multiple diffraction of a line source field by a three-part thin transmissive slab

$: Multiple Diffraction of a Line Source Field by a Three-Part Thin Transmissive Slab$
ZAMM � Z. Angew. Math. Mech. 78 (1998) 3, 183±±195

Alkumru, A.; Polat, B.

Multiple Diffraction of a Line Source Fieldby a Three-Part Thin Transmissive Slab

A uniform asymptotic high-frequency solution is presented for the problem of diffraction of a line source field by a three-part thin transmissive slab. After simulating the slab by a material plane with a set of approximate boundary conditionsused recently by Rawlins et al., the three-part boundary-value problem is transformed into a modified matrix Wiener-Hopf equation. By performing the factorization of the kernel matrix through the Daniele-Khrapkov method, the mod-ified matrix Wiener-Hopf equation is first reduced to a pair of coupled Fredholm integral equations of the second kindand then solved approximately by iterations. An interesting feature of the present solution is that the classical Wiener-Hopf arguments yield unknown constants which may be determined by means of the edge conditions.

MSC (1991): 45E10, 47B35, 30E25, 34E20, 78A45, 76Q05

1. Introduction

The present work deals with the multiple diffraction of a line source field by a three-part thin transmissive slab. Theslab is simulated by a material plane with a set of approximate boundary conditions under the assumption that thethickness of the slab is small when compared to the source wavelength. These approximate boundary conditions areused by Rawlins et al. [1] to treat the diffraction of a line source field by an acoustically transmissive half-plane. Theaim of this paper is to extend the half-plane problem considered in [1] to the more general case consisting of a three-part plane.

As is well known, diffraction problems related to a three-part plane geometry constitute a mixed boundary-valueproblem which may be transformed into a modified Wiener-Hopf equation (MWHE) of the general form

Q�a� � F1�a�Yÿ�a� � eial F2�a�Y��a� � F3�a� : �1�Here, a is the Fourier transform variable, F1�a�, F2�a�, and F3�a� are known regular functions in a certain striph� < Im�a� < hÿ. The unknown functions Yÿ�a� and Y��a� appearing in (1) are regular in the spectral half-planesIm�a� < hÿ and Im�a� > h�, respectively, whereas Q�a� is an unknown entire function, and l is the width of the stripresiding between two semi-infinite material planes.

The MWHE given by (1) is either a scalar or matrix one due to the boundary conditions to be satisfied on bothfaces of the three-part plane being symmetrical or not, respectively. The solution of this MWHE can always be reducedto the solution of a pair of coupled Fredholm integral equations of the second kind and then be obtained approxi-mately by an iterative approach introduced by Jones [2, Sec. 9.12]. This iterative method was later applied success-fully to the analysis of different kinds of material strips [3, 4, 5, 6, 7].

In order to obtain the solution of the MWHE in (1), one needs to express the kernel functions F1�a� and F2�a�as the product of the functions, say Fÿ1 �a�, F�1 �a�, and Fÿ2 �a�, F�2 �a�, respectively. The functions Fÿ1 �a�, Fÿ2 �a�, andF�1 �a�, F�2 �a� are regular, free of zeros, and of algebraic growth at infinity in the spectral half-planes Im�a� < hÿ andIm�a� < h�, respectively. This procedure is called the Wiener-Hopf factorization. If the kernel functions F1�a� andF2�a� are scalar, then the Wiener-Hopf factorization can easily be done by taking the logarithm of the kernel functionsand writing them as the sum of two functions analytic in the spectral half-planes Im�a� < hÿ and Im�a� < h� with theaid of Cauchy's Theorem. However, this procedure given for the scalar case makes use of the commutative proper-ties of the appropriate factors obtained by taking their exponentials, and of course cannot be generalized to thematrix case which involves non-commutative algebras. Because of this main difficulty in solving matrix MWHE, theWiener-Hopf factorization of an arbitrary kernel matrix still remains at present an open problem. Significant amountof progress has been achieved in the last years for a restricted class of kernel matrices satisfying some propertiessuch as having only branch-point singularities [8, 9], only pole singularities, or pole and branch-point singularities[10, 11, 12, 13].

By considering the approximate boundary conditions derived by Rawlins et al. [1] for the simulation of a trans-missive slab by a material plane and using the Fourier transform technique, the related three-part boundary-valueproblem is reduced to a modified matrix Wiener-Hopf equation (MMWHE) the solution of which requires the factoriza-tion of the kernel matrices as the product of two non-singular matrices the entries of which being regular and of algebraicgrowth at infinity in certain overlapping halves of the complex plane. After performing the Wiener-Hopf factorizationof the kernel matrices through the Daniele-Khrapkov method [10, 11], the MMWHE is first reduced to a pair ofcoupled Fredholm integral equations of the second kind and then solved by iterations. The unknown constants appear-ing in the solution as a result of the classical Wiener-Hopf arguments, are determined by analysing the behaviour of

Alkumru, A.; Polat, B.: Multiple Diffraction of a Line Source Field 183

the field near the edges. Henceforth the uniform asymptotic expressions for the diffracted fields are derived up to andincluding second-order interaction terms. Some numerical results concerning the variation of the singly and doublydiffracted fields with respect to the observation angle are presented for different values of the material parametersrelated to the three-part thin transmissive slab. Note that the boundary conditions used by [14, 15] for simulating apenetrable thin slab give rise to the excitation of surface waves. On the contrary, the approximate boundary conditionsused in this work do not allow to have surface waves.

An eÿiwt time factor, where i � ��ÿ1p

, w is the angular frequency, and t is the time variable, is assumed andsuppressed throughout the paper.

2. Formulation of the boundary-value problem

The geometrical configuration considered in the present paper is illustrated in Fig. 1. The problem consists in studyingthe line source field diffraction by the junctions O and Q of the three-part transmissive plane. The three-part materialplane at y � 0 is illuminated by a time harmonic line source located at x � x0, y � y0 > 0, z 2 �ÿ1; 1�:

The total acoustic field u�x; y� satisfies the Helmholtz equation

@2

@x2� @2

@y2� k2

� �u�x; y� � ÿd�xÿ x0� d�yÿ y0� �2�

in the free space excluding the material plane at y � 0. Here k is the free-space wave number, which is temporarilyassumed to have a small imaginary part. For analysis purposes, u�x; y� is expressed as follows:

u�x; y� �u1�x; y� ; y > y0

u2�x; y� ; 0 < y < y0

u3�x; y� ; y < 0:

8<: �3�

As a solution of (2), it is appropriate to consider the following integral representation:

u1�x; y� ��1ÿ1

A�a� eiK�a� yÿ iax da ; y > y0 ; �4a�

u2�x; y� ��1ÿ1�B�a� eiK�a� y � C�a� eÿiK�a� y� eÿiax da ; 0 < y < y0 ; �4b�

u3�x; y� ��1ÿ1

D�a� eÿiK�a� yÿ iax da ; y < 0 ; �4c�where

K�a� ��k2 ÿ a2p

; �4d�and A�a�, B�a�, C�a�, and D�a� are yet unknown spectral coefficients. The square-root function in (4d) is defined inthe complex a-plane cut as shown in Fig. 2, such that K�0� � k. A�a�, B�a�, C�a�, and D�a� appearing in (4a±c) aredetermined by using the definition of the line source at y � y0 which reads

u1�x; y0� � u2�x; y0� ; @

@yu1�x; y0� ÿ @

@yu2�x; y0� � ÿd�xÿ x0� ; �5a; b�

and the following approximate boundary conditions simulating the three-part material plane at y � 0�1�;u2�x; 0� � s1u3�x; 0� ; x < 0 ; �5c�@

@yu2�x; 0� � t1

@

@yu3�x; 0� ; x < 0 ; �5d�

u2�x; 0� � s2u3�x; 0� ; 0 < x < l ; �5e�

184 ZAMM � Z. Angew. Math. Mech. 78 (1998) 3

Fig. 1. Geometry of the diffraction problem Fig. 2. Branch-cuts and integration lines in the complex plane

@

@yu2�x; 0� � t2

@

@yu3�x; 0� ; 0 < x < l ; �5f�

u2�x; 0� � s3u3�x; 0� ; x > l ; �5g�@

@yu2�x; 0� � t3

@

@yu3�x; 0� ; x > l : �5h�

In (5c±±h) sn, tn n � 1; 2; 3, specify

sn � 1ÿ 2i�~rn=~r� khn sin j0 ; n � 1; 2; 3 ; �6a�and

tn � 1ÿ 2i�~r=~rn� khn��c=cn�2 ÿ cos2 j0�=sin j0� ; n � 1; 2; 3 ; �6b�where ~r, c and ~rn; cn are the density and the velocity of the free space and the simulated three-part thin transmissiveslab with thickness hn, respectively.

In order to obtain a unique solution it is also necessary to take into account the edge conditions at x � 0 andx � l. According to the method described in the Appendix, these conditions are given by

u�x; 0� � O�x~l1� as x! 0 ; �7a�and

u�x; 0� � O��xÿ l�~l3� as x! l : �7b�In (7a, b) ~l1 and ~l3 are

~ln � dnln ; n � 1; 3 ; �7c�where

ln � 1

2pilog

1ÿ en1� en

� �; n � 1; 3 ; �7d�

dn � sign arg1ÿ en1� en

� �� ; n � 1; 3 ; �7e�

with

en ��sn ÿ s2��tn ÿ t2��sn � t2��tn � s2�

s; n � 1; 3 : �7f�

From the Appendix one also concludes that

0 < Re�~ln� < 1

2; n � 1; 3 : �7g�

Substituting (4a, b) into (5a, b) and inverting the resulting integral equations one gets

A�a� ÿB�a� � C�a� eÿ2iK�a� y0 ; �8a�

A�a� ÿB�a� � i

2pK�a� eiax0 ÿ iK�a� y0 ÿ C�a� eÿ2iK�a� y0 : �8b�

From (8a, b) C�a� can be solved to give

C�a� � i

4pK�a� eiax0 � iK�a� y0 : �9�

Consider the Fourier transform of the boundary conditions in (5c±±h), namely

Gÿ�a� � s1Hÿ�a� ; _Gÿ�a� � t1

_Hÿ�a� ; �10a; b�G1�a� � s2H1�a� ; _G1�a� � t2

_H1�a� ; �10c; d�G��a� � s3H

��a� ; _G��a� � t3_H��a� : �10e; f�

In (10a±±f) G�, _G�, H�, _H�, G1, _G1, H1, and _H1 specify

Gÿ�a�Hÿ�a��

� 1

2p

�0ÿ1

u2�x; 0�u3�x; 0��

eiax dx ;G1�a�H1�a��

� 1

2p

�l0

u2�x; 0�u3�x; 0��

eiax dx ; �11a; b�


G��a�H��a��

� 1

2p

�1l

u2�x; 0�u3�x; 0��

eia�xÿ l� dx ;_Gÿ�a�_Hÿ�a�

" #� 1

2p

�0ÿ1

@

@yu2�x; 0�

@

@yu3�x; 0�

26643775 eiax dx ; �11c; d�

_G1�a�_H1�a�

" #� 1

2p

�l0

@

@yu2�x; 0�

@

@yu3�x; 0�

26643775 eiax dx ;

_G��a�_H��a�

" #� 1

2p

�1l

@

@yu2�x; 0�

@

@yu3�x; 0�

26643775 eia�xÿ l� dx : �11e; f�

Owing to the analytical properties of Fourier integrals, the functions Gÿ�a�, Hÿ�a�, _Gÿ�a�, and _Hÿ�a� are regular inthe lower spectral half-plane Im�a� < Im�k�, and the functions G��a�, H��a�, _G��a�, and _H��a� are regular in theupper spectral half-plane Im�a� > Im�ÿk�, while G1�a�, H1�a�, _G1�a�, and _H1�a� are entire functions. By using theedge conditions (7a, b) one can easily show that

Gÿ�a� � O�aÿ~l1 ÿ 1� ; Hÿ�a� � O�aÿ~l1 ÿ 1� ; �12a; b�_Gÿ�a� � O�aÿ~l1� ; _Hÿ�a� � O�aÿ~l1� ; �12c; d�G��a� � O�aÿ~l3 ÿ 1� ; H��a� � O�aÿ~l3 ÿ 1� ; �12e; f�_G��a� � O�aÿ~l3� ; _H��a� � O�aÿ~l3� ; �12g; h�

when a!1 in their respective regions of regularity. From (11a±±f) and (4b, c) one can write

Gÿ�a� �G1�a� � eialG��a� � B�a� � C�a� ; �13a�Hÿ�a� �H1�a� � eialH��a� � D�a� ; �13b�_Gÿ�a� � _G1�a� � eial _G��a� � iK�a��B�a� ÿ C�a�� ; �13c�_Hÿ�a� � _H1�a� � eial _H��a� � ÿiK�a�D�a� : �13d�

The elimination of B�a� and D�a� among (13a±±d) and the use of the relations in (10a±±f) yield the followingMMWHE which is valid in the spectral strip Im �ÿk� < Im�a� < Im�k� :

s2 � t2

t2P1�a� �V1�a�Pÿ�a� � eial V3�a�P��a� � F�a� ; �14a�

where

P1�a� �_G1�a�H1�a�

" #; P��a� �

_G��a�H��a�

" #; �14b; c�

Vn�a� �s2 � tn

tni�s2 ÿ sn�K�a�

i�tn ÿ t2�tnt2K�a�

sn � t2

t2

26643775 ; n � 1; 3 ; �14d�

and

F�a� � 2C�a� ÿiK�a�1=t2

� �: �14e�

3. Solution of the MMWHE

The first step in solving the MMWHE given in (14a) is to factorize the kernel matrix Vn�a� given in (14d) as theproduct of two invertible matrices, say V�n �a�, and Vÿn �a�, the entries of which being regular functions of a withalgebraic behaviour for jaj ! 1 in the upper and lower spectral half-planes, respectively. To this end one writes

Vn�a� � CnWn�a� ; n � 1; 3 ; �15a�where

Cn �s2 � tn

tn0

0sn � t2

t2

264375 ; n � 1; 3 ; �15b�


and

Wn�a� �1

itn�s2 ÿ sn�tn � s2

K�a�i�tn ÿ t2�

tn�sn � t2�K�a� 1

26643775 ; n � 1; 3 : �15c�

The matrix Wn�a� given in (15c) belongs to the class for which the Wiener-Hopf factorization can be accomplishedthrough the Daniele-Khrapkov method [10, 11] as

W�n �a� � �1ÿ e2

n�1=4cosh cn�a� tnbng�a� sinh cn�a�sinh cn�a�tnbng�a� cosh cn�a�

264375 ; n � 1; 3 ; �16a�

Wÿn �a� �W�

n �ÿa� ; n � 1; 3 ; �16b�with bn, g�a�, cn�a�, and cn�ÿa� being defined by

bn ��sn ÿ s2��sn � t2��tn � s2��tn ÿ t2�

s; n � 1; 3 ; �16c�

g�a� ��a2 ÿ k2p

� ÿiK�a� ; �16d�cn�a� � ÿiln arccos

a

k; n � 1; 3 ; �16e�

cn�ÿa� � ÿiln p ÿ arccosa

k

h i; n � 1; 3 : �16f�

Considering the known asymptotics it can be easily shown that one has

W�n �a� '

��a�~ln tnbndn��a�~ln � 1

dntnbn

��a��ln ÿ 1 ��a�~ln

264375 ; n � 1; 3 ; �17�

for jaj ! 1 in the upper and lower spectral half-planes, respectively.Now by using the method described in [2], the MMWHE given in (14a) can be reduced to the following pair of

coupled Fredholm type integral equations of the second kind:

Wÿ1 �a�Pÿ�a� �

1

2pi

�lÿ

�C1W�1 �x��ÿ1 V3�x�P��x� eixl

dx

xÿ a

ÿ 1

2pi

�lÿ

�C1W�1 �x��ÿ1 F�x� dx

xÿ a�R ; �18a�

W�3 �a�P��a� � ÿ

1

2pi

�l�

�C3Wÿ3 �x��ÿ1 V1�x�Pÿ�x� eÿixl

dx

xÿ a

� 1

2pi

�l�

�C3Wÿ3 �x��ÿ1 F�x� eÿixl

dx

xÿ a� S ; �18b�

with

R � r 1

0

� �; S � s 1

0

� �: �19a; b�

R and S appearing in (18a) and (18b) are yet unknown constant vectors resulting from the application of Liouville'stheorem in the Wiener-Hopf procedure. In (18a, b), P��a� and Pÿ�a� are regular in the upper and lower spectral half-planes, respectively, and the paths of integration, l� and lÿ, are depicted in Fig. 2.

The solutions of the integral equations (18a, b) can be obtained by iterations. When l is large, the free termsappearing on the right-hand side of (18a) and (18b) are the first order solutions. The substitution of the first ordersolutions instead of the unknown functions appearing in the integrands of (18a, b) gives the second order solutions.Thus one writes

Pÿ�a� � Pÿ�1��a� �Pÿ�2��a� � . . . ; �20a�P��a� � P��1��a� �P��2��a� � . . . ; �20b�


where

Pÿ�j��a� � �Wÿ1 �a��ÿ1 �I�j��a� �R�j�� ; j � 1; 2; . . . ; �21a�

P��j��a� � �W�3 �a��ÿ1 �J�j��a� � S�j�� ; j � 1; 2; . . . : �21b�

Here it is assumed that the constant vectors R and S can be expressed as

R � R�1� �R�2� � . . . ; S � S�1� � S�2� � . . . : �21c; d�In (21a, b), I�j��a� and J�j��a� for j � 1; 2 are given by

I�1��a� � ÿ 1

2pi

�lÿ

�C1W�1 �x��ÿ1 F�x� dx

xÿ a; �22a�

J�1��a� � 1

2pi

�l�

�C3Wÿ3 �x��ÿ1 F�x� eÿixl

dx

xÿ a; �22b�

and

I�2��a� � 1

2pi

�lÿ

�C1W�1 �x��ÿ1 V3�x�P��1��x� eixl

dx

xÿ a; �22c�

J�2��a� � ÿ 1

2pi

�l�

�C3Wÿ3 �x��ÿ1 V1�x�Pÿ�1��x� eÿixl

dx

xÿ a: �22d�

Making the substitutions x � ÿk cos h, x0 � r0 cos j0, and y0 � r0 sin j0, (22a) passes into

I�1��a� � 1

�2p�2�G

�C1W�1 �ÿk cos h��ÿ1 ÿik sin h

1=t2

� �eÿikr0 cos�h�j0�

a� k cos hdh ; �23�

where G is the integration line depicted in Fig. 3. For kr0 � 1, (23) can be evaluated asymptotically through thesaddle point method. During the deformation of G onto the steepest descent path (SDP) passing through the saddlepoint hs � p ÿ j0, one may cross the pole occurring at hp � arccos �ÿa=k� (see Fig. 3). The residue contribution fromthis pole is given by

I�1�res�a� �1

2p�C1W

�1 �a��ÿ1

1i

t2K�a�

24 35 eir0�a cos j0 �K�a� sin j0� H�aÿ k cos j0� ; �24a�

where H denotes the unit step function. The dominant contribution to the integral on the SDP comes from near thesaddle point, and one obtains

I�1�d �a� '

eÿip=4

�2p�3=2

�C1W�1 �k cos j0��ÿ1

aÿ k cos j0

ÿik sin j0

1=t2

� �eikr0��kr0

p ; kr0 � 1 : �24b�

Hence, I�1��a� can be approximated as

I�1��a� ' I�1�res�a� � I�1�d �a� : �24c�


Fig. 3. The integration line G and the steepest descent path in the complex h-plane

By proceeding similarly, the expression of J�1��a� given by (22b) can be obtained as

J�1��a� � J�1�d �a� �

1

2p�C3W

ÿ3 �a��ÿ1

1i

t2K�a�

24 35 eir0 �a cos w0 �K�a� sin w0� H�k cos w0 ÿ a� �25a�

with

J�1��d��a� ' ÿ

eÿip=4

�2p�3=2�C3W

ÿ3 �k cos w0��ÿ1

aÿ k cos w0

ÿik sin w0

1=t2

� �eikr0��kr0

p ; kr0 � 1 : �25b�

In (25a, b) �r0;w0� are the cylindrical polar coordinates measured from the junction Q (see Fig. 1). Substituting (14d)in (22c), I�2��a� can be rearranged as follows:

I�2��a� � 1

2pi

�lÿ

�C1W�1 �x��ÿ1

s2 � t3

t3i�s2 ÿ s3�K�x�

i�t3 ÿ t2�t3t2K�x�

s3 � t2

t2

26643775P��1��x� eixl

dx

�xÿ a� : �26�

Since l > 0, according to Jordan's lemma, the integration line lÿ in (26) can be shifted onto the branch cut c�1 � cÿ1lying in the upper spectral half-plane (see Fig. 2), and I�2��a� can now be expressed as

I�2��a� � 1

aÿ k cos j0

�c�1

M�x� �xÿ k�ÿ1=2

�xÿ k�1=2" #

1

xÿ aÿ 1

xÿ k cos j0

� �eixl dx ; �27a�

where

M�a� � 1

pi�C1W

�1 �a��ÿ1

0 �s3 ÿ s2��a� kp

~P��1� 2�a�t3 ÿ t2

t2t3

~P��1� 1�a��a� kp 0

264375 �27b�

is a matrix valued function regular in the upper half-plane with the definitions

~P��1� 1�a� � �aÿ k cos j0� P��1� 1�a� ; ~P��1� 2�a� � �aÿ k cos j0� P��1� 2�a� : �27c; d�

In (27c, d), P��1� 1�a� and P��1� 2�a� are the components of the vector

P��1��a� �P��1� 1�a�P��1� 2�a�

" #: �27e�

Substituting x � t� k in (27a), this integral can be transformed into the following one written along the positive realaxis to get

I�2��a� � eikl

aÿ k cos j0

�10

M�t� k� tÿ1=2

t1=2

� �1

t� �kÿ a� ÿ1

t� k�1ÿ cos j0��

eitl dt : �28�

When the acoustical width kl of the transmissive slab is large �kl� 1�, the main contribution to the integral in(28) comes from a vicinity of the end point t � 0. Thus, M�t� k� can be taken outside the integral by assigning itsvalue at t � 0. The resulting integral can be expressed in terms of the following modified Fresnel integral:

F �z� � ÿ2i��zp

eÿiz�1��zp eit

2

dt �29�

to give

I�2��a� ��kpp eikl��

klp eip=4

aÿ k cos j0

M�k�F �kl�1ÿ a=k��

kÿ aÿ F �kl�1ÿ cos j0��

k�1ÿ cos j0�F �kl�1ÿ cos j0�� ÿ F �kl�1ÿ a=k��

24 35 : �30�

By using a similar procedure, J�2��a� in (22d) can be obtained as

J�2��a� ��kpp eikl��

klp ei3p=4

aÿ k cos j0

N�ÿk�F �kl�1� a=k��

k� aÿ F �kl�1� cos w0��

k�1� cos w0�F �kl�1� a=k�� ÿ F �kl�1� cos w0��

24 35 ; �31a�


where

N�a� � ÿ 1

pi�C3W

ÿ3 �a��ÿ1

0 �s2 ÿ s1��kÿ ap

~Pÿ�1� 2�a�t1 ÿ t2

t1t2

~Pÿ�1� 1�a��kÿ ap 0

264375 �31b�

with

~Pÿ�1� 1�a� � �aÿ k cos w0� Pÿ�1� 1�a� ; ~Pÿ�1� 2�a� � �aÿ k cos w0� Pÿ�1� 2�a� : �31c; d�

In (31c, d), Pÿ�1� 1�a� and Pÿ�1� 2�a� are the components of the vector

Pÿ�1��a� �Pÿ�1� 1�a�Pÿ�1� 2�a�

" #: �31e�

4. Determination of the constants

The unknown constant vectors R�j� and S�j� appearing in (21a, b) can now be determined by using the edge conditionsgiven by (7a) and (7b). From (14c) and (12b, c, f, g) one has

Pÿ�j� '�ÿa�ÿ~l1

�ÿa�ÿ~l1 ÿ 1

24 35 ; P��j� 'aÿ~l3

aÿ~l3 ÿ 1

" #; j � 1; 2; . . . ; �32a; b�

as jaj ! 1 in their respective regions of regularity. Letting jaj ! 1 in (21a) and (21b) gives

Pÿ�j� ' Const:

�ÿa�~l1 ÿ 1 � �ÿa�~l1 �r�j� � d1t1b1~I�j�2 �

�ÿa�~l1 ÿ 2 � �ÿa�~l1 ÿ 1 d1

t1b1r�j� � ~I

�j�2

� �264375 ; j � 1; 2; . . . ; �33a�

and

P��j� ' Const:

a~l3 ÿ 1 � a

~l3 �s�j� ÿ d3t3b3~J�j�2 �

a~l3 ÿ 2 � a

~l3 ÿ 1 ÿ d3

t3b3s�j� � ~J

�j�2

� �264375 ; j � 1; 2; . . . : �33b�

In (33a, b) ~I�j�2 and ~J

�j�2 are defined by

~I�j�2 � lim

a!1 aI�j�2 �a� ; ~J

�j�2 � lim

a!1 aJ�j�2 �a� ; j � 1; 2; . . . ; �33c; d�

where I�j�2 �a� and J

�j�2 �a� are the second rows of the vectors I�j��a� and J�j��a�, respectively. Considering (7g), the

admissible asymptotic behaviour of P��j��a� given by (32a, b) is satisfied by choosing the constants r�j� and s�j� in(33a, b) as

r�j� � ÿd1t1b1~I�j�2 ; s�j� � d3t3b3

~J�j�2 ; j � 1; 2; . . . : �34a; b�

5. Analysis of the diffracted fields

The substitution of (10a, f) in (13a, d) and the elimination of the entire functions H1�a� and _G1�a� between (13a, b)and (13c, d), respectively, give the expressions of the spectral coefficients B�a�, D�a� in terms of P��j��a� as follows:

B�a�D�a��

� P1j� 1

B�j��a�D�j��a��

� P1j� 1

�L1�a�Pÿ�j��a� � eial L3�a�P��j��a�� C�a�

s2 � t2

s2 ÿ t2

2

h i; �35a�

where

Ln�a� � 1

s2 � t2

s2�tn ÿ t2�itnK�a� t2�sn ÿ s2�tn ÿ t2

itnK�a� sn ÿ s2

26643775 ; n � 1; 3 : �35b�


Once B�a� is obtained from (35a), by using (8a) or (8b) it can be shown that

A�a� � P1j� 1

A�j��a� � P1j� 1

B�j��a� � i

2K�a� eiax0 ÿ iK�a� y0 : �36�

Thus, the spectral coefficients appearing in (4a±±c) are determined. Note that in (35a) the first and second terms inparanthesis are related to the j-th diffracted fields emanating from the junctions O and Q, respectively.

5.1. Singly diffracted fields

Consider first the singly diffracted field by the edge O, say u�1�O , and replace the terms corresponding to j � 1 appear-

ing in (35a) and (36) into (4a±±c). The asymptotic evaluation of the resulting integrals through the steeped descentpath (SDP) method gives

u�1�O �r; j� �

��2pp

eÿi3p=4�1ÿ e21�ÿ1=4

s2 � t2

eikr��kr

p t�1�1 �j; j0� ;t�1�2 �j; j0� ;

0 < j < p ;

ÿp < j < 0 ;

(�37a�

where �r; j� are the cylindrical polar coordinates measured from O, and t�1�1; 2�j; j0� specify

t�1�1 �j; j0� � �I�1�d1�ÿk cos j� � r�1�� s2�t1 ÿ t2�

t1cos�l1j� ÿ it2�s1 ÿ s2�

t1b1sin�l1j�

� �� k sin jI

�1�d2�ÿk cos j� �b1s2�t1 ÿ t2� sin�l1j� � it2�s1 ÿ s2� cos�l1j�� ; �37b�

t�1�2 �j; j0� � �I�1�d1�ÿk cos j� � r�1�� t1 ÿ t2

t1cos�l1j� ÿ i�s1 ÿ s2�

t1b1sin�l1j�

� �� k sin jI

�1�d2�ÿk cos j� �b1�t1 ÿ t2� sin�l1j� � i�s1 ÿ s2� cos�l1�2p ÿ j�� : �37c�

By using (24b), I�1�d1�ÿk cos j� and I

�1�d2�ÿk cos j� appearing in (37b, c) read

I�1�d1�ÿk cos j� � t1�1ÿ e2

1�ÿ1=4 eÿip=4 sin j0

�2p�3=2�s1 � t2��s2 � t1��cos j� cos j0�eikr0��kr0

p� �i�s1 � t2� cos�l1j0� ÿ b1�s2 � t1� sin�l1j0�� ; �38a�

I�1�d2�ÿk cos j� � ÿ �1ÿ e2

1�ÿ1=4 eÿip=4

�2p�3=2 k�s1 � t2��s2 � t1��cos j� cos j0�eikr0��kr0

p� �s2 � t1� cos�l1j0� �

i�s1 � t2�b1

sin�l1j0��

: �38b�

A similar analysis shows that the singly diffracted field by the edge Q, say u�1�Q , can be obtained from (37a±±c) and

(38a, b) by making the following substitutions:

e1 ! e3 ; r! r ; j! p ÿ w ; j0 ! p ÿ w0 ; r�1� ! s�1� ;

s1 ! s3 ; t1 ! t3 ; b1 ! b3 ; l1 ! l3 ; r0 ! r0 ;

where r and w 2 �0; 2p� are the cylindrical polar coordinates measured from Q.

5.2. Doubly diffracted fields

The doubly diffracted field by the junction O, say u�2�O , can be obtained by substituting the terms corresponding to

j � 2 appearing in (35a) and (36) into (4a±±c), and then evaluating the resulting integrals asymptotically via the SDPmethod. Thus, we get

u�2�O �r; j� �

��2pp

eÿi3p=4�1ÿ e21�ÿ1=4

s2 � t2

eikr��kr

p t�2�1 �j; j0� ;t�2�2 �j; j0� ;

0 < j < p ;

ÿp < j < 0 ;

(�39a�

where t�2�1; 2�j; j0� are defined by

t�2�1 �j; j0� � �I�2�1 �ÿk cos j� � r�2�� s2�t1 ÿ t2�t1

cos�l1j� ÿ it2�s1 ÿ s2�t1b1

sin�l1j��

� k sin jI�2�2 �ÿk cos j� �b1s2�t1 ÿ t2� sin�l1j� � it2�s1 ÿ s2� cos�l1j�� ; �39b�


t�2�2 �j; j0� � �I�2�1 �ÿk cos j� � r�2�� t1 ÿ t2

t1cos�l1j� ÿ i�s1 ÿ s2�

t1b1sin�l1j�

� �� k sin jI

�2�2 �ÿk cos j� �b1�t1 ÿ t2� sin�l1j� � i�s1 ÿ s2� cos�l1j�� ; �39c�

with

I�2�1 �ÿk cos j� � ei3p=4��

2pp eikl��

klp 2kt1�s3 ÿ s2��1ÿ e2

1�ÿ1=4�1ÿ e23�ÿ1=4

s2 � t1

1ÿ cos j0

cos j� cos j0

� l3

t3b3k�J �1�d1�k� � s�1�� J�1�d2

�k��

�F �kl�1ÿ cos j0�� ÿ F �kl�1� cos j0�� ; �39d�

I�2�2 �ÿk cos j� � ei3p=4��

2pp eikl��

klp �1ÿ e2

1�ÿ1=4�1ÿ e23�ÿ1=4

�s2 � t1��s1 � t2�1ÿ cos j0

cos j� cos j0

� 2l1�s1 � t2��s3 ÿ s2�b1

l3

t3b3k�J�1�d1�k� � s�1�� J �1�d2

�k��

� �F �kl�1ÿ cos j0�� ÿ F �kl�1� cos j0�� s2 � t1��t3 ÿ t2�

t3�J�1�d1�k� � s�1��

� F �kl�1� cos j0��k�1ÿ cos j� ÿ F �kl�1ÿ cos j0��

k�1ÿ cos j0��

; �39e�

From (25b), J�1�d1�k� and J

�1�d2�k� appearing in (39d, e) are calculated as

J�1�d1�k� � �1ÿ e2

3�ÿ1=4 t3 eÿip=4 sin w0

�2p�3=2�s2 � t3��s3 � t2��1ÿ cos w0�eikr0��kr0

p

� �i�s3 � t2� cos�l3�p ÿ w0�� ÿ b3�s2 � t3� sin�l3�p ÿ w0�� ; �40a�

J�1�d2�k� � ÿ �1ÿ e2

3�ÿ1=4 eÿip=4

�2p�3=2 k�s2 � t3��s3 � t2��1ÿ cos w0�eikr0��kr0

p

� �s2 � t3� cos�l3�p ÿ w0�� i�s3 � t2�

b3sin�l3�p ÿ w0��

� �: �40b�


Fig. 4. 20 logju�1�O � u�1�Q j versus the observation angle j

As for the singly diffracted fields, the doubly diffracted field by the edge Q, say u�2�Q , can be derived from the

above results by making the following substitutions:

e1 , e3 ; l1 , l3 ; b1 , b3 ; s1 , s3 ; t1 , t3 ; j0 , p ÿ w0 ;

r�2� ! s�2� ; s�1� ! r�1� ; r! r ; r0 ! r0 ; j! p ÿ w :

Fig. 4 and Fig. 5 show the variations of the singly and doubly diffracted fields, l being the wavelength(20 logju�1�O � u�1�Q j and 20 logju�2�O � u�2�Q j), respectively, for different values of the slab densities. In these graphical solu-tions we take l � 2l, j0 � 90�, r0 � 10l, r � 20l, c0 � 1, ~r0 � 1, ~r1 � 2, ~r3 � 3, c1 � 3, c2 � 5, c3 � 4, h1 � 0:001l,h2 � 0:003l, h3 � 0:002l. The total amplitudes of the singly and doubly diffracted fields increase with the increasingvalues of the density ~r2. It is seen that the contribution of the doubly diffracted fields to the total field are negligibly small.

Appendix: Derivation of the edge conditions

In this appendix we present the derivation of the expressions given by (7a±±g) which shows the behaviour of the totalacoustic field u�x; y� near the junctions O and Q of the three-part transmissive plane (see Fig. 1). These edge condi-tions will be obtained by using the Meixner procedure [16] for a two-part plane illustrated in Fig. 6. To this end, oneat first writes the Helmholtz equation

rT@

@rTrT

@u

@rT

� �� @2u

@j2T

� �krT �2 u � 0 �A1�

satisfied by the total field u�rT ; jT � with rT and jT 2 �ÿp; p� being the cylindrical polar coordinates measured fromT (see Fig. 6). Since krT ! 0 at the neighborhood of the edge T , (A1) can be reduced to the Laplace equation

Du�rT ; jT � � rT@

@rTrT

@u

@rT

� �� @2u

@j2T

� 0 ; �A2�

which will introduce the asymptotic solution for u�rT ; jT � as krT ! 0. By using the method of separation of variablesthe solution of (A2) can be given as:

u�rT ; jT � �A1jT �B1 � �C1 cos�~ljT � �D1 sin�~ljT ��rT �

~l ; 0 < jT < p ;

A2jT �B2 � �C2 cos�~ljT � �D2 sin�~ljT ��rT �~l ; ÿp < jT < 0 ;

(�A3�


Fig. 5. 20 logju�2�O � u�2�Q j versus the observation angle j

Fig. 6. Geometry of the two-part material plane

14 Z. Angew. Math. Mech., Bd. 78, H. 3

where A1; 2, B1; 2, C1; 2, D1; 2, and ~l are the unknown coefficients which will be determined by considering the followingapproximate boundary conditions related to the two-part material plane located at y � 0 (see Fig. 6):

u�rT ; p� � sLu�rT ; ÿp� ; @

@jT

u�rT ; p� � tL@

@jT

u�rT ; ÿp� ; �A4a; b�

u�rT ; �0� � sRu�rT ; ÿ0� ; @

@jT

u�rT ; �0� � tR@

@jT

u�rT ; ÿ0� : �A4c; d�

Substituting first (A3) into (A4c, d) one gets

A1 � tRA2 ; B1 � sRB2 ; �A5a; b�and

C1 � sRC2 ; D1 � tRD2 : �A5c; d�The substitution of (A3) into (A4a, b) and using (A5a±±d) yields

p�sL � sR� sR ÿ sL

tR ÿ tL 0

� �A2

B2

� �� sR ÿ sL� cos�~lp� �tR � sL� sin�~lp�ÿ~l�sR � tL� sin�~lp� ~l�tR ÿ sL� cos�~lp�

" #C2

D2

� ��rT �

~l � 0 : �A6�

The solution of (A6) can only be obtained as

�sR ÿ sL��tR ÿ tL� � 0 or A2 � B2 � 0 ; �A7a�and

tan2�~lp� � ÿe2 or C2 � D2 � 0 �A7b�with

e ��sL ÿ sR��tL ÿ tR��sL � tR��tL � sR�

s: �A8�

Since sL 6� sR and tL 6� tR, the unique possible solution of (A7a) is the trivial one,

A2 � B2 � 0 : �A9�On the other hand, in order to have the non-trivial solution of (A6), it is required that C2 6� 0 and D2 6� 0.

Thus, by also considering that the acoustic energy in any finite domain of the edge must be bounded (Re�~l� > 0� [2],(A7b) can only be satisfied for

~l � dl ; �A10a�where

d � sign arg1ÿ e

1� e

� �� ; �A10b�

and

l � 1

2pilog

1ÿ e

1� e

� �: �A10c�

From (A10a±±c) one concludes that

0 < Re�~l� < 1

2: �A11�

Substituting (A5a±±d) and (A10a) with (A9) into the expression of u�rT ; jT � given by (A3), the behaviour of the totalacoustic field at y � 0 near the edge T can be easily obtained as

u�x; 0� � O��xÿ t�~l� as x! t : �A12�

References

1 Rawlins, A. D.; Meister, E.; Speck, F. O.: Diffraction by an acoustically transmissive or an electromagnetically dielectric halfplane. Math. Meth. Appl. Sci. 14 (1991), 387±±402.

2 Jones, D. S.: The theory of electromagnetism. Pergamon Press 1964.3 Kobayashi, K.: Plane wave diffraction by a strip: Exact and asymptotic solutions. J. Phys. Soc. Japan 60 (1991), 1891±±1905.4 B�uy�ukaksoy, A.; Uzg�oren, G.: Secondary diffraction of a plane wave by a metallic wide residing on the plane interface of two

dielectric media. Radio Sci. 22 (1987), 183±±191.


5 Serbest, A. H.; Uzg�oren, G.; B�uy�ukaksoy, A.: Diffraction of plane waves by a resistive strip residing between two impe-dance half-planes. Ann. Telecommun. 46 (1991), 359±±366.

6 B�uy�ukaksoy, A.; Alkumru, A.: Multiple diffraction of plane waves by an acoustically penetrable strip located between twosoft/hard half-planes. Internat. J. Engng. Sci. 32 (1994), 779±±789.

7 B�uy�ukaksoy, A.; Alkumru, A.: Multiple diffraction of plane waves by a soft/hard strip. J. Engng. Math. 29 (1995), 105±±120.8 Hurd, R. A.: The Wiener-Hopf-Hilbert method for diffraction problems. Canadian J. Phys. 54 (1976), 775±±780.9 Rawlins, A. D.; Williams, W. E.: Matrix Wiener-Hopf factorisation. Quart. J. Mech. Appl. Math. 34 (1981), 1±±8.

10 Daniele, V. G.: On the factorization of Wiener-Hopf matrices in problems solvable with Hurd's method. IEEE Trans. AP. 26(1978), 614±±616.

11 Khrapkov, A. A.: Certain cases of the elastic equilibrium of an infinite wedge with a nonsymmetric nodge at the vertex, sub-jected to concentrated forces. Prikl. Math. Mekh. 35 (1971), 625±±637.

12 Rawlins, A. D.: A note on the factorization of matrices occurring in Wiener-Hopf problems. IEEE Trans. AP. 28 (1980),933±±934.

13 Jones, D. S.: Factorization of a Wiener-Hopf matrix. IMA J. Appl. Math. 32 (1984), 211±±220.14 Anderson, I.: Plane wave diffraction by a thin dielectric half-plane. IEEE Trans. AP. 27 (1979), 584±±589.15 Volakis, J. L.; Senior, T. B. A.: Diffraction by a thin dielectric half-plane. IEEE Trans. AP. 35 (1987), 1483±±1487.16 Meixner, J.: The behaviour of electromagnetic fields at edges. New York University Research Report. EM-72, 1954.

Received February 5, 1996, revised January 13, 1997, accepted April 14, 1997

Addresses: Dr. Burak Polat, Electronics Engineering Department, Istanbul University, TR-34850 Avcõllar, Istanbul, Turkey;Dr. Ali Alkumru, Electronics Engineering, Gebze Institute of Technology, TR-41400 Gebze, Kocaeli, Turkey


15 Z. Angew. Math. Mech., Bd. 78, H. 3

multiple diffraction of a line source field by a three-part thin transmissive slab

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