1. Introduction

Porosity [1–3] is one of the most important parameters for describing the acoustic propagation in porous materials. This parameter intervenes in the propagation phenomena at all frequencies. Porosity is the relative fraction, by volume, of the air contained within the material. Airsaturated porous materials [1, 2, 4] as plastic foams, fibrous or granular materials are of great interest for a wide range of industrial applications. Figure 1 gives an example of air-saturated porous material commonly used in the sound absorption (passive control). These materials are frequently used in the automotive and aeronautics industries and in the building trade. Beranek [3] has developed an apparatus for measuring the porosity of air saturated porous materials. This device was based on the equation of state for ideal gases at constant temperature i.e., Boyle's law. Porosity can be determined by measuring the change in air pressure occurring with a known change in volume of the chamber containing the sample. In this apparatus, both

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© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

α ωð Þ given by Johnson et al. [4] and the dynamic compressibility of the air included in the porous material β ωð Þ given by Allard and Champoux [5]. In the frequency domain, these factors multiply the density of the fluid and its compressibility, respectively, and represent the deviation from the behavior of the fluid in free space as the frequency increases. In the time domain, they act as operators, and in the high-frequency approximation, their expressions are given by [11]

> 2 Λ

2ð Þ γ � 1 Λ0

In these equations, δð Þt is the Dirac function, Pr is the Prandtl number, η and r<sup>f</sup> are fluid viscosity and fluid density, respectively, and γ is the adiabatic constant. The most important physical parameters of the model are the medium's tortuosity α<sup>∞</sup> initially introduced by Zwikker and Kosten [2] and viscous and the thermal characteristic lengths Λ and Λ<sup>0</sup> introduced by Johnson et al. [4] and Allard and Champoux [5]. In this model, the time convolution

�1=<sup>2</sup> with a function is interpreted as a semi-derivative operator according to the definition

ð Þ <sup>t</sup> � <sup>u</sup> �ν�<sup>1</sup>

Ka ∗ ∂p

∂vx x; z; t <sup>0</sup> ð Þ=∂t

∂vz x; z; t <sup>0</sup> ð Þ=∂t

ffiffiffiffiffiffiffiffiffiffi <sup>t</sup> � <sup>t</sup> <sup>p</sup> <sup>0</sup> dt<sup>0</sup>

ffiffiffiffiffiffiffiffiffiffi <sup>t</sup> � <sup>t</sup> <sup>p</sup> <sup>0</sup> dt<sup>0</sup>

0

0

0

¼ � <sup>∂</sup>vxð Þ <sup>x</sup>; <sup>z</sup>; <sup>t</sup>

ðt 0

<sup>∂</sup><sup>t</sup> ¼ �∇ip and <sup>β</sup>~ð Þ<sup>t</sup>

where <sup>∗</sup> denotes the time convolution operation, p is the acoustic pressure, v is the particle velocity and Ka is the bulk modulus of the air. The first equation is the Euler equation, the second is a constitutive equation obtained from the equation of mass conservation associated

> ðt 0

> ðt 0

∂p x; z; t <sup>0</sup> ð Þ=∂t

ffiffiffiffiffiffiffiffiffiffi <sup>t</sup> � <sup>t</sup> <sup>p</sup> <sup>0</sup> dt<sup>0</sup>

η πr<sup>f</sup> !<sup>1</sup>=<sup>2</sup>

> η πPrr<sup>f</sup>

!<sup>1</sup>=<sup>2</sup>

t �1=2 1

1

t �1=2

A, (1)

Ultrasound Measuring of Porosity in Porous Materials http://dx.doi.org/10.5772/intechopen.72696 113

A: (2)

x uð Þdu, (3)

<sup>∂</sup><sup>t</sup> ¼ �∇:v, (4)

¼ � <sup>∂</sup>p xð Þ ; <sup>z</sup>; <sup>t</sup> <sup>∂</sup><sup>x</sup> ,

¼ � <sup>∂</sup>p xð Þ ; <sup>z</sup>; <sup>t</sup> <sup>∂</sup><sup>z</sup> ,

<sup>∂</sup><sup>x</sup> � <sup>∂</sup>vzð Þ <sup>x</sup>; <sup>z</sup>; <sup>t</sup>

<sup>∂</sup><sup>z</sup> ,

(5)

α~ðÞ¼ t α<sup>∞</sup> δð Þþ t

<sup>β</sup>~ðÞ¼ <sup>t</sup> <sup>δ</sup>ð Þþ <sup>t</sup>

of the fractional derivative of order ν given in Samko et al. [19]

½ �¼ x tð Þ <sup>1</sup>

Γð Þ �ν

In this framework, the basic equations of our model can be expressed as follows

Dν

<sup>r</sup><sup>f</sup> <sup>α</sup>~ð Þ<sup>t</sup> <sup>∗</sup> <sup>∂</sup>vi

In the plane (xoz), the constitutive equation (4) can be written as

2r<sup>f</sup> α<sup>∞</sup> Λ

2r<sup>f</sup> α<sup>∞</sup> Λ

η πr<sup>f</sup> Pr !<sup>1</sup>=<sup>2</sup>

η πr<sup>f</sup>

η πr<sup>f</sup> !<sup>1</sup>=<sup>2</sup>

> ðt 0

!<sup>1</sup>=<sup>2</sup>

where Γð Þx is the Gamma function.

with the behavior (or adiabatic) equation.

∂vxð Þ x; z; t ∂t þ

∂vzð Þ x; z; t ∂t þ

> 2ð Þ γ � 1 KaΛ<sup>0</sup>

r<sup>f</sup> α<sup>∞</sup>

r<sup>f</sup> α<sup>∞</sup>

∂p xð Þ ; z; t ∂t þ

1 Ka

of t

0 @ 0 @

Figure 1. Air saturated plastic foam.

pressure change and volume change are monitored using a U-shaped fluid-filled manometer. Leonard [6] has given an alternate dynamic method for measuring porosity. Other techniques using water as the pore-filling fluid, rather than air, are common in geophysical studies [7, 8]. Mercury has been used as the pore-filling fluid in other applications [9]. However, for many materials, the introduction of liquids into the material is not appropriate. Recently, a similar device to that of Beranek [3], involving the use of an electronic pressure transducer, was introduced by Champoux et al. [10]. This device can be used to measure very slight changes in pressure accurately, and the output can be recorded by a computer.

Generally, the most methods used for measuring the porosity cited previously do not use acoustic waves. Here, we present an ultrasonic method for measuring porosity using ultrasonic reflected waves by the porous material. The direct and inverse problem is solved in the time domain using experimental reflected data. The inverse problem is solved directly in time domain using the waveforms. The attractive feature of a time domain-based approach [11–16] is that the analysis is naturally limited by the finite duration of ultrasonic pressures and is consequently the most appropriate approach for the transient signal.
