Wave scattering by soft–hard three spaced waveguide

Applied Mathematical Modelling xxx (2014) xxx–xxx

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Applied Mathematical Modelling

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Wave scattering by soft–hard three spaced waveguide

http://dx.doi.org/10.1016/j.apm.2014.03.0040307-904X/� 2014 Elsevier Inc. All rights reserved.

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Please cite this article in press as: Mahmood-ul-Hassan, Wave scattering by soft–hard three spaced waveguide, Appl. Math.(2014), http://dx.doi.org/10.1016/j.apm.2014.03.004

Mahmood-ul-Hassan ⇑Dept. of Math, COMSATS Institute of Information Technology, Park Road, Chak Shahzad, Islamabad, Pakistan

a r t i c l e i n f o

Article history:Received 11 July 2012Received in revised form 10 October 2013Accepted 11 March 2014Available online xxxx

Keywords:DuctWaveguideEigenvaluesEigenfunctionScatteringReflection

a b s t r a c t

In this work we consider the sound radiation of a fundamental plane wave mode from asemi-infinite soft–hard duct. This duct is symmetrically located inside an infinite duct. Thisinfinite waveguide consist of soft and hard plates. The whole system constitutes a threespaced waveguide. A closed form solution is obtained by using eigenfunction expansionmatching method. This particular problem has been solved previously by Rawlins in closedform but without numerical work. Here the numerical results for reflection coefficient aregiven when the lowest mode propagates out from the semi-infinite duct. At the end wegive comparison to both methods.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

Unpleasant noise is an unwanted major pollutant in an industrialised society. The unwanted noise from exhaust systemof transportation vehicles are major sources of noise pollution. One way to reduce noise is to design novel geometries, andthe use of various sound absorbent materials for exhaust system. The scattering problems of acoustic waves by half planesatisfying soft, hard and impedance boundary conditions are important in the diffraction theory and have been investigatedby many researchers (see [1–7]). The case of non-staggered half planes in three equal space were firstly studied by Jones [8].His problem ended in 3� 3 matrix Wiener Hopf equations. The equation is reduced first to a system of 2� 2 matrix WienerHopf equations with a single scalar. In [8] an explicit solution and various properties of scattered field for a plane wave to thescattering problem are obtained. One should note that it is not always possible to solve problems involving 2� 2 matricesusing Wiener–Hopf technique because the original decomposition is not available in the literature.

Although some trifurcated waveguide problems has been solved by using standard scalar Wiener Hopf technique with theresult of symmetry. For example, Rawlins [1], a mathematical model to predict noise in an exhaust system was proposed.This model resulted in Wiener Hopf problem having a matrix to be factorized.

Mahmood-ul-Hassan and Rawlins [9] have also used the Wiener–Hopf technique very successfully for solving two prob-lems of radiation of sound in the trifurcated ducts. They have considered the compressible fluid in motion with constant fluidvelocity. They have presented their solution in the form of contour integrals which are evaluated in terms of infinite series ofmodes. These infinite series of modes are propagating in the trifurcated waveguide. Numerical results have been presentedfor reflection field amplitude for particular values of the ratio of the fluid velocity and the speed of sound. Both particularmodels can be used as the possible practical exhaust splitter plates models.

Modell.

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2 Mahmood-ul-Hassan / Applied Mathematical Modelling xxx (2014) xxx–xxx

The present trifurcated waveguide problem has been solved by Rawlins. He published this problem as a technical reportin [1] which was subsequently published as part of the paper [2]. Rawlins ended this problem in Wiener Hopf problem hav-ing non-trivial matrix. This matrix Wiener Hopf problem has been solved explicitly. He has presented closed form solution in[1,2] but has involved complicated factor/split functions. There is no numerical work for reflection coefficient in [1,2] whichmay be due to these complicated factors. In this paper, we solve this problem by using eigenfunction expansion matchingmethod. This method is straightforward and easy as compared with the Wiener–Hopf equations technique. Graphical resultsare given in terms of reflection coefficient.

We divide our problem into four regions. The value of potential in different regions can be found by using separation ofvariables since the geometry is simple. The matched eigenfunction expansions [10] can then be used to obtained the solu-tion. This method have been employed in many scattering problems (see [11–15]), and we use this method in this work. Itshould be noted that rapid convergence of this method is somewhat reduced for the sharp edges of the semi-infinite plates.At the end, we give analysis on the numerical results by plotting jRj versus the wave number ka when the lowest mode isassumed to propagate. This would be helpful in modelling the exhaust practical system.

2. Three spaced waveguide problem

We consider the acoustic diffraction of a plane wave mode which propagates out of the open end of a semi-infinite duct.Fig. 1 shows the geometry of the problem. The geometry of the trifurcated waveguide problem is such that the semi-infiniteduct consists of soft and hard plates. This semi-infinite duct is symmetrically located inside the infinite duct having soft andhard boundary conditions on the plates. The sound source field, which is located at y ¼ y0; x ¼ x0 ðx0 < 0; �a < y0 < aÞ andpropagate modes across the semi-infinite duct. We define the acoustic pressure of the potential function uðx; y; tÞ by

Please(2014

P ¼ �qo@u@t

;

where qo is the density in the equilibrium state. We define the velocity by

u!¼ gradðuÞ:

The incident sound field is assumed to have time variation e�iwt with the wave number k ¼ wc ; w is angular frequency and c is

the speed of sound.We shall remove the time from the problem by writing

uðx; y; tÞ ¼ Re½/ðx; yÞe�iwt�; ð1Þ

in the rest of the work. Now, by using (1) with the wave equation given by

r2u ¼ ð1=c2Þutt; ð2Þ

these two equations give

r2/ðx; yÞ þ k2/ðx; yÞ ¼ 0; ð3Þ

which is the two dimensional Helmholtz equation to be solved for /ðx; yÞ in the trifurcated waveguide system with theboundary conditions:

/ðx; yÞ ¼ 0; y ¼ b; �1 < x <1; ð4Þ

Fig. 1. Geometry of three spaced duct.

cite this article in press as: Mahmood-ul-Hassan, Wave scattering by soft–hard three spaced waveguide, Appl. Math. Modell.), http://dx.doi.org/10.1016/j.apm.2014.03.004


Mahmood-ul-Hassan / Applied Mathematical Modelling xxx (2014) xxx–xxx 3

Please(2014

/ðx; yÞ ¼ 0; y ¼ a; �1 < x < 0; ð5Þ

/yðx; yÞ ¼ 0; y ¼ �a; �1 < x < 0; ð6Þ

/yðx; yÞ ¼ 0; y ¼ �b; �1 < x <1; ð7Þ

where b > a. In addition to that, we have the radiation condition which insures the solution to be bounded or any wave to beout going, and the edge conditions

/ðx;þaÞ ¼ Oðx12Þ as x! 0 ð8Þ

and

/yðx;þaÞ ¼ Oðx�12Þ as x! 0 ð9Þ

at the end of the semi infinite plates.

3. Solution of the waveguide problem

We divide our problem into four regions (see Fig. 1). We will use these regions in the solution process.

3.1. Region 1 fa 6 y 6 b; x < 0g

In Region 1, we follow the standard eigenfunction expansion [10]. We define eigenvalues an by

an ¼np

b� a; n ¼ 1;2; . . . ; ð10Þ

The eigenvalues an satisfy the equation

sinaðy� bÞ ¼ 0:

We define the vertical eigenfunctions in Region 1 by

wnðyÞ ¼ffiffiffiffiffiffiffiffiffiffiffi

2b� a

rsin anðy� aÞ: ð11Þ

The functions wnðyÞ satisfy the orthonormal relation
Z b
awmðyÞwnðyÞdy ¼ dmn; ð12Þ

where dmn is the Kronecker delta defined by

dmn ¼1; m ¼ n;0; m – n

�

and functions wnðyÞ form the complete set over the interval a; b½ �. A general form for the potential /ðx; yÞ which satisfies (3)–(5) is thus

/ðx; yÞ ¼X1n¼1

Ane�ianxwnðyÞ; ð13Þ

where

a1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � p

b� a

� �2r

;

a2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � 2p

b� a

� �2s

;

..

.

an ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � np

b� a

� �2r

; n ¼ 1;2; . . .

with 0 < Rea1 < Rea2 < � � �.




3.2. Region 2 f�a 6 y 6 a; x < 0g

In Region 2, we find the potential by separating the variables

Please(2014

/ðx; yÞ ¼ eib1xf1ðyÞ þX1n¼1

Bne�ibnxfnðyÞ; ð14Þ

which satisfies (3), (5) and (6), and eib1xf1ðyÞ is the incident wave from x �!�1 travelling from left to right. In (14) b2n�1 arethe eigenvalues, given by

b2n�1 ¼ð2n� 1Þp

4a; n ¼ 1;2; . . . ; ð15Þ

which are the roots of the equation

cosð2abÞ ¼ 0:

We define the vertical eigenfunctions in Region 2, as

fnðyÞ ¼cos b2n�1ðyþ aÞffiffiffi

ap ð16Þ

and they satisfies the orthonormal relation
Z a
�afnðyÞfmðyÞdy ¼ dmn: ð17Þ

In (14)

b1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � p

4a

� �2r

;

b2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � 3p

4a

� �2s

;

..

.

b2n�1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � ð2n� 1Þp

4a

� �2s

; n ¼ 1;2; . . . ;

with 0 < Reb1 < Reb2 < . . ..If we restrict

p4a

< k <3p4a

;

then the only lowest incident and reflected mode in Region 2 can be sustained by having b1 > 0 and b2 ¼ iffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

3p4a

� 2 � k2q

. Thusfor n > 1, we have Imbn > 0 and Rebn ¼ 0.

3.3. Region 3f�b 6 y 6 �a; x < 0g

In this, we solve (3) subject to the conditions given by (6) and (7) by using the separation of variables, and we find that thepotential satisfies

/ðx; yÞ ¼ E0e�ikxg0ðyÞ þX1n¼1

Ene�icnxgnðyÞ: ð18Þ

We define the vertical eigenfunctions in Region 3, by

gnðyÞ ¼ gn cos cnðyþ bÞ n ¼ 0;1;2; . . . ; ð19Þ

where

gn ¼

ffiffiffiffiffiffi2

b�a

q; n – 0;ffiffiffiffiffiffi

1b�a

q; n ¼ 0:

8><>: ð20Þ

The eigenvalues cn are given by

cn ¼np

b� a; n ¼ 0;1;2; . . . ; ð21Þ




which are the zeros of the equation

Please(2014

sin cðb� aÞ ¼ 0: ð22Þ

The eigenfunctions gnðyÞ satisfy the relation
Z �a
�bgnðyÞgmðyÞdy ¼ dmn; ð23Þ

where

c0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � c2

o

q¼ k;

c1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � p

b� a

� �2r

;

..

.

cn ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � np

b� a

� �2r

; n ¼ 0;1;2; . . . ;

with 0 < Rec0 < Rec1 < . . ..

3.4. Region 4f�b 6 y 6 b; x > 0g

By using the separation of variables in Region 4, we find that the potential satisfies

/ðx; yÞ ¼X1n¼1

FneiknxnnðyÞ; ð24Þ

which satisfies (3), (4) and (7). The vertical eigenfunctions nnðyÞ in Region 4, are given by

nnðyÞ ¼cos k2n�1ðyþ bÞffiffiffi

bp ; ð25Þ

which satisfies the orthonormal relation
Z b
�bnnðyÞnmðyÞdy ¼ dmn: ð26Þ

The eigenvalues k2n�1 are given by

k2n�1 ¼ð2n� 1Þp

4b; ð27Þ

which satisfy the equation

cosð2bkÞ ¼ 0; ð28Þ

where

k1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � p

4b

� �2r

;

k2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � 3p

4b

� �2s

;

..

.

k2n�1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � ð2n� 1Þp

4b

� �2s

; n ¼ 1;2; . . .

with 0 < Rek1 < Rek2 < . . ..

4. The system of equations for the exhaust trifurcated duct

The continuity of the pressure across x ¼ 0 gives

X1n¼1

AnwnðyÞ ¼X1n¼1

FnnnðyÞ: ð29Þ




Taking the inner product with wmðyÞ and using (12) and (29) becomes

Please(2014

Am ¼X1n¼1

FnLmn; ð30Þ

where

Lmn ¼Z b

awmðyÞnnðyÞdy ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2bðb� aÞp � cosð2bk2n�1 þ amðb� aÞÞ

k2n�1 þ amþ cosð2bk2n�1 � amðb� aÞÞ

k2n�1 � am

þ 1k2n�1 þ am

� 1k2n�1 � am

� �cosðk2n�1ðbþ aÞÞ

�: ð31Þ

Also, we have

f1ðyÞ þX1n¼1

BnfnðyÞ ¼X1n¼1

FnnnðyÞ: ð32Þ

Again, taking the inner product with fmðyÞ and using (17) and (32) becomes

dm1 þ Bm ¼X1n¼1

FnMmn; ð33Þ

where

Mmn ¼Z a

�afmðyÞnnðyÞdy ¼ 1

2ffiffiffiffiffiffiabp sinðk2n�1ðbþ aÞ þ b2m�1ð2aÞÞ

k2n�1 þ b2m�1þ sinðk2n�1ðbþ aÞ � b2m�1ð2aÞÞ

k2n�1 � b2m�1

� 1k2n�1 þ b2m�1

� 1k2n�1 � b2m�1

� �sinðk2n�1ðbþ aÞÞ

�: ð34Þ

The continuity of the pressure across x ¼ 0, also gives

E0g0ðyÞ þX1n¼1

EngnðyÞ ¼X1n¼1

FnnnðyÞ: ð35Þ

Multiplying (35) by gmðyÞ and integrating over �b; �a½ �, and using (23) and (35) becomes

Em ¼X1n¼1

FnNmn; m ¼ 0;1;2; . . . ; ð36Þ

where

Nmn ¼Z �a

�bnnðyÞgmðyÞdy ¼ gm

2ffiffiffibp sinðk2n�1 þ cmÞðb� aÞ

k2n�1 þ cmþ sinðk2n�1 � cmÞðb� aÞ

k2n�1 � cm

�: ð37Þ

The continuity of the velocity across x ¼ 0 gives

�X1n¼1

AnanwnðyÞ ¼X1n¼1

FnknnnðyÞ: ð38Þ

Again taking the inner product with wmðyÞ and using (12) and (38) becomes

�amAm ¼X1n¼1

FnknLmn: ð39Þ

Also, we have

b1f1ðyÞ �X1n¼1

BnbnfnðyÞ ¼X1n¼1

FnknnnðyÞ: ð40Þ

Multiplying by fmðyÞ and integrating over �a; a½ �, and using (17) and (40) becomes

b1dm1 � bmBm ¼X1n¼1

FnknMmn: ð41Þ

Also, the continuity of the velocity across x ¼ 0 gives

�kE0g0ðyÞ �X1n¼1

EncngnðyÞ ¼X1n¼1

FnknnnðyÞ: ð42Þ




Multiplying by gmðyÞ and integrating over �b; �a½ �, and using (23) and (42) becomes

Please(2014

�cmEm ¼X1n¼1

FnknNmn; m ¼ 0;1;2; . . . : ð43Þ

5. Determination of expansion constants

In this section, we shall solve the matching Eqs. (30), (33), (36), (39) and (41). Thus, from (30) and (39), we have

X1n¼1

FnLmnðkn þ amÞ ¼ 0; m ¼ 1;2; . . . ; ð44Þ

Eqs. (33) and (41), give

X1n¼1

FnMmnðkn þ bmÞ ¼ ðb1 þ bmÞdm1; m ¼ 1;2; . . . ; ð45Þ

From (36) and (43), we have

X1n¼1

FnNmnðkn þ cmÞ ¼ 0; m ¼ 0;1;2; . . . ; ð46Þ

For solving these systems of equations, we restrict ourselves to n ¼ 1;2; . . . ;N and m ¼ 1;2; . . . ;M. Once the Fn are deter-mined then the other constants i.e. An; Bn, and En can be determined.

6. Numerical results

To know the intensity of acoustic power in the incident mode expðib1xÞ is of interest. This power is distributed over thedifferent waveguide regions. Particularly, the fundamental mode n ¼ 1 and the reflection coefficient which is the coefficientof the expð�ib1xÞ is given by

jRj ¼ jB1j:

The power, which is radiated from the semi-infinite waveguide is represented by ð1� jRj2Þ and which is divided over theother different regions. Thus absolute value of reflected amplitude jRj determines the amount of energy of expðib1xÞ whichis divided over the other modes of the waveguide. We have calculated jRj numerically. Here we present some graphs for jRj,which is a useful model of the transmitted power in the waveguide system. We have figured jRj against the wave number kain two situations. The value of jRj will suffer a quick change as more modes are activated in the various regions.

Case I. In the case (I), the dimensions of waveguide are such that no mode propagates in region (1), one mode propagatesin each of the region (2), and region (3), and two modes propagate in region (4). In this case, we choose b ¼ 3a=2and the range of the frequency p=6 < ka < 3p=4. Fig. 2 shows the case (I).

Fig. 2. jRj versus wave number ka for b = 3a/2.



Fig. 3. jRj versus wave number ka for b ¼ 4a.


Case II. In the case (II), the dimensions of waveguide are such that one mode can propagate ina < y < b; x < 0; ðp=3 < ka < 2p=3Þ, two possible modes can propagate in region (3), and five modes propagatein the forward direction in region (4).Fig. 3 is the graph of jRj for b ¼ 4a and p=4 < ka < 3p=4.

Case III. The case (III) describes the guide dimensions in such a way that the infinite duct is kept fixed at b ¼ 3=2. While wechoose (i) a ¼ 1 and (ii) a ¼ 0:5. Fig. 4 shows this situation. In Fig. 4 solid line is for a ¼ 1 and chained line is fora ¼ 0:5. The frequency range is p=4 < k < 3p=4.

Case IV. In case (IV), the graph of jRj is plotted by fixing b ¼ 4 and choosing (i) a ¼ 1 and (ii) a ¼ 0:5. The frequency range isp=4 < ka < 3p=4. Fig. 5 depicts this situation. In Fig. 5, solid line is for a ¼ 1 and dotted line is for a ¼ 0:5.

7. Energy balance

We can use the Green’s theorem

Please(2014

Z ZXð/�O2/� /O2/�Þdv ¼

I@X

/�@/@n� /

@/�

@n

� �ds ð47Þ

to derive a power balance relationship between the various reflection and transmission coefficients. Where / is the solutionto (3)–(7), � denotes conjugate and X is the region of the plane �b < y < b; �1 < x <1with cuts for the semi-infinite duct.Since / satisfies (3)–(7) then we have only contributions from duct at infinity. Assuming that all the decaying modes can beneglected, then it is not difficult to show that a energy balance relationship (47) gives:

b1ð1� jRj2Þ ¼ kjE0j2 þ c1jE1j2 þ a1jA1j2 þX3

j¼1

kjjFjj2; ð48Þ

Fig. 4. jRj versus wave number ka for b = 3/2.



Fig. 5. jRj versus wave number ka for b ¼ 4.


where b1ð1� jRj2Þ represents the power that exits from jyj < a; x < 0 and which is distributed among the other regions. Inthe special case of propagating of only one mode in region (3), one mode in region (2) and no mode in region (1), then thepower balance becomes:

Please(2014

b1ð1� jRj2Þ ¼ kjE0j2 þX2

j¼1

kjjFjj2: ð49Þ

8. Remarks

We considered the two dimensional problem of linear acoustic diffraction by the end of a semi-infinite duct located insidean infinite duct. Sound-soft and sound-hard conditions are used at the top and bottom boundaries of the duct respectively.This problem has been solved in closed form by Rawlins using the Wiener–Hopf technique. The solution method presented inthis paper relies on splitting the domain at the end of the semi-infinite duct into four sub-domains. We expand the velocitypotential in standard eigenfunctions in each of these four sub-domain. We match the normal velocities along the splittingboundaries. These matching yield an infinite linear system of equations for the unknown coefficients in the eigenfunctionexpansions which are solved numerically by the truncation. This method is much more straightforward and the resultingsystems of equations are easy to implement.

This is an interesting piece of work that gives an alternative to the Wiener–Hopf method to solve a known problem. Onecan consider a more general problem, for example, the arbitrary choice of soft and hard boundary conditions. The techniqueof this paper would work for other problems which may not be solved by the Wiener–Hopf technique and other methods.The interested researchers encourage to find the novel applications of this technique in the other areas.

9. Conclusion

Rawlins [1] has solved the same problem by using the matrix Wiener–Hopf factorization method in the closed form withno graphical results. We have presented the solution of the trifurcated waveguide problem by using the Matched Eigenfunc-tion Expansion. The solution does not involve the complicated factor/split functions (which arise from the Wiener–Hopfmethod). The results of the solved trifurcated waveguide problem by matched eigenfunction expansion will be of use inacoustic waveguide problems. The fundamental mode plays a very important role in the practical applications as it cariesmost of energy. For the fundamental mode propagating in the semi infinite duct, the reflection coefficient has been evaluatednumerically. Figs. 2 and 3 represent the jRj versus the wave number ka that would of use in the design of practical exhaustsystems. The singularity in the velocity near the tip of the plate revealed by (8) and (9), the convergence is slow as the trun-cation parameter N exceeds from 60, and the oscillations start occurring.

In Fig. 2, the cut-on (off) frequencies are at ka ¼ p=6; p=4, and at ka ¼ p=2. The onset, of waves travelling in forwarddirection jyj < b; x > 0 are at ka ¼ p=4 and ka ¼ p=2. The onset, of wave propagating in backward directiona < jyj < b; x < 0 is at ka ¼ p=6.

In Fig. 3, there are more abrupt changes, which reveals that the more modes are activated in the various regions. The on-set, of waves travelling in the forward direction x > 0; jyj < b are at ka ¼ p=4; 5p=16; 7p=16; 9p=16, and at ka ¼ 11p=16.The onset, of wave propagating in the backward direction a < jyj < b; x < 0 are at ka ¼ p=3 and ka ¼ 2p=3.

In Fig. 4, we can see that the reflection increases (for p=2 < k < 3p=4Þ as the dimension of the inner duct decreases (froma ¼ 1 to a ¼ 0:5) for the fixed infinite duct. The cutoff frequency is changed from p=4 to p=2. Similar behaviour is observedwhen the dimension of the semi-infinite duct is decreased for fixed infinite duct as shown in the Fig. 5. The cut-off frequen-cies are also changed.




Acknowledgements

The author would like to thank the referees and Editor-in-Chief (Professor Johann (Hans) Sienz) for their valuable andconstructive comments. The author would like to thank to Faheem Anjum for his calculation working. The author is alsograteful to Dr. S.M. Junaid Zaidi, Rector, COMSATS Institute of Information and Technology, for providing the researchfacilities.

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Please cite this article in press as: Mahmood-ul-Hassan, Wave scattering by soft–hard three spaced waveguide, Appl. Math. Modell.(2014), http://dx.doi.org/10.1016/j.apm.2014.03.004

http://refhub.elsevier.com/S0307-904X(14)00107-3/h0010

http://dx.doi.org/10.1029/2007RS003768


















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Wave scattering by soft–hard three spaced waveguide

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