material outlines their relative heat transmission.

PZT Epoxy

*Q hA T T c ce* (14)

equation is described as

exposed to the water (m2), and *T*

**3.2 Temperature experimental setup** 

overall UST design consideration.

**3.2.1 Hardware and methods** 

up to a depth of 0.45m.

Fig. 17. (a) The experimental setup used to measure the UST temperature and thrust at the same time. Thrust is generated in the normal direction and out of the page. (b) A top view of the UST temperature measurement setup.

Similar torque calibration is performed prior to each set of test as discussed in Section 2.3.2. The sensing tip of five AWG 36 T-type (copper-constantan) thermocouples (SW-TTC2-F36- CL1) is attached to the transducer surface at five locations; T1 (north), T2 (east), T3 (west), T4 (south), and T5 (center) when viewed face-on as shown in Fig. 18. Thermal grease is applied to the junction of the thermocouples for improved thermal contact and each junction is affixed to the surface with a small adhesive tape. All thermocouples measure 2m in length and each terminates at a type-T miniature plug (CN001-T) is plugged into an Eight Channel Thermocouple Logger (OctTemp, MadgeTech), which has corrected cold-junction compensation internally for improved accuracy and response time. Each of the thermocouple is wrapped with an electromagnetic interference (EMI) tape and grounded to remove any RF interference. All temperature sensors have been calibrated by the manufacturer and are prescribed with an accuracy of 0.5C. Temperature data is sampled at 4Hz for all channels. The data logger has a background temperature noise of less than 0.1C. Overall, temperature measurement uncertainty is estimated to be 1C in all cases. We consider a quasi-steady state of heat transfer to be defined by a temperature change of 0.2C in a minute for 3 minutes.

Fig. 18. Locations of thermocouples on the UST surface for average temperature measurement. Thermocouple cables are wrapped with RFI tape to ground RF interference.

### **3.3 Thermal characteristics results**

Each temperature profile at a particular applied power is repeated four times independently under ambient water conditions. The experiment begins by recording 5 seconds of ambient

Ultrasonic Thruster 167

where *T* (C) is the time-variant temperature of the epoxy with respect to time *t* (s), *Te* (C) is the averaged temperature of the epoxy surface determined as [22.9, 26.7, 30.6, 35.0, 39.2]C

(C) is the ambient tank temperature at

= 1200 kg/m3 is the

density of the epoxy, *ce* = 1110 J/kgC is the specific heat capacity of epoxy, *Ve* = 8.410-7m3 is the volume of the epoxy through which sound transmits, *h* (W/m2C) is the average convective heat transfer coefficient determined as [25, 24, 22, 18, 14]W/m2C from the stepped input voltage, and *Ae* = 5.610-3m2 is the cross-sectional area of the epoxy through

Most of this transmitted heat energy is convected away into the water as the surface of the transducer heats up. With this knowledge and the acoustic efficiency plot in Fig. 19, where efficiency decline rapidly as output power increases, we can see that the supplied electrical power has been significantly converted into heat energy while thrust increased diminutively – that net thrust begins to saturate above 90W. In the region of the saturated thrust, we also note an increase in heating of the transducer when the power applied is increased further,

Fig. 19. Thrust force generated, acoustic efficiency, and average surface temperature versus the electrical power driving the UST. Gradients of tangent at each of the clusters of thrust points indicate the UST efficiency. Low power operation generally gives higher efficiency,

0 50 100 150 Electrical power (W)

2a

UST

Next, substitute Eq. (13) into Eq. (14) to determine *Tp*, which is then substituted into Eq. (11) together with Eq. (12) to determine *Tmid*. With *C* = [11.26, 11.38, 11.81, 12.53, 13.62]nF and tan

 ≈ 0.4%, the calculated values of *Tmid* are [23.84, 31.12, 42.49, 59.35, 82.59]C at average output voltage *Erms* = [12.75, 29, 47.5, 66.5, 85.5]Vrms. The superfluous heat, *Ql* (W), can be

*mid e*

*eq T T*

(16)

10

0

Ave. transducer surface temp. (C)

20

30

Jet

Efficiency (%)

40

50

60

but at a trivial lower thrust. At high output power, thrust generally saturates.

*l*

*<sup>Q</sup> <sup>R</sup>*

is the time constant of the cooling process, where

explaining the fundamental cause of loss of thrust at high electrical power.

Efficiency Thrust Temperature

from the stepped input voltage, and *T*

*e c V h A* 

which sound transmits (note *Ae* = *Ap*).

Thrust (mN)

21.5C. *ee e*

calculated using

water temperature, after which the power amplifier is switched on for about 9mins. During this time, the transducer is observed to increase its temperature steadily and then stabilizes. 9mins into the actuation, all thermocouple achieve a quasi-steady state and the power amplifier is switched off. Cooling proceeds for another 10mins and the tank is allowed to settle. Ambient temperature of the water tank is monitored separately at the start and end of the experiment, and further cooling is allowed if the ambient water temperature increased significantly.

Generally, increasing the input voltage (in steps of 0.2Vpp to 1.0Vpp) to the amplifier increases the temperature recorded on the transducer surface. Table 3 summarizes the input voltage to the amplifier, output voltage and output power of the amplifier, the corresponding thrust, and the mean transducer surface temperature. Four sets of data are each tabulated for the output voltage, output power, and thrust, to demonstrate the consistency of the system.

Using the output power and thrust columns of Table 3, a plot of thrust versus power is shown in Fig. 19. From Eq. (2), the acoustic efficiency of the UST can be worked out, which essentially is the gradient at each cluster of points multiply by the speed of sound in the water, *c*. Generally at lower power, the efficiency is higher, however, effective thrust is also lower, which may not be practically useful. As the power increases, efficiency declines rapidly and Fig. 19 illustrates the trend. In Fig. 19, it can be seen that generated thrust starts to decline at higher electrical power level. However, as we discuss below, the acoustic efficiency is observed to vary approximately in a first-order fashion as *P* increases. Ultrasonic thrust appears to begin to saturate near 16mN.


Table 3. Table of output power from the amplifier and the corresponding thrust generated. Four sets of data from each stepped input voltage are recorded. Ambient tank temperature is maintained at 21.5C.

#### **3.3.1 Thermal losses at high voltage**

To calculate the averaged convective heat transfer, we use the lumped-capacity solution for a heated body transferring heat into the water by free convection. The solution can be solved for *T* (*t* = 0) to give (Lienhard IV & Lienhard V, 2002)

$$T = \left(T\_e - T\_{\infty}\right)e^{-\frac{t\mathcal{V}}{T}} + T\_{\infty} \tag{15}$$

water temperature, after which the power amplifier is switched on for about 9mins. During this time, the transducer is observed to increase its temperature steadily and then stabilizes. 9mins into the actuation, all thermocouple achieve a quasi-steady state and the power amplifier is switched off. Cooling proceeds for another 10mins and the tank is allowed to settle. Ambient temperature of the water tank is monitored separately at the start and end of the experiment, and further cooling is allowed if the ambient water temperature increased

Generally, increasing the input voltage (in steps of 0.2Vpp to 1.0Vpp) to the amplifier increases the temperature recorded on the transducer surface. Table 3 summarizes the input voltage to the amplifier, output voltage and output power of the amplifier, the corresponding thrust, and the mean transducer surface temperature. Four sets of data are each tabulated for the output voltage, output power, and thrust, to demonstrate the

Using the output power and thrust columns of Table 3, a plot of thrust versus power is shown in Fig. 19. From Eq. (2), the acoustic efficiency of the UST can be worked out, which essentially is the gradient at each cluster of points multiply by the speed of sound in the water, *c*. Generally at lower power, the efficiency is higher, however, effective thrust is also lower, which may not be practically useful. As the power increases, efficiency declines rapidly and Fig. 19 illustrates the trend. In Fig. 19, it can be seen that generated thrust starts to decline at higher electrical power level. However, as we discuss below, the acoustic efficiency is observed to vary approximately in a first-order fashion as *P* increases.

Output power (W) Thrust (mN)

16.43, 16.43, 16.5, 16.35,

(15)

16.5 39.2

0.2 13, 13, 12, 13 3.2, 3.2, 2.9, 3.2 4.59, 4.44, 4.59, 4.52 22.9 0.4 30, 29, 28, 29 18.0, 16.8, 15.6, 16.8 7.08, 7.01, 6.93, 6.86 26.7 0.6 48, 47, 47, 48 45.6, 43.6, 43.6, 45.6 10.85, 10.70, 10.70, 10.78 30.6 0.8 67, 66, 66, 67 90.4, 87.2, 87.2, 90.4 14.85, 14.77, 14.62, 14.85 35.0

Table 3. Table of output power from the amplifier and the corresponding thrust generated. Four sets of data from each stepped input voltage are recorded. Ambient tank temperature

To calculate the averaged convective heat transfer, we use the lumped-capacity solution for a heated body transferring heat into the water by free convection. The solution can be solved

> *<sup>t</sup> T T Te T <sup>e</sup>*

147.8

Mean surface temp. (C)

significantly.

Input voltage (Vpp)

consistency of the system.

Output voltage (Vrms)

is maintained at 21.5C.

**3.3.1 Thermal losses at high voltage** 

Ultrasonic thrust appears to begin to saturate near 16mN.

1.0 86, 84, 86, 86 147.8, 141.8, 147.8,

for *T* (*t* = 0) to give (Lienhard IV & Lienhard V, 2002)

where *T* (C) is the time-variant temperature of the epoxy with respect to time *t* (s), *Te* (C) is the averaged temperature of the epoxy surface determined as [22.9, 26.7, 30.6, 35.0, 39.2]C from the stepped input voltage, and *T* (C) is the ambient tank temperature at 21.5C. *ee e e c V h A* is the time constant of the cooling process, where = 1200 kg/m3 is the density of the epoxy, *ce* = 1110 J/kgC is the specific heat capacity of epoxy, *Ve* = 8.410-7m3 is the volume of the epoxy through which sound transmits, *h* (W/m2C) is the average convective heat transfer coefficient determined as [25, 24, 22, 18, 14]W/m2C from the stepped input voltage, and *Ae* = 5.610-3m2 is the cross-sectional area of the epoxy through which sound transmits (note *Ae* = *Ap*).

Most of this transmitted heat energy is convected away into the water as the surface of the transducer heats up. With this knowledge and the acoustic efficiency plot in Fig. 19, where efficiency decline rapidly as output power increases, we can see that the supplied electrical power has been significantly converted into heat energy while thrust increased diminutively – that net thrust begins to saturate above 90W. In the region of the saturated thrust, we also note an increase in heating of the transducer when the power applied is increased further, explaining the fundamental cause of loss of thrust at high electrical power.

Fig. 19. Thrust force generated, acoustic efficiency, and average surface temperature versus the electrical power driving the UST. Gradients of tangent at each of the clusters of thrust points indicate the UST efficiency. Low power operation generally gives higher efficiency, but at a trivial lower thrust. At high output power, thrust generally saturates.

Next, substitute Eq. (13) into Eq. (14) to determine *Tp*, which is then substituted into Eq. (11) together with Eq. (12) to determine *Tmid*. With *C* = [11.26, 11.38, 11.81, 12.53, 13.62]nF and tan ≈ 0.4%, the calculated values of *Tmid* are [23.84, 31.12, 42.49, 59.35, 82.59]C at average output voltage *Erms* = [12.75, 29, 47.5, 66.5, 85.5]Vrms. The superfluous heat, *Ql* (W), can be calculated using

$$Q\_I = \frac{\left(T\_{mid} - T\_e\right)}{R\_{eq}} \tag{16}$$

Ultrasonic Thruster 169

increase the thrust but not appreciably; instead most added energy will be converted to heat

Finally, we verify that *Tmid* is not higher than the Curie temperature of the transducer, specified by the manufacturer at 320C, to maintain its poled lattice integrity. We also validate the Biot number (Bi), which must be Bi « 1 to justify the temperature within the

0.0600, 0.0491, 0.0382]. More importantly, this condition must also be satisfied for lump-

The ultrasonic thruster technology could bring about interesting and novel attributes to robotic propulsion devices and systems. These include low cost and high robustness when applied at the centimeter scale or smaller. The robustness is due to the fact that the UST effectively has no moving parts, and will not biofoul – these are properties unavailable in the rotary and biomimetic propulsors in use today. Our experiments indicated that frequency, and voltage level can both strongly influence the behavior of the UST, in terms of wake, thrust, and efficiency. We have successfully implemented three sub-centimeter UST devices into a small robot, and made calculations showing short missions can be developed

We have also studied the heating of ultrasonic transducers under conditions of thrust production. In view of practical application of the ultrasonic transducer in the medical field, it has been reported that clinical ultrasonic probes generate considerable heat when driven at off-resonance frequencies (Duck, et al., 1989). Most medical ultrasonic devices have a safety regulation on the level of power that the transducer can produce; for example, the IEC Standard 60606-2-37 limits the surface temperature of ultrasonic transducer to 43C. In extreme cases, ultrasonic probes could reach a steady-state temperature of 80C in ambient air at 25°C; obviously this is not suitable for human contact in practice. While the UST is not subject to complying with this standard, a UST device is still limited by extreme heating which may cause physical damage, and also because at high temperature conditions it suffers a saturation in thrust. It may be possible to minimize or harness heat for recycling in the UST system architecture, or even recoup a part of it through specialized nozzle appendages so as to enhance efficiency. For example, the backing layer of the transducer can be ventilated or cooled to remove heat. (Deardorff & Diederich, 2000) demonstrated using a water-cooling system and reported not only it does not reduce the acoustic intensity or beam distribution, but also allows more than 45W additional power supplied to the

*e*

*<sup>k</sup>* = [0.0682, 0.0655,

*hx*

losses, which becomes undesirable in the UST design scheme.

capacity solution in Eq. (15) to be accurate.

**4. Conclusion** 

transducer to be relatively even. From the above parameters, Bi *<sup>e</sup>*

with such craft, despite its inherently low propulsive efficiency.

transducer. Indeed, its thrust assistive quality remains to be investigated.

Clearly UST technology would benefit from further developmental work on applicationspecific areas. The UST could, for example, complement an existing propulsor system to fine tune maneuvering, or to strategically control or manipulate a flow-field for other purposes. Propulsive efficiency could conceivably be enhanced by developing a waveguide external to the transducer. The use of DPIV for characterizing UST properties is considerably richer than velocity measurement using hot wire method alone, and could also aid new transducer designs traceable to the wake field. New applications, designed to exploit the above UST's

where the equivalent thermal resistance, <sup>2</sup> 1 1 4 4 *<sup>p</sup> <sup>e</sup> eq p p e e x <sup>x</sup> <sup>R</sup> kA kA* , for the transducer distance from *x* = 0 to the epoxy surface. We plot this heat loss against the amplifier output voltage, *Erms*, in Fig. 20. Introducing a dimensionless parameter, *Ql <sup>P</sup>* , which is the ratio of the heat loss energy Eq. (16) to the electrical power supplied Eq. (4), as

$$\frac{Q\_l}{P} = \frac{\left(T\_{mid} - T\_e\right) \text{Re}\left(\Re\right)}{R\_{eq} E\_{rms}^2} \,\text{.}\tag{17}$$

We refer to this parameter as the lossy ratio. A large lossy ratio means more electrical energy has been converted to superfluous heat and a small ratio indicates most of the electrical energy has been converted to thrust. This ratio is also plotted in Fig. 20. Note that although the minimum of the graph signifies minimal heat loss to the electrical power supplied, it does not indicate the maximum thrust.

Fig. 20. Superfluous heat loss through the surface of the UST versus output voltage, *Erms*. An increase in output voltage is generally associated with an increase in heat loss. The minimum of the lossy curve signifies the least heat loss in the system. Higher lossy ratio means more electrical energy is converted into heat losses instead of proportionately thrust.

#### **3.4 Thermal losses design considerations**

To determine the maximum thrust without significant loss due to heating, we will consider acoustic efficiency above 10% to be a practical value for this setup. From Fig. 19, the electrical power necessary to generate 10% efficiency would be less than 95W. From (4), and = 50, *Erms* is found to be 69Vrms. When this voltage is applied across the transducer, from Fig. 19, a UST thrust of about 15mN can be expected. Obviously increasing the voltage will

distance from *x* = 0 to the epoxy surface. We plot this heat loss against the amplifier output

 2

*eq rms*

*<sup>l</sup> mid e* Re

We refer to this parameter as the lossy ratio. A large lossy ratio means more electrical energy has been converted to superfluous heat and a small ratio indicates most of the electrical energy has been converted to thrust. This ratio is also plotted in Fig. 20. Note that although the minimum of the graph signifies minimal heat loss to the electrical power supplied, it

Fig. 20. Superfluous heat loss through the surface of the UST versus output voltage, *Erms*. An

0 20 40 60 80 100

Voltage (V)

To determine the maximum thrust without significant loss due to heating, we will consider acoustic efficiency above 10% to be a practical value for this setup. From Fig. 19, the electrical power necessary to generate 10% efficiency would be less than 95W. From (4), and = 50, *Erms* is found to be 69Vrms. When this voltage is applied across the transducer, from Fig. 19, a UST thrust of about 15mN can be expected. Obviously increasing the voltage will

increase in output voltage is generally associated with an increase in heat loss. The minimum of the lossy curve signifies the least heat loss in the system. Higher lossy ratio means more electrical energy is converted into heat losses instead of proportionately thrust.

UST Average temp.

Lossy ratio

Transducer heat loss

*Q T T P R E* 4

*p p e e*

. (17)

0.205

0.2

0.195

0.19

Lossy ratio

0.185

0.18

0.175

, for the transducer

*<sup>P</sup>* , which is the ratio of

*kA kA*

4 *<sup>p</sup> <sup>e</sup> eq*

*<sup>x</sup> <sup>R</sup>*

*x*

where the equivalent thermal resistance, <sup>2</sup> 1 1

voltage, *Erms*, in Fig. 20. Introducing a dimensionless parameter, *Ql*

does not indicate the maximum thrust.

**3.4 Thermal losses design considerations** 

0

5

10

15

Transducer heat loss (W)

20

25

30

the heat loss energy Eq. (16) to the electrical power supplied Eq. (4), as

increase the thrust but not appreciably; instead most added energy will be converted to heat losses, which becomes undesirable in the UST design scheme.

Finally, we verify that *Tmid* is not higher than the Curie temperature of the transducer, specified by the manufacturer at 320C, to maintain its poled lattice integrity. We also validate the Biot number (Bi), which must be Bi « 1 to justify the temperature within the

transducer to be relatively even. From the above parameters, Bi *<sup>e</sup> e hx <sup>k</sup>* = [0.0682, 0.0655,

0.0600, 0.0491, 0.0382]. More importantly, this condition must also be satisfied for lumpcapacity solution in Eq. (15) to be accurate.

## **4. Conclusion**

The ultrasonic thruster technology could bring about interesting and novel attributes to robotic propulsion devices and systems. These include low cost and high robustness when applied at the centimeter scale or smaller. The robustness is due to the fact that the UST effectively has no moving parts, and will not biofoul – these are properties unavailable in the rotary and biomimetic propulsors in use today. Our experiments indicated that frequency, and voltage level can both strongly influence the behavior of the UST, in terms of wake, thrust, and efficiency. We have successfully implemented three sub-centimeter UST devices into a small robot, and made calculations showing short missions can be developed with such craft, despite its inherently low propulsive efficiency.

We have also studied the heating of ultrasonic transducers under conditions of thrust production. In view of practical application of the ultrasonic transducer in the medical field, it has been reported that clinical ultrasonic probes generate considerable heat when driven at off-resonance frequencies (Duck, et al., 1989). Most medical ultrasonic devices have a safety regulation on the level of power that the transducer can produce; for example, the IEC Standard 60606-2-37 limits the surface temperature of ultrasonic transducer to 43C. In extreme cases, ultrasonic probes could reach a steady-state temperature of 80C in ambient air at 25°C; obviously this is not suitable for human contact in practice. While the UST is not subject to complying with this standard, a UST device is still limited by extreme heating which may cause physical damage, and also because at high temperature conditions it suffers a saturation in thrust. It may be possible to minimize or harness heat for recycling in the UST system architecture, or even recoup a part of it through specialized nozzle appendages so as to enhance efficiency. For example, the backing layer of the transducer can be ventilated or cooled to remove heat. (Deardorff & Diederich, 2000) demonstrated using a water-cooling system and reported not only it does not reduce the acoustic intensity or beam distribution, but also allows more than 45W additional power supplied to the transducer. Indeed, its thrust assistive quality remains to be investigated.

Clearly UST technology would benefit from further developmental work on applicationspecific areas. The UST could, for example, complement an existing propulsor system to fine tune maneuvering, or to strategically control or manipulate a flow-field for other purposes. Propulsive efficiency could conceivably be enhanced by developing a waveguide external to the transducer. The use of DPIV for characterizing UST properties is considerably richer than velocity measurement using hot wire method alone, and could also aid new transducer designs traceable to the wake field. New applications, designed to exploit the above UST's

Ultrasonic Thruster 171

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## **5. Acknowledgment**

This research was supported by the Singapore National Research Foundation (NRF) through the Singapore-MIT Alliance for Research and Technology (SMART) Centre, Centre for Environmental Sensing and Modeling (CENSAM). The authors are also grateful to M. Triantafyllou and B. Simpson for access to a DPIV system.

### **6. References**


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**5. Acknowledgment** 

**6. References** 

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**1. Introduction**

indoor positioning.

The fields of application for position information have been expanded along with developments in the information-driven society. Outdoor position information systems such as car navigation systems that use the global positioning system (GPS) have spread widely. On the other hand, indoor position information is also important for humans and robots for self-location and guided navigation along routes. However, as the signals from satellites seldom reach indoors, it is hard to convert a directly outdoor positioning system to an indoor one. Therefore, systems that use sensors such as pseudolites (Petrovsky et al., 2004), infrared rays (Lee et al., 2004), or ultrasonic waves (Shih et al., 2001) have been investigated for use in

**Real-Time Distance Measurement for Indoor Positioning System Using Spread Spectrum** 

Akimasa Suzuki, Taketoshi Iyota and Kazuhiro Watanabe

**9**

*Japan* 

**Ultrasonic Waves** 

*Faculty of Engineering, Soka University* 

Compared to other methods, systems using ultrasonic waves have the advantage that they can be built at low cost and have comparatively high accuracy, because the propagation speed of ultrasonic waves is slower than that of electromagnetic waves. However, these systems are generally said to be weak in terms of noise resistance, and to take a longer time to acquire data, because they utilize the time-division multiplexed method, which becomes more taxing as the number of objects to be measured is increased. To overcome these drawbacks, positioning systems with spread spectrum (SS) ultrasonic waves have been investigated (Hazas & Hopper, 2006), (Yamane et al., 2004), (Suzuki et al., 2009); they use code division multiple access (CDMA) methods and are more robust to noise, because they use spread spectrum ultrasonic signals. According to these studies, although the procedure of reception and positioning calculation is computed off-line, the systems are shown to be highly effective. Correlation calculation is one of the most important procedures when measuring a position using SS ultrasonic waves. This calculation (in air) is difficult to carry out using existing methods, which were developed either for ultrasonic signals traveling through solids and liquids or for radio signals such as GPS because of differences in speed, frequency, and susceptibility to signal loss. The process of correlation calculation in the positioning systems also requires many calculations for long signal sequences if it is to maintain CDMA performance and robustness against noise. Therefore, real-time correlation calculation must

be realized with efficient use of limited electronic circuits.

Zhou, S.-W., & Rogers, C. A. (1995). Heat generation, temperature, and thermal stress of structurally integrated piezo-actuators. *Journal of Intelligent Material Systems and Structures, 6*(Compendex), 372-379.

## **Real-Time Distance Measurement for Indoor Positioning System Using Spread Spectrum Ultrasonic Waves**

Akimasa Suzuki, Taketoshi Iyota and Kazuhiro Watanabe *Faculty of Engineering, Soka University Japan* 

### **1. Introduction**

172 Ultrasonic Waves

Zhou, S.-W., & Rogers, C. A. (1995). Heat generation, temperature, and thermal stress of

*Structures, 6*(Compendex), 372-379.

structurally integrated piezo-actuators. *Journal of Intelligent Material Systems and* 

The fields of application for position information have been expanded along with developments in the information-driven society. Outdoor position information systems such as car navigation systems that use the global positioning system (GPS) have spread widely. On the other hand, indoor position information is also important for humans and robots for self-location and guided navigation along routes. However, as the signals from satellites seldom reach indoors, it is hard to convert a directly outdoor positioning system to an indoor one. Therefore, systems that use sensors such as pseudolites (Petrovsky et al., 2004), infrared rays (Lee et al., 2004), or ultrasonic waves (Shih et al., 2001) have been investigated for use in indoor positioning.

Compared to other methods, systems using ultrasonic waves have the advantage that they can be built at low cost and have comparatively high accuracy, because the propagation speed of ultrasonic waves is slower than that of electromagnetic waves. However, these systems are generally said to be weak in terms of noise resistance, and to take a longer time to acquire data, because they utilize the time-division multiplexed method, which becomes more taxing as the number of objects to be measured is increased. To overcome these drawbacks, positioning systems with spread spectrum (SS) ultrasonic waves have been investigated (Hazas & Hopper, 2006), (Yamane et al., 2004), (Suzuki et al., 2009); they use code division multiple access (CDMA) methods and are more robust to noise, because they use spread spectrum ultrasonic signals. According to these studies, although the procedure of reception and positioning calculation is computed off-line, the systems are shown to be highly effective.

Correlation calculation is one of the most important procedures when measuring a position using SS ultrasonic waves. This calculation (in air) is difficult to carry out using existing methods, which were developed either for ultrasonic signals traveling through solids and liquids or for radio signals such as GPS because of differences in speed, frequency, and susceptibility to signal loss. The process of correlation calculation in the positioning systems also requires many calculations for long signal sequences if it is to maintain CDMA performance and robustness against noise. Therefore, real-time correlation calculation must be realized with efficient use of limited electronic circuits.

**2. Positioning system using SS ultrasonic waves**

Positioning System Using Spread Spectrum Ultrasonic Waves

the reception unit in Fig. 1 for real-time positioning.

Fig. 1. Experimental environment of the positioning system

Position calculation for the indoor positioning system with SS ultrasonic waves is outlined in Fig. 2. First, spheres are drawn with radii equal to the distance between a receiver Rc and each transmitter (at the center); 2 pairs of the 2 spheres are selected. In Fig. 2, 2 pairs of spheres centered on Tr1 and Tr3, Tr2 and Tr3 are selected. From these pairs of spheres, Plane13 and Plane23 in Fig. 2 can be solved as a simultaneous equation. Line of the intersection is also obtained from the 2 planes of Plane13 and Plane23. Last, points at the intersection of Line with an equation of an arbitrary sphere are solved. Although the 2 intersection points are obtained, in the situation in which transmitters are installed in the four corners of a room, one solution is located outside of the room. Thus, the other solution becomes the position of the receiver

**2.2 A method for position calculation**

Rc.

In our previous study, off-line positioning experiments were conducted that utilized transmitters installed on a positional environment and a receiver mounted on a positioning target (Suzuki et al., 2009). Fig. 1 shows the experimental environment for the indoor positioning system using SS ultrasonic waves. In Fig. 1, there are 4 transmitters Tr1, Tr2, Tr3, and Tr4 including sensor nodes called SPANs (smart passive / active nodes) (Nonaka et al., 2010), and 1 receiver Rc placed on a robot. This SPAN positioning system can also utilize inverse-GPS based positioning to swap a positional relation between the transmitter and the receiver. There is also a hardware component for controlling the time of transmission and the sampling frequency of the ultrasonic waves, labeled "timing generator" in Fig. 1. We can measure the position of an object with wireless because of the radio transceiver on the timing generator and the receiver unit. In this chapter, real-time correlation calculation is realized at

175

**2.1 Indoor positioning environment**

Real-Time Distance Measurement for Indoor

As the positioning system is required to control moving robots, a study of real-time positioning should be carried out. In the positioning process, correlation calculation is required for detecting an SS signal. Some positioning systems (e.g., GPS) calculate correlation values using a serial search method or using a matched filter with analog elements (e.g., the SAW convolver (Misra & Enge, 2001)). In the case of indoor positioning using SS ultrasonic waves (with approximately 1/100000th of the propagation velocity and 1/38000th of the frequency of electrical waves), it is unlikely that one could directly utilize the conventional methods that are applicable to electrical waves. Although methods of correlation calculation for ultrasonic waves traveling through liquids or solids such as metals have been investigated (Teramoto & Yamasaki, 1988), they are still difficult to apply to indoor positioning systems with ultrasonic waves through the air because of the differences of velocity, wavelength, and attenuation rate of the ultrasonic signal. Therefore, we focused on a digital-matched filter to enable the system to both acquire the signal in a relatively short time and apply ultrasound easily.

The method for correlation calculation uses a digital-matched filter; because it requires product-sum operations on received data within a cycle of SS signals in a sampling period, it needs a large amount of calculation. Compared to outdoor positioning, the phase of SS signals shift more noticeably if the object is moving; thus, measured results must be more accurate. The carrier wavelength of ultrasound used was approximately 8.5 [mm]; filtering is required at a higher frequency than the carrier frequency if one is to apply a digital-matched filter in these conditions. Therefore, correlation calculations can be considered a bottleneck to the realization of real-time processing. S/N ratios and the number of channels in the PN sequence increase, as the length of PN sequences increases. In addition, the longer the PN sequences become, the more time is required for processing. Thus, the relation of noise tolerance and CDMA ability to processing time for correlation calculations is a trade-off.

It is difficult to find research on real-time correlation calculation or real-time positioning using SS ultrasonic signals. In general, methods of correlation calculation using product-sum operations (where pipelines use a number of multipliers that corresponds to the length of the PN sequence) have been considered. In addition, there is a method for sequential calculation using one multiplier. In the former case, a result can be obtained in 1 clock cycle, so that real-time calculation can be realized easily. However, it is unrealistic because of the huge number of logic elements required. In contrast, in the latter case, correlation calculations can be realized with a minimum logic size (i.e., they utilize only one multiplier); however, high-speed computation is required to realize real-time positioning. Therefore, certain innovations were required to allow real-time correlation calculation with SS ultrasonic waves for the purpose of self-positioning of humans and robots.

To achieve this, a new algorithm for real-time correlation calculation that uses external memory is suggested. In this chapter, the effectiveness of the proposed algorithm, named SPCM (Stored Partial Correlation Method), is presented in the form of experimental results of correlation calculations using original hardware. We also describe the effectiveness of real-time indoor positioning using SS ultrasonic waves and SPCM hardware based on experimental results of distance measurement.

## **2. Positioning system using SS ultrasonic waves**

## **2.1 Indoor positioning environment**

2 Will-be-set-by-IN-TECH

As the positioning system is required to control moving robots, a study of real-time positioning should be carried out. In the positioning process, correlation calculation is required for detecting an SS signal. Some positioning systems (e.g., GPS) calculate correlation values using a serial search method or using a matched filter with analog elements (e.g., the SAW convolver (Misra & Enge, 2001)). In the case of indoor positioning using SS ultrasonic waves (with approximately 1/100000th of the propagation velocity and 1/38000th of the frequency of electrical waves), it is unlikely that one could directly utilize the conventional methods that are applicable to electrical waves. Although methods of correlation calculation for ultrasonic waves traveling through liquids or solids such as metals have been investigated (Teramoto & Yamasaki, 1988), they are still difficult to apply to indoor positioning systems with ultrasonic waves through the air because of the differences of velocity, wavelength, and attenuation rate of the ultrasonic signal. Therefore, we focused on a digital-matched filter to enable the system to both acquire the signal in a relatively short time and apply ultrasound

The method for correlation calculation uses a digital-matched filter; because it requires product-sum operations on received data within a cycle of SS signals in a sampling period, it needs a large amount of calculation. Compared to outdoor positioning, the phase of SS signals shift more noticeably if the object is moving; thus, measured results must be more accurate. The carrier wavelength of ultrasound used was approximately 8.5 [mm]; filtering is required at a higher frequency than the carrier frequency if one is to apply a digital-matched filter in these conditions. Therefore, correlation calculations can be considered a bottleneck to the realization of real-time processing. S/N ratios and the number of channels in the PN sequence increase, as the length of PN sequences increases. In addition, the longer the PN sequences become, the more time is required for processing. Thus, the relation of noise tolerance and

It is difficult to find research on real-time correlation calculation or real-time positioning using SS ultrasonic signals. In general, methods of correlation calculation using product-sum operations (where pipelines use a number of multipliers that corresponds to the length of the PN sequence) have been considered. In addition, there is a method for sequential calculation using one multiplier. In the former case, a result can be obtained in 1 clock cycle, so that real-time calculation can be realized easily. However, it is unrealistic because of the huge number of logic elements required. In contrast, in the latter case, correlation calculations can be realized with a minimum logic size (i.e., they utilize only one multiplier); however, high-speed computation is required to realize real-time positioning. Therefore, certain innovations were required to allow real-time correlation calculation with SS ultrasonic

To achieve this, a new algorithm for real-time correlation calculation that uses external memory is suggested. In this chapter, the effectiveness of the proposed algorithm, named SPCM (Stored Partial Correlation Method), is presented in the form of experimental results of correlation calculations using original hardware. We also describe the effectiveness of real-time indoor positioning using SS ultrasonic waves and SPCM hardware based on

CDMA ability to processing time for correlation calculations is a trade-off.

waves for the purpose of self-positioning of humans and robots.

experimental results of distance measurement.

easily.

In our previous study, off-line positioning experiments were conducted that utilized transmitters installed on a positional environment and a receiver mounted on a positioning target (Suzuki et al., 2009). Fig. 1 shows the experimental environment for the indoor positioning system using SS ultrasonic waves. In Fig. 1, there are 4 transmitters Tr1, Tr2, Tr3, and Tr4 including sensor nodes called SPANs (smart passive / active nodes) (Nonaka et al., 2010), and 1 receiver Rc placed on a robot. This SPAN positioning system can also utilize inverse-GPS based positioning to swap a positional relation between the transmitter and the receiver. There is also a hardware component for controlling the time of transmission and the sampling frequency of the ultrasonic waves, labeled "timing generator" in Fig. 1. We can measure the position of an object with wireless because of the radio transceiver on the timing generator and the receiver unit. In this chapter, real-time correlation calculation is realized at the reception unit in Fig. 1 for real-time positioning.

Fig. 1. Experimental environment of the positioning system

## **2.2 A method for position calculation**

Position calculation for the indoor positioning system with SS ultrasonic waves is outlined in Fig. 2. First, spheres are drawn with radii equal to the distance between a receiver Rc and each transmitter (at the center); 2 pairs of the 2 spheres are selected. In Fig. 2, 2 pairs of spheres centered on Tr1 and Tr3, Tr2 and Tr3 are selected. From these pairs of spheres, Plane13 and Plane23 in Fig. 2 can be solved as a simultaneous equation. Line of the intersection is also obtained from the 2 planes of Plane13 and Plane23. Last, points at the intersection of Line with an equation of an arbitrary sphere are solved. Although the 2 intersection points are obtained, in the situation in which transmitters are installed in the four corners of a room, one solution is located outside of the room. Thus, the other solution becomes the position of the receiver Rc.

As Plane13 crosses through Rc13 orthogonal to *l*13, we can apply Equation 5.

*l*1

*<sup>l</sup>*<sup>13</sup> <sup>+</sup> *<sup>y</sup>*<sup>31</sup>

<sup>+</sup>*z*<sup>31</sup>

The right-hand side of Equation 6 becomes a constant. Let us denote this number by *k*31.

Thus, an equation for Plane13 is obtained. In the case of the same height for all transmitters,

By solving the simultaneous Equations 8 and 9, one can obtain the x-y coordinates of Rc as

The height of receiver Rc(z) can be also obtained using the coordinates of an arbitrary

In the situation shown in Fig. 1, 4 results for position are obtained by 4 combinations of

A position is calculated from three or more TOF (time of flight, between the transmitters and the receiver) values. An architecture of measurement TOF for the positioning system is shown in Fig. 3. In a transmission unit, a D/A converter and a FPGA, which are used to generate carrier waves and M-sequences, are included. In a reception unit, an A/D converter and a FPGA, which are used to make correlation calculations, perform peak detection, and take

,

x32y31 − x31y32

 r2

transmitters. The measurement position is defined as an average of these results.

**2.3 Distance measurement hardware structure in the positioning system**

*y* − 

*z* −  *y*<sup>1</sup> + *y*<sup>31</sup>

*l*1

*x*31*x* + *y*31*y* + *z*31*z* = *k*<sup>31</sup> (7)

*x*31*x* + *y*31*y* = *k*<sup>31</sup> (8)

*x*32*x* + *y*32*y* = *k*<sup>32</sup> (9)

<sup>i</sup> − (<sup>x</sup> − xi)<sup>2</sup> − (<sup>y</sup> − yi)<sup>2</sup> (11)

y32k31 − y31k32 x31y32 <sup>−</sup> x32y31

*z*<sup>1</sup> + *z*<sup>31</sup>

*l*1 *l*13 *x*2 <sup>31</sup> <sup>+</sup> *<sup>y</sup>*<sup>2</sup>

*l*1 *<sup>l</sup>*<sup>13</sup>

*<sup>l</sup>*<sup>13</sup> <sup>=</sup> <sup>0</sup> (5)

(10)

177

<sup>31</sup> <sup>+</sup> *<sup>z</sup>*<sup>2</sup> 31

= *x*31*x*<sup>1</sup> + *y*31*y*<sup>1</sup> + *z*31*z*<sup>1</sup> + *l*1*l*<sup>13</sup> (6)

*x*<sup>1</sup> + *x*<sup>31</sup>

*x*31*x* + *y*31*y* + *z*31*z* = *x*31*x*<sup>1</sup> + *y*31*y*<sup>1</sup> + *z*31*z*<sup>1</sup> +

*z*<sup>31</sup> =0; then, the equation for Plane13 can be expressed as follow.

Rc(x, y) = x32k31 <sup>−</sup> x31k32

Rc(z) = zi −

In addition, Plane23 can be obtained from Rc23.

follows.

transmitter *Tri*(*xi*, *yi*, *zi*).

time measurements, are included.

*<sup>x</sup>*<sup>31</sup> *x* − 

Real-Time Distance Measurement for Indoor

Positioning System Using Spread Spectrum Ultrasonic Waves

Fig. 2. Diagram for explanation of positioning calculations

Our specific method of calculation is as follows. First, to obtain equations for Plane13 and Plane23, the coordinates of points Rc13 and Rc23 in their respective planes are solved. Here, we focus on Rc13, which is between Tr1 and Tr3. In the case in which (*x*1, *y*1, *z*1) and (*x*3, *y*3, *z*3) are defined as the coordinates of Tr1 and Tr3, respectively, the distance *l*<sup>13</sup> between Tr1 and Tr3 is given by Equation 1.

$$M\_{13} = \sqrt{(x\_1 - x\_3)^2 + (y\_1 - y\_3)^2 + (z\_1 - z\_3)^2} \tag{1}$$

In the case of defining *l*<sup>1</sup> as a distance from Tr1 to the Rc13 at the center of Plane13, we can express this as follow from equations for obtaining radii of Plane13 using Pythagorean theorem and variables *r*<sup>1</sup> and *r*2.

$$r\_1^2 - l\_1^2 = r\_3^2 - \left(l\_{13} - l\_1\right)^2\tag{2}$$

Also, Equation 3 can be expressed as follow.

$$l\_1 = \frac{r\_1^2 - r\_3^2 + l\_{13}^2}{2l\_{13}}\tag{3}$$

Here, we define *x*31, *z*31, and *z*<sup>31</sup> as *x*<sup>31</sup> = *x*<sup>3</sup> − *x*1, *y*<sup>31</sup> = *y*<sup>3</sup> − *y*1, and *z*<sup>31</sup> = *z*<sup>3</sup> − *z*1, respectively. *Rc*<sup>13</sup> can be defined as in Equation 4.

$$Rc\_{13}(x,y,z) = \left(x\_1 + x\_{31}\frac{l\_1}{l\_{13}}, y\_1 + y\_{31}\frac{l\_1}{l\_{13}}, z\_1 + z\_{31}\frac{l\_1}{l\_{13}}\right) \tag{4}$$

4 Will-be-set-by-IN-TECH

Our specific method of calculation is as follows. First, to obtain equations for Plane13 and Plane23, the coordinates of points Rc13 and Rc23 in their respective planes are solved. Here, we focus on Rc13, which is between Tr1 and Tr3. In the case in which (*x*1, *y*1, *z*1) and (*x*3, *y*3, *z*3) are defined as the coordinates of Tr1 and Tr3, respectively, the distance *l*<sup>13</sup> between Tr1 and Tr3

In the case of defining *l*<sup>1</sup> as a distance from Tr1 to the Rc13 at the center of Plane13, we can express this as follow from equations for obtaining radii of Plane13 using Pythagorean

> <sup>1</sup> <sup>−</sup> *<sup>r</sup>*<sup>2</sup> <sup>3</sup> + *l* 2 13

Here, we define *x*31, *z*31, and *z*<sup>31</sup> as *x*<sup>31</sup> = *x*<sup>3</sup> − *x*1, *y*<sup>31</sup> = *y*<sup>3</sup> − *y*1, and *z*<sup>31</sup> = *z*<sup>3</sup> − *z*1, respectively.

*l*1 *l*13

2*l*<sup>13</sup>

, *y*<sup>1</sup> + *y*<sup>31</sup>

*l*1 *l*13

, *z*<sup>1</sup> + *z*<sup>31</sup>

*l*1 *l*13 

<sup>3</sup> − (*l*<sup>13</sup> − *l*1)

(*x*<sup>1</sup> − *<sup>x</sup>*3)<sup>2</sup> + (*y*<sup>1</sup> − *<sup>y</sup>*3)<sup>2</sup> + (*z*<sup>1</sup> − *<sup>z</sup>*3)<sup>2</sup> (1)

<sup>2</sup> (2)

(3)

(4)

Fig. 2. Diagram for explanation of positioning calculations

*l*<sup>13</sup> = 

> *r* 2 <sup>1</sup> − *l* 2 <sup>1</sup> = *r* 2

*<sup>l</sup>*<sup>1</sup> <sup>=</sup> *<sup>r</sup>*<sup>2</sup>

*x*<sup>1</sup> + *x*<sup>31</sup>

is given by Equation 1.

theorem and variables *r*<sup>1</sup> and *r*2.

Also, Equation 3 can be expressed as follow.

*Rc*13(*x*, *y*, *z*) =

*Rc*<sup>13</sup> can be defined as in Equation 4.

As Plane13 crosses through Rc13 orthogonal to *l*13, we can apply Equation 5.

$$\begin{aligned} \left\{ \mathbf{x}\_{31} \left\{ \mathbf{x} - \left( \mathbf{x}\_{1} + \mathbf{x}\_{31} \frac{l\_{1}}{l\_{13}} \right) \right\} + y\_{31} \left\{ y - \left( y\_{1} + y\_{31} \frac{l\_{1}}{l\_{13}} \right) \right\} \\ + z\_{31} \left\{ z - \left( z\_{1} + z\_{31} \frac{l\_{1}}{l\_{13}} \right) \right\} = 0 \end{aligned} \tag{5}$$

$$\begin{aligned} x\_{31}x + y\_{31}y + z\_{31}z &= x\_{31}x\_1 + y\_{31}y\_1 + z\_{31}z\_1 + \frac{l\_1}{l\_{13}} \left( x\_{31}^2 + y\_{31}^2 + z\_{31}^2 \right) \\ &= x\_{31}x\_1 + y\_{31}y\_1 + z\_{31}z\_1 + l\_1l\_{13} \end{aligned} \tag{6}$$

The right-hand side of Equation 6 becomes a constant. Let us denote this number by *k*31.

$$x\_{31}x + y\_{31}y + z\_{31}z = k\_{31} \tag{7}$$

Thus, an equation for Plane13 is obtained. In the case of the same height for all transmitters, *z*<sup>31</sup> =0; then, the equation for Plane13 can be expressed as follow.

$$x\_{\overline{3}1}x + y\_{\overline{3}1}y = k\_{\overline{3}1} \tag{8}$$

In addition, Plane23 can be obtained from Rc23.

$$x\_{32}x + y\_{32}y = k\_{32} \tag{9}$$

By solving the simultaneous Equations 8 and 9, one can obtain the x-y coordinates of Rc as follows.

$$\text{Rc}(\mathbf{x}, \mathbf{y}) = \left(\frac{\mathbf{x}\_{32}\mathbf{k}\_{31} - \mathbf{x}\_{31}\mathbf{k}\_{32}}{\mathbf{x}\_{32}\mathbf{y}\_{31} - \mathbf{x}\_{31}\mathbf{y}\_{32}}, \frac{\mathbf{y}\_{32}\mathbf{k}\_{31} - \mathbf{y}\_{31}\mathbf{k}\_{32}}{\mathbf{x}\_{31}\mathbf{y}\_{32} - \mathbf{x}\_{32}\mathbf{y}\_{31}}\right) \tag{10}$$

The height of receiver Rc(z) can be also obtained using the coordinates of an arbitrary transmitter *Tri*(*xi*, *yi*, *zi*).

$$\text{Rc(z)} = \mathbf{z\_i} - \sqrt{\mathbf{r\_i^2} - (\mathbf{x} - \mathbf{x\_i})^2 - (\mathbf{y} - \mathbf{y\_i})^2} \tag{11}$$

In the situation shown in Fig. 1, 4 results for position are obtained by 4 combinations of transmitters. The measurement position is defined as an average of these results.

#### **2.3 Distance measurement hardware structure in the positioning system**

A position is calculated from three or more TOF (time of flight, between the transmitters and the receiver) values. An architecture of measurement TOF for the positioning system is shown in Fig. 3. In a transmission unit, a D/A converter and a FPGA, which are used to generate carrier waves and M-sequences, are included. In a reception unit, an A/D converter and a FPGA, which are used to make correlation calculations, perform peak detection, and take time measurements, are included.

29 <sup>−</sup> <sup>1</sup> <sup>=</sup> 511 [chip] due to 9-bit shift register used to generate the M-sequence in our system.

179

From the peak detector in Fig. 3, peaks in the correlation values are obtained. Fig. 5 gives the example of a distribution of correlation values obtained from a distance measurement. In this figure, sample values connected by lines are plotted with height as the vertical axis and sampling number as the horizontal axis. A peak value from among the correlation values is obtained for the situation in which the replica signals match the received signals. Here, *stc* is defined as the number of samples corresponding to a chip length *tc*. From *stc* before the peak, the waveform in Fig. 5 becomes am upward sloping line because of the transitional intergradation of the correlation values. The high correlation value of a reflected wave is also shown in Fig. 5, arriving from some indirect pathway, as did the multi pass. The correlation value of the reflected waves can be higher than that of the direct waves; therefore, the sampling with the highest correlation values cannot be decided simply in terms of TOF. We require the detection of measurement time to be defined as the time from the start of a

transmission to the first peak, because direct ultrasonic waves have the shortest path.

In this system, the frequency of the carrier waves was set to 40.2 [kHz].

Fig. 4. Received SS signals on the reception unit in Fig. 3

**2.5 Real-time peak detection**

Real-Time Distance Measurement for Indoor

Positioning System Using Spread Spectrum Ultrasonic Waves

Fig. 5. A method for peak detection

In this system, a SS signal is generated for the multiplication of carrier waves by M-sequences. The SS signal is dynamically generated by the transmission unit shown in Fig. 3 and is outputted by a transducer, after D/A conversion. At the same time as the transmission starts, the time counter is started so as to measure the TOF. Additionally, correlation values are calculated from the sound data by the correlation calculator and the A/D convertor, shown in Fig. 3, which constitutes the on-line, real-time hardware processing. The next peak detector shown in Fig. 3 finds a peak from among the correlation values. The time counter measures the TOF by counting the number of sampling times that elapse from the beginning of a transmission to the arrival of a peak. Thereafter, the 3D position of the receiver can be calculated from three or more distances using the TOF between the transmitters and the receiver. If the correlation calculation part is installed on the hardware, as shown in Fig. 3, real-time positioning is permissible, because other processing such as positioning calculation from distances can be calculated comparatively quickly in software using optimization expressions.

Fig. 3. System architecture for TOF measurement using SS ultrasonic signals

#### **2.4 SS signal**

In our indoor positioning system, SS signals are modulated by BPSK (binary phase shift keying) using an M-sequence, a pseudorandom code sequence, as used in the DS (direct sequence) method. Although an M-sequence of '0' or '1' is generated by the shift register, we used the value of '-1' in place of '0' to facilitate signal processing. Fig. 4 shows received SS signals. In Fig. 4, the signals corresponding '1' and '-1' are plotted by solid and dashed lines, respectively. Each dot in Fig. 4 called a 'sampling'; the number of samples including 1 period worth of carrier waves was selected to be 4 [samples]. Here, chip length *tc* is defined as the required time to describe 1-chip worth of M-sequence. The chip length can be also defined as *tc* = 4/ *f* , using carrier frequency *f* = 1/*tcr*. The length of the SS ultrasonic signals becomes 29 <sup>−</sup> <sup>1</sup> <sup>=</sup> 511 [chip] due to 9-bit shift register used to generate the M-sequence in our system. In this system, the frequency of the carrier waves was set to 40.2 [kHz].

Fig. 4. Received SS signals on the reception unit in Fig. 3

#### **2.5 Real-time peak detection**

6 Will-be-set-by-IN-TECH

In this system, a SS signal is generated for the multiplication of carrier waves by M-sequences. The SS signal is dynamically generated by the transmission unit shown in Fig. 3 and is outputted by a transducer, after D/A conversion. At the same time as the transmission starts, the time counter is started so as to measure the TOF. Additionally, correlation values are calculated from the sound data by the correlation calculator and the A/D convertor, shown in Fig. 3, which constitutes the on-line, real-time hardware processing. The next peak detector shown in Fig. 3 finds a peak from among the correlation values. The time counter measures the TOF by counting the number of sampling times that elapse from the beginning of a transmission to the arrival of a peak. Thereafter, the 3D position of the receiver can be calculated from three or more distances using the TOF between the transmitters and the receiver. If the correlation calculation part is installed on the hardware, as shown in Fig. 3, real-time positioning is permissible, because other processing such as positioning calculation from distances can be calculated comparatively quickly in software

Fig. 3. System architecture for TOF measurement using SS ultrasonic signals

In our indoor positioning system, SS signals are modulated by BPSK (binary phase shift keying) using an M-sequence, a pseudorandom code sequence, as used in the DS (direct sequence) method. Although an M-sequence of '0' or '1' is generated by the shift register, we used the value of '-1' in place of '0' to facilitate signal processing. Fig. 4 shows received SS signals. In Fig. 4, the signals corresponding '1' and '-1' are plotted by solid and dashed lines, respectively. Each dot in Fig. 4 called a 'sampling'; the number of samples including 1 period worth of carrier waves was selected to be 4 [samples]. Here, chip length *tc* is defined as the required time to describe 1-chip worth of M-sequence. The chip length can be also defined as *tc* = 4/ *f* , using carrier frequency *f* = 1/*tcr*. The length of the SS ultrasonic signals becomes

using optimization expressions.

**2.4 SS signal**

From the peak detector in Fig. 3, peaks in the correlation values are obtained. Fig. 5 gives the example of a distribution of correlation values obtained from a distance measurement. In this figure, sample values connected by lines are plotted with height as the vertical axis and sampling number as the horizontal axis. A peak value from among the correlation values is obtained for the situation in which the replica signals match the received signals. Here, *stc* is defined as the number of samples corresponding to a chip length *tc*. From *stc* before the peak, the waveform in Fig. 5 becomes am upward sloping line because of the transitional intergradation of the correlation values. The high correlation value of a reflected wave is also shown in Fig. 5, arriving from some indirect pathway, as did the multi pass. The correlation value of the reflected waves can be higher than that of the direct waves; therefore, the sampling with the highest correlation values cannot be decided simply in terms of TOF. We require the detection of measurement time to be defined as the time from the start of a transmission to the first peak, because direct ultrasonic waves have the shortest path.

Fig. 5. A method for peak detection

the pre-correlation value results. To produce a large number of pre-correlation values, we

181

Hardware for the correlation calculation (using SPCM and FPGA) is shown in Fig. 6. The hardware is operated at 50 [MHz]. It is mounted on a FPGA of FLEX10KA (1728 LEs) and has an external memory of PB-SRAM. In this hardware, the transmission of ultrasonic signals triggers the start of correlation calculations. Also, the hardware shown in Fig. 6 consists of a multiplier into which the carrier waves and received signal are inputted, accumulators for 1 chip length worth of M-sequence, a 1-chip data memory to save the result of the previous accumulated result, a pre-correlator, a 4-chip data memory, and a correlator with which a whole correlation value is calculated using the 4-chip data. Although 1-chip memory is installed on the FPGA, 4-chip data is installed on the external memory because of the large amount of pre-correlation data gathered. Block 1, 2, and 3 in Fig. 6 are defined as a part for generating 1-chip data, a part for pre-correlation calculation, and a part for correlation calculation using the obtained pre-correlation values, respectively. Processing of these blocks

Fig. 7 describes correlation calculation using SPCM including the procedures at (a) block 1, (b) block 2, and (c) block 3. In block 1, received signals are multiplied only by the carrier waves, which are the elements of replica signals. Multiplied results shown in the third row of Fig. 7 (a) are obtained from both the idealized versions of waves of received signals that are shown in the first row of Fig. 7 (a), and the carrier waves generated by a reception unit shown in the second row of Fig. 7 (a). The multiplied data shown in a third row of Fig. 7 (a) are

utilize a general-purpose, active memory system.

Positioning System Using Spread Spectrum Ultrasonic Waves

Real-Time Distance Measurement for Indoor

in parallel is conducted with SPCM.

Fig. 6. Hardware layout on the real-time processor

**3.2 Generation of 1-chip data**

Fig. 5 also describes an algorithm for peak detection. First, a threshold level is selected, based on the noise level without SS signals. Next, after starting the transmission, the first correlation value that is over the threshold is detected. In this procedure, the sampling number of this correlation value is treated as a central sample *s*center, and a line is traced around a sample population near to *s*center. The slope of the line is calculated from the sample population using the least-squares method. In the group of samples, *s*max and *s*min refer to the minimum and a maximum sampling number, respectively, and are defined as having the same distance from *s*center. After this step, the slopes are calculated repeatedly, using the sample population shifted forward at increments of one sample. If a slope becomes negative, because we can consider *s*center as having reached a peak sample, the sampling number of *s*center is outputted as the peak position.

In the hardware component, a shift register is used for storage of the sample population. The slope *a* is obtained as follows. First, Let *N* and *xi*(*i* = 1, 2, 3...) denote the length of the shift register and the sampling distance from *s*center, respectively. Here, *N* is restricted to odd numbers. *x*1, *x*2... are named in ascending order (of sample number) from the minimum onward; *x*<sup>1</sup> is the sampling distance between *s*min and *s*center. *xi* is given by Equation 12.

$$-\frac{N-1}{2} \le x\_i \le \frac{N-1}{2} \tag{12}$$

Next, *yi* is defined as the correlation value of *xi*. The slope *a* is calculated using the least-squares method as follows.

$$a = \frac{N\sum\_{i=1}^{N} x\_i y\_i - \sum\_{i=1}^{N} x\_i \sum\_{i=1}^{N} y\_i}{N\sum\_{i=1}^{N} x\_i^2 - \left(\sum\_{i=1}^{N} x\_i\right)^2} \tag{13}$$

In Equation 13, a summation of *xi* <sup>2</sup> will become a constant number for a fixed bit length *N*. As *K* is defined to be a constant and the total of *xi* becomes 0, Equation 13 can be given as follows.

$$a = K \sum\_{i=1}^{N} x\_i y\_i \tag{14}$$

We select *N* = 5 for the register length. Peak detection hardware for utilizing this method can be realized comparatively easily using Equation 14.

#### **3. Real-time correlation calculation with SPCM**

#### **3.1 Hardware of real-time correlator**

To obtain a correlation value, one must perform product-sum operations for all samples within an M-sequence cycle. In the proposed positioning system, 8184 product-sum operations are required within 6.25 [microseconds]. We realize a real-time correlation calculation to construct a hardware-utilizing SPCM. In the SPCM, a preliminary part, calculated from received signals, is processed as a pre-correlation value and saved first. Thereafter, the amount of parallelism for the processing of the correlation calculation is improved by repeatedly using 8 Will-be-set-by-IN-TECH

Fig. 5 also describes an algorithm for peak detection. First, a threshold level is selected, based on the noise level without SS signals. Next, after starting the transmission, the first correlation value that is over the threshold is detected. In this procedure, the sampling number of this correlation value is treated as a central sample *s*center, and a line is traced around a sample population near to *s*center. The slope of the line is calculated from the sample population using the least-squares method. In the group of samples, *s*max and *s*min refer to the minimum and a maximum sampling number, respectively, and are defined as having the same distance from *s*center. After this step, the slopes are calculated repeatedly, using the sample population shifted forward at increments of one sample. If a slope becomes negative, because we can consider *s*center as having reached a peak sample, the sampling number of *s*center is outputted

In the hardware component, a shift register is used for storage of the sample population. The slope *a* is obtained as follows. First, Let *N* and *xi*(*i* = 1, 2, 3...) denote the length of the shift register and the sampling distance from *s*center, respectively. Here, *N* is restricted to odd numbers. *x*1, *x*2... are named in ascending order (of sample number) from the minimum onward; *x*<sup>1</sup> is the sampling distance between *s*min and *s*center. *xi* is given by Equation 12.

<sup>2</sup> <sup>≤</sup> *xi* <sup>≤</sup>

Next, *yi* is defined as the correlation value of *xi*. The slope *a* is calculated using the

*<sup>i</sup>*=<sup>1</sup> *xiyi* <sup>−</sup> <sup>∑</sup>*<sup>N</sup>*

*K* is defined to be a constant and the total of *xi* becomes 0, Equation 13 can be given as follows.

We select *N* = 5 for the register length. Peak detection hardware for utilizing this method can

To obtain a correlation value, one must perform product-sum operations for all samples within an M-sequence cycle. In the proposed positioning system, 8184 product-sum operations are required within 6.25 [microseconds]. We realize a real-time correlation calculation to construct a hardware-utilizing SPCM. In the SPCM, a preliminary part, calculated from received signals, is processed as a pre-correlation value and saved first. Thereafter, the amount of parallelism for the processing of the correlation calculation is improved by repeatedly using

*N* ∑ *i*=1

*a* = *K*

*N* − 1

*<sup>i</sup>*=<sup>1</sup> *xi* <sup>∑</sup>*<sup>N</sup>*

*<sup>i</sup>*=<sup>1</sup> *yi*

<sup>2</sup> will become a constant number for a fixed bit length *N*. As

<sup>2</sup> (12)

<sup>2</sup> (13)

*xiyi* (14)

<sup>−</sup> *<sup>N</sup>* <sup>−</sup> <sup>1</sup>

*N* ∑*<sup>N</sup> <sup>i</sup>*=<sup>1</sup> *<sup>x</sup>*<sup>2</sup> *i* − ∑*<sup>N</sup> <sup>i</sup>*=<sup>1</sup> *xi*

*<sup>a</sup>* <sup>=</sup> *<sup>N</sup>* <sup>∑</sup>*<sup>N</sup>*

as the peak position.

least-squares method as follows.

In Equation 13, a summation of *xi*

**3.1 Hardware of real-time correlator**

be realized comparatively easily using Equation 14.

**3. Real-time correlation calculation with SPCM**

the pre-correlation value results. To produce a large number of pre-correlation values, we utilize a general-purpose, active memory system.

Hardware for the correlation calculation (using SPCM and FPGA) is shown in Fig. 6. The hardware is operated at 50 [MHz]. It is mounted on a FPGA of FLEX10KA (1728 LEs) and has an external memory of PB-SRAM. In this hardware, the transmission of ultrasonic signals triggers the start of correlation calculations. Also, the hardware shown in Fig. 6 consists of a multiplier into which the carrier waves and received signal are inputted, accumulators for 1 chip length worth of M-sequence, a 1-chip data memory to save the result of the previous accumulated result, a pre-correlator, a 4-chip data memory, and a correlator with which a whole correlation value is calculated using the 4-chip data. Although 1-chip memory is installed on the FPGA, 4-chip data is installed on the external memory because of the large amount of pre-correlation data gathered. Block 1, 2, and 3 in Fig. 6 are defined as a part for generating 1-chip data, a part for pre-correlation calculation, and a part for correlation calculation using the obtained pre-correlation values, respectively. Processing of these blocks in parallel is conducted with SPCM.

## **3.2 Generation of 1-chip data**

Fig. 7 describes correlation calculation using SPCM including the procedures at (a) block 1, (b) block 2, and (c) block 3. In block 1, received signals are multiplied only by the carrier waves, which are the elements of replica signals. Multiplied results shown in the third row of Fig. 7 (a) are obtained from both the idealized versions of waves of received signals that are shown in the first row of Fig. 7 (a), and the carrier waves generated by a reception unit shown in the second row of Fig. 7 (a). The multiplied data shown in a third row of Fig. 7 (a) are

Fig. 8 (b).

Fig. 8. Production of 4-chip data from 1-chip data

Real-Time Distance Measurement for Indoor

Positioning System Using Spread Spectrum Ultrasonic Waves

**3.4 Correlation calculation using 4-chip data**

the obtained fcdg in the combination shown in Fig. 8 (1) to (4).

In block 3, the correlation calculation is conducted using the part correlation values. For example, Fig. 8 and Fig. 9 describe calculation processes in block 2 and block 3, respectively, in the case in which we use a SS signal consisting of 12 chips as 1 cycle of the M-sequence. Fig. 8 explains the procedure inside the 4-chip data memory. 16 data entries regarding fcd, generated in 1 sampling time, are shown in Fig. 8 as a group of 4-chip data (viz. 'fcdg'). A part of the product-sum operation in Fig. 8 is also shown as a pre-correlation calculation in

183

In Fig. 8, 1-chip data is generated in order from right to left, i.e., fcd1, fcd2,... When 1-chip data is obtained, a group of 4-chip data is also generated using the newest 4-data of ocd in the following order: fcdg1, fcdg2,... In Fig. 8, fcdg5 (i.e., a group of 4-chip data) is generated after a time corresponding to 4 chips from ocd4. A part correlation value corresponding to 8 chips can be generated if we lay out each fcdg1 and fcdg5 using different ocd values, as shown in Fig. 8. Also, fcdg2 and fcdg6, fcdg3 and fcdg7, and fcdg4 and fcdg8 can be created from other part correlation values at each 1-chip time. Therefore, Fig. 8 describes 4 groups of (1) fcdg1, fcdg5,... to (4) fcdg4, fcdg8,... Finally, a correlation value for all chips can be calculated using

In block 3, a correlation value is obtained by accumulating 128 continuous, memorized chips of 4-chip data corresponding to replica signals. Fig. 9 depicts the process of correlation calculation on block 3 using 4-chip data. Replica signals of the M-sequence are shown in Fig. 9 (a). The replica signals are divided into 4 sequences from the upper right-most signal in Fig. 9 (a); combinations of 4-sequences {-1,-1,-1,-1} to {1, 1, 1, 1} are named RP0 to RP15. For example, '1, 1, 1, 1', shown in the top group of replica signals, is called RP15. Fig. 9 (b) describes the arriving scenes of pre-correlation values in chronological order. Here, although pre-correlated signals are generated as fcdg1, fcdg2 in order, Fig. 9 (b) only describes a certain group (1) fcdg1, fcdg5,... in Fig. 8. In Fig. 9, first, because of the correlation calculation with RP15 from a replica signal, fcd1{1,1,1,1} in fcdg1 generated by block 2 is loaded. After 16 sampling times have elapsed, fcdg5 is generated. At this time, fcdg1 must deal with a part correlation value

accumulated as 1-chip data (i.e., ocd in Fig. 7). In block 1, segments of M-sequences cannot be detected, because the time for 16 samples is spent on 1 chip. Therefore, 16 accumulators are installed on the hardware to allow the generation of the 1-chip data.

Fig. 7. Work flow of the real-time correlation

#### **3.3 Generation of 4-chip data with pre-correlation calculation**

In block 2, the part correlation value for the *n*-chip time (viz. *n*-chip data) is generated as a pre-correlation calculation using continuous 1-chip data of *n* samples for each sampling clock. In this process, the more '*n*' increases, the more memory required, and the less time required for calculation. In our hardware, 4-chip data is generated, as *n* = 4. This is chosen in consideration of the operating frequency of the FPGA and the amount of installed memory. 4-chip data are labeled 'fcd' in Fig. 7 (b). In block 2, first, the 4 generated 1-chip data are multiplied by '1' or '-1' following an M-sequence. Next, the sum of the 4 results obtained from this calculation is saved as 'fcd'. In block 2, 24 = 16 patterns of 4-chip data (viz. fcd{−1,−1,−1,−1} to fcd{1,1,1,1}) are created based on all combinations of '1' or '-1' for 4 samples of 1-chip data. These 16 results for fcd, shown in Fig. 7, are to be part correlations values for 4 chips corresponding to 4 partial replica signals of {-1,-1,-1,-1} to {1,1,1,1}. All parts of the correlation values for a cycle of one M-sequence are saved in 4-chip data memory. Amount of memory *dext* can be expressed as

$$d\_{\rm ext} = \mathcal{Z}^{\rm n} s\_M l\_{\rm pc} \tag{15}$$

In equation 15, *sM* is to be the sampling number in the cycle of the M-sequence, 2*<sup>n</sup>* is the number of patterns in the *n*-chip data generated by pre-correlation calculation, and *lpc* is the data width of the part correlation value. The amount of memory then becomes 261888 words, because of the 32-bit data width of our hardware.

10 Will-be-set-by-IN-TECH

accumulated as 1-chip data (i.e., ocd in Fig. 7). In block 1, segments of M-sequences cannot be detected, because the time for 16 samples is spent on 1 chip. Therefore, 16 accumulators are

In block 2, the part correlation value for the *n*-chip time (viz. *n*-chip data) is generated as a pre-correlation calculation using continuous 1-chip data of *n* samples for each sampling clock. In this process, the more '*n*' increases, the more memory required, and the less time required for calculation. In our hardware, 4-chip data is generated, as *n* = 4. This is chosen in consideration of the operating frequency of the FPGA and the amount of installed memory. 4-chip data are labeled 'fcd' in Fig. 7 (b). In block 2, first, the 4 generated 1-chip data are multiplied by '1' or '-1' following an M-sequence. Next, the sum of the 4 results obtained from this calculation is saved as 'fcd'. In block 2, 24 = 16 patterns of 4-chip data (viz. fcd{−1,−1,−1,−1} to fcd{1,1,1,1}) are created based on all combinations of '1' or '-1' for 4 samples of 1-chip data. These 16 results for fcd, shown in Fig. 7, are to be part correlations values for 4 chips corresponding to 4 partial replica signals of {-1,-1,-1,-1} to {1,1,1,1}. All parts of the correlation values for a cycle of one M-sequence are saved in 4-chip data memory. Amount of

In equation 15, *sM* is to be the sampling number in the cycle of the M-sequence, 2*<sup>n</sup>* is the number of patterns in the *n*-chip data generated by pre-correlation calculation, and *lpc* is the data width of the part correlation value. The amount of memory then becomes 261888 words,

*dext* = 2*nsMlpc* (15)

installed on the hardware to allow the generation of the 1-chip data.

Fig. 7. Work flow of the real-time correlation

memory *dext* can be expressed as

because of the 32-bit data width of our hardware.

**3.3 Generation of 4-chip data with pre-correlation calculation**

In block 3, the correlation calculation is conducted using the part correlation values. For example, Fig. 8 and Fig. 9 describe calculation processes in block 2 and block 3, respectively, in the case in which we use a SS signal consisting of 12 chips as 1 cycle of the M-sequence. Fig. 8 explains the procedure inside the 4-chip data memory. 16 data entries regarding fcd, generated in 1 sampling time, are shown in Fig. 8 as a group of 4-chip data (viz. 'fcdg'). A part of the product-sum operation in Fig. 8 is also shown as a pre-correlation calculation in Fig. 8 (b).

Fig. 8. Production of 4-chip data from 1-chip data

In Fig. 8, 1-chip data is generated in order from right to left, i.e., fcd1, fcd2,... When 1-chip data is obtained, a group of 4-chip data is also generated using the newest 4-data of ocd in the following order: fcdg1, fcdg2,... In Fig. 8, fcdg5 (i.e., a group of 4-chip data) is generated after a time corresponding to 4 chips from ocd4. A part correlation value corresponding to 8 chips can be generated if we lay out each fcdg1 and fcdg5 using different ocd values, as shown in Fig. 8. Also, fcdg2 and fcdg6, fcdg3 and fcdg7, and fcdg4 and fcdg8 can be created from other part correlation values at each 1-chip time. Therefore, Fig. 8 describes 4 groups of (1) fcdg1, fcdg5,... to (4) fcdg4, fcdg8,... Finally, a correlation value for all chips can be calculated using the obtained fcdg in the combination shown in Fig. 8 (1) to (4).

## **3.4 Correlation calculation using 4-chip data**

In block 3, a correlation value is obtained by accumulating 128 continuous, memorized chips of 4-chip data corresponding to replica signals. Fig. 9 depicts the process of correlation calculation on block 3 using 4-chip data. Replica signals of the M-sequence are shown in Fig. 9 (a). The replica signals are divided into 4 sequences from the upper right-most signal in Fig. 9 (a); combinations of 4-sequences {-1,-1,-1,-1} to {1, 1, 1, 1} are named RP0 to RP15. For example, '1, 1, 1, 1', shown in the top group of replica signals, is called RP15. Fig. 9 (b) describes the arriving scenes of pre-correlation values in chronological order. Here, although pre-correlated signals are generated as fcdg1, fcdg2 in order, Fig. 9 (b) only describes a certain group (1) fcdg1, fcdg5,... in Fig. 8. In Fig. 9, first, because of the correlation calculation with RP15 from a replica signal, fcd1{1,1,1,1} in fcdg1 generated by block 2 is loaded. After 16 sampling times have elapsed, fcdg5 is generated. At this time, fcdg1 must deal with a part correlation value

those of the conventional off-line correlation calculation that uses a sequential computation,

185

In these experiments, the ultrasonic signals were transmitted by a super tweeter to a condenser microphone to measure with the distance *d* between the receiver and the transmitter installed at 1.2 [m] intervals from 1.2 [m] to 12.0 [m]. Additionally, the TOF (from transmitter to receiver) using the results of real-time correlation calculation and temperatures was measured.

Calculated correlation values for different sampling times are obtained at the shortest *d* =1.2 [m] and the longest *d* =12 [m] distances; they are shown in Fig. 10(a) and Fig. 10(b), respectively. Each figure represents the differences between (1) off-line sequential computation using software, and (2) the real-time method using hardware. In addition, each figure is arranged on the x-axis from low to high sampling time, and on the y-axis from low

(a) Setting distance *d* =1.2m (b) Setting distance *d* =12.0m

Fig. 10. Difference between off-line sequential computation using software and the real-time

Comparing (1) to (2), the experimental results in Fig. 10(a) and Fig. 10(b), the peak values are plotted at the same time of 566 [samples] and 5609 [samples], respectively. These figures also show that the peak can be detected without the influence of noise from indirect signals, especially at a distance of *d* = 12 [m] where the lowest S/N ratios were obtained in this

experiment; this effect was due to the reduction of signal strength.

experiments were conducted using the received sound data.

Positioning System Using Spread Spectrum Ultrasonic Waves

Real-Time Distance Measurement for Indoor

**4.2 Discussion of results of correlation calculation**

to high correlation values.

method using the SPCM hardware

Distance errors in the data in this experiment were calculated in real-time.

corresponding to RP3. Therefore, fcd1 {-1,-1,1,1} in fcdg1 is loaded. In this case, fcd5 {1,1,1,1} is loaded from fcdg5 from within the latest 4-chip data. Part correlation values for 8 chips are obtained by accumulating fcd1 {-1,-1,1,1} and fcd5 {1,1,1,1}. Similar processes to those described above occurred for every generation of a 4-chip data group. Finally, the complete correlation values are obtained by accumulating 4 fcds corresponding to RP15, RP8, RP3, and RP15 from 4-chip data groups of fcdg1 to fcdg13, as shown at the bottom of Fig. 9 (b).

Fig. 9. Selection of 4-chip data using replica signals in block 3

In this method, a part correlation value generated once and saved previously can be reused in the calculation of the total correlation value. The SPCM is only required to calculate the latest 4-chip data as a part correlation value, represented by the gray block in Fig. 9 (b), for obtaining the whole correlation value of the SS signal. SPCM reduces the calculation to 1/16 of its previous value; however, the amount of saved data increases because of the pre-correlation calculation in block 2. Therefore, a real-time correlation calculation can be conducted comparatively easily using SPCM.

In this algorithm, two kinds of carrier waves (sine and cosine) are utilized for the detection of orthogonal components. The whole correlation values are obtained using a root-mean-square value of both correlation values using sine and cosine waves in block 3.

## **4. Behavior of real-time correlation hardware**

#### **4.1 Experiments of correlation calculation using actual received signals**

The accuracies and real-time performance of correlation values with SPCM must be verified to allow a discussion of the effectiveness of this method in indoor positioning applications. The proposed real-time correlation mechanism will be mounted on conventional signal reception hardware. Thus, measuring experiments were conducted using the hardware shown in Fig. 3; the received sound data was inputted into the real-time correlation hardware. Additionally, to compare the resultant correlation values gathered by the real-time processing method to 12 Will-be-set-by-IN-TECH

corresponding to RP3. Therefore, fcd1 {-1,-1,1,1} in fcdg1 is loaded. In this case, fcd5 {1,1,1,1} is loaded from fcdg5 from within the latest 4-chip data. Part correlation values for 8 chips are obtained by accumulating fcd1 {-1,-1,1,1} and fcd5 {1,1,1,1}. Similar processes to those described above occurred for every generation of a 4-chip data group. Finally, the complete correlation values are obtained by accumulating 4 fcds corresponding to RP15, RP8, RP3, and

In this method, a part correlation value generated once and saved previously can be reused in the calculation of the total correlation value. The SPCM is only required to calculate the latest 4-chip data as a part correlation value, represented by the gray block in Fig. 9 (b), for obtaining the whole correlation value of the SS signal. SPCM reduces the calculation to 1/16 of its previous value; however, the amount of saved data increases because of the pre-correlation calculation in block 2. Therefore, a real-time correlation calculation can be

In this algorithm, two kinds of carrier waves (sine and cosine) are utilized for the detection of orthogonal components. The whole correlation values are obtained using a root-mean-square

The accuracies and real-time performance of correlation values with SPCM must be verified to allow a discussion of the effectiveness of this method in indoor positioning applications. The proposed real-time correlation mechanism will be mounted on conventional signal reception hardware. Thus, measuring experiments were conducted using the hardware shown in Fig. 3; the received sound data was inputted into the real-time correlation hardware. Additionally, to compare the resultant correlation values gathered by the real-time processing method to

RP15 from 4-chip data groups of fcdg1 to fcdg13, as shown at the bottom of Fig. 9 (b).

Fig. 9. Selection of 4-chip data using replica signals in block 3

value of both correlation values using sine and cosine waves in block 3.

**4.1 Experiments of correlation calculation using actual received signals**

conducted comparatively easily using SPCM.

**4. Behavior of real-time correlation hardware**

those of the conventional off-line correlation calculation that uses a sequential computation, experiments were conducted using the received sound data.

In these experiments, the ultrasonic signals were transmitted by a super tweeter to a condenser microphone to measure with the distance *d* between the receiver and the transmitter installed at 1.2 [m] intervals from 1.2 [m] to 12.0 [m]. Additionally, the TOF (from transmitter to receiver) using the results of real-time correlation calculation and temperatures was measured. Distance errors in the data in this experiment were calculated in real-time.

## **4.2 Discussion of results of correlation calculation**

Calculated correlation values for different sampling times are obtained at the shortest *d* =1.2 [m] and the longest *d* =12 [m] distances; they are shown in Fig. 10(a) and Fig. 10(b), respectively. Each figure represents the differences between (1) off-line sequential computation using software, and (2) the real-time method using hardware. In addition, each figure is arranged on the x-axis from low to high sampling time, and on the y-axis from low to high correlation values.

Fig. 10. Difference between off-line sequential computation using software and the real-time method using the SPCM hardware

Comparing (1) to (2), the experimental results in Fig. 10(a) and Fig. 10(b), the peak values are plotted at the same time of 566 [samples] and 5609 [samples], respectively. These figures also show that the peak can be detected without the influence of noise from indirect signals, especially at a distance of *d* = 12 [m] where the lowest S/N ratios were obtained in this experiment; this effect was due to the reduction of signal strength.

Fig. 11 shows that the errors in distance are within ±1.5 [cm] for all distances measured. We require a 10 [cm] or smaller positioning error for accurate self-location recognition for robots and humans; the obtained experimental measurement error in distance is within this range.

187

This chapter discussed the real-time correlation calculation for SS ultrasonic signals for use in an indoor positioning system. The experimental results were gathered using an original SPCM hardware system with external memory. In our method, positioning processing, signal receiving, correlation calculation, peak detection, and distance measurement are realized by

Because of the real-time correlation calculation, which requires the most time during hardware processing, we proposed and installed a SPCM system with external memory and comparatively small logic elements. SPCM can be divided into 3 blocks: extraction of M-sequence signal by multiplying a received signal by carrier waves and accumulation of the signal in block 1, pre-correlation calculation with 4 chips in block 2, and calculation for whole correlation values in block 3 using pre-correlation values. The amount of correlation

A distance measurement experiment was conducted to evaluate both the correlation values and the real-time performances. Experimental results using the original hardware with SPCM are compared to the results of off-line sequential computation using software. Hardware processing time was measured using a counter based on the hardware clock. Additionally, the measurement distance was calculated from TOF data utilizing the SPCM hardware in

From the experiments, we found that the real-time hardware computed correctly within 5.76 [microseconds], which was less than the sampling time; the distance errors obtained were within ±1.5 [cm]. Thus, the effectiveness of this hardware has been shown. In the case of 3D indoor positioning with SS ultrasonic signals, more than 3 signals (made by different channels from transmitters using CDMA) must be used. As a correlator is required when calculating signals on 1 channel, downsizing of the logic elements is important for creating useful receivers. At this time, indoor positioning using SPCM is suggested for use in real-time positioning using comparatively small logic elements. This chapter also shows the possibility of real-time position sensing for mobile robots and humans using SS ultrasonic waves.

Hazas, M. & Hopper, A. (2006). Broadband ultrasonic location systems for improved indoor positioning, *IEEE Transaction on Mobile Computing* Vol. 5(No. 5): 536 – 547. Lee, C., Chang, Y., Park, G., Ryu, J., Jeong, S.-G., Park, S., Park, J. W., Lee, H. C., shik Hong, K.

Misra, P. & Enge, P. (2001). *Global Positioning System Signals, Measurements and Performance*,

& Lee, M. H. (2004). Indoor positioning system based on incident angles of infrared emitters, *30th Annual Conference of IEEE Industrial Electronics Society*, Vol. 3, pp. 2218

hardware processing on a FPGA mounted on a PCI board.

Real-Time Distance Measurement for Indoor

Positioning System Using Spread Spectrum Ultrasonic Waves

calculation can be reduced by 1/64 by the pre-correlation calculation.

**5. Conclusion**

real-time.

**6. References**

– 2222.

Ganga-Jamuna Press.

#### **4.3 Discussion of real-time performances**

Table 1 describes the time required to execute each block in the whole process of the real-time correlation hardware, shown in Fig. 8. For measuring the time required in each block, a counter installed on the FPGA is counted utilizing 50 [MHz] of oscillator. The counted values are outputted to the PC via the PCI bus after real-time processing.

Table 1 shows the time required in each block as the counted values of oscillated frequency. The experimental results show that the required time for the whole process is 5.76 [microseconds], including the required times of 0.44, 1.78, and 5.12 [microseconds] for block 1, 2, and 3, respectively. The required time for the whole process depends on the longest time taken by the block processing, as each block is processed in parallel. Thus, the sum of the required time of block 3 and data transfer time among the blocks yields the overall time required by the whole process, as given in Table 1. Additionally, as shown in Table 1, the time for the whole process is less than the sampling cycle of 6.25 [microseconds]; therefore, the experimental results show that an indoor real-time positioning system using SS ultrasonic signals could be created by the correlation calculation hardware.


Table 1. Required time for each block

#### **4.4 Measurement errors of the experiments using SS ultrasonic waves**

Fig. 11 shows the errors in the distances obtained in real-time at setting distances *d* between 1.2 [m] and 12.0 [m]. Fig. 11 has an x-axis from *d* = 1.2 [m] to 12.0 [m] and a y-axis from low to high distance error. The distance errors *derr* in Fig. 11 are approximated using Equation 16 (from sampling time *sd* and temperature *T*).

#### Fig. 11. Experiment errors in the distance measurement using SS ultrasonic waves

$$d\_{err} = d - 6.25 \text{s}\_d (331.5 + 0.60714T) \tag{16}$$

Fig. 11 shows that the errors in distance are within ±1.5 [cm] for all distances measured. We require a 10 [cm] or smaller positioning error for accurate self-location recognition for robots and humans; the obtained experimental measurement error in distance is within this range.

## **5. Conclusion**

14 Will-be-set-by-IN-TECH

Table 1 describes the time required to execute each block in the whole process of the real-time correlation hardware, shown in Fig. 8. For measuring the time required in each block, a counter installed on the FPGA is counted utilizing 50 [MHz] of oscillator. The counted values

Table 1 shows the time required in each block as the counted values of oscillated frequency. The experimental results show that the required time for the whole process is 5.76 [microseconds], including the required times of 0.44, 1.78, and 5.12 [microseconds] for block 1, 2, and 3, respectively. The required time for the whole process depends on the longest time taken by the block processing, as each block is processed in parallel. Thus, the sum of the required time of block 3 and data transfer time among the blocks yields the overall time required by the whole process, as given in Table 1. Additionally, as shown in Table 1, the time for the whole process is less than the sampling cycle of 6.25 [microseconds]; therefore, the experimental results show that an indoor real-time positioning system using SS ultrasonic

> block 1 0.44 block 2 1.78 block 3 5.12 whole process 5.76

Fig. 11 shows the errors in the distances obtained in real-time at setting distances *d* between 1.2 [m] and 12.0 [m]. Fig. 11 has an x-axis from *d* = 1.2 [m] to 12.0 [m] and a y-axis from low to high distance error. The distance errors *derr* in Fig. 11 are approximated using Equation 16

Fig. 11. Experiment errors in the distance measurement using SS ultrasonic waves

*derr* = *d* − 6.25*sd*(331.5 + 0.60714*T*) (16)

required time [*μ*s]

**4.3 Discussion of real-time performances**

Table 1. Required time for each block

(from sampling time *sd* and temperature *T*).

are outputted to the PC via the PCI bus after real-time processing.

signals could be created by the correlation calculation hardware.

**4.4 Measurement errors of the experiments using SS ultrasonic waves**

This chapter discussed the real-time correlation calculation for SS ultrasonic signals for use in an indoor positioning system. The experimental results were gathered using an original SPCM hardware system with external memory. In our method, positioning processing, signal receiving, correlation calculation, peak detection, and distance measurement are realized by hardware processing on a FPGA mounted on a PCI board.

Because of the real-time correlation calculation, which requires the most time during hardware processing, we proposed and installed a SPCM system with external memory and comparatively small logic elements. SPCM can be divided into 3 blocks: extraction of M-sequence signal by multiplying a received signal by carrier waves and accumulation of the signal in block 1, pre-correlation calculation with 4 chips in block 2, and calculation for whole correlation values in block 3 using pre-correlation values. The amount of correlation calculation can be reduced by 1/64 by the pre-correlation calculation.

A distance measurement experiment was conducted to evaluate both the correlation values and the real-time performances. Experimental results using the original hardware with SPCM are compared to the results of off-line sequential computation using software. Hardware processing time was measured using a counter based on the hardware clock. Additionally, the measurement distance was calculated from TOF data utilizing the SPCM hardware in real-time.

From the experiments, we found that the real-time hardware computed correctly within 5.76 [microseconds], which was less than the sampling time; the distance errors obtained were within ±1.5 [cm]. Thus, the effectiveness of this hardware has been shown. In the case of 3D indoor positioning with SS ultrasonic signals, more than 3 signals (made by different channels from transmitters using CDMA) must be used. As a correlator is required when calculating signals on 1 channel, downsizing of the logic elements is important for creating useful receivers. At this time, indoor positioning using SPCM is suggested for use in real-time positioning using comparatively small logic elements. This chapter also shows the possibility of real-time position sensing for mobile robots and humans using SS ultrasonic waves.

## **6. References**


**10** 

*Turkey* 

*1Dokuz Eylul University* 

**Ultrasonic Waves in Mining Application** 

This chapter is aimed to introduce ways of beneficiation from ultrasonic waves in earth science, especially in mining practices. Since rocks are non-homogenous, elasto-plastic material, it has always been difficult to predict the behaviour of rock under any stress loaded environment. Unless removing uncertainties in the rock masses, designers can face to highly surprising and costly operational results in mining practices, so reducing the risk factor becomes vital element of underground constructions. To reduce risks may only be possible by knowing the surroundings where you work in very well. Sometimes it becomes costly to make the mining environment clear, so some practical methods have been trying to develop over years. One of them is acoustic methods based on the theory of elasticity. The elastic properties of substances are characterized by elastic module or constants that specify the relationship between stress and strain. The strains in a body are deformations, which produce restoring forces opposed to the stress. Tensile and compressive stresses give rise to longitudinal and volume strains, which are measured as unit changes in length and volume under pressure. Shear strains are measured by deformation angles. It is usually assumed that the strains are small and reversible, that is, a body resumes its original shape and size when the stresses are relieved. If the stress in an elastic medium is released suddenly, the

The principle of the ultrasonic testing method is to create waves at a point and determine the time of arrival at a number of other points for the energy that is travelling within different rock masses. The velocity of ultrasonic pulses travelling in a solid material depends on the density and elastic properties of that material. The quality of some rock masses is sometimes related to their elastic stiffness and rock mass structure, such that the measurement of ultrasonic pulse velocity in these materials can often be used to indicate their quality, as well as to determine their elastic properties. Travelling velocities of ultrasonic pulses are high in homogenous rock masses with high mechanical properties (UCS, tensile strength, cohesion, internal friction angle), which can be used as identification method of the quality of any rock structure. Some methods had been developed to measure rock diggability, stress distribution near a mine opening, bench blasting efficiency due to structural identification of rock masses by comparing the ultrasonic pulse travelling velocities in a reference sample with real in-situ measurements. In laboratory scale, available

condition of strain propagates within the medium as an elastic wave.

techniques and measurable rock mass properties are given in Table 1.

**1. Introduction** 

Ahmet Hakan Onur1, Safa Bakraç2 and Doğan Karakuş<sup>1</sup>

*2Turkish General Directorate of Mineral Research and Exploration* 


## **Ultrasonic Waves in Mining Application**

Ahmet Hakan Onur1, Safa Bakraç2 and Doğan Karakuş<sup>1</sup>

*1Dokuz Eylul University 2Turkish General Directorate of Mineral Research and Exploration Turkey* 

## **1. Introduction**

16 Will-be-set-by-IN-TECH

188 Ultrasonic Waves

Nonaka, J., Kon, T., Choi, Y. & Watanabe, K. (2010). Implementation of task processing

Petrovsky, I., Ishii, M., Asako, M., Okano, K., Torimoto, H. & Suzuki, K. (2004). Pseudelite

Shih, S., Minami, M., Morikawa, H. & Aoyama, T. (2001). An implementation and evaluation of indoor ultrasonic tracking system, *IEIC Technical Report* Vol. 101(No. 71): 1 – 8. Suzuki, A., Yamane, A., Iyota, T., Choi, Y., Kubota, Y. & Watanabe, K. (2009). Measurement

Yamane, A., Iyoda, T., Choi, Y., Kubota, Y. & Watanabe, K. (2004). A study on propagation

transducer, *Journal of Robotics and Mechatronics* Vol. 16(No. 3): 333 – 341.

application for its, *IEICE Technical Report* Vol. 230(No. 104): 13 –18.

pp. 67–69.

7): 655 – 661.

modules to a sensor node of span for offering services in an indoor positioning sensor network, *2010 International Symposium on Smart Sensing and Actuator System*,

accuracy on indoor positioning system using spread spectrum ultrasonic waves, *4th International Conference on Autonomous Robots and Agents 2009*, IEEE, pp. 294 – 297. Teramoto, K. & Yamasaki, H. (1988). Circular holographic sonar utilizing an inverse problem

solution, *Transactions of the Society of Instrument and Control Engineers* Vol. 24(No.

characteristics of spread spectrum sound waves using a band-limited ultrasonic

This chapter is aimed to introduce ways of beneficiation from ultrasonic waves in earth science, especially in mining practices. Since rocks are non-homogenous, elasto-plastic material, it has always been difficult to predict the behaviour of rock under any stress loaded environment. Unless removing uncertainties in the rock masses, designers can face to highly surprising and costly operational results in mining practices, so reducing the risk factor becomes vital element of underground constructions. To reduce risks may only be possible by knowing the surroundings where you work in very well. Sometimes it becomes costly to make the mining environment clear, so some practical methods have been trying to develop over years. One of them is acoustic methods based on the theory of elasticity. The elastic properties of substances are characterized by elastic module or constants that specify the relationship between stress and strain. The strains in a body are deformations, which produce restoring forces opposed to the stress. Tensile and compressive stresses give rise to longitudinal and volume strains, which are measured as unit changes in length and volume under pressure. Shear strains are measured by deformation angles. It is usually assumed that the strains are small and reversible, that is, a body resumes its original shape and size when the stresses are relieved. If the stress in an elastic medium is released suddenly, the condition of strain propagates within the medium as an elastic wave.

The principle of the ultrasonic testing method is to create waves at a point and determine the time of arrival at a number of other points for the energy that is travelling within different rock masses. The velocity of ultrasonic pulses travelling in a solid material depends on the density and elastic properties of that material. The quality of some rock masses is sometimes related to their elastic stiffness and rock mass structure, such that the measurement of ultrasonic pulse velocity in these materials can often be used to indicate their quality, as well as to determine their elastic properties. Travelling velocities of ultrasonic pulses are high in homogenous rock masses with high mechanical properties (UCS, tensile strength, cohesion, internal friction angle), which can be used as identification method of the quality of any rock structure. Some methods had been developed to measure rock diggability, stress distribution near a mine opening, bench blasting efficiency due to structural identification of rock masses by comparing the ultrasonic pulse travelling velocities in a reference sample with real in-situ measurements. In laboratory scale, available techniques and measurable rock mass properties are given in Table 1.

Ultrasonic Waves in Mining Application 191

spectrum and their analysis locates the anomalies within the structure. The resonant frequency test defines dynamic property of a sample and in generally used in laboratory environments. An oscillator outputs vibrations, which is analysed into the materials transverse, longitudinal and torsional frequencies of the material. Sonic tomography analyses seismic P-wave velocities to image sections of a material. The idea behind the previous technique and the ultrasonic pulse velocity method is similar; however sonic tomography can use a large number of transmitters and receiver at the same time. Its sensivity allows this method to analyse between different anomalies within a structure and

There has been big development on measurement devices, since their first introduction by Leslie & Cheesman (1949). Then, measurement techniques pervaded in rock mechanic application given rise to ASTM D2845-08 Standard Test Method for Laboratory Determination of Pulse Velocities and Ultrasonic Constants of Rock. This test method describes equipment and procedures for laboratory measurements of the pulse velocities of compression waves and shear waves in rock and the determination of ultrasonic elastic

This chapter is about to introduce a practical application of ultrasonic waves in marble industry. Miners working in the marble industry have always been interested in identifying structural weaknesses in marble blocks before they are cut in a marble quarry and transported to marble processing plants. To achieve this difficult task, several simple methods have been developed among miners but observation-based methods do not consistently provide satisfactory results. A nondestructive method developed for testing concrete could be used for this purpose. In this chapter, this simple method based on differences in ultrasonic wave propagation in different rock masses will be presented, and the test results performed both in the laboratory and a marble quarry will be discussed.

When ultrasonic testing is applied to marble blocks, its objective is to detect internal flaws that send echoes back in the direction of the incident beam. These echoes are detected by a receiving transducer. The measurement of the time taken for the pulse to travel from the surface to a flaw and back again enables the position of the flaw to be located. This ultrasonic testing technique was originally developed for assessing the quality and condition of concrete. One instrument used for this purpose is known as PUNDIT. The apparatus has been designed especially for field testing, being light, portable and simple to use. Simple correlations between concrete strength, concrete aggregate gradation, water-

The possibility of identifying these structural defects using ultrasonic pulses will be discussed and results obtained from these measurements will be introduced in the scope of this chapter. The shape and size of any abnormality in a block can be determined by direct measurements taken from suitably spaced grids. It is important to find the exact position of the surface in marble blocks so that precautions can be taken before the cutting process starts. As stated before, if any discontinuity surface lies in the pulse path, the measured time corresponds to the pulse that follows the shortest path. This is important because any discontinuity causes a time delay compared with the travel time of pulses in homogenous blocks. This study concentrated on the relationship between structural discontinuities and

process a sectional view of a sample (Moozar, 1993).

constants of an isotropic rock or one exhibiting slight anisotropy.

cement ratio and curing time have been analyzed using PUNDIT.


Table 1. Non-destructive methods and applications (Moozar, 1993)

Fig. 1. Pulse velocity test (Malhotra & Carino, 1991)

Discontinuities and their dimensions within any material can be determined by the techniques given in Table 1 (Hasani et al.). There are four main measuring methods, namely; pulse velocity, pulse echo, resonant frequency and sonic tomography. Pulse velocity measures the time of an ultrasonic pulse within a material, hence finding the pulse velocity of the medium. The instruments used with this system include piezoelectric transducers, coupling agents, pulse generator, signal amplifier and an analyses system (Fig. 1.). The Pulse Echo system uses the transmission of low stress pulse energy to its sensor to delineate defects of material. The echoes received from defects are captured on a time domain

Water Content

Discontinuities and their dimensions within any material can be determined by the techniques given in Table 1 (Hasani et al.). There are four main measuring methods, namely; pulse velocity, pulse echo, resonant frequency and sonic tomography. Pulse velocity measures the time of an ultrasonic pulse within a material, hence finding the pulse velocity of the medium. The instruments used with this system include piezoelectric transducers, coupling agents, pulse generator, signal amplifier and an analyses system (Fig. 1.). The Pulse Echo system uses the transmission of low stress pulse energy to its sensor to delineate defects of material. The echoes received from defects are captured on a time domain

Hardness

Density

Reinforcement

Delimitation

Other

Non-destructive Methods

Acoustic Emission Electrical Electromagnetic Nuclear Ultrasonic Mechanical

Crack Location

Thickness

Table 1. Non-destructive methods and applications (Moozar, 1993)

Fig. 1. Pulse velocity test (Malhotra & Carino, 1991)

spectrum and their analysis locates the anomalies within the structure. The resonant frequency test defines dynamic property of a sample and in generally used in laboratory environments. An oscillator outputs vibrations, which is analysed into the materials transverse, longitudinal and torsional frequencies of the material. Sonic tomography analyses seismic P-wave velocities to image sections of a material. The idea behind the previous technique and the ultrasonic pulse velocity method is similar; however sonic tomography can use a large number of transmitters and receiver at the same time. Its sensivity allows this method to analyse between different anomalies within a structure and process a sectional view of a sample (Moozar, 1993).

There has been big development on measurement devices, since their first introduction by Leslie & Cheesman (1949). Then, measurement techniques pervaded in rock mechanic application given rise to ASTM D2845-08 Standard Test Method for Laboratory Determination of Pulse Velocities and Ultrasonic Constants of Rock. This test method describes equipment and procedures for laboratory measurements of the pulse velocities of compression waves and shear waves in rock and the determination of ultrasonic elastic constants of an isotropic rock or one exhibiting slight anisotropy.

This chapter is about to introduce a practical application of ultrasonic waves in marble industry. Miners working in the marble industry have always been interested in identifying structural weaknesses in marble blocks before they are cut in a marble quarry and transported to marble processing plants. To achieve this difficult task, several simple methods have been developed among miners but observation-based methods do not consistently provide satisfactory results. A nondestructive method developed for testing concrete could be used for this purpose. In this chapter, this simple method based on differences in ultrasonic wave propagation in different rock masses will be presented, and the test results performed both in the laboratory and a marble quarry will be discussed.

When ultrasonic testing is applied to marble blocks, its objective is to detect internal flaws that send echoes back in the direction of the incident beam. These echoes are detected by a receiving transducer. The measurement of the time taken for the pulse to travel from the surface to a flaw and back again enables the position of the flaw to be located. This ultrasonic testing technique was originally developed for assessing the quality and condition of concrete. One instrument used for this purpose is known as PUNDIT. The apparatus has been designed especially for field testing, being light, portable and simple to use. Simple correlations between concrete strength, concrete aggregate gradation, watercement ratio and curing time have been analyzed using PUNDIT.

The possibility of identifying these structural defects using ultrasonic pulses will be discussed and results obtained from these measurements will be introduced in the scope of this chapter. The shape and size of any abnormality in a block can be determined by direct measurements taken from suitably spaced grids. It is important to find the exact position of the surface in marble blocks so that precautions can be taken before the cutting process starts. As stated before, if any discontinuity surface lies in the pulse path, the measured time corresponds to the pulse that follows the shortest path. This is important because any discontinuity causes a time delay compared with the travel time of pulses in homogenous blocks. This study concentrated on the relationship between structural discontinuities and

Ultrasonic Waves in Mining Application 193

site. (Renaud V, et al., 1990) deals with the determination of the excavated damaged zone around a nuclear waste storage cavity using borehole ultrasonic imaging. This analysis is based on a method that is able to sound and image the rock mass velocity field. Another interesting work was published by (Jones, et al., 2010) to monitor and assess the structural health of draglines. They had announced that, by using ultrasonic waves and by studying both the diffraction pattern and the reflected waves, it is possible to detect and size cracking in a typical weld cluster. In the work of Deliormanli at al. 2007, laboratory measurements of P and S wave velocities of marbles from different origins were presented and their anisotropic performances at pressures up to 300 MPa were calculated and compared with

**3. Determination of discontinuities in marble blocks via a non-destructive** 

ultrasonic wave propagation (Dereman et al., 1998, Zhang et al, 2006).

This ultrasonic testing technique was originally developed for assessing the quality and condition of concrete. One instrument used for this purpose is known as PUNDIT. The apparatus has been designed especially for field testing, being light, portable and simple to use. Simple correlations between concrete strength, concrete aggregate gradation, watercement ratio and curing time have been analyzed using PUNDIT (Saad & Qudais, 2005). Several articles have been published on the subject of defining the mechanical properties of several different materials apart from rock by nondestructive test methods based on

There has been a rapid increase in the demand for natural materials to be used in construction engineering, interior decoration, and urban fitting. Over the years, there has been no shortage of quarried blocks, but problems have been encountered in providing sufficient numbers of high quality marble blocks. Blocks of commercial size are directly extracted from the massif. In the case of homogeneous rocks having constant features, structural discontinuities affect the marketability of the blocks. It is important to identify such abnormalities in the marble before the cutting process is performed in order to save money and time. The possibility of finding these structural defects using ultrasonic pulses has been studied, and promising results were obtained. This study concentrated on the relationship between structural discontinuities and ultrasonic pulse travelling velocities in non-homogenous marble blocks. Mathematical formulations were developed to find the

exact locations of the surfaces that cause a separation during the cutting process.

The principle of the method is to create wave at a point and determine at a number of other points the time of arrival of the energy that is reflected by the discontinuities between different rock surfaces. This then enables the position of the discontinuities to be deduced. The basis of the seismic methods is the theory of elasticity. The elastic properties of substances are characterized by elastic moduli or constants, which specify the relation between the stress and strain. The strains in a body are deformations, which produce restoring forces opposed to the stress. Tensile and compressive stresses give rise to longitudinal and volume strains which are measured as the measured as the change in

the elastic properties.

**ultrasonic technique** 

**3.1 Elastic constants and waves** 

ultrasonic pulse travelling velocities in non-homogeneous marble blocks. Mathematical formulations were developed to find the exact locations of the surfaces that cause a separation during the cutting process. To verify the mathematical model explained above, a cubic homogenous marble block with a certain cut inside was prepared in laboratory. This chapter also covers the results obtained from model marble block in laboratory as well as the in-situ measurements obtained from industrial size marble blocks. Block subjected to testing of mathematical modelling in the marble plant was observed in detail before and after slice cut and results will be discussed.

## **2. Ultrasonic waves in mining application**

There are two main mining applications of ultrasonic waves: one is to define the mechanical properties of intact rock and other is to define geological structures of the rock masses. There are several studies on determination of rock characterisation such as uni-axial compressive strength, static modulus of elasticity via non-destructive techniques especially after development of Portable Non-Destructive Digital Indicating Tester (PUNDIT) (Bray & McBride 1992, Green, 1991, ISRM, 1979, Mix, 1987, Vary, 1991, Chary et all, 2006, D'Andrea et all, 1965, Kahraman, 2007, Vasconles et all, 2008, Bakhorji, 2010, Khandelwala & Ranjithb 1996)

The other area of beneficiation of ultrasonic waves is to achieve rock mass classification based on rock mass structures. Several researchers had done work on indirect in-situ measurements to obtain practical data for the rock mass classification studies, since classical methods are expensive and time taking processes (Lockner, 1993, Galdwin, 1982, Karpuz & Pasamehmetoglu, 1997, Ondera, 1963).

There are some works reported on mineral processing industry about ultrasonic waves such as discharging feeding chutes, vibro-acoustics crushers, increasing shaking table performance, ore washing, milling, screening and optimizing bulb performance in flotation (Ozkan, 2004, Stoev & Martin, 1992, Asai & Sasaki, 1958, Kowalski & Kowalska, 1978, Nicol et all, 1986, Slaczka, 1987, Yerkovic et all, 1993, Djendova and Mehandjski, 1986).

Ultrasonic velocity measurements have previously proven valuable tool in measuring the development of stiffness of cement mixtures, so an engineered mine cemented paste backfill material were tested by ultrasonic wave and it is reported that measurements can be used as a non-destructive test to be correlated with other forms of laboratory testing (Galaa, et al. 2011). Grouted rock bolts are widely used to reinforce excavated ground in mining and civil engineering structures. A research was performed to find opportunities for testing the quality of the grout in grouted rock bolts by using ultrasonic methods instead of destructive, time-consuming and costly pull-out tests and over-coring methods (Madenga, et al., 2009, Zou, et al., 2010, Madenga, et al., 2009). A valuable work was performed by Lee, 2010 to predict ground conditions ahead of the tunnel face. This study investigates the development and application of a high resolution ultrasonic wave imaging system, which captures the multiple reflections of ultrasonic waves at the interface, to detect discontinuities at laboratory scale rock mass model. Another application of ultrasonic waves was introduced by Gladwin, 2011 to determine stress changes induced in a large underground support pillar by mining development at Mt Isa Mine. They introduced an ultrasonic stress monitoring device and compared the results with continuous strain recordings at a nearby

ultrasonic pulse travelling velocities in non-homogeneous marble blocks. Mathematical formulations were developed to find the exact locations of the surfaces that cause a separation during the cutting process. To verify the mathematical model explained above, a cubic homogenous marble block with a certain cut inside was prepared in laboratory. This chapter also covers the results obtained from model marble block in laboratory as well as the in-situ measurements obtained from industrial size marble blocks. Block subjected to testing of mathematical modelling in the marble plant was observed in detail before and

There are two main mining applications of ultrasonic waves: one is to define the mechanical properties of intact rock and other is to define geological structures of the rock masses. There are several studies on determination of rock characterisation such as uni-axial compressive strength, static modulus of elasticity via non-destructive techniques especially after development of Portable Non-Destructive Digital Indicating Tester (PUNDIT) (Bray & McBride 1992, Green, 1991, ISRM, 1979, Mix, 1987, Vary, 1991, Chary et all, 2006, D'Andrea et all, 1965, Kahraman, 2007, Vasconles et all, 2008, Bakhorji, 2010, Khandelwala & Ranjithb 1996) The other area of beneficiation of ultrasonic waves is to achieve rock mass classification based on rock mass structures. Several researchers had done work on indirect in-situ measurements to obtain practical data for the rock mass classification studies, since classical methods are expensive and time taking processes (Lockner, 1993, Galdwin, 1982, Karpuz &

There are some works reported on mineral processing industry about ultrasonic waves such as discharging feeding chutes, vibro-acoustics crushers, increasing shaking table performance, ore washing, milling, screening and optimizing bulb performance in flotation (Ozkan, 2004, Stoev & Martin, 1992, Asai & Sasaki, 1958, Kowalski & Kowalska, 1978, Nicol

Ultrasonic velocity measurements have previously proven valuable tool in measuring the development of stiffness of cement mixtures, so an engineered mine cemented paste backfill material were tested by ultrasonic wave and it is reported that measurements can be used as a non-destructive test to be correlated with other forms of laboratory testing (Galaa, et al. 2011). Grouted rock bolts are widely used to reinforce excavated ground in mining and civil engineering structures. A research was performed to find opportunities for testing the quality of the grout in grouted rock bolts by using ultrasonic methods instead of destructive, time-consuming and costly pull-out tests and over-coring methods (Madenga, et al., 2009, Zou, et al., 2010, Madenga, et al., 2009). A valuable work was performed by Lee, 2010 to predict ground conditions ahead of the tunnel face. This study investigates the development and application of a high resolution ultrasonic wave imaging system, which captures the multiple reflections of ultrasonic waves at the interface, to detect discontinuities at laboratory scale rock mass model. Another application of ultrasonic waves was introduced by Gladwin, 2011 to determine stress changes induced in a large underground support pillar by mining development at Mt Isa Mine. They introduced an ultrasonic stress monitoring device and compared the results with continuous strain recordings at a nearby

et all, 1986, Slaczka, 1987, Yerkovic et all, 1993, Djendova and Mehandjski, 1986).

after slice cut and results will be discussed.

Pasamehmetoglu, 1997, Ondera, 1963).

**2. Ultrasonic waves in mining application** 

site. (Renaud V, et al., 1990) deals with the determination of the excavated damaged zone around a nuclear waste storage cavity using borehole ultrasonic imaging. This analysis is based on a method that is able to sound and image the rock mass velocity field. Another interesting work was published by (Jones, et al., 2010) to monitor and assess the structural health of draglines. They had announced that, by using ultrasonic waves and by studying both the diffraction pattern and the reflected waves, it is possible to detect and size cracking in a typical weld cluster. In the work of Deliormanli at al. 2007, laboratory measurements of P and S wave velocities of marbles from different origins were presented and their anisotropic performances at pressures up to 300 MPa were calculated and compared with the elastic properties.

## **3. Determination of discontinuities in marble blocks via a non-destructive ultrasonic technique**

This ultrasonic testing technique was originally developed for assessing the quality and condition of concrete. One instrument used for this purpose is known as PUNDIT. The apparatus has been designed especially for field testing, being light, portable and simple to use. Simple correlations between concrete strength, concrete aggregate gradation, watercement ratio and curing time have been analyzed using PUNDIT (Saad & Qudais, 2005). Several articles have been published on the subject of defining the mechanical properties of several different materials apart from rock by nondestructive test methods based on ultrasonic wave propagation (Dereman et al., 1998, Zhang et al, 2006).

There has been a rapid increase in the demand for natural materials to be used in construction engineering, interior decoration, and urban fitting. Over the years, there has been no shortage of quarried blocks, but problems have been encountered in providing sufficient numbers of high quality marble blocks. Blocks of commercial size are directly extracted from the massif. In the case of homogeneous rocks having constant features, structural discontinuities affect the marketability of the blocks. It is important to identify such abnormalities in the marble before the cutting process is performed in order to save money and time. The possibility of finding these structural defects using ultrasonic pulses has been studied, and promising results were obtained. This study concentrated on the relationship between structural discontinuities and ultrasonic pulse travelling velocities in non-homogenous marble blocks. Mathematical formulations were developed to find the exact locations of the surfaces that cause a separation during the cutting process.

#### **3.1 Elastic constants and waves**

The principle of the method is to create wave at a point and determine at a number of other points the time of arrival of the energy that is reflected by the discontinuities between different rock surfaces. This then enables the position of the discontinuities to be deduced. The basis of the seismic methods is the theory of elasticity. The elastic properties of substances are characterized by elastic moduli or constants, which specify the relation between the stress and strain. The strains in a body are deformations, which produce restoring forces opposed to the stress. Tensile and compressive stresses give rise to longitudinal and volume strains which are measured as the measured as the change in

Ultrasonic Waves in Mining Application 195

Fig. 2. shows how the transducers may be arranged on the surface of the specimen tested.

The direct transmission arrangement is the most satisfactory one since the longitudinal pulses leaving the transmitter are propagated mainly in the direction normal to the transducer face. The indirect arrangement is possible because the ultrasonic beam of energy is scattered by discontinuities within the material tested but the strength of the pulse detected in this case is only about 1 or 2 % of that defected for the same path length when the direct transmission arrangement is used. The purpose of the study was to develop a model in stone quarry so semi-direct and indirect transmissions were taken as the main measurement technique since it is sometimes very difficult to find free faces to place

(a) (b) (c)

Pulses are not transmitted through large air voids in a material and, if such a void or discontinuity surface lies directly in the pulse path, the instrument will indicate the time taken by the pulse that circumvents the void by quickest route. It is thus possible to detect large voids when a grid of pulse velocity measurements is made over a region in which these voids are located. By using this behaviour, method can be used to test rock strata and

A concrete model was designed in laboratory to find out the behaviour of ultrasonic pulses travelling through a simulated discontinuity surface inside a concrete block. A wooden

Before assessing the effects of simulated discontinuity on pulse velocity, first pulse velocity measurements made nearby simulated surface. This gives the real pulse velocity for prepared concrete block. For this purpose three direct measurements from three free

T 2.52L 3.39 s

In equation 4, 2.52 is the slope of the direct line given in Fig. 4. and –3.39 is the value T takes

(4)

The transmission can either be direct (a), semi-direct (b) or indirect (c).

transducers on the production bench.

Fig. 2. Methods of propagating ultrasonic pulses

provide useful data for geological survey work.

surfaces were obtained. The result is given in Fig. 3.

when the length value L is equal to 0.

surface was settled in the block with the dimension shown in Fig. 3.

For later use, a linear equation was set for the line shown in Fig. 4.

**3.3 Laboratory works on simulated model** 

length per unit length or change in volume per unit volume. Shear strains are measured as angles of deformations. It is usually assumed that the strains are small and reversible, that is, a body resumes its original shape and size when the stresses are relieved. If the stress applied to an elastic medium is released suddenly the condition of strain propagates within the medium as an elastic wave. There are several kinds of elastic waves:

In the longitudinal, compressional of P waves the motion of the medium is in the same direction as the direction of wave propagation. These are ordinary sound waves. Their velocity is given by (New, 1985):

$$V\_{\mathbb{P}} = \left(\frac{k4\,\mu/3}{\rho}\right)^{1/2} \tag{1}$$

Where is density of the medium and k bulk modulus, shear modulus of the medium respectively. In the transverse, shear or S waves the particles of the medium move at right angles to the direction of wave propagation and the velocity is given by (Tomsett, 1976):

$$V\_s = \left(\frac{\mu}{\rho}\right)^{1/2} \tag{2}$$

If a medium has a free surface there are also surface waves in addition to the body waves. In the Rayleight waves the particles describe ellipses in the vertical plane that contains the direction of propagation.

Another type of surface waves the Love waves. These are observed when the S wave velocity in the top layer of a medium is less than in the substratum. The particles oscillate transversely to the direction of the wave and in a parallel to the surface. The Love waves are thus essentially shear waves.

The frequency spectrum of body waves in the earth extends from about 15 Hz to about 100 Hz; the surface waves have frequencies lower than about 15 Hz (Parasnis, 1994). For the method described in this study P waves are of importance as exploration tools. For the materials like concrete, marble necessary frequency range changes from 20 – 250 KHz.

#### **3.2 Application of pulse velocity testing**

For assessing the quality of marble blocks from ultrasonic pulse velocity measurement, it is necessary to use an apparatus that generates suitable pulses and accurately measures the time of their transmission through the material tested. The instrument indicates the time taken for the earliest part of the pulse the transmitting transducer when these transducers are placed at suitable points on the surface of the material tested. The distance that the pulse travels in the material must be measured to determine the pulse velocity.

$$\text{Pulse velocity} = \frac{\text{Path length}}{\text{Transit time}} \tag{3}$$

length per unit length or change in volume per unit volume. Shear strains are measured as angles of deformations. It is usually assumed that the strains are small and reversible, that is, a body resumes its original shape and size when the stresses are relieved. If the stress applied to an elastic medium is released suddenly the condition of strain propagates within the medium as an elastic wave. There are several kinds of elastic

In the longitudinal, compressional of P waves the motion of the medium is in the same direction as the direction of wave propagation. These are ordinary sound waves. Their

> *<sup>k</sup>*4 /3 *<sup>V</sup>*

Where is density of the medium and k bulk modulus, shear modulus of the medium respectively. In the transverse, shear or S waves the particles of the medium move at right angles to the direction of wave propagation and the velocity is given by (Tomsett, 1976):

If a medium has a free surface there are also surface waves in addition to the body waves. In the Rayleight waves the particles describe ellipses in the vertical plane that contains the

Another type of surface waves the Love waves. These are observed when the S wave velocity in the top layer of a medium is less than in the substratum. The particles oscillate transversely to the direction of the wave and in a parallel to the surface. The Love waves are

The frequency spectrum of body waves in the earth extends from about 15 Hz to about 100 Hz; the surface waves have frequencies lower than about 15 Hz (Parasnis, 1994). For the method described in this study P waves are of importance as exploration tools. For the materials like concrete, marble necessary frequency range changes from 20 – 250 KHz.

For assessing the quality of marble blocks from ultrasonic pulse velocity measurement, it is necessary to use an apparatus that generates suitable pulses and accurately measures the time of their transmission through the material tested. The instrument indicates the time taken for the earliest part of the pulse the transmitting transducer when these transducers are placed at suitable points on the surface of the material tested. The distance that the pulse

Path length Pulse velocity= Transit time (3)

travels in the material must be measured to determine the pulse velocity.

1/2

p

*V*s

1/2

(1)

(2)

waves:

velocity is given by (New, 1985):

direction of propagation.

thus essentially shear waves.

**3.2 Application of pulse velocity testing** 

Fig. 2. shows how the transducers may be arranged on the surface of the specimen tested. The transmission can either be direct (a), semi-direct (b) or indirect (c).

The direct transmission arrangement is the most satisfactory one since the longitudinal pulses leaving the transmitter are propagated mainly in the direction normal to the transducer face. The indirect arrangement is possible because the ultrasonic beam of energy is scattered by discontinuities within the material tested but the strength of the pulse detected in this case is only about 1 or 2 % of that defected for the same path length when the direct transmission arrangement is used. The purpose of the study was to develop a model in stone quarry so semi-direct and indirect transmissions were taken as the main measurement technique since it is sometimes very difficult to find free faces to place transducers on the production bench.

Fig. 2. Methods of propagating ultrasonic pulses

Pulses are not transmitted through large air voids in a material and, if such a void or discontinuity surface lies directly in the pulse path, the instrument will indicate the time taken by the pulse that circumvents the void by quickest route. It is thus possible to detect large voids when a grid of pulse velocity measurements is made over a region in which these voids are located. By using this behaviour, method can be used to test rock strata and provide useful data for geological survey work.

#### **3.3 Laboratory works on simulated model**

A concrete model was designed in laboratory to find out the behaviour of ultrasonic pulses travelling through a simulated discontinuity surface inside a concrete block. A wooden surface was settled in the block with the dimension shown in Fig. 3.

Before assessing the effects of simulated discontinuity on pulse velocity, first pulse velocity measurements made nearby simulated surface. This gives the real pulse velocity for prepared concrete block. For this purpose three direct measurements from three free surfaces were obtained. The result is given in Fig. 3.

For later use, a linear equation was set for the line shown in Fig. 4.

$$\text{T = 2.52L - 3.39 \ (\mu\text{s})}\tag{4}$$

In equation 4, 2.52 is the slope of the direct line given in Fig. 4. and –3.39 is the value T takes when the length value L is equal to 0.

Ultrasonic Waves in Mining Application 197

129.5 130.8 133.4 133.4 130.8 125.5 124.7 125.3 126.3 126.8 130.0 130.0 132.5 130.0 126.0 124.4 124.0 125.0 125.7 125.7 126.0 129.0 128.5 127.0 124.5 123.5 124.0 124.6 124.5 125.5 127.0 129.4 127.0 126.5 126.0 125.3 125.3 125.0 126.0 127.2 127.2 127.4 127.0 127.5 128.5 125.1 125.2 125.9 126.0 126.3 126.4 126.6 126.7 128.5 122.0 124.0 124.7 124.7 125.1 125.3 125.5 126.7 126.5 119.5 120.8 121.5 122.3 123.0 122.8 123.0 123.0 123.5 **1 2 3 4 5 6 7 8 9** 

65.4 65.5 68.5 65.1 66.8 65.0 63.6 62.5 62.3 62.5 62.2 62.4 62.3 62.2 62.3 63.2 63.5 64.4 63.7 63.9 64.2 64.8 64.1 65.3 66.0 65.5 63.2 61.8 61.1 61.9 60.4 60.9 60.5 60.4 60.4 60.4 61.9 62.5 63.0 62.6 62.3 61.8 62.0 63.7 66.4 66.0 66.0 62.9 62.3 60.7 61.2 60.2 59.7 59.8 59.8 60.4 61.3 62.5 61.4 62.0 61.1 60.8 60.5 62.3 63.5 65.0 65.4 65.3 66.0 62.2 60.5 60.7 59.7 59.8 60.2 61.7 62.3 61.4 61.3 60.6 60.5 60.5 61.1 60.9 61.3 62.1 62.0 62.4 63.0 62.4 61.6 61.2 61.3 61.6 63.0 61.2 61.1 61.2 61.1 60.8 61.3 61.2 61.0 61.0 61.1 60.7 60.7 61.8 60.4 61.3 62.0 63.0 64.0 63.7 60.5 60.5 61.0 61.3 61.3 61.7 61.4 61.0 61.0 60.2 60.2 60.4 60.4 60.3 60.4 61.4 62.4 63.0 63.5 59.0 59.5 60.0 60.0 59.1 60.3 59.5 60.4 60.0 60.1 60.1 59.6 59.4 60.1 59.4 59.2 59.1 59.2 60.4 **1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19** 

1 57.1 56.6 56.4 55.7 55.4 55.9 55.6 56.0 56.1 56.7 56.2 55.8 55.2 54.2 54.7 55.2 55.9 55.9 56.4 56.4 56.2 55.3 55.0 55.3 56.1 55.6 56.0 56.0 56.9 57.2 57.1 58.4 54.8 54.9 55.0 55.5 55.9 56.4 56.0 56.2 55.8 55.0 55.5 56.3 57.0 58.0 58.0 59.5 59.8 60.4 61.0 58.0 55.4 54.6 54.7 55.9 56.4 56.7 56.4 57.0 57.0 58.3 59.8 62.0 62.5 63.2 66.5 66.0 66.0 64.2 60.6 56.0 54.5 54.4 55.2 56.4 58.0 59.4 62.4 63.0 66.0 66.0 68.0 68.0 70.0 67.1 73.0 74.0 68.2 63.3 58.4 56.1 56.0 55.4 55.7 57.0 56.3 58.0 67.2 73.4 74.4 76.4 75.0 71.0 67.0 67.0 65.5 62.2 61.4 61.2 58.3 57.5 55.8 55.8 56.8 55.8 56.1 68.3 82.5 77.0 71.0 66.3 63.0 61.5 60.0 59.5 57.6 56.9 57.4 56.1 55.9 55.7 55.8 57.0 55.8 56.0 56.4 71.3 95.0 64.8 60.0 57.0 56.0 55.3 56.0 55.0 54.7 54.9 55.3 55.1 55.1 55.4 56.1 56.1 56.1 56.0 58.0 62.0 60.0 56.3 55.0 54.6 54.8 55.5 54.5 54.6 54.8 55.2 55.3 55.2 56.0 **1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19** 

Measurement of pulse velocities at points that are not affected by the simulated surface provides a reliable method of assessing the pulse velocity behaviour of the homogenous concrete block. It is useful to plot a diagram of pulse velocity contours from the result obtained since this gives a clear picture of the extent of variations. It should be appreciated that the path length can influence the extent of the variations recorded because the pulse velocity measurements correspond to the average quality of the concrete. When an ultrasonic pulse travelling through concrete meets a simulated surface, there is a negligible transmission of energy across this interface so that any air-filled

Table 2. Transmission times taken from the right face of the concrete block (s)

Table 3. Transmission times taken from the front face of the concrete block (s)

Table 4. Transmission times taken from the top of the concrete block (s)

Fig. 3. Prepared concrete block and the dimension of the surface

Fig. 4. Pulse velocity determination for homogenous concrete block

Fig. 3. Prepared concrete block and the dimension of the surface

Fig. 4. Pulse velocity determination for homogenous concrete block

**0 100 200 300 400 500 600 X (mm)**

**T**

s)




Table 3. Transmission times taken from the front face of the concrete block (s)


Table 4. Transmission times taken from the top of the concrete block (s)

Measurement of pulse velocities at points that are not affected by the simulated surface provides a reliable method of assessing the pulse velocity behaviour of the homogenous concrete block. It is useful to plot a diagram of pulse velocity contours from the result obtained since this gives a clear picture of the extent of variations. It should be appreciated that the path length can influence the extent of the variations recorded because the pulse velocity measurements correspond to the average quality of the concrete. When an ultrasonic pulse travelling through concrete meets a simulated surface, there is a negligible transmission of energy across this interface so that any air-filled

Ultrasonic Waves in Mining Application 199

Fig. 7. Contour plotting of transmission times (s) taken from top of the model

defined boundary surrounded by uniformly dense concrete.

**3.4 Modeling the boundary of discontinuities** 

Fig. 8. The cause of time delay of pulses.

As shown in Fig. 5., Fig. 6. and Fig. 7., it is possible to detect the size and position of the simulated surface. Such estimates are more reliable if the discontinuity surface has a well-

The shape and size of any abnormality in a block can be determined by direct measurements taken from satisfactory grids. It becomes important to find the exact position of the surface in marble blocks so that to take all precautions before cutting process starts. As stated before, if any discontinuity surface lies in the pulse path, the measured time belongs to the pulse that follows the shortest path. This is important because it causes a time delay comparing to travel time of pulses in homogenous blocks. This case is shown in Fig. 8.

Before doing any measurement, pulse velocity behaviour of homogenous material must be obtained as discussed in laboratory work. By doing so, it becomes possible to estimate pulse travel time if no abnormality exists in the block. It is necessary to measure direct distance between transmitter to receiver in order to estimate travel time of pulses. The pulse velocity measuring device gives the minimum travelling time between two points. The pulse

cracks or void directly between the transducers will obstruct the direct beam of ultrasound when the void has a projected area larger than the area of the transducer faces. The first pulse to arrive at the receiving transducer will have been diffracted around the periphery of the defect and the transit time will be longer than in similar concrete with no defect.

In order to detect the simulated surface, pulse velocity measurements were performed over three different directions of concrete block with a grid of 2.5 cm x 2.5 cm and results are given in Table 2, 3, 4.

Fig. 5. Contour plotting of transmission times (s) taken from right face

Fig. 6. Contour plotting of transmission times (s) taken from front face of the model

cracks or void directly between the transducers will obstruct the direct beam of ultrasound when the void has a projected area larger than the area of the transducer faces. The first pulse to arrive at the receiving transducer will have been diffracted around the periphery of the defect and the transit time will be longer than in similar concrete with no

In order to detect the simulated surface, pulse velocity measurements were performed over three different directions of concrete block with a grid of 2.5 cm x 2.5 cm and results are

Fig. 5. Contour plotting of transmission times (s) taken from right face

Fig. 6. Contour plotting of transmission times (s) taken from front face of the model

defect.

given in Table 2, 3, 4.

Fig. 7. Contour plotting of transmission times (s) taken from top of the model

As shown in Fig. 5., Fig. 6. and Fig. 7., it is possible to detect the size and position of the simulated surface. Such estimates are more reliable if the discontinuity surface has a welldefined boundary surrounded by uniformly dense concrete.

#### **3.4 Modeling the boundary of discontinuities**

The shape and size of any abnormality in a block can be determined by direct measurements taken from satisfactory grids. It becomes important to find the exact position of the surface in marble blocks so that to take all precautions before cutting process starts. As stated before, if any discontinuity surface lies in the pulse path, the measured time belongs to the pulse that follows the shortest path. This is important because it causes a time delay comparing to travel time of pulses in homogenous blocks. This case is shown in Fig. 8.

Fig. 8. The cause of time delay of pulses.

Before doing any measurement, pulse velocity behaviour of homogenous material must be obtained as discussed in laboratory work. By doing so, it becomes possible to estimate pulse travel time if no abnormality exists in the block. It is necessary to measure direct distance between transmitter to receiver in order to estimate travel time of pulses. The pulse velocity measuring device gives the minimum travelling time between two points. The pulse

Ultrasonic Waves in Mining Application 201

Lets take s as the length of crack, h is the vertical distance from surface, A + B is the path of pulse travelling from transmitter to receiver and L is the shortest distance between two

Some simple linear algebra can be used to obtain h and s those are the correct places of the

2 2 h (L h) A B

2 2 h (L h) Ls <sup>A</sup>

2 2 h (L h) Ls <sup>2</sup> S h 2.Ls 2

Equation 13 is a function of vertical distance h. In this formula both s and h are unknowns. The purpose of all formulations above is to define h and s. In equation 13, if h changes from 0 to L and plotted, Fig. 11. can be obtained. In this process only the measurements that can be obtained are directly measured L, A+B that is estimated from the linear equation set

H

h h

Fig. 11. The curve that shows possible path obtained from a single measurement.

S

Ls

2.Ls 2

Ls A B (5)

2 22 SAh (6)

22 2 S B (L h) (7)

222 2 A h B (L h) (8)

22 2 2 A B h (L h) (9)

(11)

(12)

(13)

2 2 (A B).(A B) h (L h) (10)

probs. The pulse travels the distance A + B instead of L.

boundary of discontinuity.

before the homogenous material.

velocity is obtained by dividing the path length to transit time. There will be a difference between the measured pulse velocity and the velocity obtaining from equation 3 by applying path length L in the formula. This difference is an indicator of a time delay caused by longer travelling distances due to obstacle in the pulse travel path (dashed lines in Fig. 8.). The time delay will be used later to find the correct position of surface in 3D.

Fig. 9. explains the situation clearly. In this figure, there are two types of curves in the graphics. The linear one represents the direct distance from transmitter to receiver; the parabolic one represents the longer path of pulses following the boundary of discontinuity. Both figures can be obtained from the type of measurement so that receiver moves away from position 1 to position 7 while transmitter stays stable.

Fig. 9. The difference between direct distance and the pulse travel distance.

The first curve in Fig. 9 is the distance obtained from pulse travel times (shown as L in Fig. 9.) from receiver, the second one is the actual distance (shown as P in Fig. 9.) according to receiver position from 1 to 5. There has been a wide opening between two lines that shows the position of receiver 1 is away from the discontinuity surface. While the receiver moves along a line with equal distance from position 1 to 5 the gap between two lines narrows steadily. This behaviour gives a very important clue about the boundary of the surface. In the second region of the graph started from receiver position 5 to 7, two lines meet and behave in the same way. Because, at receiver position 5 the modelled surface has been passed that indicates there is no surface between transmitter and receiver. This kind of work defines the boundary in longitudinal location (s) but not in vertical one. As shown in Fig. 10. discontinuity can be at any position of (h) that is the vertical distance of the surface from top of the block.

Fig. 10. Pulses moving around the crack

velocity is obtained by dividing the path length to transit time. There will be a difference between the measured pulse velocity and the velocity obtaining from equation 3 by applying path length L in the formula. This difference is an indicator of a time delay caused by longer travelling distances due to obstacle in the pulse travel path (dashed lines in Fig.

Fig. 9. explains the situation clearly. In this figure, there are two types of curves in the graphics. The linear one represents the direct distance from transmitter to receiver; the parabolic one represents the longer path of pulses following the boundary of discontinuity. Both figures can be obtained from the type of measurement so that receiver moves away

8.). The time delay will be used later to find the correct position of surface in 3D.

Fig. 9. The difference between direct distance and the pulse travel distance.

The first curve in Fig. 9 is the distance obtained from pulse travel times (shown as L in Fig. 9.) from receiver, the second one is the actual distance (shown as P in Fig. 9.) according to receiver position from 1 to 5. There has been a wide opening between two lines that shows the position of receiver 1 is away from the discontinuity surface. While the receiver moves along a line with equal distance from position 1 to 5 the gap between two lines narrows steadily. This behaviour gives a very important clue about the boundary of the surface. In the second region of the graph started from receiver position 5 to 7, two lines meet and behave in the same way. Because, at receiver position 5 the modelled surface has been passed that indicates there is no surface between transmitter and receiver. This kind of work defines the boundary in longitudinal location (s) but not in vertical one. As shown in Fig. 10. discontinuity can be at any position of (h) that is the vertical distance of the surface from top

from position 1 to position 7 while transmitter stays stable.

of the block.

Fig. 10. Pulses moving around the crack

Lets take s as the length of crack, h is the vertical distance from surface, A + B is the path of pulse travelling from transmitter to receiver and L is the shortest distance between two probs. The pulse travels the distance A + B instead of L.

$$\text{Ls } = \text{A } + \text{B} \tag{5}$$

Some simple linear algebra can be used to obtain h and s those are the correct places of the boundary of discontinuity.

$$\mathbf{S}^2 = \mathbf{A}^2 - \hbar^2 \tag{6}$$

$$\mathbf{S}^2 = \mathbf{B}^2 - (\mathbf{L} - \mathbf{h})^2\tag{7}$$

$$\mathbf{A}^2 - \mathbf{h}^2 = \mathbf{B}^2 - (\mathbf{L} - \mathbf{h})^2\tag{8}$$

$$\mathbf{A}^2 - \mathbf{B}^2 = \mathbf{h}^2 - (\mathbf{L} - \mathbf{h})^2\tag{9}$$

$$(\mathbf{A} + \mathbf{B}). (\mathbf{A} - \mathbf{B}) = \mathbf{h}^2 - (\mathbf{L} - \mathbf{h})^2 \tag{10}$$

$$\mathbf{A} - \mathbf{B} = \frac{\mathbf{h}^2 - (\mathbf{L} - \mathbf{h})^2}{\mathbf{L}\mathbf{s}} \tag{11}$$

$$\mathbf{A} = \frac{\mathbf{h}^2 - (\mathbf{L} - \mathbf{h})^2}{2 \,\mathrm{L} \,\mathrm{s}} + \frac{\mathrm{L} \,\mathrm{s}}{2} \tag{12}$$

$$\mathbf{S} = \sqrt{\frac{\mathbf{h}^2 - (\mathbf{L} - \mathbf{h})^2}{2\mathbf{L}\mathbf{s}} + \frac{\mathbf{L}\mathbf{s}}{2} - \mathbf{h}^2} \tag{13}$$

Equation 13 is a function of vertical distance h. In this formula both s and h are unknowns. The purpose of all formulations above is to define h and s. In equation 13, if h changes from 0 to L and plotted, Fig. 11. can be obtained. In this process only the measurements that can be obtained are directly measured L, A+B that is estimated from the linear equation set before the homogenous material.

Fig. 11. The curve that shows possible path obtained from a single measurement.

Ultrasonic Waves in Mining Application 203

With an iteration of equation 13 for both measurements on the same plane, only one point equals the h and s. The problem is to find out this point that is the boundary of the surface.

To verify the model explained in previous section, a cubic homogenous marble block with a certain cut inside was prepared. The dimension of the block and cut is shown in Fig. 14.

The transmitter is placed on the top of the block from the left side and 13 cm away from the front face. The receiver was moved along the front face of the block with a 2.5cm x 2.5cm

Considering Fig. 13, the values those can be obtained from measurements are:

L1 = The direct distance from transmitter to receiver position 1 L2 = The direct distance from transmitter to receiver position 2 LS1 = The length of pulse travelling path for receiver position 1 LS2 = The length of pulse travelling path for receiver position 2

Art = The distance between receiver 1 and 2

**3.5 Verification of the model** 

Fig. 14. The block model dimension

Fig. 15. The position of transmitter and receiver

grid patterns (Fig 15).

When the receiver moves to position 2 in Fig. 12., the same measurements are made to plot second curve. Both figures are combined to obtain an intersection point that gives the correct position of the boundary, in another saying h and s can be determined (Fig. 12.).

Fig. 12. The combined curves giving h and s

Obtaining h and s is very important because if the receiver moves in four different directions that mean four different h and s can be obtained in different directions. In common, receiver moves in such a way that enables us to find the exact shape of boundary of the discontinuity in 3D.

Finding h and s values are a time taken process so a computer program has been written to analyse and to plot the entire finding. For better understanding, Fig. 13. must be explained first.

Fig. 13. Pulses travel paths for both receiver positions.

When the receiver moves to position 2 in Fig. 12., the same measurements are made to plot second curve. Both figures are combined to obtain an intersection point that gives the correct position of the boundary, in another saying h and s can be determined (Fig. 12.).

Obtaining h and s is very important because if the receiver moves in four different directions that mean four different h and s can be obtained in different directions. In common, receiver moves in such a way that enables us to find the exact shape of boundary

Finding h and s values are a time taken process so a computer program has been written to analyse and to plot the entire finding. For better understanding, Fig. 13. must be explained

Fig. 12. The combined curves giving h and s

Fig. 13. Pulses travel paths for both receiver positions.

of the discontinuity in 3D.

first.

Considering Fig. 13, the values those can be obtained from measurements are:


With an iteration of equation 13 for both measurements on the same plane, only one point equals the h and s. The problem is to find out this point that is the boundary of the surface.

#### **3.5 Verification of the model**

To verify the model explained in previous section, a cubic homogenous marble block with a certain cut inside was prepared. The dimension of the block and cut is shown in Fig. 14.

Fig. 14. The block model dimension

The transmitter is placed on the top of the block from the left side and 13 cm away from the front face. The receiver was moved along the front face of the block with a 2.5cm x 2.5cm grid patterns (Fig 15).

Fig. 15. The position of transmitter and receiver

Ultrasonic Waves in Mining Application 205

measured block showed directional anisotropies those affect the liability of the measurements. The second one was that the existence of more than one discontinuity between two probs. The model was developed to search only one discontinuity or cave in the block so better results could be obtained with the moving transmitter. Besides of all these difficulties, a main discontinuity surface was detected and located in the block. From

Before the measurements were taken, first transmitter was located in such a position that it could indicate some visible cracks from the surface. After locating the transmitter on such an area, the opposite side of the marble block was divided into grids with 20 cm. There are 12 measurement points in longitudinal and 5 measurements in vertical direction. The linear

1 275 279 282 280 280 296 312 334 351 404 390 412 2 248 250 258 296 280 297 311 329 347 363 386 410 3 332 342 300 296 278 287 306 324 347 367 389 411 4 391 384 378 310 280 289 306 328 346 372 393 417 5 416 394 380 286 292 299 316 334 354 375 409 510 1 2 3 4 5 6 7 8 9 10 11 12

1 223 227 235 246 260 277 295 316 337 360 384 409 2 225 229 237 248 262 278 297 317 339 361 385 410 3 231 235 242 253 267 283 301 321 342 365 389 413 4 241 244 251 262 275 291 308 328 349 371 394 418 5 253 257 263 273 286 301 318 337 358 379 402 426 1 2 3 4 5 6 7 8 9 10 11 12

1 150 153 158 166 175 186 198 212 227 242 258 275 2 152 154 159 167 176 187 199 213 228 243 259 275 3 156 158 163 170 179 190 202 216 230 245 261 277 4 162 164 169 176 185 195 207 220 235 249 265 281 5 170 173 177 184 192 202 214 227 240 255 270 286 1 2 3 4 5 6 7 8 9 10 11 12

1 184 187 189 188 188 198 209 224 235 271 261 276 2 166 167 173 198 188 199 208 220 233 243 259 275 3 222 229 201 198 186 192 205 217 233 246 261 275 4 262 257 253 208 188 194 205 220 232 249 263 279 5 279 264 255 192 196 200 212 224 237 251 274 342 1 2 3 4 5 6 7 8 9 10 11 12

T 1.49 x 0.56 (ms) (15)

behaviour of homogenous block was measured with indirect method (Equation 15).

several block measurements only one of them will be given in detail.

Table 6. Measured travelling times (s)

Table 7. Standard travelling times (s)

Table 8. Distances between transmitter and receiver (cm)

Table 9. Calculated pulse path lengths (cm)

Before the pulse travelling times are taken, standard linear equation for homogenous marble block was obtained by direct measurements as the following equation:

$$\text{T = } 0.877 \times \text{L + } 7.287 \text{ (ms)}\tag{14}$$


Table 5 shows the measured times for 9 different receiver position according to Fig. 15.

Table 5. The results of measurements

Table 5 is made up of 4 sections. The first one is the direct measurements taken from the instrument. By using linear behaviour of homogenous block given in formula 12, standard travelling times are calculated for a case if no cut exists in the path of the probes and they are given in the second section of Table 5. The exact distances are measured and given in the third section. The last section is calculated pulse path length obtained from measured travelling time as indicated before. As long as receiver position (1, 1) is concerned, this gives the highest measured travelling time showing (the first section in Table 5) the pulse travels the longest path to reach the receiver. This indicates a crack exists between two probs. When the receiver is moved to position (3, 3), there is no difference between measured travelling time and standard travelling time that indicates no obstacle between two probs. Measured travelling times and distances between probes are given to the computer as data so that all other values can be calculated in order to show the exact place of the cut. The output of the computer located crack position is given in Fig. 16.

Fig. 16. The computed output of the crack

#### **3.6 In-situ measurements**

The same tests were performed on several quarried blocks obtained from a marble factory and a good match was found with the estimated discontinuity surface.

The location of the transmitter and the mesh of receiver are shown in Fig. 17. Working with a block dimension of 155cm x 249cm x 98 cm brought about some difficulties. First, the

Before the pulse travelling times are taken, standard linear equation for homogenous marble

Table 5 shows the measured times for 9 different receiver position according to Fig. 15.

1 27 23 23 20 21 23 15 16 18 18 22 18 2 25 23 24 21 22 24 16 17 19 20 17 18 3 25 24 25 23 24 25 19 21 20 20 19 21 1 2 3 1 2 3 1 2 3 1 2 3

Table 5 is made up of 4 sections. The first one is the direct measurements taken from the instrument. By using linear behaviour of homogenous block given in formula 12, standard travelling times are calculated for a case if no cut exists in the path of the probes and they are given in the second section of Table 5. The exact distances are measured and given in the third section. The last section is calculated pulse path length obtained from measured travelling time as indicated before. As long as receiver position (1, 1) is concerned, this gives the highest measured travelling time showing (the first section in Table 5) the pulse travels the longest path to reach the receiver. This indicates a crack exists between two probs. When the receiver is moved to position (3, 3), there is no difference between measured travelling time and standard travelling time that indicates no obstacle between two probs. Measured travelling times and distances between probes are given to the computer as data so that all other values can be calculated in order to show the exact place of the cut. The output of the

The same tests were performed on several quarried blocks obtained from a marble factory

The location of the transmitter and the mesh of receiver are shown in Fig. 17. Working with a block dimension of 155cm x 249cm x 98 cm brought about some difficulties. First, the

and a good match was found with the estimated discontinuity surface.

T 0.877 x L 7.287 (ms) (14)

Calculated pulse path length (cm)

Distance between the transmitter and the receiver (cm)

block was obtained by direct measurements as the following equation:

Standard travelling time (μs)

Measured travelling time (μs)

Table 5. The results of measurements

computer located crack position is given in Fig. 16.

Fig. 16. The computed output of the crack

**3.6 In-situ measurements** 

measured block showed directional anisotropies those affect the liability of the measurements. The second one was that the existence of more than one discontinuity between two probs. The model was developed to search only one discontinuity or cave in the block so better results could be obtained with the moving transmitter. Besides of all these difficulties, a main discontinuity surface was detected and located in the block. From several block measurements only one of them will be given in detail.

Before the measurements were taken, first transmitter was located in such a position that it could indicate some visible cracks from the surface. After locating the transmitter on such an area, the opposite side of the marble block was divided into grids with 20 cm. There are 12 measurement points in longitudinal and 5 measurements in vertical direction. The linear behaviour of homogenous block was measured with indirect method (Equation 15).

$$\text{T = 1.49 \times - } \begin{pmatrix} 0.56 \end{pmatrix} \text{m/s} \tag{15}$$


Table 6. Measured travelling times (s)


Table 7. Standard travelling times (s)


Table 8. Distances between transmitter and receiver (cm)


Table 9. Calculated pulse path lengths (cm)

Ultrasonic Waves in Mining Application 207

waves is the same as seismic wave propagation in a way that they both have P and S waves in the same form. Measuring the waves reflected from different bodies beneath earth surface and interpreting the data obtained from those measurements give very useful information

The importance of determining the marble blocks those are affected by any discontinuity such as void, crack, caves et., in quarry has been presented through this chapter. It is also important to notice that any abnormalities in marble blocks must be pointed out by using a simple method without giving any damage to the main body. The method of testing the quality of concrete is perfectly well adapted to the determination the marble block because of its simplicity. First, experiments performed in the laboratory on simulated block gave promising results that encourage the possibility of using such a technique on marble blocks. But it must be bear in mind that a prepared concrete block differs from the natural stones concerning with homogeneity. Direct measurements give better understanding of the structure of any block measured because it clearly shows the boundary of a surface in the body. Nevertheless, as far as field investigation is concerned, direct measurements become very hard to apply depending on the number of free faces. To develop a measurement technique in field semi-direct and indirect applications have been developed on the blocks obtained from a marble factory. Mathematical model was applied on the blocks but results showed that fixing the transmitter in a stable position does not give a picture of the body if there is a complicated structure as far as discontinuities concern. Although it is possible to find exact positions of the discontinuities, the number of measurements increases in logarithmic scale with the moving transmitter that enables a practical method. However, statistical analysis would better give a high reliability for spotting out this kind of structures, instead of finding the correct location. Authors of this chapter suggests to develop a new measurement technique to allow multiple receivers located on several positions of the block to be analysed and moving transmitter along with the surface to be measured. Much more

Ш

Bakhorji, A.M., *Laboratory Measurements of Static and Dynamic Elastic Properties in Carbonate*,

Bray, D. E. & McBride, D., (1992), *Nondestructive Testing Technique*, John Wiley and Sons

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D'Andrea, D.V., Fischer, R.L., & Fogelson, D.E. (1965) Prediction Of Compressive Strength

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From Other Rock Properties. *United States Bureau of Mines, Report of Investigation:*

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*. International Coal Preparation* 

precise results could be obtained from mobile transmitter unit.

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**5. References** 

Fig. 17. Dimension and measurement grids of the block.

Table 6 gives measured travelling times taken from the experiment applied on commercial block in the factory before it was sent to sawing machine. Standard expected travelling times representing the block mass behaviour in Table 7 can be obtained by accommodating direct distances between transmitter and receiver (given in Table 8) in to equation 15. Calculated pulse path lengths in Table 9 were obtained from measured travelling times.

The location of transmitter and receiver movements are given in Fig. 17 (d). Because of the necessity of locating the transmitter only one stable position on the block, the experiment was aimed to focus on one discontinuity that could be observed from surface. Several reflection points (Fig. 17 (b)) were observed when formulations given in the text are applied to the model. Fig. 17 (a) gives a diagrammatical illustration of the surface located within the block obtained by connecting the edge points of the discontinuity surface. Better organisation of transmitter position could have been done to obtain better 3D view of the surface within the block by using a single device that have several receivers connecting to same device, but the aim of this study was to show the possibility of exposing hidden surfaces in an enclosed environment.

### **4. Conclusion**

Ultrasonic wave velocity measurements have proven to be a valuable tool in mining industry, since successful applications of this technique have been introducing widely in earth science. Predicting earth conditions before any engineering practices has been one of the most important requirements of mining industry. This is because of difficulties in predicting earth structures before they are reached. In this chapter, some practical techniques have been given related with ultrasonic wave propagation to provide helpful tools to remove discrepancies in mining applications. People dealing with earth science are very familiar with the techniques of seismicity in answering questions such as what it is made up of, how deep it is, what is the position, how big it is. Logic behind the ultrasonic

Table 6 gives measured travelling times taken from the experiment applied on commercial block in the factory before it was sent to sawing machine. Standard expected travelling times representing the block mass behaviour in Table 7 can be obtained by accommodating direct distances between transmitter and receiver (given in Table 8) in to equation 15. Calculated pulse path lengths in Table 9 were obtained from measured travelling times.

The location of transmitter and receiver movements are given in Fig. 17 (d). Because of the necessity of locating the transmitter only one stable position on the block, the experiment was aimed to focus on one discontinuity that could be observed from surface. Several reflection points (Fig. 17 (b)) were observed when formulations given in the text are applied to the model. Fig. 17 (a) gives a diagrammatical illustration of the surface located within the block obtained by connecting the edge points of the discontinuity surface. Better organisation of transmitter position could have been done to obtain better 3D view of the surface within the block by using a single device that have several receivers connecting to same device, but the aim of this study was to show the possibility of exposing hidden

Ultrasonic wave velocity measurements have proven to be a valuable tool in mining industry, since successful applications of this technique have been introducing widely in earth science. Predicting earth conditions before any engineering practices has been one of the most important requirements of mining industry. This is because of difficulties in predicting earth structures before they are reached. In this chapter, some practical techniques have been given related with ultrasonic wave propagation to provide helpful tools to remove discrepancies in mining applications. People dealing with earth science are very familiar with the techniques of seismicity in answering questions such as what it is made up of, how deep it is, what is the position, how big it is. Logic behind the ultrasonic

Fig. 17. Dimension and measurement grids of the block.

surfaces in an enclosed environment.

**4. Conclusion** 

waves is the same as seismic wave propagation in a way that they both have P and S waves in the same form. Measuring the waves reflected from different bodies beneath earth surface and interpreting the data obtained from those measurements give very useful information for engineers.

The importance of determining the marble blocks those are affected by any discontinuity such as void, crack, caves et., in quarry has been presented through this chapter. It is also important to notice that any abnormalities in marble blocks must be pointed out by using a simple method without giving any damage to the main body. The method of testing the quality of concrete is perfectly well adapted to the determination the marble block because of its simplicity. First, experiments performed in the laboratory on simulated block gave promising results that encourage the possibility of using such a technique on marble blocks. But it must be bear in mind that a prepared concrete block differs from the natural stones concerning with homogeneity. Direct measurements give better understanding of the structure of any block measured because it clearly shows the boundary of a surface in the body. Nevertheless, as far as field investigation is concerned, direct measurements become very hard to apply depending on the number of free faces. To develop a measurement technique in field semi-direct and indirect applications have been developed on the blocks obtained from a marble factory. Mathematical model was applied on the blocks but results showed that fixing the transmitter in a stable position does not give a picture of the body if there is a complicated structure as far as discontinuities concern. Although it is possible to find exact positions of the discontinuities, the number of measurements increases in logarithmic scale with the moving transmitter that enables a practical method. However, statistical analysis would better give a high reliability for spotting out this kind of structures, instead of finding the correct location. Authors of this chapter suggests to develop a new measurement technique to allow multiple receivers located on several positions of the block to be analysed and moving transmitter along with the surface to be measured. Much more precise results could be obtained from mobile transmitter unit.

### **5. References**


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**11** 

*1,2Malaysia* 

**Design and Development of** 

*1Universiti Malaysia Perlis (UniMAP) 2Universiti Teknologi Malaysia (UTM)* 

 **Ultrasonic Process Tomography** 

Mohd Hafiz Fazalul Rahiman1, Ruzairi Abdul Rahim2, Herlina Abdul Rahim2 and Nor Muzakkir Nor Ayob2

Process tomography is a process of obtaining the plane-section images of a threedimensional object. Process tomography techniques produce cross-section images of the distribution of flow components in a pipeline, and it offers great potential for the

The measurement of two-component flow such as liquid or oil flow through a pipe is increasingly important in a wide range of applications, for example, pipeline control in oil exploitation, and chemical process monitoring. Knowledge of the flow component distribution is required for the determination of flow parameters such as the void fraction,

Real-time reconstruction of the flow image is needed in order to estimate the flow regime when it is continuously evolving. This flow image is important in many areas of industry, and scientific research concerning liquid/gas two-phase flow. The operation efficiency of such a process is closely related to accurate measurement, and control of hydrodynamic parameters such as flow regime, and flow rate (Rahiman et al., 2010). Commonly, the monitoring in the process industry is limited to either visual inspection or a single-point product sampling assuming the product is uniformed. This approach for the determination

Ultrasonic sensors have been successfully applied in flow measurement, non-destructive testing, and it is widely used in medical imaging. The method involves in using ultrasonic is through transmitting, and receiving sensors that are axially spaced along the flow stream. The sensors do not obstruct the flow. As the suspended solids' concentration fluctuates, the ultrasonic beam is scattered, and the received signal fluctuates in a random manner about a mean value. This type of sensor can be used for measuring the flow velocity. Two pairs of sensors are required in order to obtain the velocity using a cross- correlation method.

development, and verification of flow models, and also for process diagnostic.

of fluid flow parameters of two-component flow is also known as flow imaging.

Ultrasonic sensor propagates acoustic waves within the range of 18 kHz to 20 MHz.

**1. Introduction** 

and the flow regime.

**1.1 Principle overview** 

