**An Operational Approach to the Acoustic Analogy Equations**

Dorel Homentcovschi and Ronald Miles

*Department of Mechanical Engineering State University of New York at Binghamton USA*

### **1. Introduction**

48 Acoustic Waves – From Microdevices to Helioseismology

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Great progress has been made in the last sixty years in the study of the important problem of noise generated by the interaction of flow with stationary or mobile bodies such as occurs in jets, rotating blade propulsion machinery (propellers, turbofans helicopter rotors) and last but not least in aircraft at all ranges of flight and speed. An important part of this progress was based on a rigorous theory known as the Acoustic Analogy initiated by Sir James Lighthill in (Lighthill, 1952) and (Lighthill, 1954). Lighthill considered a free flow, as for example with a jet engine, and the nonstationary fluctuations of the stream represented by a distribution of quadrupole sources in the same volume. The flow parameters such as the surface pressure and the Lighthill tensor *Tij* are assumed known from solving the aerodynamic problem in the region of sound generation or furnished by measurements. For the first time, this revealed a clear distinction between Aerodynamic Theory, meant to determine mainly the aerodynamic parameters as the lift and damping on the moving object (and also supplying the data for the noise determination) and the Aeroacoustic Theory needed for studying the noise produced, generally at large distances, by the flying (or moving) objects. A primary aim of the following is to show that by using an operational calculus based on the multidimensional Fourier Transform all the theory involved in obtaining the Ffowcs Williams-Hawkings formula (Ffowcs Williams and Hawkings, 1969) can be performed using only classical mathematical analysis.

Curle's contribution (Curle, 1955) is a formal solution of the Acoustic Analogy which takes stationary hard surfaces into consideration. The theory developed by Ffowcs Williams and Hawkings (FW-H) (Ffowcs Williams and Hawkings, 1969) is valid for aeroacoustic sources in relative motion with respect to a hard surface, as is the case in many technical applications for example in the automotive industry or in air travel. The calculation involves quadrupole, dipole and monopole terms. An important point is that FW-H theory, developed in (Ffowcs Williams and Hawkings, 1969) assumed that the boundary surface coincides with the physical body surface and is impenetrable. In both Aerodynamic and Aeroacoustic theories the domain was the same: the infinite air domain external to the moving body.

When the Aeroacoustic Theory was developed by Lighthill, Curle, Ffowcs Williams and Hawkings there were not a lot of experimental or theoretical data to be used as input to their aeroacoustic theoretical work. For this reason, they derived mainly qualitative results

operational properties. Thus, for example, discontinuous solutions of linear equations using the Green's function are easily obtained by posing the problem in generalized function spaces. In the following we show that the theory connected with the FW-H formula can be made much simpler, without manipulating multidimensional generalized functions, by using an operational calculus based on the multidimensional Fourier Transform. The approach based on using the Fourier Transform preserves all the good operational properties of the generalized functions without the need to introduce a new sophisticated mathematical tool. The method was used previously by us (Homentcovschi and Singler, 1999) for a direct

An Operational Approach to the Acoustic Analogy Equations 51

In the case of permeable surfaces an alternate method for solving the Aeroacoustic problem for the infinite external domain is based on the Kirchhoff formula for the wave equation. Due to its use in Aeroacoustic theory we included also in Section 6 the Kirchhoff formulation for the solution of the wave equation in the case of mobile surfaces. The proof is based again on 3-D Fourier Transform of discontinuous functions and the final formula includes the volume sources and the surface sources as well. As a comparison of the two approaches (that based on FW-K equation and that using Kirchhoff's equation) we notice that Kirchhoff's method requires less memory because fewer quantities on the control surface needed to be stored. On the other hand, Brentner and Farassat in (Brentner & Farassat, 1998) have shown that the FW-H equation is superior to the Kirchhoff formula for aeroacoustic problems because it is based on conservation laws of fluid mechanics rather than on the wave equation. Thus, the FW-H equation is valid even if the integration surface is in the nonlinear region being therefore more robust with the choice of control surface. Another advantage of the FW-H method is that

it does not require computation of the normal derivatives on the permeable surface.

A comprehensive review of the use of Kirchhoff's method in computational aeroacoustics was given by Lyrintzis in (Lyrintzis, 1994). The same author reviewed the advances in the use of surface integral methods in aeroacoustics, including Kirchhoff's method and permeable Ffowcs-Williams Hawkings methods in (Lyrintzis, 2003). Morino in (Morino, 2003) addresses commonalities and differences between aeroacoustics and aerodynamics. A discussion about the acoustic analogy and alternative theories for jet noise prediction is the subject of (Morris and Farassat, 2002). Finally we note some interesting work about this

It is our hope that the elementary derivations included in this chapter will make the application of the FW-H equation more clear, avoiding in the future comments such as those generated by (Zinoviev and Bies, 2004) (See (Farassat, 2005), (Zinoviev and Bies, 2005),

In order to apply the 3-D Fourier Transform to a certain physical variable this has to be defined in the whole space. Thus, to utilize the Fourier Transform to examine the sound field in a finite physical domain, it is necessary to imbed this domain within an infinite space. For example, in the case of studying the air motion around a finite body, the interior of the body is considered as an air-filled domain separated from the exterior, infinite domain by an impermeable surface, *Sb* enclosing the body. This introduces a first class of discontinuity surfaces of the motion between two air-filled regions. The second class contains the natural discontinuity surfaces *Ss* inside the flow domain as shock fronts, wakes, etc. Finally, to aid computations, it is often helpful to also define a virtual permeable surface, *Sp*, enclosing body along with the portion of the air domain where viscous effects and the nonlinear terms in

**2. The equations of the acoustic analogy and their operational form**

introduction to the Boundary Element Method.

subject included in the book edited by (Raman, 2009).

(Farassat and Myers, 2006), (Zinoviev and Bies, 2006)).

which were quite useful in guiding many significant acoustic experiments and in designing low noise propulsion machinery.

The situation has changed dramatically in the last 20 years. The rapid growth in high speed digital computer technology, the availability of turbulent flow simulation codes as well as high quality measured fluid dynamic data and advances in the theory of partial differential equations, resulted in obtaining the needed data for many important problems of aeroacoustics. However, the development of reliable Computational Fluid Dynamics (CFD) methods made them also useful in the evaluation of the near-field aerodynamic parameters. Unfortunately, a fully CFD-based computational aeroacoustic methodology is so far too inefficient and beyond the capability of supercomputers of today.

To avoid these computational difficulties, the philosophy of approaching the Aeroacoustic problem has changed by introducing a surface *S* as a "permeable" control surface. The surface *S* is assumed to include inside, in the volume *V*, all the nonlinear flow effects and noise sources. This splitting of the problem into a linear problem for an infinite domain and a nonlinear setting for a bounded region allows the use of the most appropriate numerical methodology for each of them. In the bounded domain *V* the CFD methods or advanced measurement techniques will be used for obtaining the aerodynamic near-field and providing the data on the surface *S* needed for the external, infinite domain modelling. The analysis of the flow information inside *V* is, in general, expensive either using experiments or CFD. Therefore, it is advantageous to make the volume *V* as small as possible.

The FH-W equation involving a permeable surface is the proper model for determining the far-field pressure in the infinite domain. The case of permeable surfaces was analyzed by Ffowcs Williams in (Dowling and Ffowcs Williams, 1982) and (Crighton, et al.), by Francescantonio in (Francescantonio, 1997) who called it the KFWH (Kirchhoff FW-H) formula, by Pilon and Lyrintzis in (Pilon and Lyrintzis, 1997) calling it an improved Kirchhoff method and by Brentner and Farassat in (Brentner & Farassat, 1998). Besides the accessibility to the surface data the advantage of the methods using a permeable control surface is that the surface integrals and the first derivative needed can be evaluated more easily than the volume integrals and the second derivatives necessary for the calculation of the quadrupole terms when the traditional Acoustic Analogy is used.

The Acoustic Analogy approach and especially the theory based on the FW-H equation is the most widely used tool for deterministic noise sources. The beauty (and the power) of this model is that all the manipulations are completely rigorous without any *ad hoc* reasoning.

Besides the approach based on generalized functions, used in most papers approaching the FW-H formula, we note the work by Goldstein (Goldstein, 1976) where all the formulas are obtained starting with a generalized Green's formula. A similar approach was used in (Wu and Akay, 1992) . However, the algebra involved in their construction is substantial.

In the second of the two reports by Farassat (Farassat, 1994),(Farassat, 1996), which covers the details of the mathematics used for the wave equation with sources on a moving surface, the author correctly claims that the Ffowcs Williams-Hawkings famous paper published in 1969 used a level of mathematical sophistication including multidimensional generalized functions (distributions) and differential geometry unfamiliar to most researchers and designers working in the field. Many people use Dirac's *δ* (generalized) function starting with its integral definition. On the other hand, to learn about more complicated operations such as the derivative of a generalized function (distribution) we need a change of paradigm in the way we look at ordinary functions. For some people involved in practical applications this is not a simple task. The power of the theory of generalized functions stems from its 2 Will-be-set-by-IN-TECH

which were quite useful in guiding many significant acoustic experiments and in designing

The situation has changed dramatically in the last 20 years. The rapid growth in high speed digital computer technology, the availability of turbulent flow simulation codes as well as high quality measured fluid dynamic data and advances in the theory of partial differential equations, resulted in obtaining the needed data for many important problems of aeroacoustics. However, the development of reliable Computational Fluid Dynamics (CFD) methods made them also useful in the evaluation of the near-field aerodynamic parameters. Unfortunately, a fully CFD-based computational aeroacoustic methodology is so far too

To avoid these computational difficulties, the philosophy of approaching the Aeroacoustic problem has changed by introducing a surface *S* as a "permeable" control surface. The surface *S* is assumed to include inside, in the volume *V*, all the nonlinear flow effects and noise sources. This splitting of the problem into a linear problem for an infinite domain and a nonlinear setting for a bounded region allows the use of the most appropriate numerical methodology for each of them. In the bounded domain *V* the CFD methods or advanced measurement techniques will be used for obtaining the aerodynamic near-field and providing the data on the surface *S* needed for the external, infinite domain modelling. The analysis of the flow information inside *V* is, in general, expensive either using experiments or CFD.

The FH-W equation involving a permeable surface is the proper model for determining the far-field pressure in the infinite domain. The case of permeable surfaces was analyzed by Ffowcs Williams in (Dowling and Ffowcs Williams, 1982) and (Crighton, et al.), by Francescantonio in (Francescantonio, 1997) who called it the KFWH (Kirchhoff FW-H) formula, by Pilon and Lyrintzis in (Pilon and Lyrintzis, 1997) calling it an improved Kirchhoff method and by Brentner and Farassat in (Brentner & Farassat, 1998). Besides the accessibility to the surface data the advantage of the methods using a permeable control surface is that the surface integrals and the first derivative needed can be evaluated more easily than the volume integrals and the second derivatives necessary for the calculation of the quadrupole

The Acoustic Analogy approach and especially the theory based on the FW-H equation is the most widely used tool for deterministic noise sources. The beauty (and the power) of this model is that all the manipulations are completely rigorous without any *ad hoc* reasoning. Besides the approach based on generalized functions, used in most papers approaching the FW-H formula, we note the work by Goldstein (Goldstein, 1976) where all the formulas are obtained starting with a generalized Green's formula. A similar approach was used in (Wu and Akay, 1992) . However, the algebra involved in their construction is substantial. In the second of the two reports by Farassat (Farassat, 1994),(Farassat, 1996), which covers the details of the mathematics used for the wave equation with sources on a moving surface, the author correctly claims that the Ffowcs Williams-Hawkings famous paper published in 1969 used a level of mathematical sophistication including multidimensional generalized functions (distributions) and differential geometry unfamiliar to most researchers and designers working in the field. Many people use Dirac's *δ* (generalized) function starting with its integral definition. On the other hand, to learn about more complicated operations such as the derivative of a generalized function (distribution) we need a change of paradigm in the way we look at ordinary functions. For some people involved in practical applications this is not a simple task. The power of the theory of generalized functions stems from its

inefficient and beyond the capability of supercomputers of today.

Therefore, it is advantageous to make the volume *V* as small as possible.

terms when the traditional Acoustic Analogy is used.

low noise propulsion machinery.

operational properties. Thus, for example, discontinuous solutions of linear equations using the Green's function are easily obtained by posing the problem in generalized function spaces. In the following we show that the theory connected with the FW-H formula can be made much simpler, without manipulating multidimensional generalized functions, by using an operational calculus based on the multidimensional Fourier Transform. The approach based on using the Fourier Transform preserves all the good operational properties of the generalized functions without the need to introduce a new sophisticated mathematical tool. The method was used previously by us (Homentcovschi and Singler, 1999) for a direct introduction to the Boundary Element Method.

In the case of permeable surfaces an alternate method for solving the Aeroacoustic problem for the infinite external domain is based on the Kirchhoff formula for the wave equation. Due to its use in Aeroacoustic theory we included also in Section 6 the Kirchhoff formulation for the solution of the wave equation in the case of mobile surfaces. The proof is based again on 3-D Fourier Transform of discontinuous functions and the final formula includes the volume sources and the surface sources as well. As a comparison of the two approaches (that based on FW-K equation and that using Kirchhoff's equation) we notice that Kirchhoff's method requires less memory because fewer quantities on the control surface needed to be stored. On the other hand, Brentner and Farassat in (Brentner & Farassat, 1998) have shown that the FW-H equation is superior to the Kirchhoff formula for aeroacoustic problems because it is based on conservation laws of fluid mechanics rather than on the wave equation. Thus, the FW-H equation is valid even if the integration surface is in the nonlinear region being therefore more robust with the choice of control surface. Another advantage of the FW-H method is that it does not require computation of the normal derivatives on the permeable surface.

A comprehensive review of the use of Kirchhoff's method in computational aeroacoustics was given by Lyrintzis in (Lyrintzis, 1994). The same author reviewed the advances in the use of surface integral methods in aeroacoustics, including Kirchhoff's method and permeable Ffowcs-Williams Hawkings methods in (Lyrintzis, 2003). Morino in (Morino, 2003) addresses commonalities and differences between aeroacoustics and aerodynamics. A discussion about the acoustic analogy and alternative theories for jet noise prediction is the subject of (Morris and Farassat, 2002). Finally we note some interesting work about this subject included in the book edited by (Raman, 2009).

It is our hope that the elementary derivations included in this chapter will make the application of the FW-H equation more clear, avoiding in the future comments such as those generated by (Zinoviev and Bies, 2004) (See (Farassat, 2005), (Zinoviev and Bies, 2005), (Farassat and Myers, 2006), (Zinoviev and Bies, 2006)).

### **2. The equations of the acoustic analogy and their operational form**

In order to apply the 3-D Fourier Transform to a certain physical variable this has to be defined in the whole space. Thus, to utilize the Fourier Transform to examine the sound field in a finite physical domain, it is necessary to imbed this domain within an infinite space. For example, in the case of studying the air motion around a finite body, the interior of the body is considered as an air-filled domain separated from the exterior, infinite domain by an impermeable surface, *Sb* enclosing the body. This introduces a first class of discontinuity surfaces of the motion between two air-filled regions. The second class contains the natural discontinuity surfaces *Ss* inside the flow domain as shock fronts, wakes, etc. Finally, to aid computations, it is often helpful to also define a virtual permeable surface, *Sp*, enclosing body along with the portion of the air domain where viscous effects and the nonlinear terms in

Finally, on solid (deformable) impermeable surfaces *Sb* inside the flow the obvious

An Operational Approach to the Acoustic Analogy Equations 53

Since the density *ρ*<sup>0</sup> and pressure *p*<sup>0</sup> at large distances (in the unperturbed fluid) are different from zero we introduce the perturbation of the density *ρ*� = *ρ* − *ρ*<sup>0</sup> and the perturbation of the

> *∂ ∂xj ρuj*

> > <sup>2</sup> *∂ρ*� *∂xj*

*<sup>j</sup><sup>κ</sup>* <sup>+</sup> *<sup>ρ</sup>uju<sup>κ</sup>* <sup>−</sup> *<sup>c</sup>*2*ρ*�

[*ρ* (*un* − *vn*)]*e*

According to the previous discussion, these equations are valid within the flow domains in the whole space. Assuming, as in Appendix A, that there is a discontinuity surface *S*, separating the inner domain *D*(*i*) and the external domain *D*(*e*), by applying the Fourier transform to equations (6) and (7) and using the formulas (53) and (68) there results *the operational equations*

Here the overhead tilde denotes a Fourier Transform (see Appendix A), **k**(*k*1, *k*2, *k*3) is the wave vector and **y**(*y*1, *y*2, *y*3) is the position vector of the integration (source) point. The *operational* form of the conservation equations contains in the left-hand sides the Fourier transform of corresponding terms in (1) and (2) and in the right-hand sides the integrals accounting for the influence of the discontinuity surfaces. As was noted in Remark 1 in Appendix A, in the general case, in the right-hand side of equations (8) and (9) the sum of

**Remark 2.** *The square brackets in equations (8) and (9) can also be written as* [*ρ*0*un* + *ρ*� (*un* − *vn*)]

In the case of a discontinuity surface *Ss* (of shock-type) inside the fluid domain the junction conditions (3) and (4) will cancel out the integral terms in equations (8) and (9). Consequently the shock-type discontinuity surfaces are not introducing any supplementary terms in the

 *S*

*<sup>j</sup><sup>κ</sup>* + *S* 

*<sup>j</sup><sup>κ</sup>* = *Pj<sup>κ</sup>* − *p*0*δj<sup>κ</sup>* (where *δj<sup>κ</sup>* is the Kronecker delta) and will write equations

<sup>=</sup> <sup>−</sup> *<sup>∂</sup>T*� *jκ ∂x<sup>κ</sup>*

*δjκ*

*ρuj* (*un* − *vn*) + *Pκjn<sup>κ</sup>*

*un* − *vn* = 0. (5)

= 0 (6)

, (7)

<sup>−</sup>*i***k**·**y***dS* (8)

<sup>−</sup>*i***k**·**y***dS* (9)

 *e*

non-penetration condition proves true

stress tensor as *P*�

(1) and (2) as

where

*and*

**2.2 The operational form of the acoustic analogy equations**

*∂ρ*� *<sup>∂</sup><sup>t</sup>* <sup>+</sup>

*∂ ρuj <sup>∂</sup><sup>t</sup>* <sup>+</sup> *<sup>c</sup>*

*T*� *<sup>j</sup><sup>κ</sup>* = *P*�

*dt* <sup>+</sup> *ikj<sup>ρ</sup>uj* <sup>=</sup>

� = −*ikκT*�

*respectively.*

*d ρ* �

the contributions of all discontinuity surfaces will appear.

*<sup>d</sup> <sup>ρ</sup>uj*

*ρu<sup>κ</sup>* (*un* − *vn*) + *P*�

**2.3 Some particular cases**

operational equations.

*dt* <sup>+</sup> *ikjc*<sup>2</sup>*<sup>ρ</sup>*

*κj nj* 

**2.3.1 Shock-type discontinuity surfaces** (*Ss*)

the Navier-Stokes equations are important. It is convenient to evaluate the aerodynamic field in this region numerically using CFD, due to the difficulties imposed by viscosity and nonlinear effects. These calculations furnish data describing the hydrodynamic state on the virtual permeable surface, *Sp*. In the domain exterior to the permeable surface the acoustical analogy is applied to predict the sound field. In other words, the field inside the virtual permeable surface is assumed to be strongly influenced by hydrodynamic effects, while the field in the external, infinite domain can be modeled according to the usual assumptions of linear acoustics.

### **2.1 The equations of the acoustic analogy**

In the case where the whole space **R**<sup>3</sup> is filled by a compressible viscous fluid, containing several discontinuity surfaces, the flow inside each domain is governed by the following equations:

the continuity equation (mass conservation)

$$
\frac{\partial \rho}{\partial t} + \frac{\partial}{\partial x\_j} \left( \rho u\_j \right) = 0,\tag{1}
$$

and the conservation of momentum in the form written by Lighthill in Refs. (Lighthill, 1952) and (Lighthill, 1954)

$$\frac{\partial \left(\rho u\_i\right)}{\partial t} + c^2 \frac{\partial \rho}{\partial x\_i} = -\frac{\partial T\_{ij}}{\partial x\_j}.\tag{2}$$

Here *ρ* is the density, *c* is the velocity of sound in the uniform medium, *uj* is the component of fluid velocity in the direction *xj* (*j* = 1, 2, 3), and a repeated index implies a summation over these values, and,

$$T\_{ij} = P\_{ij} + \rho u\_i u\_j - c^2 \rho \,\delta\_{ij\prime}$$

is Lighthill's stress tensor. Also, we denoted by

$$P\_{ij} = p\delta\_{ij} + \mu \left( -\frac{\partial u\_j}{\partial \mathbf{x}\_j} - \frac{\partial u\_j}{\partial \mathbf{x}\_i} + \frac{2}{3} \left( \frac{\partial u\_\kappa}{\partial \mathbf{x}\_\kappa} \right) \delta\_{ij} \right) \rho$$

the compressive stress tensor, *δij* is the Kronicker delta function, *p* is the pressure and *μ* is the viscosity.

**Remark 1.** *The acoustical analogy equations (1) and (2) are in fact the exact fluid flow equations in the form written by Lighthill. This means that if one solves these equations correctly for a problem satisfying the Lighthill assumptions, then one will get the correct answer to the aerodynamic and aeroacoustic problems simultaneously.*

In the case where inside the flow domain there are discontinuity surfaces of type *Ss* (as shock fronts) the conservation laws for mass and momentum yield the Rankine-Hugoniot type junction conditions

$$\left[\rho \left(u\_n - v\_n\right)\right] = 0\tag{3}$$

$$\left[\rho u\_{\mathbb{K}} \left(u\_{\mathbb{m}} - v\_{\mathbb{m}}\right) + P\_{j\mathbb{k}} n\_{j}\right] = 0.\tag{4}$$

We denote by square brackets the jump of its content across the discontinuity surface (see also formula (54) in Appendix A). *un* is the velocity projection on the normal to the surface, **n**, and *vn* is the normal velocity of the surface.

Finally, on solid (deformable) impermeable surfaces *Sb* inside the flow the obvious non-penetration condition proves true

$$
\mu\_{\text{ll}} - \upsilon\_{\text{ll}} = 0.\tag{5}
$$

### **2.2 The operational form of the acoustic analogy equations**

Since the density *ρ*<sup>0</sup> and pressure *p*<sup>0</sup> at large distances (in the unperturbed fluid) are different from zero we introduce the perturbation of the density *ρ*� = *ρ* − *ρ*<sup>0</sup> and the perturbation of the stress tensor as *P*� *<sup>j</sup><sup>κ</sup>* = *Pj<sup>κ</sup>* − *p*0*δj<sup>κ</sup>* (where *δj<sup>κ</sup>* is the Kronecker delta) and will write equations (1) and (2) as

$$\frac{\partial \rho'}{\partial t} + \frac{\partial}{\partial \mathfrak{x}\_j} \left( \rho u\_j \right) = 0 \tag{6}$$

$$\frac{\partial \left(\rho u\_{\dot{j}}\right)}{\partial t} + c^2 \frac{\partial \rho'}{\partial x\_{\dot{j}}} = -\frac{\partial T'\_{j\kappa}}{\partial x\_{\kappa}}.\tag{7}$$

where

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

the Navier-Stokes equations are important. It is convenient to evaluate the aerodynamic field in this region numerically using CFD, due to the difficulties imposed by viscosity and nonlinear effects. These calculations furnish data describing the hydrodynamic state on the virtual permeable surface, *Sp*. In the domain exterior to the permeable surface the acoustical analogy is applied to predict the sound field. In other words, the field inside the virtual permeable surface is assumed to be strongly influenced by hydrodynamic effects, while the field in the external, infinite domain can be modeled according to the usual assumptions of

In the case where the whole space **R**<sup>3</sup> is filled by a compressible viscous fluid, containing several discontinuity surfaces, the flow inside each domain is governed by the following

and the conservation of momentum in the form written by Lighthill in Refs. (Lighthill, 1952)

<sup>2</sup> *∂ρ ∂xi*

Here *ρ* is the density, *c* is the velocity of sound in the uniform medium, *uj* is the component of fluid velocity in the direction *xj* (*j* = 1, 2, 3), and a repeated index implies a summation over

*Tij* <sup>=</sup> *Pij* <sup>+</sup> *<sup>ρ</sup>uiuj* <sup>−</sup> *<sup>c</sup>*2*ρ δij*,

<sup>−</sup> *<sup>∂</sup>uj ∂xi* + 2 3

the compressive stress tensor, *δij* is the Kronicker delta function, *p* is the pressure and *μ* is the

**Remark 1.** *The acoustical analogy equations (1) and (2) are in fact the exact fluid flow equations in the form written by Lighthill. This means that if one solves these equations correctly for a problem satisfying the Lighthill assumptions, then one will get the correct answer to the aerodynamic and*

In the case where inside the flow domain there are discontinuity surfaces of type *Ss* (as shock fronts) the conservation laws for mass and momentum yield the Rankine-Hugoniot

*ρu<sup>κ</sup>* (*un* − *vn*) + *Pjκnj*

We denote by square brackets the jump of its content across the discontinuity surface (see also formula (54) in Appendix A). *un* is the velocity projection on the normal to the surface, **n**, and

<sup>=</sup> <sup>−</sup> *<sup>∂</sup>Tij ∂xj*

> *∂u<sup>κ</sup> ∂x<sup>κ</sup>*

 *δij* ,

[*ρ* (*un* − *vn*)] = 0 (3)

= 0. (4)

= 0, (1)

. (2)

*∂ ∂xj ρuj* 

*∂ρ <sup>∂</sup><sup>t</sup>* <sup>+</sup>

*∂* (*ρui*) *<sup>∂</sup><sup>t</sup>* <sup>+</sup> *<sup>c</sup>*

> <sup>−</sup> *<sup>∂</sup>uj ∂xj*

linear acoustics.

equations:

and (Lighthill, 1954)

these values, and,

viscosity.

**2.1 The equations of the acoustic analogy**

the continuity equation (mass conservation)

is Lighthill's stress tensor. Also, we denoted by

*aeroacoustic problems simultaneously.*

*vn* is the normal velocity of the surface.

type junction conditions

*Pij* = *pδij* + *μ*

$$T'\_{j\kappa} = P'\_{j\kappa} + \rho u\_j u\_\kappa - c^2 \rho' \delta\_{j\kappa}$$

According to the previous discussion, these equations are valid within the flow domains in the whole space. Assuming, as in Appendix A, that there is a discontinuity surface *S*, separating the inner domain *D*(*i*) and the external domain *D*(*e*), by applying the Fourier transform to equations (6) and (7) and using the formulas (53) and (68) there results *the operational equations*

$$\frac{d\rho'}{dt} + ik\_{\!\!\!/\rho} \widetilde{u}\_{\!\!\!/} = \int\_{S} \left[ \rho \left( u\_{n} - v\_{n} \right) \right] e^{-i\mathbf{k} \cdot \mathbf{y}} dS \tag{8}$$

$$\frac{d\rho\bar{u}\_{\rangle}}{dt} + ik\_{\rangle}c^{2}\tilde{\rho}' = -ik\_{\kappa}\tilde{T}'\_{j\kappa} + \int\_{S} \left[\rho u\_{j} \left(u\_{n} - v\_{n}\right) + P\_{\kappa j} u\_{\kappa}\right] e^{-i\mathbf{k}\cdot\mathbf{y}} d\mathbf{S} \tag{9}$$

Here the overhead tilde denotes a Fourier Transform (see Appendix A), **k**(*k*1, *k*2, *k*3) is the wave vector and **y**(*y*1, *y*2, *y*3) is the position vector of the integration (source) point. The *operational* form of the conservation equations contains in the left-hand sides the Fourier transform of corresponding terms in (1) and (2) and in the right-hand sides the integrals accounting for the influence of the discontinuity surfaces. As was noted in Remark 1 in Appendix A, in the general case, in the right-hand side of equations (8) and (9) the sum of the contributions of all discontinuity surfaces will appear.

**Remark 2.** *The square brackets in equations (8) and (9) can also be written as* [*ρ*0*un* + *ρ*� (*un* − *vn*)] *and ρu<sup>κ</sup>* (*un* − *vn*) + *P*� *κj nj respectively.*

### **2.3 Some particular cases**

### **2.3.1 Shock-type discontinuity surfaces** (*Ss*)

In the case of a discontinuity surface *Ss* (of shock-type) inside the fluid domain the junction conditions (3) and (4) will cancel out the integral terms in equations (8) and (9). Consequently the shock-type discontinuity surfaces are not introducing any supplementary terms in the operational equations.

the components of the vector **<sup>w</sup>** can be written as

 *t* −∞

+*ikj t*

**3.2 The equation for the operational density**

**3.3 The case of an impermeable surface** (*Sb*)

*<sup>b</sup>*(**k**,*t*) = *<sup>d</sup>*

the form modified in (Brentner & Farassat, 1998)

*<sup>F</sup>*(**k**,*t*) = *<sup>d</sup>*

traction, which take into account the flow across *S*.

*F*

equation.

*<sup>ρ</sup>uj* (**k**, *<sup>t</sup>*) <sup>=</sup> <sup>−</sup>

*ρ* � (**k**, *t*) =  *t* −∞

*ikκT<sup>j</sup>κ*(**k**, *τ*)*dτ* +

representation of the operational density is given in the next section.

operational equation satisfied by the density perturbation becomes

*ρ* �

formula (17). In this case the nonhomogeneous term is

*dt S ρ*0*une*

coincides with that corresponding to an impermeable case

*dt S* *Uj* = <sup>1</sup> <sup>−</sup> *<sup>ρ</sup> ρ*0 

*Lj* = *P*�

*<sup>ρ</sup>*0*Une*−*i***k**·**y***dS* <sup>−</sup> *ikj*

*d*<sup>2</sup> *ρ* � *dt*<sup>2</sup> <sup>+</sup> *<sup>c</sup>*

<sup>−</sup><sup>∞</sup> *<sup>F</sup>*(**k**, *<sup>τ</sup>*)

*F*(**k**, *τ*)

An Operational Approach to the Acoustic Analogy Equations 55

 *t* −∞ *dτ S*(*τ*) 

cos (*ck* (*t* − *τ*)) − 1

The method given in this section has the advantage of furnishing the integral representations for the operational density and operational velocity as well. A simpler deduction of the

By eliminating the Fourier transform of the momentum *<sup>ρ</sup>uj* between equations (8) and (9) the

<sup>2</sup>*k*2*ρ*

The relationship (20) is the operational form of the nonhomogeneous wave equation. The general solution of the homogeneous wave equation in operational form can be written as

*<sup>h</sup>* = *A*(*k*) cos(*ckt*) + *B*(*k*) sin(*ckt*)) and by using Lagrange's method of variation of parameters there results the same representation formula (18) for the operational density as the solution of equation (20).

In the case where the surface *Sb* is impermeable, the condition (5) cancels out some terms in

 *S P*� *je*

*vj* <sup>+</sup> *<sup>ρ</sup>uj ρ*0

> *S*

<sup>−</sup>*i***k**·**y***dS* <sup>−</sup> *ikj*

which coincides with the nonhomogeneous term in the operational form of the FW-H

By introducing the new variables suggested by Francescantonio (Francescantonio, 1997), in

the nonhomogeneous term of the operational wave equation in the case of a permeable surface

The terms *Uj* and *Lj* can be interpreted respectively as a modified velocity and a modified

*<sup>k</sup>*<sup>2</sup> *<sup>d</sup><sup>τ</sup>*

sin (*ck* (*t* − *τ*))

*ck <sup>d</sup><sup>τ</sup>* (18)

*j e*

<sup>−</sup>*i***k**·**y***dS* (19)

*<sup>j</sup><sup>κ</sup>* (21)

(22)

*ρuj* (*un* − *vn*) + *P*�

� = *F*(**k**, *t*). (20)

<sup>−</sup>*i***k**·**y***dS* <sup>+</sup> *ikjikκ<sup>T</sup>*�

*<sup>j</sup>* + *ρuj* (*un* − *vn*) (23)

*Lje*−*i***k**·**y***dS* <sup>+</sup> *ikjikκ<sup>T</sup><sup>j</sup><sup>κ</sup>* (24)

## **2.3.2 Impermeable solid deformable surfaces** (*Sb*)

In the case of a solid with a deformable and impermeable boundary surface the condition (5) will cancel out the integral term in the operational form of the continuity equation (8) and a part of the integral in the momentum equation (9). In this case, the Fourier transform of the traction **P**=(*P*1, *P*2, *P*3) of the surface on the fluid enters in the integral term of the momentum equations as *Pj* = *Pκjnκ*. The operational momentum equation now contains in the right-hand side the action of the solid surface *Sb* on the fluid flow.

### **3. Solution of the operational form of the acoustical analogy equations.**

#### **3.1 The case of a permeable surface** � *Sp* �

Let *Sp* be a discontinuity surface of the flow variables inside the external fluid flow such that in the domain *D*(*i*) we have *p*(*i*) = *p*0, *ρ*(*i*) = *ρ*0, **v**(*<sup>i</sup>*)= **0**. We write the system of equations (8), (9) as

$$\frac{d\tilde{\mathbf{w}}}{dt} = \tilde{\mathbf{A}}\tilde{\mathbf{w}} + \tilde{\mathbf{f}} \tag{10}$$

where

$$
\widetilde{\mathbf{w}}^{T} = (\widetilde{\rho'}, \widetilde{\rho}\overline{\mu\_1}, \widetilde{\rho}\overline{\mu\_2}, \widetilde{\rho}\overline{\mu\_3})\_{\prime} \tag{11}
$$

$$
\tilde{\mathbb{A}} = \begin{bmatrix}
0 & -ik\_1 & -ik\_2 & -ik\_3 \\
\end{bmatrix} \prime \tag{12}
$$

$$\begin{array}{c} \widetilde{f}\_{1} = \int\_{\mathcal{S}} \left[ \rho\_{0} u\_{n} + \rho' \left( u\_{n} - v\_{n} \right) \right] e^{-i\mathbf{k} \cdot \mathbf{y}} d\mathbf{S} \\ \widetilde{f}\_{j+1} = -i k\_{\mathbf{k}} \widetilde{T}\_{j\mathbf{k}} + \int\_{\mathcal{S}} \left[ \rho u\_{j} \left( u\_{n} - v\_{n} \right) + P\_{j} \right] e^{-i\mathbf{k} \cdot \mathbf{y}} d\mathbf{S}, \quad j = 1, 2, 3. \end{array} \tag{13}$$

The solution of equation (10) can be obtained by using the exponential **H**� (*t*) of the matrix **A**� as

$$\tilde{\mathbf{w}} = \int\_{-\infty}^{t} \tilde{\mathbf{H}} \left( t - t' \right) \tilde{\mathbf{f}} \left( t' \right) dt' \tag{14}$$

The exponential of the matrix **A**� can be written as

$$\mathbf{H} = \begin{bmatrix} \tilde{h}\_{j\kappa} \end{bmatrix} \tag{15}$$

$$\begin{aligned} \tilde{h}\_{1,1} &= \cos\left(ckt\right), \quad \tilde{h}\_{1,\kappa+1} = -\frac{ik\_{\kappa}\sin\left(ckt\right)}{ck}, \kappa = 1,2,3\\ \tilde{h}\_{j+1,1} &= -\frac{ik\_{j}c\sin\left(ckt\right)}{k}, \quad j = 1,2,3\\ \tilde{h}\_{j+1,\kappa+1} &= \delta\_{j\kappa} + \frac{k\_{j}k\_{\kappa}\left(\cos\left(ckt\right)-1\right)}{k^{2}}, \quad j,\kappa = 1,2,3 \end{aligned} \tag{16}$$

where *k*<sup>2</sup> = *k*<sup>2</sup> <sup>1</sup> <sup>+</sup> *<sup>k</sup>*<sup>2</sup> <sup>2</sup> <sup>+</sup> *<sup>k</sup>*<sup>2</sup> <sup>3</sup> = |**k**| 2 . By introducing the function

$$\widetilde{F}(\mathbf{k},t) = \frac{d}{dt} \int\_{S} \left\{ \rho\_0 u\_{\mathrm{n}} + \rho' \left( u\_{\mathrm{n}} - v\_{\mathrm{n}} \right) \right\} e^{-i\mathbf{k}\cdot\mathbf{y}} dS \tag{17}$$

$$ -ik\_{\dot{j}} \int\_{S} \left\{ \rho u\_{\dot{j}} \left( u\_{\mathrm{n}} - v\_{\mathrm{n}} \right) + P\_{\dot{j}}' \right\} e^{-i\mathbf{k}\cdot\mathbf{y}} dS + ik\_{\dot{j}} ik\_{\mathrm{k}} \widetilde{T}\_{\mathrm{jk}}' $$

the components of the vector **<sup>w</sup>** can be written as

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

In the case of a solid with a deformable and impermeable boundary surface the condition (5) will cancel out the integral term in the operational form of the continuity equation (8) and a part of the integral in the momentum equation (9). In this case, the Fourier transform of the traction **P**=(*P*1, *P*2, *P*3) of the surface on the fluid enters in the integral term of the momentum equations as *Pj* = *Pκjnκ*. The operational momentum equation now contains in the right-hand

Let *Sp* be a discontinuity surface of the flow variables inside the external fluid flow such that

0 −*i k*<sup>1</sup> −*i k*<sup>2</sup> −*i k*<sup>3</sup> <sup>−</sup>*c*2*i k*<sup>1</sup> <sup>000</sup> <sup>−</sup>*c*2*i k*<sup>2</sup> <sup>000</sup> <sup>−</sup>*c*2*i k*<sup>3</sup> <sup>000</sup>

*<sup>S</sup>* [*ρ*0*un* <sup>+</sup> *<sup>ρ</sup>*� (*un* <sup>−</sup> *vn*)]*e*−*i***k**·**y***dS*

�

*<sup>k</sup>*<sup>2</sup> , *<sup>j</sup>*, *<sup>κ</sup>* <sup>=</sup> 1, 2, 3

�

*j* � *e*

*ρuj* (*un* − *vn*) + *Pj*

The solution of equation (10) can be obtained by using the exponential **H**� (*t*) of the matrix **A**�

**H**� = � �*hjκ* �

�*h*1,1 <sup>=</sup> cos (*ckt*), �*h*1,*κ*+<sup>1</sup> <sup>=</sup> <sup>−</sup>*i k<sup>κ</sup>* sin (*ckt*)

*kjk<sup>κ</sup>* (cos (*ckt*) − 1)

. By introducing the function

*ρuj* (*un* − *vn*) + *P*�

*ρ*0*un* + *ρ*� (*un* − *vn*)

*dt* <sup>=</sup> **<sup>A</sup>**� **<sup>w</sup>**� <sup>+</sup>�**<sup>f</sup>** (10)

��, *<sup>ρ</sup>*�*u*1, *<sup>ρ</sup>*�*u*2, *<sup>ρ</sup>*�*u*3), (11)

*ck* , *<sup>κ</sup>* <sup>=</sup> 1, 2, 3

*<sup>k</sup>* , *<sup>j</sup>* <sup>=</sup> 1, 2, 3 (16)

<sup>−</sup>*i***k**·**y***dS* <sup>+</sup> *ikjikκT*��

<sup>⎦</sup> , (12)

*<sup>e</sup>*−*i***k**·**y***dS*, *<sup>j</sup>* <sup>=</sup> 1, 2, 3. (13)

*dt*� (14)

*e*−*i***k**·**y***dS* (17)

*jκ*

(15)

⎤ ⎥ ⎥

**3. Solution of the operational form of the acoustical analogy equations.**

*Sp* �

*d***w**�

**<sup>w</sup>**� *<sup>T</sup>* = (*<sup>ρ</sup>*

**A**� =

*f* � <sup>1</sup> = �

*<sup>j</sup>*+<sup>1</sup> <sup>=</sup> <sup>−</sup>*ikκT*�*j<sup>κ</sup>* <sup>+</sup> �

The exponential of the matrix **A**� can be written as

�*hj*<sup>+</sup>1,*κ*+<sup>1</sup> = *δj<sup>κ</sup>* +

<sup>3</sup> = |**k**| 2

> *dt* � *S* �

−*ikj* � *S* �

⎡ ⎢ ⎢ ⎣

> *S* �

**<sup>w</sup>**� <sup>=</sup> � *t* −∞ **<sup>H</sup>**� � *t* − *t* � ��**f** � *t* � �

�*hj*+1,1 <sup>=</sup> <sup>−</sup>*i kjc* sin (*ckt*)

**2.3.2 Impermeable solid deformable surfaces** (*Sb*)

side the action of the solid surface *Sb* on the fluid flow.

in the domain *D*(*i*) we have *p*(*i*) = *p*0, *ρ*(*i*) = *ρ*0, **v**(*<sup>i</sup>*)= **0**.

**3.1 The case of a permeable surface** �

*f* �

where

as

where *k*<sup>2</sup> = *k*<sup>2</sup>

<sup>1</sup> <sup>+</sup> *<sup>k</sup>*<sup>2</sup>

<sup>2</sup> <sup>+</sup> *<sup>k</sup>*<sup>2</sup>

*<sup>F</sup>*�(**k**, *<sup>t</sup>*) = *<sup>d</sup>*

We write the system of equations (8), (9) as

$$\tilde{\rho'}\left(\mathbf{k},t\right) = \int\_{-\infty}^{t} \tilde{F}(\mathbf{k},\tau) \frac{\sin\left(ck\left(t-\tau\right)\right)}{ck}d\tau\tag{18}$$

$$\begin{split} \rho \widetilde{\boldsymbol{\mu}}\_{j}(\mathbf{k}, t) &= -\int\_{-\infty}^{t} ik\_{\mathbf{k}} \widetilde{\boldsymbol{T}}\_{j\mathbf{k}}(\mathbf{k}, \tau) d\tau + \int\_{-\infty}^{t} d\tau \int\_{S(\tau)} \left\{ \rho u\_{j} \left( u\_{\mathrm{n}} - v\_{\mathrm{n}} \right) + \boldsymbol{P}\_{j}^{\prime} \right\} e^{-i\mathbf{k} \cdot \mathbf{y}} dS \end{split} \tag{19}$$
 
$$\begin{split} + ik\_{\dot{\boldsymbol{\eta}}} \int\_{-\infty}^{t} \widetilde{\boldsymbol{F}}(\mathbf{k}, \tau) \frac{\cos \left( ck \left( t - \tau \right) \right) - 1}{k^{2}} d\tau \end{split} \tag{10}$$

The method given in this section has the advantage of furnishing the integral representations for the operational density and operational velocity as well. A simpler deduction of the representation of the operational density is given in the next section.

### **3.2 The equation for the operational density**

By eliminating the Fourier transform of the momentum *<sup>ρ</sup>uj* between equations (8) and (9) the operational equation satisfied by the density perturbation becomes

$$\frac{d^2\bar{\rho'}}{dt^2} + c^2k^2\tilde{\rho'} = \tilde{F}(\mathbf{k}, t). \tag{20}$$

The relationship (20) is the operational form of the nonhomogeneous wave equation. The general solution of the homogeneous wave equation in operational form can be written as

$$
\rho'\_h = A(k)\cos(c\,k\,t) + B(k)\sin(c\,k\,t),
$$

and by using Lagrange's method of variation of parameters there results the same representation formula (18) for the operational density as the solution of equation (20).

### **3.3 The case of an impermeable surface** (*Sb*)

In the case where the surface *Sb* is impermeable, the condition (5) cancels out some terms in formula (17). In this case the nonhomogeneous term is

$$\tilde{F}\_{\mathbf{b}}(\mathbf{k},t) = \frac{d}{dt} \int\_{S} \rho\_{0} u\_{\mathrm{il}} e^{-i\mathbf{k}\cdot\mathbf{y}} dS - i k\_{\mathrm{j}} \int\_{S} P\_{\mathbf{j}}' e^{-i\mathbf{k}\cdot\mathbf{y}} dS + i k\_{\mathrm{j}} i k\_{\mathrm{k}} \tilde{T}\_{\mathrm{j} \mathbf{k}}^{\mathrm{l}} \tag{21}$$

which coincides with the nonhomogeneous term in the operational form of the FW-H equation.

By introducing the new variables suggested by Francescantonio (Francescantonio, 1997), in the form modified in (Brentner & Farassat, 1998)

$$\mathcal{U}\_{j} = \left(1 - \frac{\rho}{\rho\_0}\right) v\_j + \frac{\rho u\_j}{\rho\_0} \tag{22}$$

$$L\_{\hat{\jmath}} = P\_{\hat{\jmath}}' + \rho u\_{\hat{\jmath}} \left( u\_n - v\_n \right) \tag{23}$$

the nonhomogeneous term of the operational wave equation in the case of a permeable surface coincides with that corresponding to an impermeable case

$$\tilde{F}(\mathbf{k},t) = \frac{d}{dt} \int\_{S} \rho\_0 \mathbf{U}\_{\mathrm{rel}} e^{-i\mathbf{k}\cdot\mathbf{y}} d\mathbf{S} - i\mathbf{k}\_{\mathrm{j}} \int\_{S} \mathbf{L}\_{\mathrm{j}} e^{-i\mathbf{k}\cdot\mathbf{y}} d\mathbf{S} + i\mathbf{k}\_{\mathrm{j}} i\mathbf{k}\_{\mathrm{k}} \tilde{T}\_{\mathrm{jk}} \tag{24}$$

The terms *Uj* and *Lj* can be interpreted respectively as a modified velocity and a modified traction, which take into account the flow across *S*.

In other worlds, the value of the density at the observation point **x** at the moment *t* is determined by the noise sources at the *emission (radiating) time τe* on the *emission (radiating)*

An Operational Approach to the Acoustic Analogy Equations 57

It is now necessay to consider the coordinate systems. Let a fixed point *P*<sup>0</sup> reside on a material

a) A frame *η* fixed relative to the moving material surface. This is the frame used by a designer to describe the structure geometry for purposes of fabrication. The variable *η* will be called the Lagrangian variable of the point *P*<sup>0</sup> and in the case supposed here (nondeformable material

b) A coordinate frame fixed with respect to the undisturbed medium having the origin *O* (see Fig.1). The position of the observation point is given by the position vector **x**. The position of the point *P*<sup>0</sup> is described by the position vector **y** (*η*, *τ*). The position vectors **x** and **y** will give the Eulerian coordinates of the observation and source terms, respectively. The formula

η

*dτ*� (30)

η

surface such as an airplane wing or a blade etc. We consider two coordinate frames:

**y** = *η* +

 *τ c***M** *η*, *τ*� 

moment *c***M***e*, the radiation vector **r***<sup>e</sup>* = **x** − **y***e*, and the emission distance *re* = |**r***e*| .

gives the connection between the Lagrangian and Eulerian coordinates of the point *P*0. For *η* fixed the equation **y** = **y** (*η*, *τ*) gives the trajectory of the point *P*<sup>0</sup> and *c***M** =*d***y**/*dτ* is the velocity of the source point with respect to the undisturbed medium (source convection velocity). Formula (30) can be viewed as a transformation between the Lagrangian and Eulerian coordinates of the point *P*0. The inverse transformation will be denoted by *η* = *η* (**y**, *τ*). In the case where the transformation involves only translations and rotations we have det(*∂***y**/*∂η*) = det(*∂η*/*∂***y**) = 1. Fig.1 shows the observer's position **x** at time *t*, the emission (radiation) position of the material surface (*Se* ≡ *Sτe*), the position of the same surface at the observation time (*St*), the position vector **y***<sup>e</sup>* at the emission time and at the observation time **y** (*η*, *t*), the trajectory of the point *P*0, the convection velocity of the source at the emission

*surface Se* ≡ *Sτe*.

surface) is independent of time.

Fig. 1. Coordinate frame

### **4. Determination of velocity**

### **4.1 The lift component of velocity**

The lift component of velocity is given by inverse Fourier transform of the term

$$\left(\widetilde{\rho u\_{\dot{j}}}\right)\_{L} = \mathrm{ik}\_{\dot{j}} \mathrm{ik}\_{\mathcal{r}} \int\_{-\infty}^{t} d\tau \int\_{S\_{\mathbb{T}}} P'\left(\mathbf{y},\tau\right) n'\_{r}\left(\mathbf{y},\tau\right) e^{-i\mathbf{k}\cdot\mathbf{y}} \frac{\cos\left(ck\left(t-\tau\right)\right) - 1}{k^{2}} dS$$

Therefore, by using formula (73) we obtain

$$\left(\rho u\_{\dot{j}}\right)\_{L} = \frac{\partial^{2}}{\partial x\_{\dot{j}}\partial x\_{r}} \int\_{-\infty}^{t} d\tau \int\_{S\_{\tau}} P' n\_{r}^{\prime} \frac{H\left(t - \tau - r/c\right)}{4\pi r} dS\_{\tau}$$

where

$$r = |\mathbf{x} - \mathbf{y}| \equiv \sqrt{(\mathbf{x}\_1 - y\_1)^2 + (\mathbf{x}\_2 - y\_2)^2 + (\mathbf{x}\_3 - y\_3)^2}.$$

**x**(*x*1, *x*2, *x*3) being the position vector of the observation point.

### **5. Determination of density**

Since the general permeable case can be studied by means of an equivalent impermeable case we shall determine the density in the case where the nonhomogeneous term is (21).

### **5.1 The case of sources on a surface**

Consider the simpler case when the noise sources are on a rigid surface having only translation and rotation motions. Then,

$$\tilde{F}\_{\mathbf{S}}\left(\mathbf{k},t\right) = \int\_{\mathbf{S}} \mathcal{Q}\left(\mathbf{y},t\right) e^{-i\mathbf{k}\cdot\mathbf{y}} dS \tag{25}$$

where *Q* (**y**,*t*) is the surface intensity. In this case we have

$$\tilde{\rho}\_{\rm S}^{\prime}(\mathbf{k}, t) = \int\_{-\infty}^{t} d\tau \int\_{S\_{\rm r}} \mathcal{Q}\left(\mathbf{y}, \tau\right) e^{-i\mathbf{k}\cdot\mathbf{y}} \frac{\sin\left(ck\left(t - \tau\right)\right)}{ck} dS \tag{26}$$

Hence,

$$\begin{split} \rho'\_{\mathbf{S}}(\mathbf{x},t) &= \frac{1}{\left(2\pi\right)^{3}} \int e^{i\mathbf{k}\cdot\mathbf{x}} d\mathbf{k} \int\_{-\infty}^{t} d\tau \int\_{S\_{\mathrm{r}}} Q\left(\mathbf{y},\tau\right) e^{-i\mathbf{k}\cdot\mathbf{y}} \frac{\sin\left(ck\left(t-\tau\right)\right)}{ck} d\mathbf{S} \\ &= \int\_{-\infty}^{t} d\tau \int\_{S\_{\mathrm{r}}} Q\left(\mathbf{y},\tau\right) d\mathcal{S} \int \frac{\sin\left(ck\left(t-\tau\right)\right)}{\left(2\pi\right)^{3}ck} e^{i\mathbf{k}\cdot\left(\mathbf{x}-\mathbf{y}\right)} d\mathbf{k} \\ &= \int\_{-\infty}^{t} d\tau \int\_{S\_{\mathrm{r}}} Q\left(\mathbf{y},\tau\right) \frac{\delta\left(\tau-\left(t-r/c\right)\right)}{4\pi c^{2}r} d\mathbf{S} \end{split} \tag{27}$$

Here the relationship (71) and the property *δ* (−*t*) = *δ* (*t*) have been used. The only contribution in the last integral in formula (27) comes from the time *τe* which is the solution of the equation

$$g(\mathbf{r}, \mathbf{y}, t, \mathbf{x}) = 0 \tag{28}$$

where

$$g(\mathbf{r}, \mathbf{y}, t, \mathbf{x}) = \mathbf{r} - t + \frac{|\mathbf{x} - \mathbf{y}|}{c} \tag{29}$$

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

The lift component of velocity is given by inverse Fourier transform of the term

 *t* −∞ *dτ Sτ P*� *n*� *r*

(*x*<sup>1</sup> − *y*1)

we shall determine the density in the case where the nonhomogeneous term is (21).

<sup>2</sup> <sup>+</sup> (*x*<sup>2</sup> <sup>−</sup> *<sup>y</sup>*2)

*<sup>Q</sup>* (**y**, *<sup>τ</sup>*) *<sup>e</sup>*−*i***k**·**<sup>y</sup>** sin (*ck* (*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*))

 sin (*ck* (*t* − *τ*)) (2*π*) <sup>3</sup> *ck*

Since the general permeable case can be studied by means of an equivalent impermeable case

Consider the simpler case when the noise sources are on a rigid surface having only translation

 *S*

*P*� (**y**, *τ*) *n*�

*<sup>r</sup>* (**y**, *<sup>τ</sup>*)*e*−*i***k**·**<sup>y</sup>** cos (*ck* (*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*)) <sup>−</sup> <sup>1</sup>

*H* (*t* − *τ* − *r*/*c*) 4*πr*

<sup>2</sup> <sup>+</sup> (*x*<sup>3</sup> <sup>−</sup> *<sup>y</sup>*3)

*<sup>k</sup>*<sup>2</sup> *dS*

*dS*

2 .

*Q* (**y**,*t*) *e*−*i***k**·**y***dS* (25)

*<sup>Q</sup>* (**y**, *<sup>τ</sup>*)*e*−*i***k**·**<sup>y</sup>** sin (*ck* (*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*))

*dS*

*ei***k**·(**x**−**y**)

*g*(*τ*, **y**, *t*, **x**) =0 (28)

*ck dS* (26)

*ck dS*

*<sup>c</sup>* (29)

*d***k** (27)

**4. Determination of velocity**

**5. Determination of density**

and rotation motions. Then,

*ρ*�

solution of the equation

**5.1 The case of sources on a surface**

*ρ*�

*<sup>S</sup>* (**x**, *<sup>t</sup>*) <sup>=</sup> <sup>1</sup>

= *t* −∞ *dτ Sτ*

= *t* −∞ *dτ Sτ*

 *ρuj* 

where

Hence,

where

**4.1 The lift component of velocity**

*<sup>L</sup>* <sup>=</sup> *ikjikr*

Therefore, by using formula (73) we obtain

 *ρuj* 

 *t* −∞ *dτ Sτ*

*r* = |**x** − **y**| ≡

*<sup>L</sup>* <sup>=</sup> *<sup>∂</sup>*<sup>2</sup> *∂xj∂xr*

**x**(*x*1, *x*2, *x*3) being the position vector of the observation point.

*F*

 *t* −∞ *dτ Sτ*

*ei***k**·**x***d***k**

 *t* −∞ *dτ Sτ*

*Q* (**y**, *τ*) *dS*

Here the relationship (71) and the property *δ* (−*t*) = *δ* (*t*) have been used.

*<sup>Q</sup>* (**y**, *<sup>τ</sup>*) *<sup>δ</sup>* (*<sup>τ</sup>* <sup>−</sup> (*<sup>t</sup>* <sup>−</sup> *<sup>r</sup>*/*c*)) 4*πc*2*r*

The only contribution in the last integral in formula (27) comes from the time *τe* which is the

*<sup>g</sup>*(*τ*, **<sup>y</sup>**, *<sup>t</sup>*, **<sup>x</sup>**) =*<sup>τ</sup>* <sup>−</sup> *<sup>t</sup>* <sup>+</sup> <sup>|</sup>**<sup>x</sup>** <sup>−</sup> **<sup>y</sup>**<sup>|</sup>

where *Q* (**y**,*t*) is the surface intensity. In this case we have

*<sup>S</sup>* (**k**, *t*) =

(2*π*) 3  *<sup>S</sup>* (**k**,*t*) =

In other worlds, the value of the density at the observation point **x** at the moment *t* is determined by the noise sources at the *emission (radiating) time τe* on the *emission (radiating) surface Se* ≡ *Sτe*.

It is now necessay to consider the coordinate systems. Let a fixed point *P*<sup>0</sup> reside on a material surface such as an airplane wing or a blade etc. We consider two coordinate frames:

a) A frame *η* fixed relative to the moving material surface. This is the frame used by a designer to describe the structure geometry for purposes of fabrication. The variable *η* will be called the Lagrangian variable of the point *P*<sup>0</sup> and in the case supposed here (nondeformable material surface) is independent of time.

b) A coordinate frame fixed with respect to the undisturbed medium having the origin *O* (see Fig.1). The position of the observation point is given by the position vector **x**. The position of the point *P*<sup>0</sup> is described by the position vector **y** (*η*, *τ*). The position vectors **x** and **y** will give the Eulerian coordinates of the observation and source terms, respectively. The formula

Fig. 1. Coordinate frame

$$\mathbf{y} = \eta + \int^{\tau} c\mathbf{M} \left(\eta\_{\prime}\pi^{\prime}\right) d\tau^{\prime} \tag{30}$$

gives the connection between the Lagrangian and Eulerian coordinates of the point *P*0. For *η* fixed the equation **y** = **y** (*η*, *τ*) gives the trajectory of the point *P*<sup>0</sup> and *c***M** =*d***y**/*dτ* is the velocity of the source point with respect to the undisturbed medium (source convection velocity). Formula (30) can be viewed as a transformation between the Lagrangian and Eulerian coordinates of the point *P*0. The inverse transformation will be denoted by *η* = *η* (**y**, *τ*). In the case where the transformation involves only translations and rotations we have det(*∂***y**/*∂η*) = det(*∂η*/*∂***y**) = 1. Fig.1 shows the observer's position **x** at time *t*, the emission (radiation) position of the material surface (*Se* ≡ *Sτe*), the position of the same surface at the observation time (*St*), the position vector **y***<sup>e</sup>* at the emission time and at the observation time **y** (*η*, *t*), the trajectory of the point *P*0, the convection velocity of the source at the emission moment *c***M***e*, the radiation vector **r***<sup>e</sup>* = **x** − **y***e*, and the emission distance *re* = |**r***e*| .

where *D*(*e*)

*ρ*�

where

*thickness* and *ρ*�

formula (71) there results

*ρ*�

the relationship (36) becomes

*<sup>q</sup>* (**x**, *<sup>t</sup>*) <sup>=</sup> *<sup>∂</sup>*<sup>2</sup>

*ρ*�

**6. The Kirchhoff method in Aeroacoustics**

form of the wave equation (38)

increases the source strength by |1 − *Mr*|

*∂xj∂x<sup>κ</sup>*

*<sup>q</sup>* (**x**, *<sup>t</sup>*) <sup>=</sup> *<sup>∂</sup>*<sup>2</sup>

 *t* −∞ *dτ D*(*e*) *τ*

*∂xj∂x<sup>κ</sup>*

*loading* completely specify the density field.

nonhomogeneous wave equation for the pressure perturbation

 <sup>Δ</sup> <sup>−</sup> <sup>1</sup> *c*2 *∂*2 *∂t*<sup>2</sup> 

*d*<sup>2</sup> *p* � *d t*<sup>2</sup> <sup>+</sup> *<sup>c</sup>*2*k*<sup>2</sup> *<sup>p</sup>*

> *St c* 2 *∂ p*� *∂ n* + *vn ∂ p*� *∂ t*

*p*� 

*<sup>c</sup>*2*<sup>i</sup>* **<sup>k</sup>** · **<sup>n</sup>**

 *t* −∞ 

*G* (**k**,*t*) = −

− *St*

*p* � (**k**, *t*) = −1

*D*(*<sup>e</sup>*)(*τ*)

where the effect of source convection is revealed by the Doppler factor; convection effectively

for its numerical implementation were made by Farassat and Brentner (Farassat and Brentner, 1988) and by Brentner in (Brentner, 1997). In the case where the discontinuity surface is permeable (of type *Sp* ) this term is missing, the surface being usually chosen outside the space containing the quadruple sources. In the general case the sum of the solutions *ρ*�

Besides the Acoustic Analogy approach for the solution of the Aeroacoustic noise, another widely used method is based on Kirchhoffs' solution of the wave equation. We start with the

where *g* (**x**, *t*) represents the density of pressure sources. By applying the Fourier transform with respect to the spatial variables and using formulas (58) and (70) we obtain the operational

*<sup>e</sup>*−*i***k**·**y***dS* <sup>−</sup> *<sup>d</sup>*

<sup>−</sup>*c*2*g*(**k**, *<sup>τ</sup>*) <sup>+</sup> *<sup>G</sup>* (**k**, *<sup>τ</sup>*)

Equation (39) is similar to equation (20). Consequently, its solution can be written as

*d t St vn p*� 

*<sup>τ</sup>* denotes the 3-D domain occupied by volume sources at the moment *τ*. The last

 *Tj<sup>κ</sup>* (**y**, *<sup>τ</sup>*) <sup>4</sup>*πc*2*<sup>r</sup>* |<sup>1</sup> − *Mr*|

*Tj<sup>κ</sup>* (**y**, *<sup>τ</sup>*) *<sup>δ</sup>* (*<sup>τ</sup>* <sup>−</sup> (*<sup>t</sup>* <sup>−</sup> *<sup>r</sup>*/*c*))

4*πc*2*r*

*ret*

. Further transformations of the formula (37) useful

*p*� (**x**, *t*) = *g* (**x**, *t*) (38)

� <sup>=</sup> <sup>−</sup>*c*2*g*(**k**, *<sup>t</sup>*) <sup>+</sup> *<sup>G</sup>* (**k**,*t*) (39)

*e*−*i***k**·**y***dS*

sin (*c k* (*t* − *τ*))

*<sup>e</sup>*−*i***k**·**y***dS* (40)

*c k <sup>d</sup><sup>τ</sup>* (41)

*d***y**, (36)

*dη*. (37)

*<sup>q</sup>* (**x**, *t*),

integral in formula (35) was calculated in Appendix B. Introducing its expression given by

An Operational Approach to the Acoustic Analogy Equations 59

To change the integration variable in the last integral in formula (27) from *τ* to *g*, we calculate

$$\frac{d\,\,\mathcal{g}}{d\tau} = 1 - \frac{1}{c} \nabla r \cdot \frac{\partial \,\mathbf{y}}{\partial \tau} = 1 - \frac{\mathbf{r}}{r} \cdot \frac{\mathbf{v}}{c} = 1 - M\_r \tag{31}$$

where *Mr* is the Mach number at the point *η* in the radiation direction at the time *τ*. The density perturbation can therefore be written as

$$\rho\_S^{\prime} \left( \mathbf{x}, t \right) = \int\_{S\_\varepsilon} \left[ \frac{\mathbb{Q} \left( \mathbf{y}, \tau \right)}{4 \pi c^2 r \left| 1 - M\_r \right|} \right]\_{ret} dS\_\eta \equiv \int\_{S^\* \left( \tau\_\varepsilon \right)} \frac{\mathbb{Q}^\* \left( \eta\_\prime \,\tau\_\varepsilon \right)}{4 \pi c^2 r\_\varepsilon \left| 1 - M\_{r\_\varepsilon} \right|} dS\_\eta \tag{32}$$

Here, |1 − *Mre*| is the *Doppler factor.* The square brackets []*ret* imply that the contents are to be evaluated at the retarded (emission or radiating time) *τ<sup>e</sup>* given implicitly by *g* (*τ*) = 0. The emission position is **y***<sup>e</sup>* = **y** (*η*, *τe*), the emission distance *re* of the source point *η* to the observer position **x** is *re* = |**x** − **y** (*ηe*, *τe*)| , and *Q*<sup>∗</sup> (*η*, *τe*) = *Q* (**y***e*, *τe*).

### **5.1.1 The thickness noise**

The thickness noise is given by the term

$$\tilde{F}\_{\text{thickness}}(\mathbf{k}, t) = \frac{d}{dt} \int\_{S} \rho\_0 \mu\_n e^{-i\mathbf{k} \cdot \mathbf{y}} dS$$

An analysis similar with that of the section 5.1 yields

$$\rho\_{\text{thickness}}'(\mathbf{x}, t) = \frac{\partial}{\partial t} \int\_{\mathcal{S}} \left[ \frac{\rho\_0 u\_n}{4\pi c^2 r \left| 1 - M\_r \right|} \right]\_{ret} dS\_\eta \tag{33}$$

### **5.1.2 The loading noise**

The last term in relationship (21) describes the loading noise

$$\tilde{F}\_{loading}(\mathbf{k},t) = -i k\_{\circ} \int\_{S} P\_{\circ}^{\prime} e^{-i\mathbf{k}\cdot\mathbf{y}} dS$$

Its contribution to the perturbed density *ρ*� is

$$\rho\_{loading}'(\mathbf{x},t) = -\frac{\partial}{\partial \mathbf{x}\_j} \int\_{\mathcal{S}} \left[ \frac{P\_j'}{4\pi c^2 r \left| 1 - M\_r \right|} \right]\_{ret} dS\_\eta \tag{34}$$

We have given here a very short presentation of formulas for thickness noise and loading noise. A more complete presentation about these formulas and their implementation can be found in (Farassat, 2007).

### **5.2 The quadrupole noise term**

The last term in formula (21) corresponds to a quadrupole noise source:

$$\begin{split} \rho\_q'(\mathbf{x}, t) &= \mathcal{F}^{-1}\left\{ ik\_j ik\_\mathbf{k} \tilde{T}\_{j\mathbf{k}} \right\} \\ &= \int\_{-\infty}^t d\tau \int ik\_j ik\_\mathbf{k} \tilde{T}\_{j\mathbf{k}} \frac{\sin\left(ck\left(t - \tau\right)\right)}{\left(2\pi\right)^3 ck} e^{i\mathbf{k}\cdot\mathbf{x}} d\mathbf{k} \\ &= \frac{\hat{\partial}^2}{\partial \mathbf{x}\_j \partial \mathbf{x}\_\mathbf{k}} \int\_{-\infty}^t d\tau \int\_{D\_\tau^{(\leq)}} T\_{j\mathbf{k}}\left(\mathbf{y}, \tau\right) d\mathbf{y} \int \frac{\sin\left(ck\left(t - \tau\right)\right)}{\left(2\pi\right)^3 ck} e^{i\mathbf{k}\cdot\left(\mathbf{x} - \mathbf{y}\right)} d\mathbf{k}, \end{split} \tag{35}$$

where *D*(*e*) *<sup>τ</sup>* denotes the 3-D domain occupied by volume sources at the moment *τ*. The last integral in formula (35) was calculated in Appendix B. Introducing its expression given by formula (71) there results

$$\rho\_{\boldsymbol{q}}^{\prime}\left(\mathbf{x},t\right) = \frac{\partial^2}{\partial \mathbf{x}\_{\boldsymbol{j}} \partial \mathbf{x}\_{\boldsymbol{k}}} \int\_{-\infty}^{t} d\tau \int\_{D\_{\tau}^{(\varepsilon)}} T\_{\rm j\x}\left(\mathbf{y},\tau\right) \frac{\delta\left(\tau - \left(t - r/c\right)\right)}{4\pi c^2 \tau} d\mathbf{y},\tag{36}$$

the relationship (36) becomes

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

To change the integration variable in the last integral in formula (27) from *τ* to *g*, we calculate

*r* · **v**

*S*∗(*τe*)

*ρ*0*une*−*i***k**·**y***dS*

*<sup>j</sup>e*−*i***k**·**y***dS*

*ret*

*ret*

*<sup>c</sup>* <sup>=</sup> <sup>1</sup> <sup>−</sup> *Mr* (31)

*dS<sup>η</sup>* (32)

*dS<sup>η</sup>* (33)

*dS<sup>η</sup>* (34)

*<sup>i</sup>***k**·**x***d***k** (35)

*ei***k**·(**x**−**y**)

*d***k**,

*Q*∗ (*η*, *τe*) <sup>4</sup>*πc*2*re* |<sup>1</sup> − *Mre*|

*d g <sup>d</sup><sup>τ</sup>* <sup>=</sup> <sup>1</sup> <sup>−</sup> <sup>1</sup>

The density perturbation can therefore be written as

 *Se*

*ρ*�

**5.1.1 The thickness noise**

**5.1.2 The loading noise**

found in (Farassat, 2007).

*ρ*�

**5.2 The quadrupole noise term**

*<sup>q</sup>* (**x**, *<sup>t</sup>*) <sup>=</sup> <sup>F</sup> <sup>−</sup><sup>1</sup>

= *t* −∞ *dτ* 

<sup>=</sup> *<sup>∂</sup>*<sup>2</sup> *∂xj∂x<sup>κ</sup>*

*ikjikκT<sup>j</sup><sup>κ</sup>*

 *t* −∞ *dτ D*(*e*) *τ*

*<sup>S</sup>* (**x**, *t*) =

The thickness noise is given by the term

*c* ∇*r* · *∂* **y** *∂τ* <sup>=</sup> <sup>1</sup> <sup>−</sup> **<sup>r</sup>**

 *Q* (**y**, *τ*) <sup>4</sup>*πc*2*<sup>r</sup>* |<sup>1</sup> − *Mr*|

observer position **x** is *re* = |**x** − **y** (*ηe*, *τe*)| , and *Q*<sup>∗</sup> (*η*, *τe*) = *Q* (**y***e*, *τe*).

*thickness*(**k**,*t*) = *<sup>d</sup>*

*∂t S*

*loading*(**k**, *t*) = −*ikj*

*∂xj S*

We have given here a very short presentation of formulas for thickness noise and loading noise. A more complete presentation about these formulas and their implementation can be

> sin (*ck* (*t* − *τ*)) (2*π*) <sup>3</sup> *ck*

*Tj<sup>κ</sup>* (**y**, *τ*) *d***y**

*e*

 sin (*ck* (*t* − *τ*)) (2*π*) <sup>3</sup> *ck*

*F*

The last term in relationship (21) describes the loading noise

*F*

*loading* (**x**, *<sup>t</sup>*) <sup>=</sup> <sup>−</sup> *<sup>∂</sup>*

The last term in formula (21) corresponds to a quadrupole noise source:

*ikjikκT<sup>j</sup><sup>κ</sup>*

*thickness* (**x**, *<sup>t</sup>*) <sup>=</sup> *<sup>∂</sup>*

An analysis similar with that of the section 5.1 yields

*ρ*�

Its contribution to the perturbed density *ρ*� is

*ρ*�

where *Mr* is the Mach number at the point *η* in the radiation direction at the time *τ*.

*ret*

Here, |1 − *Mre*| is the *Doppler factor.* The square brackets []*ret* imply that the contents are to be evaluated at the retarded (emission or radiating time) *τ<sup>e</sup>* given implicitly by *g* (*τ*) = 0. The emission position is **y***<sup>e</sup>* = **y** (*η*, *τe*), the emission distance *re* of the source point *η* to the

> *dt S*

*ρ*0*un*

 *S P*�

 *P*� *j* <sup>4</sup>*πc*2*<sup>r</sup>* |<sup>1</sup> − *Mr*|

<sup>4</sup>*πc*2*<sup>r</sup>* |<sup>1</sup> − *Mr*|

*dS<sup>η</sup>* ≡

$$\rho\_q' \left( \mathbf{x}, t \right) = \frac{\partial^2}{\partial \mathbf{x}\_j \partial \mathbf{x}\_\mathbf{x}} \int\_{D^{(\varepsilon)}(\mathbf{r})} \left[ \frac{T\_{j\mathbf{k}} \left( \mathbf{y}, \mathbf{r} \right)}{4\pi c^2 r \left| 1 - M\_r \right|} \right]\_{ret} d\eta. \tag{37}$$

where the effect of source convection is revealed by the Doppler factor; convection effectively increases the source strength by |1 − *Mr*| −1 . Further transformations of the formula (37) useful for its numerical implementation were made by Farassat and Brentner (Farassat and Brentner, 1988) and by Brentner in (Brentner, 1997). In the case where the discontinuity surface is permeable (of type *Sp* ) this term is missing, the surface being usually chosen outside the space containing the quadruple sources. In the general case the sum of the solutions *ρ*� *<sup>q</sup>* (**x**, *t*), *ρ*� *thickness* and *ρ*� *loading* completely specify the density field.

### **6. The Kirchhoff method in Aeroacoustics**

Besides the Acoustic Analogy approach for the solution of the Aeroacoustic noise, another widely used method is based on Kirchhoffs' solution of the wave equation. We start with the nonhomogeneous wave equation for the pressure perturbation

$$
\left[\Delta - \frac{1}{c^2} \frac{\partial^2}{\partial t^2}\right] p'\left(\mathbf{x}, t\right) = \mathbf{g}\left(\mathbf{x}, t\right) \tag{38}
$$

where *g* (**x**, *t*) represents the density of pressure sources. By applying the Fourier transform with respect to the spatial variables and using formulas (58) and (70) we obtain the operational form of the wave equation (38)

$$\frac{d^2\bar{p'}}{dt^2} + c^2k^2\tilde{p'} = -c^2\tilde{\mathcal{g}}\left(\mathbf{k},t\right) + \tilde{\mathcal{G}}\left(\mathbf{k},t\right) \tag{39}$$

where

$$
\widetilde{\mathbf{G}}\left(\mathbf{k},t\right) = -\int\_{S\_l} \left( c^2 \left[ \frac{\partial \mathbf{\bar{p'}}}{\partial n} \right] + v\_n \left[ \frac{\partial \mathbf{\bar{p'}}}{\partial t} \right] \right) e^{-i\mathbf{k}\cdot\mathbf{y}} dS \tag{40}
$$

$$
$$

Equation (39) is similar to equation (20). Consequently, its solution can be written as

$$\tilde{p'}\left(\mathbf{k},t\right) = \int\_{-\infty}^{t} \left(-c^2 \tilde{g}\left(\mathbf{k},\tau\right) + \tilde{G}\left(\mathbf{k},\tau\right)\right) \frac{\sin\left(c\,\mathbf{k}\,\left(t-\tau\right)\right)}{c\,\mathbf{k}} d\tau \tag{41}$$

Hence the contribution of the nonhomogeneous term in equation (38) can be written as

$$\begin{split} p'\_{\mathcal{S}}(\mathbf{x},t) &= -c^2 \int\_{-\infty}^t d\tau \int \widetilde{\mathbf{g}}(\mathbf{k},\tau) \frac{\sin\left(ck\left(t-\tau\right)\right)}{\left(2\pi\right)^3 c\lambda} e^{i\mathbf{k}\cdot\mathbf{x}} d\mathbf{k} \\ &= -\int\_{-\infty}^t d\tau \int \mathbf{g}\left(\mathbf{y},\tau\right) \frac{\delta\left(t-\tau-|\mathbf{x}-\mathbf{y}|/c\right)}{4\pi\left|\mathbf{x}-\mathbf{y}\right|} d\mathbf{y} \\ &= -\int \mathbf{g}\left(\mathbf{y},t-|\mathbf{x}-\mathbf{y}|/c\right) \frac{d\mathbf{y}}{4\pi\left|\mathbf{x}-\mathbf{y}\right|}\end{split} \tag{42}$$

Finally, the contribution of the terms corresponding to the boundary conditions on the mobile surface *S* can be written as

$$\begin{split} p'\_G(\mathbf{x}, t) &= -\frac{\partial}{\partial \mathbf{x}\_i} \int\_{S\_i} \left[ \frac{p' n\_i}{4 \pi r \left| 1 - M\_r \right|} \right]\_{ret} dS \\ &- \frac{\partial}{\partial t} \int\_{S\_l} \left[ \frac{p' M\_{ll}}{4 \pi r \left| 1 - M\_r \right|} \right]\_{ret} dS \\ &- \int\_{S\_l} \left[ \left( \frac{\partial}{\partial \mathbf{n}} \frac{p'}{\mathbf{n}} + M\_{ll} \frac{\partial}{\partial \tau} \right) \frac{1}{4 \pi r \left| 1 - M\_r \right|} \right]\_{ret} dS \end{split} \tag{43}$$

Fig. 2. Boundary Surface

with respect to space variables

formula can be written as

**differentiable function 9.1.1 The first basic formula**

> F *∂ϕ ∂x*<sup>1</sup> = *D*(*i*)

*∂ϕ*(*i*) *∂x*<sup>1</sup>

*e*−*i***k**·**x***d***x** =

 *D*(*i*)

 *D*(*i*)

We write

But

Generally, the function *ϕ* (**x**, *t*) is discontinuous across the surface *S*. We call it a piecewise differentiable function (pdf). Assuming that the function *ϕ*(*e*) (**x**, *t*) is decreasing sufficiently rapidly at infinity (for more precise conditions about the function *ϕ*(*e*) (**x**, *t*) see (Homentcovschi and Singler, 1999)) we can take the Fourier Transform of the function *ϕ* (**x**, *t*)

An Operational Approach to the Acoustic Analogy Equations 61

Here **x** = (*x*1,*x*2,*x*3), **k** = (*k*1,*k*2,*k*3), **k** · **x** = *k*1*x*<sup>1</sup> + *k*2*x*<sup>2</sup> + *k*3*x*<sup>3</sup> is the inner product of the two vectors, *d***x** = *dx*1*dx*2*dx*<sup>3</sup> and the integral is extended over the whole **R**<sup>3</sup> space. The inversion

*<sup>D</sup>*(*i*) *<sup>ϕ</sup>*(*i*) (**x**, *<sup>t</sup>*) *<sup>e</sup>*−*i***k**·**x***d***<sup>x</sup>** <sup>+</sup>

**9.1 Fourier transform of the derivative with respect to a spatial variable of a piecewise**

*e*−*i***k**·**x***d***x** +

(2*π*) 3 

 *D*(*e*)

*d***x** + *ik*<sup>1</sup>

*∂ϕ*(*e*) (**x**, *t*) *∂x*<sup>1</sup>

*<sup>D</sup>*(*i*) *<sup>ϕ</sup>*(*i*) (**x**, *<sup>t</sup>*)*<sup>e</sup>*

*ϕ* (**x**, *t*)*e*−*i***k**·**x***d***x** (45)

*<sup>ϕ</sup>* (**k**, *<sup>t</sup>*)*ei***x**·**k***d***<sup>k</sup>** (46)

*<sup>D</sup>*(*e*) *<sup>ϕ</sup>*(*e*) (**x**, *<sup>t</sup>*)*e*−*i***k**·**x***d***<sup>x</sup>** (47)

*e*−*i***k**·**x***d***x** (48)

<sup>−</sup>*i***k**·**x***d***x** (49)

*<sup>ϕ</sup>* (**k**,*t*) ≡ F {*<sup>ϕ</sup>* (**x**, *<sup>t</sup>*)} <sup>=</sup>

*<sup>ϕ</sup>* (**x**, *<sup>t</sup>*) ≡ F <sup>−</sup><sup>1</sup> {*<sup>ϕ</sup>* (**k**,*t*)} <sup>=</sup> <sup>1</sup>

*∂ϕ*(*i*) (**x**, *t*) *∂x*<sup>1</sup>

> *∂ ∂x*<sup>1</sup> *ϕ*(*i*) *e* −*i***k**·**x**

where *d***k** =*dk*1*dk*2*dk*3. Accounting for relationship (44) we can write

F {*ϕ* (**x**, *t*)} =

which coincides with the relationship (5.3) given in (Ffowcs Williams and Hawkings, 1969).

### **7. Concluding remarks**

Acoustic Analogy is one of the greatest contributions to the field of acoustics of the previous century. It is a major extension of acoustics made by Sir M. J. Lighthill (and other contributors) who formulated for the first time the science of how sound is created by fluid motion. This theory completes the previous work by famous researchers in the field of acoustics who had discovered how sound propagates through various media and across surrounding surfaces. In this chapter we have attempted to simplify the application of the Acoustic Analogy by showing how to apply it using only classical mathematical analysis tools.

### **8. Acknowledgment**

This work has been supported by the National Institute on Deafness and Other Communication Disorders grant R01 DC009429 to RNM. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institute on Deafness and Other Communication Disorders or the National Institutes of Health.

### **9. Appendix A: Fourier Transform of piecewise differentiable functions**

Let *D*(*i*) (*t*) be a bounded mobile domain in **R**<sup>3</sup> having a smooth boundary surface *St*. Denote by *<sup>D</sup>*(*e*) (*t*) the domain external to the surface *St* (See Fig.2): *<sup>D</sup>*(*i*) <sup>∪</sup> *<sup>S</sup>* <sup>∪</sup> *<sup>D</sup>*(*e*) <sup>=</sup> **<sup>R</sup>**3, *<sup>D</sup>*(*i*) <sup>∩</sup> *<sup>D</sup>*(*e*) <sup>=</sup> *φ*. Let also *ϕ*(*i*) (**x**, *t*) be a continuous differentiable function defined in the closed domain *<sup>D</sup>*(*i*) <sup>×</sup> **<sup>R</sup>** and *<sup>ϕ</sup>*(*e*) (**x**, *<sup>t</sup>*) a continuous differentiable function defined in *<sup>D</sup>*(*e*) <sup>×</sup> **<sup>R</sup>**. We define also

$$\varphi\left(\mathbf{x},t\right) = \begin{cases} \varphi^{(i)}\left(\mathbf{x},t\right) \text{, in } \mathcal{D}^{(i)}\\ \varphi^{(\varepsilon)}\left(\mathbf{x},t\right) \text{, in } \mathcal{D}^{(\varepsilon)} \end{cases} \tag{44}$$

Fig. 2. Boundary Surface

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

*<sup>g</sup>* (**y**,*<sup>t</sup>* <sup>−</sup> <sup>|</sup>**<sup>x</sup>** <sup>−</sup> **<sup>y</sup>**<sup>|</sup> /*c*) *<sup>d</sup>***<sup>y</sup>**

Finally, the contribution of the terms corresponding to the boundary conditions on the mobile

*p*�

 *ret*

*∂ p*� *∂ τ*

which coincides with the relationship (5.3) given in (Ffowcs Williams and Hawkings, 1969).

Acoustic Analogy is one of the greatest contributions to the field of acoustics of the previous century. It is a major extension of acoustics made by Sir M. J. Lighthill (and other contributors) who formulated for the first time the science of how sound is created by fluid motion. This theory completes the previous work by famous researchers in the field of acoustics who had discovered how sound propagates through various media and across surrounding surfaces. In this chapter we have attempted to simplify the application of the Acoustic Analogy by

This work has been supported by the National Institute on Deafness and Other Communication Disorders grant R01 DC009429 to RNM. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institute on Deafness and Other Communication Disorders or the National Institutes

Let *D*(*i*) (*t*) be a bounded mobile domain in **R**<sup>3</sup> having a smooth boundary surface *St*. Denote by *<sup>D</sup>*(*e*) (*t*) the domain external to the surface *St* (See Fig.2): *<sup>D</sup>*(*i*) <sup>∪</sup> *<sup>S</sup>* <sup>∪</sup> *<sup>D</sup>*(*e*) <sup>=</sup> **<sup>R</sup>**3, *<sup>D</sup>*(*i*) <sup>∩</sup> *<sup>D</sup>*(*e*) <sup>=</sup> *φ*. Let also *ϕ*(*i*) (**x**, *t*) be a continuous differentiable function defined in the closed domain *<sup>D</sup>*(*i*) <sup>×</sup> **<sup>R</sup>** and *<sup>ϕ</sup>*(*e*) (**x**, *<sup>t</sup>*) a continuous differentiable function defined in *<sup>D</sup>*(*e*) <sup>×</sup> **<sup>R</sup>**. We define

*ϕ*(*i*) (**x**, *t*), in *D*(*i*)

*<sup>ϕ</sup>*(*e*) (**x**, *<sup>t</sup>*), in *<sup>D</sup>*(*e*) (44)

*ni* 4*πr* |1 − *Mr*|

1

4*πr* |1 − *Mr*|

 *ret dS*

> *ret dS*

*<sup>g</sup>*(**k**, *<sup>τ</sup>*) sin (*ck* (*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*)) (2*π*) <sup>3</sup> *c k*

*<sup>g</sup>* (**y**, *<sup>τ</sup>*) *<sup>δ</sup>* (*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>* <sup>−</sup> <sup>|</sup>**<sup>x</sup>** <sup>−</sup> **<sup>y</sup>**<sup>|</sup> /*c*)

*ei***k**·**x***d***k**

<sup>4</sup>*<sup>π</sup>* <sup>|</sup>**<sup>x</sup>** <sup>−</sup> **<sup>y</sup>**<sup>|</sup> (42)

*dS* (43)

<sup>4</sup>*<sup>π</sup>* <sup>|</sup>**<sup>x</sup>** <sup>−</sup> **<sup>y</sup>**<sup>|</sup> *<sup>d</sup>***<sup>y</sup>**

Hence the contribution of the nonhomogeneous term in equation (38) can be written as

 *t* −∞ *dτ* 

*p*�

surface *S* can be written as

**7. Concluding remarks**

**8. Acknowledgment**

of Health.

also

*<sup>g</sup>* (**x**, *<sup>t</sup>*) <sup>=</sup> <sup>−</sup>*c*<sup>2</sup>

= − *t* −∞ *dτ* 

= − 

*<sup>G</sup>* (**x**, *<sup>t</sup>*) <sup>=</sup> <sup>−</sup> *<sup>∂</sup>*

 *∂ p*� *∂ n*

*p*�

showing how to apply it using only classical mathematical analysis tools.

**9. Appendix A: Fourier Transform of piecewise differentiable functions**

*ϕ* (**x**, *t*) =

*∂ xi St*

*Mn* 4*πr* |1 − *Mr*|

+ *Mn*

*p*�

− *∂ ∂t St*

− *St*

Generally, the function *ϕ* (**x**, *t*) is discontinuous across the surface *S*. We call it a piecewise differentiable function (pdf). Assuming that the function *ϕ*(*e*) (**x**, *t*) is decreasing sufficiently rapidly at infinity (for more precise conditions about the function *ϕ*(*e*) (**x**, *t*) see (Homentcovschi and Singler, 1999)) we can take the Fourier Transform of the function *ϕ* (**x**, *t*) with respect to space variables

$$\tilde{\boldsymbol{\varphi}}\left(\mathbf{k},t\right) \equiv \mathcal{F}\left\{\boldsymbol{\varphi}\left(\mathbf{x},t\right)\right\} = \int \boldsymbol{\varphi}\left(\mathbf{x},t\right) e^{-i\mathbf{k}\cdot\mathbf{x}} d\mathbf{x} \tag{45}$$

Here **x** = (*x*1,*x*2,*x*3), **k** = (*k*1,*k*2,*k*3), **k** · **x** = *k*1*x*<sup>1</sup> + *k*2*x*<sup>2</sup> + *k*3*x*<sup>3</sup> is the inner product of the two vectors, *d***x** = *dx*1*dx*2*dx*<sup>3</sup> and the integral is extended over the whole **R**<sup>3</sup> space. The inversion formula can be written as

$$\log \left( \mathbf{x}, t \right) \equiv \mathcal{F}^{-1} \left\{ \tilde{\boldsymbol{\varphi}} \left( \mathbf{k}, t \right) \right\} = \frac{1}{\left( 2\pi \right)^{3}} \int \tilde{\boldsymbol{\varphi}} \left( \mathbf{k}, t \right) e^{i\mathbf{x} \cdot \mathbf{k}} d\mathbf{k} \tag{46}$$

where *d***k** =*dk*1*dk*2*dk*3. Accounting for relationship (44) we can write

$$\mathcal{F}\left\{\boldsymbol{\varrho}\left(\mathbf{x},t\right)\right\} = \int\_{D^{(i)}} \boldsymbol{\varrho}^{(i)}\left(\mathbf{x},t\right) \boldsymbol{\varepsilon}^{-i\mathbf{k}\cdot\mathbf{x}} d\mathbf{x} + \int\_{D^{(\epsilon)}} \boldsymbol{\varrho}^{(\epsilon)}\left(\mathbf{x},t\right) \boldsymbol{\varepsilon}^{-i\mathbf{k}\cdot\mathbf{x}} d\mathbf{x} \tag{47}$$

### **9.1 Fourier transform of the derivative with respect to a spatial variable of a piecewise differentiable function**

**9.1.1 The first basic formula**

$$\text{We write}$$

$$\mathcal{F}\left\{\frac{\partial\boldsymbol{\varrho}}{\partial\mathbf{x}\_{1}}\right\} = \int\_{D^{(\boldsymbol{\varrho})}} \frac{\partial\boldsymbol{\varrho}^{(\boldsymbol{i})}\left(\mathbf{x}\_{\boldsymbol{\varrho}}\boldsymbol{t}\right)}{\partial\mathbf{x}\_{1}} e^{-i\mathbf{k}\cdot\mathbf{x}} d\mathbf{x} + \int\_{D^{(\boldsymbol{\varrho})}} \frac{\partial\boldsymbol{\varrho}^{(\boldsymbol{\varrho})}\left(\mathbf{x}\_{\boldsymbol{\varrho}}\boldsymbol{t}\right)}{\partial\mathbf{x}\_{1}} e^{-i\mathbf{k}\cdot\mathbf{x}} d\mathbf{x} \tag{48}$$

But

$$\int\_{D^{(i)}} \frac{\partial \boldsymbol{\varrho}^{(i)}}{\partial \boldsymbol{\chi}\_{1}} e^{-i\mathbf{k}\cdot\mathbf{x}} d\mathbf{x} = \int\_{D^{(i)}} \frac{\partial}{\partial \boldsymbol{\chi}\_{1}} \left(\boldsymbol{\varrho}^{(i)} e^{-i\mathbf{k}\cdot\mathbf{x}}\right) d\mathbf{x} + ik\_{1} \int\_{D^{(i)}} \boldsymbol{\varrho}^{(i)} \left(\mathbf{x}, t\right) e^{-i\mathbf{k}\cdot\mathbf{x}} d\mathbf{x} \tag{49}$$

Moreover, we can write

**Δ**

where *k*<sup>2</sup> = *k*<sup>2</sup>

*St*<sup>0</sup> we can write

relation (60) yields

*D*(*i*)(*t*)

*D*(*<sup>i</sup>*)(*t*)

*D*(*i*)(*t*)

*D*(*i*)(*t*)

Finally,

= *i***k**· 

<sup>1</sup> <sup>+</sup> *<sup>k</sup>*<sup>2</sup>

the points **x** (*t*0) and **x** (*t*). But,

**9.2.2 Reynolds' transport theorem**

*<sup>ϕ</sup>*(*i*) (**x**, *<sup>t</sup>*)*e*−*i***k**·**x***d***x**<sup>−</sup>

*<sup>ϕ</sup>*(*i*) (**x**, *<sup>t</sup>*0) *<sup>e</sup>*−*i***k**·**x***d***x**<sup>−</sup>

*<sup>ϕ</sup>*(*i*) (**x**, *<sup>t</sup>*) <sup>−</sup> *<sup>ϕ</sup>*(*i*) (**x**, *<sup>t</sup>*0)

<sup>−</sup>*i***k**·**x***d***x**<sup>−</sup>

*ϕ*(*i*) (**x**, *t*) *e*

*<sup>ϕ</sup>* <sup>=</sup> <sup>−</sup>*k*2*<sup>ϕ</sup>* (**k**, *<sup>t</sup>*) <sup>−</sup>

<sup>2</sup> <sup>+</sup> *<sup>k</sup>*<sup>2</sup>

F {**Δ** *ϕ*} ≡ F {∇·∇ *ϕ*} = *i***k**·∇

 *St*

 *St*

*<sup>i</sup>***<sup>k</sup>** *<sup>ϕ</sup>* (**k**,*t*) <sup>−</sup>

<sup>3</sup> = |**k**| 2 .

**9.2.1 The displacement velocity of a mobile surface**

 *∂S ∂y*<sup>1</sup>

*dy*<sup>1</sup> *dt* <sup>+</sup>

*<sup>n</sup>*<sup>1</sup> <sup>=</sup> *<sup>∂</sup>S*/*∂y*<sup>1</sup> |*grad S*|

> *dy*<sup>1</sup> *dt* <sup>=</sup> *<sup>v</sup>*1,

are the components of the external normal unit vector **n** and

*ϕ* (**k**, *t*) −

An Operational Approach to the Acoustic Analogy Equations 63

**<sup>n</sup>** [*<sup>ϕ</sup>* (**y**, *<sup>t</sup>*)]*e*−*i***k**·**y***dS*

*<sup>i</sup>***<sup>k</sup>** · **<sup>n</sup>** [*<sup>ϕ</sup>* (**y**, *<sup>t</sup>*)]*e*−*i***k**·**y***dS* <sup>−</sup>

Let *S*(*y*1, *y*2, *y*3, *t*) = 0 be the equation of the mobile surface *St*. Then, for *S*(*y*01, *y*02, *y*03, *t*0) on

the partial derivatives of the function *S* being calculated at a certain point **x**� lying between

, *<sup>n</sup>*<sup>2</sup> <sup>=</sup> *<sup>∂</sup>S*/*∂y*<sup>2</sup> |*grad S*|

are the projections on the velocity vector of a point on the surface *St* on the three axes. The

*vn* <sup>=</sup> <sup>−</sup> *<sup>∂</sup>S*/*∂<sup>t</sup>*

which is the displacement velocity of the geometric surface *St*. We mention that the displacement velocity of a surface has the direction of the normal vector to this surface.

For calculating the Fourier Transform of a time derivative of a certain function we write

*D*(*i*)(*t*0)

*D*(*<sup>i</sup>*)(*t*)

*D*(*i*)(*t*<sup>0</sup> )

*e*−*i***k**·**x***d***x**+

*dy*<sup>2</sup> *dt* <sup>=</sup> *<sup>v</sup>*2,

*∂S ∂y*<sup>3</sup> *dy*<sup>3</sup> *dt* <sup>+</sup>

*dy*<sup>3</sup>

*ϕ*(*i*) (**x**, *t*0) *e*−*i***k**·**x***d***x**+

*ϕ*(*i*) (**x**, *t*0) *e*−*i***k**·**x***d***x** =

*<sup>D</sup>*(*i*)(*t*)−*D*(*i*)(*t*<sup>0</sup> )

**9.2 Fourier transform of the time derivative of a piecewise differentiable function**

*∂S ∂y*<sup>2</sup> *dy*<sup>2</sup> *dt* <sup>+</sup>  *St*

> − *St*

> > *St*

0 = *S*(*y*1, *y*2, *y*3, *t*) − *S*(*y*01, *y*02, *y*03, *t*0) = (60)

*∂S ∂t* 

, *<sup>n</sup>*<sup>3</sup> <sup>=</sup> *<sup>∂</sup>S*/*∂y*<sup>3</sup>

(*t* − *t*0),

**n**· [∇ (**y**,*t*) *ϕ*]*e*

 *e* <sup>−</sup>*i***k**·**y***dS*

 *e*

<sup>|</sup>*grad S*<sup>|</sup> (61)

*dt* <sup>=</sup> *<sup>v</sup>*3, (62)

<sup>|</sup>*grad S*<sup>|</sup> (63)

*ϕ*(*i*) (**x**, *t*0)*e*−*i***k**·**x***d***x** = (64)

*ϕ*(*i*) (**x**, *t*0)*e*−*i***k**·**x***d***x**

 *∂ϕ <sup>∂</sup><sup>n</sup>* (**y**,*t*)

 *∂ϕ <sup>∂</sup><sup>n</sup>* (**y**, *<sup>t</sup>*)

<sup>−</sup>*i***k**·**y***dS* (58)

<sup>−</sup>*i***k**·**y***dS* (59)

The first integral in the right-hand side of relationship (49) can be replaced, by using the divergence theorem by an integral over the boundary surface *St*.

$$\int\_{D^{(i)}} \frac{\partial}{\partial \mathbf{x}\_1} \left( \boldsymbol{\varrho}^{(i)} \boldsymbol{e}^{-i\mathbf{k} \cdot \mathbf{x}} \right) d\mathbf{x} = \int\_{S\_l} n\_1 \boldsymbol{\varrho}^{(i)} \left( \mathbf{y}, t \right) \boldsymbol{e}^{-i\mathbf{k} \cdot \mathbf{y}} d\mathbf{S} \tag{50}$$

where **n** = (*n*1, *n*2, *n*3) is the external unit normal to *St*. Therefore,

$$\int\_{D^{(i)}} \frac{\partial \boldsymbol{\varrho}^{(i)}}{\partial \mathbf{x}\_{1}} e^{-i\mathbf{k}\cdot\mathbf{x}} d\mathbf{x} = i\mathbf{k}\_{1} \int\_{D^{(i)}} \boldsymbol{\varrho}^{(i)}\left(\mathbf{x},t\right) e^{-i\mathbf{k}\cdot\mathbf{x}} d\mathbf{x} + \int\_{S\_{l}} n\_{1} \boldsymbol{\varrho}^{(i)}\left(\mathbf{y},t\right) e^{-i\mathbf{k}\cdot\mathbf{y}} d\mathbf{S} \tag{51}$$

Similarly,

$$\int\_{D^{(\varepsilon)}} \frac{\partial \boldsymbol{\varrho}^{(\varepsilon)}}{\partial \boldsymbol{\chi}\_{1}} e^{-i\mathbf{k}\cdot\mathbf{x}} d\mathbf{x} = ik\_{1} \int\_{D^{(\varepsilon)}} \boldsymbol{\varrho}^{(\varepsilon)}\left(\mathbf{x}, t\right) e^{-i\mathbf{k}\cdot\mathbf{x}} d\mathbf{x} - \int\_{S\_{l}} n\_{1} \boldsymbol{\varrho}^{(\varepsilon)}\left(\mathbf{y}, t\right) e^{-i\mathbf{k}\cdot\mathbf{y}} d\mathbf{S} \tag{52}$$

Finally, the relationships (49), (51), and (52) give *the first basic formula*:

$$\mathcal{F}\left\{\frac{\partial\boldsymbol{\varrho}}{\partial\mathbf{x}\_{1}}\right\} = ik\_{1}\,\widetilde{\boldsymbol{\varrho}}\left(\mathbf{k},t\right) - \int\_{S} n\_{1}\left[\boldsymbol{\varrho}\left(\mathbf{y},t\right)\right] e^{-i\mathbf{k}\cdot\mathbf{y}}d\mathbf{S} \tag{53}$$

Here we denoted by square bracket the jump of the function *ϕ* (**x**, *t*) across the surface *St*

$$\mathbb{E}\left[\boldsymbol{\varrho}\left(\mathbf{y},t\right)\right] = \lim\_{\mathbf{x}^{(\varepsilon)} \to \mathbf{y}} \boldsymbol{\varrho}^{(\varepsilon)}\left(\mathbf{x}^{(\varepsilon)},t\right) - \lim\_{\mathbf{x}^{(i)} \to \mathbf{y}} \boldsymbol{\varrho}^{(i)}\left(\mathbf{x}^{(i)},t\right) \tag{54}$$

for **<sup>y</sup>** <sup>∈</sup> *St*, **<sup>x</sup>**(*i*) <sup>∈</sup> *<sup>D</sup>*(*i*) and **<sup>x</sup>**(*e*) <sup>∈</sup> *<sup>D</sup>*(*e*). Similar relationships to (53) can be proved for the derivatives in the directions *x*<sup>2</sup> and *x*3.

**Remark 3.** *It is clear that the obtained relationships can be extended immediately to the case where there are more discontinuity surfaces of the given function. The resulting formulas will contain sums of integrals corresponding to each discontinuity surface.*

### **9.1.2 Other formulas**

The relationship (53) gives also

$$\mathcal{F}\left\{\nabla\boldsymbol{\varphi}\right\} \equiv \widetilde{\nabla\boldsymbol{\varphi}} = i\mathbf{k}\,\widetilde{\boldsymbol{\varphi}}\left(\mathbf{k},t\right) - \int\_{S\_l} \mathbf{n}\left[\boldsymbol{\varphi}\left(\mathbf{y},t\right)\right] e^{-i\mathbf{k}\cdot\mathbf{y}}d\mathbf{S} \tag{55}$$

In the case where we write **V** (**x**, *t*) = (*V*<sup>1</sup> (**x**, *t*), *V*<sup>2</sup> (**x**, *t*), *V*<sup>3</sup> (**x**, *t*)) where *Vj* (**x**, *t*) is a piecewise differentiable function defined by a relationship similar to (44) we can write also the formulas

$$\mathcal{F}\left\{\nabla \cdot \mathbf{V}\right\} \equiv \widehat{\nabla \cdot \mathbf{V}} = i\mathbf{k} \cdot \tilde{\mathbf{V}}\left(\mathbf{k}, t\right) - \int\_{\mathcal{S}\_l} \mathbf{n} \cdot \left[\mathbf{V}\left(\mathbf{y}, t\right)\right] e^{-i\mathbf{k} \cdot \mathbf{y}} d\mathbf{S} \tag{56}$$

$$\mathcal{F}\left\{\nabla \times \mathbf{V}\right\} \equiv \widehat{\nabla \times \mathbf{V}} = i\mathbf{k} \times \tilde{\mathbf{V}}\left(\mathbf{k}, t\right) - \int\_{S\_l} \mathbf{n} \times \left[\mathbf{V}\left(\mathbf{y}, t\right)\right] e^{-i\mathbf{k} \cdot \mathbf{y}} dS \tag{57}$$

The formulas (55), (56) and (57) permit the calculation of the Fourier Transforms of a gradient of a scalar field of a divergence and a curl of a vector field in the case of piecewise differentiable scalar and vector fields.

*St*

Moreover, we can write

$$\mathcal{F}\left\{\mathbf{A}\,\boldsymbol{\varrho}\right\} \equiv \mathcal{F}\left\{\nabla \cdot \nabla \,\boldsymbol{\varrho}\right\} = i\mathbf{k} \cdot \widehat{\nabla \,\boldsymbol{\varrho}}\left(\mathbf{k},t\right) - \int\_{S\_{l}} \mathbf{n} \cdot \left[\nabla\left\left(\mathbf{y},t\right)\,\boldsymbol{\varrho}\right] e^{-i\mathbf{k}\cdot\mathbf{y}} dS \tag{58}$$

$$\mathbf{j} = i\mathbf{k} \cdot \left(i\mathbf{k}\,\boldsymbol{\varrho}\left(\mathbf{k},t\right) - \int\_{S\_{l}} \mathbf{n}\left[\boldsymbol{\varrho}\left(\mathbf{y},t\right)\right] e^{-i\mathbf{k}\cdot\mathbf{y}} dS\right) - \int\_{S\_{l}} \left[\frac{\partial \boldsymbol{\varrho}}{\partial n}\left(\mathbf{y},t\right)\right] e^{-i\mathbf{k}\cdot\mathbf{y}} dS$$

−

*St*

Finally,

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

The first integral in the right-hand side of relationship (49) can be replaced, by using the

*n*1*ϕ*(*i*) (**y**, *t*)*e*−*i***k**·**y***dS* (50)

*n*1*ϕ*(*i*) (**y**, *t*)*e*−*i***k**·**y***dS* (51)

*n*1*ϕ*(*e*) (**y**,*t*)*e*−*i***k**·**y***dS* (52)

<sup>−</sup>*i***k**·**y***dS* (53)

**n** [*ϕ* (**y**,*t*)]*e*−*i***k**·**y***dS* (55)

**<sup>n</sup>**<sup>×</sup> [**<sup>V</sup>** (**y**,*t*)]*e*−*i***k**·**y***dS* (57)

<sup>−</sup>*i***k**·**y***dS* (56)

(54)

divergence theorem by an integral over the boundary surface *St*.

where **n** = (*n*1, *n*2, *n*3) is the external unit normal to *St*. Therefore,

Finally, the relationships (49), (51), and (52) give *the first basic formula*:

**<sup>x</sup>**(*e*)→**<sup>y</sup>**

<sup>=</sup> *ik*<sup>1</sup> *<sup>ϕ</sup>* (**k**, *<sup>t</sup>*) <sup>−</sup>

*ϕ*(*e*) **x**(*e*) , *t* 

*<sup>ϕ</sup>* <sup>=</sup> *<sup>i</sup>***<sup>k</sup>** *<sup>ϕ</sup>* (**k**,*t*) <sup>−</sup>

**<sup>V</sup>** <sup>=</sup> *<sup>i</sup>***k**·**V** (**k**,*t*) <sup>−</sup>

<sup>F</sup> {∇ × **<sup>V</sup>**} <sup>≡</sup> ∇ ×**<sup>V</sup>** <sup>=</sup> *<sup>i</sup>***k**×**V** (**k**,*t*) <sup>−</sup>

Here we denoted by square bracket the jump of the function *ϕ* (**x**, *t*) across the surface *St*

for **<sup>y</sup>** <sup>∈</sup> *St*, **<sup>x</sup>**(*i*) <sup>∈</sup> *<sup>D</sup>*(*i*) and **<sup>x</sup>**(*e*) <sup>∈</sup> *<sup>D</sup>*(*e*). Similar relationships to (53) can be proved for the

**Remark 3.** *It is clear that the obtained relationships can be extended immediately to the case where there are more discontinuity surfaces of the given function. The resulting formulas will contain sums*

In the case where we write **V** (**x**, *t*) = (*V*<sup>1</sup> (**x**, *t*), *V*<sup>2</sup> (**x**, *t*), *V*<sup>3</sup> (**x**, *t*)) where *Vj* (**x**, *t*) is a piecewise differentiable function defined by a relationship similar to (44) we can write also the formulas

The formulas (55), (56) and (57) permit the calculation of the Fourier Transforms of a gradient of a scalar field of a divergence and a curl of a vector field in the case of piecewise differentiable

 *St*

> *St*

> > *St*

**n**· [**V** (**y**, *t*)]*e*

[*ϕ* (**y**,*t*)] = lim

*of integrals corresponding to each discontinuity surface.*

F {∇*ϕ*} ≡ ∇

F {∇ · **V**} ≡ ∇ ·

*<sup>D</sup>*(*i*) *<sup>ϕ</sup>*(*i*) (**x**, *<sup>t</sup>*)*<sup>e</sup>*

*<sup>D</sup>*(*e*) *<sup>ϕ</sup>*(*e*) (**x**, *<sup>t</sup>*)*e*−*i***k**·**x***d***<sup>x</sup>** <sup>−</sup>

 *S*

<sup>−</sup>*i***k**·**x***d***x** +

 *St*

 *St*

*ϕ*(*i*) **x**(*i*) , *t* 

*n*<sup>1</sup> [*ϕ* (**y**, *t*)]*e*

− lim **<sup>x</sup>**(*i*)→**<sup>y</sup>**

*∂ ∂x*<sup>1</sup> *ϕ*(*i*) *e* −*i***k**·**x** *d***x** = *St*

<sup>−</sup>*i***k**·**x***d***x** = *ik*<sup>1</sup>

*e*−*i***k**·**x***d***x** = *ik*<sup>1</sup>

F *∂ϕ ∂x*<sup>1</sup>

derivatives in the directions *x*<sup>2</sup> and *x*3.

 *D*(*i*)

 *D*(*i*)

 *D*(*e*)

**9.1.2 Other formulas**

scalar and vector fields.

The relationship (53) gives also

Similarly,

*∂ϕ*(*i*) *∂x*<sup>1</sup> *e*

*∂ϕ*(*e*) *∂x*<sup>1</sup>

$$\begin{split} \widetilde{\Delta \boldsymbol{\tilde{\varrho}} } &= -k^2 \widetilde{\boldsymbol{\varphi}}(\mathbf{k}, t) - \int\_{S\_l} i \mathbf{k} \cdot \mathbf{n} \left[ \boldsymbol{\varphi} \left( \mathbf{y}, t \right) \right] e^{-i \mathbf{k} \cdot \mathbf{y}} d\mathbf{S} - \int\_{S\_l} \left[ \frac{\partial \boldsymbol{\varrho}}{\partial n} \left( \mathbf{y}, t \right) \right] e^{-i \mathbf{k} \cdot \mathbf{y}} d\mathbf{S} \\\ \mathbf{z}^2 &= k\_1^2 + k\_2^2 + k\_3^2 = |\mathbf{k}|^2. \end{split} \tag{59}$$

where *k*<sup>2</sup> = *k*<sup>2</sup> <sup>1</sup> <sup>+</sup> *<sup>k</sup>*<sup>2</sup> <sup>2</sup> <sup>+</sup> *<sup>k</sup>*<sup>2</sup> <sup>3</sup> = |**k**|

### **9.2 Fourier transform of the time derivative of a piecewise differentiable function 9.2.1 The displacement velocity of a mobile surface**

Let *S*(*y*1, *y*2, *y*3, *t*) = 0 be the equation of the mobile surface *St*. Then, for *S*(*y*01, *y*02, *y*03, *t*0) on *St*<sup>0</sup> we can write

$$\begin{aligned} 0 &= S(y\_1, y\_2, y\_3, t) - S(y\_{01}, y\_{02}, y\_{03}, t\_0) = \\ \left(\frac{\partial S}{\partial y\_1}\frac{dy\_1}{dt} + \frac{\partial S}{\partial y\_2}\frac{dy\_2}{dt} + \frac{\partial S}{\partial y\_3}\frac{dy\_3}{dt} + \frac{\partial S}{\partial t}\right)(t - t\_0) \end{aligned} \tag{60}$$

the partial derivatives of the function *S* being calculated at a certain point **x**� lying between the points **x** (*t*0) and **x** (*t*). But,

$$n\_1 = \frac{\partial S / \partial y\_1}{|\operatorname{grad} S|}, n\_2 = \frac{\partial S / \partial y\_2}{|\operatorname{grad} S|}, n\_3 = \frac{\partial S / \partial y\_3}{|\operatorname{grad} S|}\tag{61}$$

are the components of the external normal unit vector **n** and

$$\frac{dy\_1}{dt} = v\_{1\prime} \frac{dy\_2}{dt} = v\_{2\prime} \frac{dy\_3}{dt} = v\_{3\prime} \tag{62}$$

are the projections on the velocity vector of a point on the surface *St* on the three axes. The relation (60) yields

$$v\_n = -\frac{\left|\partial S/\partial t\right|}{\left|grad \, S\right|}\tag{63}$$

which is the displacement velocity of the geometric surface *St*. We mention that the displacement velocity of a surface has the direction of the normal vector to this surface.

### **9.2.2 Reynolds' transport theorem**

For calculating the Fourier Transform of a time derivative of a certain function we write

$$\begin{split} &\int\_{D^{(\boldsymbol{0})}(t)} \boldsymbol{\varrho}^{(\boldsymbol{i})}\left(\mathbf{x},t\right) \boldsymbol{e}^{-i\mathbf{k}\cdot\mathbf{x}} d\mathbf{x} - \int\_{D^{(\boldsymbol{0})}(t\_{0})} \boldsymbol{\varrho}^{(\boldsymbol{i})}\left(\mathbf{x},t\_{0}\right) \boldsymbol{e}^{-i\mathbf{k}\cdot\mathbf{x}} d\mathbf{x} = \\ &\int\_{D^{(\boldsymbol{0})}(t)} \boldsymbol{\varrho}^{(\boldsymbol{i})}\left(\mathbf{x},t\right) \boldsymbol{e}^{-i\mathbf{k}\cdot\mathbf{x}} d\mathbf{x} - \int\_{D^{(\boldsymbol{0})}(t)} \boldsymbol{\varrho}^{(\boldsymbol{i})}\left(\mathbf{x},t\_{0}\right) \boldsymbol{e}^{-i\mathbf{k}\cdot\mathbf{x}} d\mathbf{x} + \\ &\int\_{D^{(\boldsymbol{0})}(t)} \boldsymbol{\varrho}^{(\boldsymbol{i})}\left(\mathbf{x},t\_{0}\right) \boldsymbol{e}^{-i\mathbf{k}\cdot\mathbf{x}} d\mathbf{x} - \int\_{D^{(\boldsymbol{0})}(t\_{0})} \boldsymbol{g}^{(\boldsymbol{i})}\left(\mathbf{x},t\_{0}\right) \boldsymbol{e}^{-i\mathbf{k}\cdot\mathbf{x}} d\mathbf{x} = \\ &\int\_{D^{(\boldsymbol{0})}(t)} \left\{\boldsymbol{\varrho}^{(\boldsymbol{i})}\left(\mathbf{x},t\right) - \boldsymbol{g}^{(\boldsymbol{i})}\left(\mathbf{x},t\_{0}\right)\right\} \boldsymbol{e}^{-i\mathbf{k}\cdot\mathbf{x}} d\mathbf{x} + \int\_{D^{(\boldsymbol{0})}(t) - D^{(\boldsymbol{0})}(t\_{0})} \boldsymbol{g}^{(\boldsymbol{i})}\left(\mathbf{x},t\_{0}\right) \boldsymbol{e}^{-i\mathbf{k}\cdot\mathbf{x}} d\mathbf{x} \end{split}$$

**10. Appendix B: Determination of the Greens' function for the wave equation**

An Operational Approach to the Acoustic Analogy Equations 65

By taking the derivative with respect to parameter *t* in the both sides of relationship (71) there

<sup>F</sup> <sup>−</sup><sup>1</sup> {cos (*ckt*)} <sup>=</sup> *<sup>δ</sup>*� (*<sup>t</sup>* <sup>−</sup> *<sup>r</sup>*/*c*)

Brentner, K. S., An efficient and robust method for predicting helicopter high-speed impulsive

Brentner, K. S. & Farassat, F. 1998 Analytical comparison of the acoustic analogy and Kirchhoff

Crighton, D. G., Dowling, A. P., Ffowcs Williams, J. E., Heckl, M., and Leppington, F. G.,

Curle, N., The influence of solid boundaries upon aerodynamic sound, Proceedings of the

Farassat, F., Introduction to generalized functions with applications in aerodynamics and aeroacoustics, Corrected Copy (April 1996), NASA Technical Paper 3428, 1994, Farassat, F., The Kirchhoff formulas for moving surfaces in aeroacoustics—the subsonic and

Farassat, F., Comments on the paper by Zinoviev and Bies "On acoustic radiation by a rigid object in a fluid flow", Journal of Sound and Vibration 281 (2005) 1217–1223. Farassat, F., Derivation of Formulations 1 and 1A of Farassat, NASA Technical Memorandum

Farassat, F., and Brentner, K. S., The uses and abuses of the acoustic analogy in helicopter rotor noise prediction, Journal of the American Helicopter Society, 1988, 33, 29-36 Farassat, F., Myers, M.K., Further comments on the paper by Zinoviev and Bies, "On acoustic

Ffowcs Williams, J.E., and Hawkings, D.L., Sound generation by turbulence and surfaces

radiation by a rigid object in a fluid flow", Journal of Sound and Vibration 290 (2006),

in arbitrary motion, Philosophical Transactions of the Royal Society A 264 (1969)

Currie, I. G., Fundamental Mechanics of Fluids, 3rd edition, Marcel Dekker, 2003, pg.12. Dowling, A. P., and Ffowcs Williams, J.E., Sound and Sources of Sound, Wiley &Sons, New

supersonic cases, NASA Technical Memorandum 110285, 1996.

Modern Methods in Analytical Acoustics: Lecture Notes, Springer–Verlag, London,

<sup>=</sup> *<sup>δ</sup>* (*<sup>t</sup>* <sup>−</sup> *<sup>r</sup>*/*c*) 4*πc r*

4*πc*2*r*

<sup>=</sup> *<sup>H</sup>* (*<sup>t</sup>* <sup>−</sup> *<sup>r</sup>*/*c*) 4*πr*

(71)

(72)

(73)

By using formula 11, given at page 364 in (Gelfand and Shilov, 1964) we can write

sin (*ckt*) *k*

<sup>1</sup> <sup>−</sup> cos (*ckt*) *k*2

noise, Journal of Sound and Vibration, 1997, 203(1), 87-100.

formulation for moving surfaces. AIAA J. 36(8), 1998, 1379–1386.

<sup>F</sup> <sup>−</sup><sup>1</sup>

Similarly, integrating the formula (71) over the interval (0, *t*) we obtain

<sup>F</sup> <sup>−</sup><sup>1</sup>

*H* (*x*) being the Heaviside's function.

1992. Chap.11, Sec. 10.

York, 1982. Chap. 9, Sec. 2.

214853, 2007.

538-547.

321–342.

Royal Society A 231 (1955) 505–514.

where *r* = |**x**| .

**11. References**

results

Now, dividing by (*t* − *t*0) and taking the limit for *t* → *t*<sup>0</sup> the first term gives the Fourier transform of the time derivative of the function *ϕ*(*i*) (**x**, *t*) while in the second integral we can write *d***x** =*vn* (*t* − *t*0) *dS* (see (Jacob, 1959), (Currie, 2003)). Finally, we obtain the following form of the Reynolds' transport theorem

$$\begin{split} \frac{d}{dt} \int\_{D^{(i)}(t)} \boldsymbol{\varrho}^{(i)} \left( \mathbf{x}, t \right) \boldsymbol{e}^{-i\mathbf{k} \cdot \mathbf{x}} d\mathbf{x} &= \int\_{D^{(i)}(t)} \frac{\partial \boldsymbol{\varrho}^{(i)} \left( \mathbf{x}, t \right)}{\partial t} \boldsymbol{e}^{-i\mathbf{k} \cdot \mathbf{x}} d\mathbf{x} \\ &+ \int\_{S\_{l}} \boldsymbol{v}\_{\text{ll}} \left( \mathbf{y}, t \right) \boldsymbol{\varrho}^{(i)} \left( \mathbf{y}, t \right) \boldsymbol{e}^{-i\mathbf{k} \cdot \mathbf{y}} d\mathbf{S} \end{split} \tag{65}$$

*vn* being the displacement velocity of the surface *S* (*t*).

### **9.2.3 The second basic formula**

We calculate now

$$\mathcal{F}\left\{\frac{\partial\boldsymbol{\varrho}\left(\mathbf{x},t\right)}{\partial t}\right\} = \int\_{D^{(\boldsymbol{\varrho})}(t)} \frac{\partial\boldsymbol{\varrho}^{(\boldsymbol{\varrho})}\left(\mathbf{x},t\right)}{\partial t} e^{-i\mathbf{k}\cdot\mathbf{x}} d\mathbf{x} + \int\_{D^{(\boldsymbol{\varrho})}(t)} \frac{\partial\boldsymbol{\varrho}^{(\boldsymbol{\varrho})}\left(\mathbf{x},t\right)}{\partial t} e^{-i\mathbf{k}\cdot\mathbf{x}} d\mathbf{x}$$

By using formula (65) we get

$$\int\_{D^{(i)}(t)} \frac{\partial \boldsymbol{\varrho}^{(i)}\left(\mathbf{x}, t\right)}{\partial t} \boldsymbol{e}^{-i\mathbf{k}\cdot\mathbf{x}} d\mathbf{x} = \frac{d}{dt} \int\_{D^{(i)}(t)} \boldsymbol{\varrho}^{(i)} \boldsymbol{e}^{-i\mathbf{k}\cdot\mathbf{x}} d\mathbf{x} - \int\_{S\_l} \boldsymbol{v}\_{ll} \boldsymbol{\varrho}^{(i)}\left(\mathbf{y}, t\right) \boldsymbol{e}^{-i\mathbf{k}\cdot\mathbf{y}} d\mathbf{S} \tag{66}$$

$$\int\_{D^{(\varepsilon)}(t)} \frac{\partial \varphi^{(\varepsilon)}\left(\mathbf{x}, t\right)}{\partial t} e^{-i\mathbf{k}\cdot\mathbf{x}} d\mathbf{x} = \frac{d}{dt} \int\_{D^{(\varepsilon)}(t)} \varphi^{(\varepsilon)} e^{-i\mathbf{k}\cdot\mathbf{x}} d\mathbf{x} + \int\_{S\_l} v\_{ll} \varphi^{(\varepsilon)}\left(\mathbf{y}, t\right) e^{-i\mathbf{k}\cdot\mathbf{y}} d\mathbf{S} \tag{67}$$

The sum of formulas (66) and (67) gives the *second basic formula*

$$\mathcal{F}\left\{\frac{\partial\boldsymbol{\varrho}\left(\mathbf{x},t\right)}{\partial t}\right\} = \frac{d\,\boldsymbol{\tilde{\varrho}\left(\mathbf{k},t\right)}}{dt} + \int\_{S\_{l}} \boldsymbol{v}\_{\text{ll}}\left(\mathbf{y},t\right) \left[\boldsymbol{\varrho}\left(\mathbf{y},t\right)\right] e^{-i\mathbf{k}\cdot\mathbf{y}} d\mathbf{S}.\tag{68}$$

Formula (68) permits us to calculate the Fourier transform of the time derivative of a piecewise differentiable function. For the second time derivative we can write

$$\mathcal{F}\left\{\frac{\partial^2 \varrho\left(\mathbf{x},t\right)}{\partial t^2}\right\} = \frac{d}{dt}\frac{\widetilde{d\boldsymbol{\varphi}}}{dt} + \int\_{S\_l} v\_{ll}\left(\mathbf{y},t\right) \left[\frac{\partial \varrho\left(\mathbf{y},t\right)}{\partial t}\right] e^{-i\mathbf{k}\cdot\mathbf{y}} dS \tag{69}$$

By using again formula (68) we obtain finally,

$$\frac{\partial^2 \varrho}{\partial t^2} = \frac{d^2 \,\tilde{\varrho}}{dt^2} + \frac{d}{dt} \int\_{S\_l} v\_{ll} \left[ \varrho \right] e^{-i\mathbf{k} \cdot \mathbf{y}} dS + \int\_{S\_l} v\_{ll} \left[ \frac{\partial \varrho}{\partial t} \right] e^{-i\mathbf{k} \cdot \mathbf{y}} dS \tag{70}$$

such that in the Fourier transform of second time derivative of the piecewise differentiable function *ϕ* enters the jump of the function *ϕ* across the discontinuity and the jump of the first time derivative of *ϕ* as well.

### **10. Appendix B: Determination of the Greens' function for the wave equation**

By using formula 11, given at page 364 in (Gelfand and Shilov, 1964) we can write

$$\mathcal{F}^{-1}\left\{\frac{\sin\left(ckt\right)}{k}\right\}=\frac{\delta\left(t-r/c\right)}{4\pi c r}\tag{71}$$

where *r* = |**x**| .

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

Now, dividing by (*t* − *t*0) and taking the limit for *t* → *t*<sup>0</sup> the first term gives the Fourier transform of the time derivative of the function *ϕ*(*i*) (**x**, *t*) while in the second integral we can write *d***x** =*vn* (*t* − *t*0) *dS* (see (Jacob, 1959), (Currie, 2003)). Finally, we obtain the following

+ *St*

*e*

*D*(*<sup>i</sup>*)(*t*)

*D*(*e*)(*t*)

 *St*

Formula (68) permits us to calculate the Fourier transform of the time derivative of a piecewise

*vn* [*ϕ*]*e*−*i***k**·**y***dS* +

such that in the Fourier transform of second time derivative of the piecewise differentiable function *ϕ* enters the jump of the function *ϕ* across the discontinuity and the jump of the first

*vn* (**y**, *t*)

 *St vn ∂ϕ ∂t e*

*∂ϕ* (**y**,*t*) *∂t*

<sup>−</sup>*i***k**·**x***d***x** +

*ϕ*(*i*)

*<sup>e</sup>*−*i***k**·**x***d***<sup>x</sup>** <sup>−</sup>

*ϕ*(*e*)*e*−*i***k**·**x***d***x** +

*D*(*<sup>e</sup>*)(*t*)

 *St*

> *St*

*D*(*<sup>i</sup>*)(*t*)

*∂ϕ*(*i*) (**x**, *t*) *∂t*

*e*

*vn* (**y**, *t*) *ϕ*(*i*) (**y**,*t*) *e*−*i***k**·**y***dS*

*∂ϕ*(*e*) (**x**, *t*) *∂t*

*vn* (**y**, *t*) [*ϕ* (**y**, *t*)]*e*−*i***k**·**y***dS*. (68)

*e* <sup>−</sup>*i***k**·**x***d***x**

*vn ϕ*(*i*) (**y**,*t*) *e*−*i***k**·**y***dS* (66)

*vnϕ*(*e*) (**y**,*t*) *e*−*i***k**·**y***dS* (67)

*e*−*i***k**·**y***dS* (69)

<sup>−</sup>*i***k**·**y***dS* (70)

<sup>−</sup>*i***k**·**x***d***x** (65)

form of the Reynolds' transport theorem

*D*(*<sup>i</sup>*)(*t*)

*vn* being the displacement velocity of the surface *S* (*t*).

*D*(*<sup>i</sup>*)(*t*)

<sup>−</sup>*i***k**·**x***d***<sup>x</sup>** <sup>=</sup> *<sup>d</sup>*

*<sup>e</sup>*−*i***k**·**x***d***<sup>x</sup>** <sup>=</sup> *<sup>d</sup>*

The sum of formulas (66) and (67) gives the *second basic formula*

differentiable function. For the second time derivative we can write

 <sup>=</sup> *<sup>d</sup> dt d ϕ dt* <sup>+</sup> *St*

*d dt St*

*∂ϕ*(*i*) (**x**, *t*) *∂t*

> *dt*

> *dt*

<sup>=</sup> *<sup>d</sup> <sup>ϕ</sup>* (**k**,*t*) *dt* <sup>+</sup>

 = 

*e*

 *∂ϕ* (**x**, *t*) *∂t*

 *∂*2*ϕ* (**x**, *t*) *∂t*<sup>2</sup>

*dt*<sup>2</sup> <sup>+</sup>

By using again formula (68) we obtain finally,

*ϕ*(*i*) (**x**, *t*)*e*−*i***k**·**x***d***x** =

*d dt* 

**9.2.3 The second basic formula**

 *∂ϕ* (**x**, *t*) *∂t*

> *∂ϕ*(*i*) (**x**, *t*) *∂t*

*∂ϕ*(*e*) (**x**, *t*) *∂t*

F

F

*∂* 2*ϕ <sup>∂</sup>t*<sup>2</sup> <sup>=</sup> *<sup>d</sup>*<sup>2</sup> *<sup>ϕ</sup>*

time derivative of *ϕ* as well.

We calculate now

F

*D*(*<sup>i</sup>*)(*t*)

*D*(*e*)(*t*)

By using formula (65) we get

By taking the derivative with respect to parameter *t* in the both sides of relationship (71) there results

$$\mathcal{F}^{-1}\left\{\cos\left(ckt\right)\right\} = \frac{\delta'\left(t - r/c\right)}{4\pi c^2 r} \tag{72}$$

Similarly, integrating the formula (71) over the interval (0, *t*) we obtain

$$\mathcal{F}^{-1}\left\{\frac{1-\cos\left(ckt\right)}{k^2}\right\}=\frac{H\left(t-r/c\right)}{4\pi r}\tag{73}$$

*H* (*x*) being the Heaviside's function.

### **11. References**


**0**

**4**

*Egypt*

*<sup>i</sup>*=<sup>0</sup> *AiF<sup>i</sup>*

(*ξ*)

**Exact Solutions Expressible in Hyperbolic and**

**Jacobi Elliptic Functions of Some Important**

<sup>1</sup> *Mathematics Department, Faculty of Science, Beni-Suef University, Beni-Suef* <sup>2</sup> *Mathematics Department, Faculty of Science, Minia University, El-Minia*

Many phenomena in physics and other fields are often described by nonlinear partial differential equations (NLPDEs). The investigation of exact and numerical solutions, in particular, traveling wave solutions, for NLPDEs plays an important role in the study of nonlinear physical phenomena. These exact solutions when they exist can help one to well understand the mechanism of the complicated physical phenomena and dynamical processes modeled by these nonlinear evolution equations (NLEEs). The ion-acoustic solitary wave is one of the fundamental nonlinear wave phenomena appearing in fluid dynamics [1] and plasma physics [2, 3]. It has recently became more interesting to obtain exact analytical solutions to NLPDEs by using appropriate techniques and symbolical computer programs such as Maple or Mathematica. The capability and power of these software have increased dramatically over the past decade. Hence, direct search for exact solutions is now much more viable. Several important direct methods have been developed for obtaining traveling wave solutions to NLEEs such as the inverse scattering method [3], the tanh-function method [4], the extended tanh-function method [5] and the homogeneous balance method [6]. We assume

that the exact solution is expressed by a simple expansion *u*(*x*,*t*) = *U*(*ξ*) = ∑*<sup>N</sup>*

The main steps of the F-expansion method [13] are outlined as follows:

and reduce a given NLPDE, say in two independent variables,

where *Ai* are constants to be determined and the function *F*(*ξ*) is defined by the solution of an auxiliary ordinary differential equation (ODE). The tanh-function method is the well known method as a direct selection of the function *F*(*ξ*) = *tanh*( *ξ*). Recently, many exact solutions expressed by various Jacobi elliptic functions (JEFs) of many NLEEs have been obtained by Jacobi elliptic function expansion method [7-10], mapping method [11, 12], F-expansion method [13], extended F-expansion method [14], the generalized Jacobi elliptic function method [15] and other methods [16-20]. Various exact solutions were obtained by using these methods, including the solitary wave solutions, shock wave solutions and periodic

Step 1. Use the transformation *u*(*x*, *t*) = *u*(*ξ*); *ξ* = *k*(*x* − *ωt*) + *ξ*0, *ξ*<sup>0</sup> is an arbitrary constant,

*F*(*u*, *ut*, *ux*, *utt*, *uxx*, ...) = 0, (1.1)

**1. Introduction**

wave solutions.

**Equations of Ion-Acoustic Waves**

A. H. Khater<sup>1</sup> and M. M. Hassan<sup>2</sup>

