acoustic diffraction from an absorbing half … · acoustic diffraction from an absorbing half...

CANADIAN APPLIED

MATHEMATICS QUARTERLY

Volume 16, Number 3, Fall 2008

ACOUSTIC DIFFRACTION FROM AN

ABSORBING HALF PLANE DUE TO A POINT

SOURCE IN A MOVING FLUID

BASHIR AHMAD AND LAILA NAYEL ALSHARIEF

ABSTRACT. We discuss the diffraction of a spherical acous-tic wave (emanating due to a point source) from a semi-infiniteplane satisfying Myers’ impedance condition on both sides of itssurface in a moving fluid. The introduction of Myers’ impedancecondition improves the diffracted field. The method of solutionconsists of Fourier transform, Wiener-Hopf technique and theasymptotic methods.

1 Introduction Diffraction theory has been effectively applied tothe context of reducing traffic noise through the introduction of physicalbarriers in heavily populated areas [6, 13, 15, 20]. In this methodol-ogy, one needs to line one or both faces of the barrier with absorbentmaterial. The presence of an acoustically absorbing lining on a surfaceis described by an impedance relationship between the acoustic pressureand normal acoustic velocity fluctuation on the lining surface. For thenoise radiated from aero-engines and inside wind tunnels, it is desir-able to discuss the acoustic diffraction in the presence of moving fluids.Rawlins [31] explained this model by considering the acoustic diffractiondue to a cylindrical acoustic wave from an absorbing half plane satisfy-ing Ingard’s condition [14] in a moving fluid. Later on, Asghar et al.

[3, 4] extended Rawlins’ idea to calculate the diffraction of a sphericalacoustic wave from an absorbing half plane. There has been a consid-erable discussion and even some controversy in the literature regardingthe proper boundary conditions on the acoustic field in the presence ofa base flow. Perhaps the first correct treatment of the effect of a mov-ing medium was given concurrently by Miles [24] and Ribner [32] fora plane interface of relative motion. Ingard [14] later determined theeffect of flow on the boundary condition at a plane impedance surface.

Keywords: Diffraction, Myers’ impedance condition, half plane, Wiener-Hopftechnique, asymptotic methods.

Copyright c©Applied Mathematics Institute, University of Alberta.

219

220 B. AHMAD AND L. N. ALSHARIEF

A good discussion regarding the apparent disparities in the conditiondepending upon whether the moving acoustic medium is treated as vis-cous or inviscid can be seen in the work of Nayfeh, Kaiser and Telionis[27]. Later on, Myers [26] obtained the acoustic boundary conditionin the presence of flow using a matched asymptotic expansion type ap-proach for smaller values of the dimensionless parameter characterizingthe magnitude of the acoustic disturbances. This analysis representedthe generalization of the Ingard’s condition for flow around arbitrarycurved surfaces. Currently, Myers’ condition is the accepted form of theboundary condition for impedance walls with flow. Recently, Ahmad [2]has presented an improved model for noise barriers in a moving fluidincorporating Myers’ condition. For some recent work on the subject,see [5, 9, 12, 21, 22, 23, 25, 35, 36, 37] and the references therein.

In this paper, we consider the diffraction of acoustic waves in threedimensions (emanating due to a point source) from a semi-infinite planesatisfying Myers’ impedance condition on both sides of its surface in amoving fluid. The importance of the present work lies in the fact thatMyers’ condition involves an additional term (perturbation term) in con-trast to the impedance condition of Ingard and consequently improvesthe result of the diffraction analysis from an absorbing half plane ob-tained in [3]. In relation to the source of incident waves, point sourcesare regarded as better substitutes for the real sources and the field for afinite cylindrical source (finite distribution of point sources) can be ob-tained using the point source results. The present study also extends theline source results of [2] to point source diffraction results. Further, theconsideration of the whole system in a moving fluid extends the scopeof the work to aeronautical engineering.

Historically, there has been two different approaches to deal with highfrequency scattering: 1) full analytical solution by transform methodsand then saddle point asymptotic evaluation of complicated integrals,2) use of Keller’s geometrical theory of diffraction based on ray theoryand WKB-type asymptotic expansions at the level of the original PDEsystem. At high frequencies, the deterministic approach based on theanalytical and asymptotic methods such as the geometrical theory ofdiffraction (GTD), ray method and WKB method has been proposedfor prediction of acoustic and vibrational responses since early times[7, 8, 11, 17, 18, 34]. The main drawbacks of the infinite frequencyapproximation of geometrical optics are that diffraction effects at bound-aries are lost, and that the approximation breaks down at caustics, wherethe predicted amplitude is unbounded. For these situations, more de-tailed models such as the geometrical theory of diffraction (GTD) are

ACOUSTIC DIFFRACTION 221

needed. The geometrical theory of diffraction, viewed as a generaliza-tion of geometrical optics, was pioneered by Keller [17] in the 1960s,and provides a technique for adding diffraction effects to the geometri-cal optics approximation. The GTD is often used in scattering problemsin computational electromagnetics, where boundary effects are of majorimportance, for example in radar cross section calculations and in theoptimization of base station locations for cell phones in a city. The WKBmethod is generally applicable to problems of wave propagation in whichthe frequency of the wave is very high or, equivalently, the wavelengthof the wave is very short. The WKB approximation can also be usedto solve problems in which the functional behavior is rapidly growingor rapidly decaying other than rapidly oscillatory, an example being theproblems of boundary layer. Typically this method can determine theasymptotic solution ψ up to certain constants C, that is, ψ = C/r1/2,etc. In case of scattering problems, the ray method can also be appliedto determine the form of the asymptotic solution up to some diffractionconstants. For more examples and details, see [1, 10, 29, 30, 33] andreferences therein.

We follow the first approach to solve our problem. The application ofFourier transform to the boundary value problem results into a Wiener-Hopf functional equation [28]. In order to solve this equation, we en-counter a new kernel function which needs to be factorized to obtain theunknown quantities involved in the Wiener-Hopf equation. This factor-ization has been accomplished in Appendix B. The Wiener-Hopf proce-dure yields an integral representation of the diffracted field. These inte-grals are normally difficult to handle because of the presence of branchpoints and poles, and are only amenable to solution using asymptoticapproximation. The asymptotic representation of these integrals in thehigh frequency limit is obtained using the modified saddle point method(the modification is required in view of the fact that the saddle pointmay come close to the pole [16]) and the diffracted field is presented.For more details on the asymptotic approximation of integrals, see thetext [38].

2 Formulation and solution of the problem We consider thediffraction of an acoustic wave emanating from a point source locatedat the position (xo, yo, zo) by a semi-infinite plane occupying a spacey = 0, x ≥ 0, placed in a fluid moving with subsonic velocity U parallelto the x-axis as shown in Figure 1. The time dependence is taken tobe of harmonic type e−ift (f is the angular frequency) and the plane is


assumed to be satisfying the Myers’ impedance condition [26]

(1) un = − p

Za+

U

ifZa

∂p

∂x,

where p is surface pressure and un the normal derivative of the pertur-bation velocity, Za is the acoustic impedance of the surface and −n anormal pointing from the fluid into the surface. In passing, we remark

FIGURE 1

that the second term of equation (1) was not there in Ingard’s condition.The perturbation velocity u of the irrotational sound wave can be writ-ten in terms of the velocity potential ψ as u = ∇ψ. Then, the resultingpressure p of the sound field is

(2) p = −ρ0

(∂

∂t+ U

∂

∂x

)ψ,

where ρ0 is the density of the undisturbed stream. Thus, we have tosolve the following boundary value problem

[(1 −M2)

∂2

∂x2+ 2ikM

∂

∂x+

∂2

∂y2+

∂2

∂z2+ k2

]ψ(x, y, z)(3)

= δ(x− xo)δ(y − yo)δ(z − zo),[∂

∂y∓ 2βM

∂

∂x± ikβ ∓ iβM2

k

∂2

∂x2

]ψ(x, 0±, z) = 0 (x > 0),(4)

∂

∂yψ(x, 0+, z) =

∂

∂yψ(x, 0−, z)

ψ(x, 0+, z) = ψ(x, 0−, z)(x < 0),(5)


where k = f /c, β = ρoc/Za , c is the velocity of sound and M = U/c isthe Mach number. Here, we remark that equation (4) is obtained using(1) and (2). We shall assume that the flow is subsonic, that is, |M | < 1and Reβ > 0, a necessary condition for an absorbing surface. Moreover,we require that the field ψ represents an outward traveling wave asr =

√(x2 + y2 + z2) → ∞, that is, the field decays exponentially at

infinity.Let us define the Fourier transform and its inverse over the variable

z as

(6)

φ(x, y, ω) =

∫ ∞

−∞ψ(x, y, z)e−ikωz dz,

ψ(x, y, z) =k

2π

∫ ∞

−∞φ(x, y, ω)eikωz dω,

where kω (ω is non-dimensional) is taken to be the transform parameter.In order to satisfy the outgoing radiation condition, k is assumed to becomplex and has a small positive imaginary part.

Now, transforming equations (3)–(5) with respect to z via equa-tion (6), we obtain

[(1 −M2)

∂2

∂x2+ 2ikM

∂

∂x+

∂2

∂y2− k2ω2 + k2

]φ(x, y, ω)(7)

= δ(x − xo)δ(y − yo)e−ikωz0

[∂

∂y∓ 2βM

∂

∂x± ikβ ∓ iβM2

k

∂2

∂x2

]φ(x, 0±, ω) = 0 (x > 0),(8)

∂

∂yφ(x, 0+, ω) =

∂

∂yφ(x, 0−, ω)

φ(x, 0+, ω) = φ(x, 0−, ω)(x < 0).(9)

Since we are dealing with subsonic flow (|M | < 1), we can introduce thefollowing real substitutions

(10)

x =√

1 −M2X, x0 =√

1 −M2X0, y = Y, y0 = Y0,

z = Z, z0 = Z0, k =√

1 −M2K, β =√

1 −M2B,

φ(x, y, ω) = Ψ(X,Y, ω)e−iKMX .

Using these substitutions, equations (7)–(9) take the form


(11)

[∂2

∂X2+

∂2

∂Y 2+K2γ2

]Ψ(X,Y, ω)

=δ(X −Xo)δ(Y − Yo)√

1 −M2eiKMXo−i

√1−M2KωZ0 ,

where

γ2 = 1 − ω2(1 −M2)[∂

∂Y∓ 2BM

∂

∂X± iKB(1 +M2)(12)

∓ iBM2

(1 −M2)K

∂2

∂X2

]Ψ(X, 0±, ω) = 0 (X > 0),

∂Ψ

∂Y(X, 0+, ω) =

∂Ψ

∂Y(X, 0−, ω)

Ψ(X, 0+, ω) = Ψ(X, 0−, ω)(X < 0).(13)

It is observed that the problem represented by equations (11)–(13) in thetransformed ω-plane is identical to the two dimensional problem solvedin reference [26] except that K2γ2 replaces K2 and a multiplicative fac-

tor e−i√

1−M2KωZ0 appears on the right hand side in the wave equation.Thus, in order to avoid re-deriving useful previous formula, we employthe results of reference [26] directly to write the diffracted field φ(x, y, ω)as

φ(x, y, ω) = C̃1

[B(1 +M2)

γ√Kγ

(14)

− 2BM cosΘ0 + 2 sin(Θ/2) sin(Θ0/2)√Kγ

+BM2√γ cos2 Θ0

(1 −M2)√K

]

× exp{iK[γ(R+R0) −

√1 −M2ωZ0

]}

L+(Kγ cosΘ)L−(−Kγ cosΘ0)F (|Q̃|),

where

C̃1 =e−iKM(X−X0)

2π sinΘ√

2R0(1 −M2),


X = R cosΘ, Y = R sin Θ, X0 = R0 cosΘ0, Y0 = R0 sin Θ0,

x = r cos θ, y = r sin θ, x0 = r0 cos θ0, y0 = r0 sin θ0,

R = r

(1 −M2sin2θ

1−M2

) 12

, cosΘ =cos θ

(1 −M2 sin2 θ

) 12

,

|Q̃| =

[1

2RK[1− (1 −M2)ω2]

12

] 12 cosΘ − cosΘ0

sin Θ,

F (Q̃) = e−iQ̃2

∫ ∞

Q̃

eit2 dt,

L(ν) = 1 +B

κ

[K(1 +M2) + 2νM +

M2ν2

(1 −M2)K

],

(the factorization L(ν) = L+(ν)L−(ν) has been accomplished in Ap-pendix B). The field ψ(x, y, z) can now be calculated by taking the in-verse Fourier transform of (14), which will then complete our problem.Thus, the inverse Fourier transform of equation (14) gives

ψ(x, y, z) = C̃11

∫ ∞

−∞

exp[iKγ(R+R0) + iK√

1 −M2ω(z − z0)]

[1 − (1 −M2)ω2]12 {K[1− (1 −M2)ω2]

12 } 1

2

× F (|Q̃|)L+(Kγ cosΘ)L−(−Kγ cosΘ0)

dω

+ C̃22

∫ ∞

−∞

exp[iKγ(R+R0) + iK√

1 −M2ω(z − z0)]

(K[1 − (1 −M2)ω2]12 )

12

× F (|Q̃|)L+(Kγ cosΘ)L−(−Kγ cosΘ0)

dω

+ C̃33

∫ ∞

−∞exp[iKγ(R+R0) + iK

√1 −M2ω(z − z0)]

× F (|Q̃|)[1 − (1 −M2)ω2]14 dω

L+(Kγ cosΘ)L−(−Kγ cosΘ0),

(15)

where

C̃11 =B(1 +M2)K(1 −M2)

12

2πC̃1,


C̃22 =−K(1−M2)

12

π(BM cosΘ0 + sin(Θ/2) sin(Θ0/2))C̃1,

C̃33 =

√K(1 −M2)−

12

2π(BM2 cos2 Θ0)C̃1.

The integrals on the right-hand side of (15) are denoted by I1, I2 andI3, respectively. These integrals can be evaluated asymptotically (seeAppendix A) and the far field is given by

ψ(x, y, z) = −[B(1 +M2)

R12

(R +R0)

− 2

(BM cosΘ0 + sin

Θ

2sin

Θ0

2

)

+ (BM2 cos2 Θ0)(R +R0)

(1 −M2)R12

]

× exp[−iKM(X −X0)] exp[iKR12 − 14 iπ]

4π√πR0 sinΘ(1 −M2)

12 [R12(R11 +R12)]

12

× F (τR12 )(R+R0 +A1)12

L+(Kζ2 cosΘ)L−(−Kζ2 cosΘ0),

(16)

where

R212 = (z − z0)

2 + (R+R0)2, R2

11 = (z − z0)2 +A2

1,

A1 = R+R0 − µ2, µ =cosΘ + cosΘ0

sin Θ

(R

2

) 12

,

τR12 =[K(R+R0 +A1)]

12

(R12 +R11)12

µ, ζ2 =R+R0

R12.

3 Discussion We have presented the diffracted field from a semi-infinite plane satisfying Myers’ impedance condition on both sides of itssurface in a moving fluid. An important and new feature of the diffractedfield given by (16) is that its amplitude contains the terms upto orderM2 in contrast to the earlier results dealing with the diffracted fieldfrom the absorbing half planes, which included the terms upto order M[3], where M is the Mach number. This improvement in the diffractedfield arises due to the Myers’ impedance condition considered in this


paper. In the earlier modelling of noise reduction problems, the Ingard’simpedance condition was considered, which did not fully incorporate theabsorbing phenomenon in a moving fluid. This discrepancy can be betterunderstood by comparing the absorbing condition assumed in [3], thatis,

(17)

(∂

∂y∓ βM

∂

∂x± ikβ

)ψ(x, 0±, z) = 0.

Comparing (17) with (8), we find that the fourth term in (8), i.e.,

∓ iβM2

k

∂2ψ

∂x2(x, 0±, z)

is altogether missing in (17), and the second term of (17) differs fromthat of (8) by a multiplicative factor of 2. As a matter of fact, the fourthterm in (8) contributes to the novelty of the present results and im-provement of the earlier results [3]. In the literature, this term is knownas correction or perturbation term. Moreover, due to this perturbationterm, a new form of the kernel function appears in Wiener-Hopf func-tional equation that needs to be factorized to find the known functions.This factorization makes the analysis more complicated. The effect ofthe absorption parameter on the diffracted field is shown explicitly. It isinteresting to note that point source diffraction from a rigid half-planefollows by taking absorption parameter β = 0 and the non-convective(still fluid case) field can be recovered by letting Mach number M to bezero in equation (16).

Appendix A This appendix deals with the evaluation of the inte-grals appearing in equation (15). Combining the integrals I2 and I3 in(15), we obtain I4 given by

(A.1) I4 = C̃22

∫ ∞

−∞

exp[iK[ω̃(z − z0) + (1 − ω̃2)12 (R +R0)]]

[K(1− w̃2)12 ]

12L+(Kγ̃ cosΘ)

× F (|Q̃|)[1 + C̃44

√K[1 − ω̃2] ] dω̃

L−(−Kγ̃ cosΘ0)(1 −M2)12

,

where

C̃44 =C̃33

C̃22

, ω̃ = ω(1 −M2)12 , γ̃ = (1 − ω̃2)

12 ,


F (|Q̃|) = F (µ[K(1 − ω̃2)12 ]

12 ), µ =

cosΘ + cosΘ0

sin Θ

(R

2

) 12

.

Making use of the result

∫ ∞

ν

eiλt2 dt = eiλν2 F (λ12 ν)

λ12

,

equation (A.1) can be written as

(A.2) I4 = C̃22

×∫ ∞

µ

∫ ∞

−∞

exp[iK[ω̃(z − z0) + (1 − ω̃2)12 (t2 − µ2 +R+R0)]]

L+(Kγ̃ cosΘ)L−(−Kγ̃ cosΘ0)(1 −M2)12

× (1 + C̃44

√K(1 − ω̃2)) dω̃ dt.

Now, consider the integral

I ′4 =

∫ ∞

−∞

exp(iK[ω̃(z − z0) + (1 − ω̃2)12P ])

L+(Kγ̃ cosΘ)L−(−Kγ̃ cosΘ0)

[1 + C̃44

√K(1 − ω̃2)

]dω̃.

By the substitutions

ω̃ = cos ξ, z − z0 = R1 cosΦ, P = R1 sin Φ, (1 − ω̃2)12 = sin ξ,

I ′4 takes the form

I ′4 =

∫ ∞

−∞f(ξ) exp[iKR1 cos(ξ − Φ)] dξ,

where

f(ξ) =− sin ξ[1 + C̃44K

12 sin ξ]

L+(K sin ξ cosΘ)L−(−K sin ξ cosΘ0).

We apply the method of steepest descent to solve the integral I ′4. Forthis, we deform the contour of integration so as to pass through thepoint of steepest descent ξ = Φ, so that the major part of the integrandis given by integration over the part of the deformed contour near Φ,with f(ξ) slowly varying around it.

Hence, we can write


(A.3) I ′4 ' πf(Φ)H(1)0 (KR1) ' −πH(1)

0 (K[(z − z0)2 + P 2]

12 )

L+(Kζ cosΘ)L−(−Kζ cosΘ0)ζ,

where

ζ =P

[(z − z0)2 + P 2]12

(1 + C̃44

K12P

[(z − z0)2 + P 2]12

).

Using equation (A.3), we can write equation (A.2) as

(A.4)

I4 =−πC̃22

(1 −M2)12

∫ ∞

µ

H(1)0 (K[(z − z0)

2 + (t2 +R+R0 − µ2)2]12 )

L+(Kζ cosΘ)L−(−Kζ cosΘ0)

× t2 +R+R0 − µ2

[(z − z0)2 + (t2 +R +R0 − µ2)2]12

×[1 +

C̃44K12 (t2 + R+R0 − µ2)

[(z − z0)2 + (t2 +R+R0 − µ2)2]12

]dt.

If we make the substitutions

t2 = −A1 + (A21 +R2

11 sinh2 u)12 ,

R211 = (z − z0)

2 + A21,

A1 = R+R0 − µ2,

in (A.4), we obtain

(A.5) I4 =−π.C̃22

2(1 −M2)12

∫ ∞

η

H(1)0 (KR11 coshu)

L+(Kζ̃ cosΘ)L−(−Kζ̃ cosΘ0)

× [(A21 +R2

11 sinh2 u)12 +A1]

12

×[1 +

C̃44.K12 (A2

1 +R211 sinh2 u)

12

R11 coshu

]du,

where

ζ̃ =(R2

11 sinh2 u+A21)

12

R11 coshu, η = sinh−1[(µ2 + 2A1)

12µ/R11].


The integral (A.5) can be solved asymptotically by taking

KR11 coshu >> 1.

Therefore, we can replace the Hankel function by the first term of itsasymptotic expansion to have

I4 = − C̃22√πe−

14 iπ

[2KR11(1 −M2)]12

×∫ ∞

η

exp[iKR11 coshu][(R211 sinh2 u+A2

1)12 +A1]

12

L+(Kζ̃ cosΘ)L−(−Kζ̃ cosΘ0) cosh12 u

×[1 +

C̃44K12 (A2

1 +R211 sinh2 u)

12

R11 coshu

]du.

If we let τ = (2KR11)12 sinh(u/2), then

I4 = − C̃22(2π)12 exp{iKR11 − 1

4 iπ}[K(1 −M2)]

12

∫ ∞

τR12

eiτ2

f̃2(τ) dτ,

where

f̃2(τ) =

([τ2(τ2 + 2KR11) +A2

1K2]

12 +A1K

(τ2 +KR11)(τ2 + 2KR11)

) 12

× 1

L+(Kζ̃ cosΘ)L−(−Kζ̃ cosΘ0)

×[1 +

C̃44K12 (τ2(τ2 + 2KR11) +K2A2

1)12

(τ2 +KR11)

],

ζ̃ =[(τ2 + 2KR11)τ

2 +A21K

2]12

τ2 +KR11,

τR12 = [K(R12 −R11)]12 ,

R12 = [(z − z0)2 + (R +R0)

2]12 .

An asymptotic expansion of I4 then follows by putting τ equal to itslower limit value in the non exponential factor of the integrand plus thecontribution from τ = 0 if zero lies in the interval of integration. Hence


I4 = −√

2π

[K(1−M2)]12

eiKR11I0H(−ε1) − ε1exp[iKR12 − 1

4 iπ]

[K(1 −M2)]12

F (τR12 )

×[C̃22(R+R0 +A1)

12

[KR12(R12 +R11)]12

+ C̃33(R+R0)(R+R0 +A1)

12

R12[R12(R12 +R11)]12

]

×√

2π


where H(·) is the usual Heaviside function,

I0 =C̃22

2

(πA1)12

K12R11

[1 + C̃44

K12A1

R11

]1


ε1 = sgn τR12 , ζ2 =R+R0

R12, ζ1 =

A1

R11.

The integral I1 can be evaluated similarly.

Appendix B In this appendix, we present the factorization of

(B.1) L(ν) = 1 +B√

K2γ2 − ν2

[K(1 +M2) + 2νM +

M2ν2

(1 −M2)K

].

From equation (B.1), we get

d

dνlnL(ν) =

L′(ν)

L(ν)

=KB[2MKγ2 + ν(1+2M2γ2)

(1−M2) − M2ν3

K2(1−M2) ]

(K2γ2 − ν2)[√K2γ2 − ν2 +B(K(1 +M2) + 2νM + M2ν2

K(1−M2) )].

(B.2)

If we write

(B.3)L′(ν)

L(ν)=L′

+(ν)

L+(ν)+L′−(ν)

L−(ν)= G(ν) = G+(ν) +G−(ν),

then for −ImKγ < c < Im ν < d < ImKγ, using Cauchy’s integrals[28], we have

G+(ν) =1

2πi

∫ ∞+ic

−∞+ic

G(ξ)

(ξ − ν)dξ,(B.4)


G−(ν) = − 1

2πi

∫ ∞+id

−∞+id

G(ξ)

(ξ − ν)dξ,(B.5)

where G+(ν) is regular for Im ν > −ImKγ and G−(ν) is regular forIm ν < ImKγ. Before proceeding further, we investigate the zeros ofG(ν) because these, if they exist, will be poles of the integrand of G±(ν).The zeros of G(ν) occur at the roots of the equation

Λ(ξ) =√K2γ2 − ξ2 +B

(K(1 +M2) + 2ξM +

M2ξ2

K(1 −M2)

)= 0.

In order to locate the position of the roots, a change of variable is intro-duced in accordance with the sheet on which

√K2γ2 − ξ2

∣∣∣ξ=0

= Kγ,

that is, ξ = Kγ cos(σ + iτ), −∞ < τ <∞, 0 < σ < π. Then

(B.6) Λ(ξ) = Kγ sin(σ + iτ) +BK(1 +M 2) + 2MBKγ cos(σ + iτ)

+BM2Kγ2 cos2(σ + iτ)

(1 −M2)= 0,

if and only if Re Λ(ξ) = 0 and Im Λ(ξ) = 0. Letting B = Br + iBi andkeeping in mind that Br > 0, which is a necessary requirement for anabsorbing surface, equation (B.6) gives

γ sinσ cosh τ +Br(1 +M2) + 2γMBr cosσ cosh τ(B.7)

+ 2γMBi sinσ sinh τ +BrM

2γ2

2(1 −M2)(1 + cos 2σ cosh 2τ)

+BiM

2γ2

2(1−M2)sin 2σ sinh 2τ = 0,

γ cosσ sinh τ +Bi(1 +M2) + 2MBiγ cosσ cosh τ(B.8)

− 2MBrγ sinσ sinh τ +BiM

2γ2

2(1 −M2)(1 + cos 2σ cosh 2τ)

− BrM2γ2

2(1−M2)sin 2σ sinh 2τ = 0.


From (B.8), we find that

Bi =Br

(M2γ2

2(1−M2) sin 2σ sinh 2τ + 2Mγ sinσ sinh τ)− γ cosσ sinh τ

(1 +M2) + 2Mγ cosσ cosh τ + M2γ2

2(1−M2)(1 + cos 2σ cosh 2τ),

which, on substituting in equation (B.7) gives

(B.9) Br = − sinσ

(1 −M2)∆

([1 +M2γ3(cos2 σ − sinh2 τ)] cosh τ

+ 2Mγ2 cosσ),

where

∆ = ∆21 +

(M2γ2

2(1 −M2)sin 2σ sinh 2τ + 2Mγ sinσ sinh τ

)2

,

∆1 = (1 +M2) + 2Mγ cosσ cosh τ

+M2γ2

2(1 −M2)(1 + cos 2σ cosh 2τ).

Now, for Br ≥ 0, we must have

cosh τ +2Mγ2 cosσ

1 +M2γ3(cos2 σ − sinh2 τ)> 0,

where |M | < 1, 0 < σ < 1 (sinσ > 0), and

(B.10) ∆ = ∆21 +

(M2γ2

2(1 −M2)sin 2σ sinh 2τ + 2Mγ sinσ sinh τ

)2

< 0.

In view of the fact that equations (B.9) and (B.10) are never satisfied, weconclude that no zeros occur on the cut sheet chosen. Now, we evaluateintegrals (B.4) and (B.5) which on substituting (B.3) and (B.2) take theform

(B.11)d

dνlnL+(ν)

=KB

2πi

∫ ∞+ic

−∞+ic

[2MKγ2 + ξ(1+2M2γ2)(1−M2) − M2ξ3

K2(1−M2) ]

[√K2γ2 − ξ2 +B(K(1 +M2) + 2ξM + M2ξ2

K(1−M2) )]


× dξ

(K2γ2 − ξ2)(ξ − ν), (Im ν > −e)

(B.12)d

dνlnL−(ν)

=KB

2πi

∫ ∞+ic

−∞+ic

−[2MKγ2 + ξ(1+2M2γ2)(1−M2) − M2ξ3

K2(1−M2) ]

[√K2γ2 − ξ2 +B(K(1 +M2) + 2ξM + M2ξ2

K(1−M2) )]

× dξ

(K2γ2 − ξ2)(ξ − ν), (Im ν < −e)

where e = min(|c|, |d|). Normally, we calculate only one of the above in-tegrals and then using the parity property of G(ν), the value of the otherintegral can be obtained. Here, we notice that the parity is not with re-spect to ν alone but also depends onM, that is, G(ν) = −G(−ν)|M=−M .Therefore, from (B.4) and (B.5), we find that

G−(−ν)∣∣M=−M

= − 1

2πi

∫ ∞+ie

−∞+ie

G(ξ)|M=−M

(ξ + ν)dξ,

= − 1

2πi

∫ ∞+ie

−∞+ie

G(ξ)

(ξ − ν)dξ = −G+(ν).

(B.13)

In equation (B.11), the contour of integration is deformed into the lowerhalf plane until it runs around the branch cut −Kγ to −∞Kγ (for moredetails, see [31]). Thus,

G+(ν) = − 1

2(ν +Kγ)

+KB

2πi

∫

Γ

−[2MKγ2 + ξ(1+2M2γ2)(1−M2) − M2ξ3

K2(1−M2) ]

[√K2γ2 − ξ2 +B(K(1 +M2) + 2ξM + M2ξ2

K(1−M2) )]

× dξ

(K2γ2 − ξ2)(ξ − ν),

where the first term comes from the simple pole contribution at ξ = −Kγand the contribution from the infinite semicircle is zero. Hence we can


write

G+(ν) = − 1

2(ν +Kγ)

− KB

2πi

∫ −Kγ

−∞Kγ

(2MKγ2 + ξ(1+2M2γ2)(1−M2) − M2ξ3

K2(1−M2) )

[(1 + 6M2B2

1−2M2 )ξ2 + 4MB2K1−2M2 ξ +K2( B2

1−2M2 − γ2)]

×[i√ξ2 −K2γ2 −B(K(1 +M2) + 2ξM + M2ξ2

K(1−M2) )] dξ

(K2γ2 − ξ2)(ξ − ν)

− KB

2πi

∫ −∞Kγ

−Kγ

(2MKγ2 + ξ(1+2M2γ2)(1−M2) − M2ξ3

K2(1−M2) )

[(1 + 6M2B2

1−2M2 )ξ2 + 4MB2K1−2M2 ξ +K2( B2

1−2M2 − γ2)]

×[−i

√ξ2 −K2γ2 −B(K(1 +M2) + 2ξM + M2ξ2

K(1−M2) )] dξ

(K2γ2 − ξ2)(ξ − ν)

= − 1

2(ν +Kγ)

+KB

(1 + 6M2B2

1−2M2 )π

∫ ∞Kγ

Kγ

(2MKγ2 + ξ(1+2M2γ2)(1−M2) − M2ξ3

K2(1−M2) ) dξ√ξ2 −K2γ2(ξ +Kν1)(ξ +Kν2)(ξ + ν)

,

(B.14)

where

ν1 =

(1 +

6M2B2

1 − 2M2

)

×(− 2MB2 +

√(1 − 4M2)γ2 −B2(1 − 2M2 + 2M2B2 − 6M2γ2)

),

ν2 =

(1 +

6M2B2

1 − 2M2

)

×(− 2MB2 −

√(1 − 4M2)γ2 −B2(1 − 2M2 + 2M2B2 − 6M2γ2)

).

Expanding the integrals of equation (B.14) into partial fractions andusing the result

f(p) =

∫ ∞Kγ

Kγ

dξ√ξ2 −K2γ2(ξ + p)

=cos−1(p/Kγ)√K2γ2 − p2

, Re (p+Kγ) > 0,

we get

G+(ν) = − 1

2(ν +Kγ)


+KB[2Mγ2 − ν1{(1+2M2γ2)−M2ν2

1}(1−M2) ]

π(1 + 6M2B2

1−2M2 )(ν1 − ν2)F (ν,Kν1)

−KB[2Mγ2 − ν2{(1+2M2γ2)−M2ν2

2}(1−M2) ]

π(1 + 6M2B2

1−2M2 )(ν1 − ν2)F (ν,Kν2),

where

F (ν,Kνo) =f(ν) − f(νo)

(ν − νo).

Thus,

lnL+(ν) − lnL+(0) =

∫ ν

0

G+(ν) dν,

or

L+(ν) = L+(0) exp

∫ ν

0

G+(ν) dν.

Using the asymptotic formula

f(ν) =π

2Kγ− πν

K2γ2+O(ν2), ν → 0,

it is easy to show that

F (ν, νo) =1

νo

[f(νo) −

π

2Kγ

]+O(ν), ν → 0.

Hence

L+(ν)

L−(ν)= exp

( ∫ ν

o

(constt +O(ν)) dν

)→ 1 as ν → 0,

so that L+(0) = L−(0), which coupled with the fact that

L(0) = L+(0)L−(0) = 1 +B(1 +M2)

γ,

gives

L+(0) = L−(0) =

√

1 +B(1 +M2)

γ.

Therefore,

L+(ν) =

√1 + [B(1 +M2)/γ]

(1 + ν/Kγ)exp

[− KB

π(1 + 6M2B2

1−2M2 )


×{

[2Mγ2 − ν2{(1+2M2γ2)−M2ν22}

(1−M2) ]

(ν1 − ν2)

∫ ν

0

F (ν,Kν2) dν

−[2Mγ2 − ν1{(1+2M2γ2)−M2ν2

1}(1−M2) ]

(ν1 − ν2)

∫ ν

0

F (ν,Kν1) dν

}].

To obtain an asymptotic estimate for |L±(ν)| as ν → ∞, we use atheorem due to Kranzer and Radlow [19] which is a slight modificationof the theorem given in Noble [28]. Thus, it has been found that

|L±(ν)| ≈ O(|ν|∓(1/2π) arg[limη→∞

L(η)L(−η) ]) as |ν| → ∞,

which, on using (B.1), yields

|L±(ν)| ≈ O(1).

This completes the factorization.

Acknowledgment The authors are very grateful to the refereesand Professor Michael J. Ward (co-managing editor) for their valuablesuggestions and comments that led to the improvement of the originalmanuscript.

REFERENCES

1. J. A. Addison, S. D. Howison and J. R. King, Ray methods for free boundaryproblems, Quart. Appl. Math. 64 (2006), 41–59.

2. B. Ahmad, An improved model for noise barriers in a moving fluid, J. Math.Anal. Appl. 321 (2006), 609–620.

3. S. Asghar, B. Ahmad and M. Ayub, Point source diffraction by an absorbinghalf plane, IMA J. Appl. Math. 46 (1991), 217–224.

4. S. Asghar, B. Ahmad and M. Ayub, Sub-Mach-1 sound from apoint sourcenear an absorbing half plane, J. Acoust. Soc. Am. 93 (1993), 66–70.

5. Yu. I. Bobrovnitskii, A new impedance-based approach to analysis and controlof sound scattering, J. Sound Vib. 297 (2006), 743–760.

6. G. F. Butler, A note on improving the attenuation given by a noise barrier, J.Sound Vibr. 32 (1974), 367–369.

7. P. Filippi, D. Habault, J. P. Lefebvre and A. Bergassoli,Acoustics: BasicPhysics, Theory and Methods, Academic Press, New York, 1999.

8. L. J. Fradkin and A. P. Kiselev, The two-component representation of time-harmonic elastic body waves in the high- and intermediatefrequency regimes,J. Acoust. Soc. Amer. 101 (1997), 52–65.


9. A. K. Gautesen, Diffraction of plane waves by a wedge with impedance bound-ary conditions, Wave Motion 41 (2005), 239–246.

10. E. Giladi and J. B. Keller, A hybrid numerical asymptotic method for scatteringproblems, J. Computat. Phys. 174 (2001), 226–247.

11. R. Gunda, S. M. Vijayakar and R. Singh, Method of images for the harmonicresponse of beams and rectangular plates, J. Sound Vib. 185 (1995), 791–808.

12. M. Hornikx and J. Forssn, Noise abatement schemes for shielded canyons,Appl. Acoust. 70 (2009), 267–283.

13. M. S. Howea, A. M. Colgana and T. A. Brungart, On self-noise at the nose ofa supercavitating vehicle, J. Sound Vibr. 322 (2009), 772–784.

14. U. Ingard, it Influence of fluid motion past a plane boundary on sound reflec-tion, absorption and transmission, J. Acoust. Soc. Am. 31 (1959), 1035–1036.

15. D. S. Jones, Aerodynamic sound due to a source near a half plane, J. Inst.Math. Appl. 9 (1972), 114–122.

16. D. S. Jones, Acoustic and Electromagnetic Waves, Clarendon Press, Oxford1986.

17. J. B. Keller, Geometrical theory of diffraction, J. Opt. Soc. Amer. 52 (1962),116–130.

18. J. B. Keller, Wave Propagation and Underwater Acoustics, Lecture NotesPhysics 70, Springer, New York, 1977.

19. H. C. Kranzer and J. Radlow, An asymptotic method for solving perturbedWiener-Hopf problems, J. Math. Mech. 14 (1965), 41–59.

20. U. J. Kurze, Noise reduction by barriers, J. Acoust. Soc. Am. 55 (1974), 504–518.

21. S. K. Lau and S. K. Tang, Performance of a noise barrier within an enclosedspace, Appl. Acoust. 70 (2009), 50–57.

22. K. M. Li, M. K. Law and M. P. Kwok, Absorbent parallel noise barriers inurban environments, J. Sound Vib. 315 (2008), 239–257.

23. R. Makarewicz and M. Zoltowski, Variations of road traffic noise in residentialareas, J. Acoust. Soc. Am. 124 (2008), 3568–3575.

24. J. W. Miles, On the reflection of sound at an interface of relative motion, J.Acoust. Soc. Am. 29 (1957), 226–228.

25. F. G. Mitri, Acoustic radiation force acting on absorbing spherical shells, WaveMotion 43 (2005), 12–19.

26. M. K. Myers, On the acoustic boundary condition in the presence of flow, J.Sound Vib. 71 (1980), 429–434.

27. A. Nayfeh, J. Kaiser and D. Telionis, Acoustics of aircraft engine-duct systems,American Inst. Aeronautics and Astronautics J. 13 (1975), 130–153.

28. B. Noble, Methods Based on the Wiener-Hopf Technique, Pergamon, London,1958.

29. Young-Ho Park and Suk-Yoon Hong, Vibrational power flow models for trans-versely vibrating finite Mindlin plate, J. Sound Vib. 317 (2008), 800–840.

30. N. Peake, On applications of high-frequency asymptotics in aeroacoustics, Phil.Trans. R. Soc. Lond. A 362 (2004), 673–696.

31. A. D. Rawlins, Acoustic diffraction by an absorbing semi-infinite half plane ina moving fluid, Proc. Roy. Soc. Edinburgh Sect. A 72 (1975), 337–357.

32. H. S. Ribner, Reflection, transmission and amplification of sound by a movingmedium, J. Acoust. Soc. Am. 29 (1957), 435–441.

33. O. Runborg, Mathematical models and numerical methods for high frequencywaves, Commun. Comput. Phys. 2 (2007), 827–880.

34. B. D. Seckler and J. B. Keller, Geometrical theory of diffraction in inhomoge-neous media, J. Acoust. Soc. Amer. 31 (1959), 192–205.

35. S. K. Tang and G. C. Y. Lam, On sound propagation from a slanted side branchinto an infinitely long rectangular duct, J. Acoust. Soc. Am. 124 (2008), 1921–1929.


36. M. Tarabini and A. Roure, Modeling of influencing parameters in active noisecontrol on an enclosure wall, J. Sound Vib. 311 (2008), 1325–1339.

37. Y. Z. Umul and U. Yalcin, The effect of impedance boundary conditions onthe potential function of the boundary diffraction wave theory, Opt. Commun.281 (2008), 23–27.

38. R. Wong, Asymptotic Approximations of Integrals, Academic Press, San Diego,CA, 1989.

Department of Mathematics, Faculty of Science,

King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia.

E-mail address: bashir [email protected]

Department of Mathematics, Faculty of Science,

King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia.

E-mail address: dr [email protected]

acoustic diffraction from an absorbing half … · acoustic diffraction from an absorbing half...

Documents