An Operational Approach to the Acoustic Analogy Equations

The concept of acoustic wave is a pervasive one, which emerges in any type of medium, from solids to plasmas, at length and time scales ranging from sub-micrometric layers in microdevices to seismic waves in the Sun's interior. This book presents several aspects of the active research ongoing in this field. Theoretical efforts are leading to a deeper understanding of phenomena, also in complicated environments like the solar surface boundary. Acoustic waves are a flexible probe to investigate the properties of very different systems, from thin inorganic layers to ripening cheese to biological systems. Acoustic waves are also a tool to manipulate matter, from the delicate evaporation of biomolecules to be analysed, to the phase transitions induced by intense shock waves. And a whole class of widespread microdevices, including filters and sensors, is based on the behaviour of acoustic waves propagating in thin layers. The search for better performances is driving to new materials for these devices, and to more refined tools for their analysis.

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 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)).

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, S b 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 S s inside the flow domain as shock fronts, wakes, etc. Finally, to aid computations, it is often helpful to also define a virtual permeable surface, S p , enclosing body along with the portion of the air domain where viscous effects and the nonlinear terms in 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, S p . 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.

The equations of the acoustic analogy
In the case where the whole space R 3 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) and the conservation of momentum in the form written by Lighthill in Refs. (Lighthill, 1952) and (Lighthill, 1954) Here ρ is the density, c is the velocity of sound in the uniform medium, u j is the component of fluid velocity in the direction x j (j = 1, 2, 3), and a repeated index implies a summation over these values, and, is Lighthill's stress tensor. Also, we denoted by the compressive stress tensor, δ ij is the Kronicker delta function, p is the pressure and μ is the viscosity. (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.

Remark 1. The acoustical analogy equations
In the case where inside the flow domain there are discontinuity surfaces of type S s (as shock fronts) the conservation laws for mass and momentum yield the Rankine-Hugoniot type junction conditions We denote by square brackets the jump of its content across the discontinuity surface (see also formula (54) in Appendix A). u n is the velocity projection on the normal to the surface, n,and v n is the normal velocity of the surface.
Finally, on solid (deformable) impermeable surfaces S b inside the flow the obvious non-penetration condition proves true

The operational form of the acoustic analogy equations
Since the density ρ 0 and pressure p 0 at large distances (in the unperturbed fluid) are different from zero we introduce the perturbation of the density ρ ′ = ρ − ρ 0 and the perturbation of the stress tensor as P ′ jκ = P jκ − p 0 δ jκ (where δ jκ is the Kronecker delta) and will write equations (1) and (2) as where 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 the contributions of all discontinuity surfaces will appear. (8) and (9) can also be written as

Remark 2. The square brackets in equations
and ρu κ (u n − v n ) + P ′ κj n j respectively.

Shock-type discontinuity surfaces (S s )
In the case of a discontinuity surface S s (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.

Impermeable solid deformable surfaces (S b )
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 P j = P κj n κ . The operational momentum equation now contains in the right-hand side the action of the solid surface S b on the fluid flow.
3. Solution of the operational form of the acoustical analogy equations.

The case of a permeable surface S p
Let S p be a discontinuity surface of the flow variables inside the external fluid flow such that in the domain We write the system of equations (8), (9) as The solution of equation (10) can be obtained by using the exponential H (t) of the matrix A as The exponential of the matrix A canbewrittenas where k 2 = k 2 1 + k 2 2 + k 2 3 = |k| 2 . By introducing the function 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.

The equation for the operational density
By eliminating the Fourier transform of the momentum ρu j between equations (8) and (9) the operational equation satisfied by the density perturbation becomes 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 (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).

The case of an impermeable surface (S b )
In the case where the surface S b is impermeable, the condition (5) cancels out some terms in formula (17). In this case the nonhomogeneous term is 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) the nonhomogeneous term of the operational wave equation in the case of a permeable surface coincides with that corresponding to an impermeable case The terms U j and L j can be interpreted respectively as a modified velocity and a modified traction, which take into account the flow across S.

The lift component of velocity
The lift component of velocity is given by inverse Fourier transform of the term Therefore, by using formula (73) we obtain x(x 1 , x 2 , x 3 ) being the position vector of the observation point.

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).

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, where Q (y, t) is the surface intensity. In this case we have Hence, 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 τ e which is the solution of the equation g(τ, y, t, x)=0 where 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 τ e on the emission (radiating) surface S e ≡ S τ e . It is now necessay to consider the coordinate systems. Let a fixed point P 0 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 0 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 gives the connection between the Lagrangian and Eulerian coordinates of the point P 0 .F o r η fixed the equation y = y (η, τ) gives the trajectory of the point P 0 and cM =dy/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 (S e ≡ S τ e ), the position of the same surface at the observation time (S t ), the position vector y e 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 cM e , the radiation vector r e = x − y e , and the emission distance r e = |r e | .
To change the integration variable in the last integral in formula (27) from τ to g, we calculate where M r is the Mach number at the point η in the radiation direction at the time τ.
The density perturbation can therefore be written as Here, |1 − M r e | is the Doppler factor. The square brackets [] ret imply that the contents are to be evaluated at the retarded (emission or radiating time) τ e given implicitly by g (τ) = 0. The emission position is y e = y (η, τ e ) , the emission distance r e of the source point η to the observer position x is r e = |x − y (η e , τ e )| ,andQ * (η, τ e ) = Q (y e , τ e ) .

The thickness noise
The thickness noise is given by the term An analysis similar with that of the section 5.1 yields

The loading noise
The last term in relationship (21) describes the loading noise Its contribution to the perturbed density ρ ′ is 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).

The quadrupole noise term
The last term in formula (21) corresponds to a quadrupole noise source: where D (e) τ denotes the 3-D domain occupied by volume sources at the moment τ.T h el a s t integral in formula (35) was calculated in Appendix B. Introducing its expression given by formula (71) there results the relationship (36) becomes where the effect of source convection is revealed by the Doppler factor; convection effectively increases the source strength by |1 − M r | −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 S p ) 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 ′ q (x, t) , ρ ′ thickness and ρ ′ loading completely specify the density field.

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 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) where Equation (39) is similar to equation (20). Consequently, its solution can be written as Hence the contribution of the nonhomogeneous term in equation (38) can be written as Finally, the contribution of the terms corresponding to the boundary conditions on the mobile surface S canbewrittenas which coincides with the relationship (5.3) given in (Ffowcs Williams and Hawkings, 1969).

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.

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.

Appendix A: Fourier Transform of piecewise differentiable functions
Let D (i) (t) be a bounded mobile domain in R 3 having a smooth boundary surface S t .Denote by D (e) (t) the domain external to the surface S t (See Fig.2): .L e t a l s oϕ (i) (x, t) be a continuous differentiable function defined in the closed domain D (i) × R and ϕ (e) (x, t) a continuous differentiable function defined in D (e) × R.W ed e fi n e also

Fig. 2. Boundary Surface
Generally, the function ϕ (x, t) is discontinuous across the surface S.W ec a l li ta 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 Here x = (x 1 ,x 2 ,x 3 ) , k = (k 1 ,k 2 ,k 3 ) , k · x = k 1 x 1 + k 2 x 2 + k 3 x 3 is the inner product of the two vectors, dx = dx 1 dx 2 dx 3 and the integral is extended over the whole R 3 space. The inversion formula can be written as where dk =dk 1 dk 2 dk 3 . Accounting for relationship (44) we can write 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 We write But 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 S t .
where n = (n 1 , n 2 , n 3 ) is the external unit normal to S t . Therefore, Similarly, Finally, the relationships (49), (51), and (52) give the first basic formula: Here we denoted by square bracket the jump of the function ϕ (x, t) across the surface S t for y ∈ S t , x (i) ∈ D (i) and x (e) ∈ D (e) . Similar relationships to (53) can be proved for the derivatives in the directions x 2 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.

Other formulas
The relationship (53) gives also In the case where we write 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 scalar and vector fields.

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 S t . Then, for S(y 01 , y 02 , y 03 , t 0 ) on S t 0 we can write 0 = S(y 1 , y 2 , y 3 , t) − S(y 01 , y 02 , y 03 , t 0 )= (60) ∂S ∂y 1 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, are the components of the external normal unit vector n and are the projections on the velocity vector of a point on the surface S t on the three axes. The relation (60) yields which is the displacement velocity of the geometric surface S t . We mention that the displacement velocity of a surface has the direction of the normal vector to this surface.

Reynolds' transport theorem
For calculating the Fourier Transform of a time derivative of a certain function we write Now, dividing by (t − t 0 ) and taking the limit for t → t 0 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 dx =v n (t − t 0 ) dS (see (Jacob, 1959), (Currie, 2003)). Finally, we obtain the following form of the Reynolds' transport theorem + S t v n (y, t) ϕ (i) (y, t) e −ik·y dS v n being the displacement velocity of the surface S (t) .

The second basic formula
We calculate now By using formula (65) we get The sum of formulas (66) and (67) By using again formula (68) we obtain finally, 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.

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 where r = |x| . By taking the derivative with respect to parameter t in the both sides of relationship (71) there results Similarly, integrating the formula (71) over the interval (0, t) we obtain H (x) being the Heaviside's function.