Surface wave diffraction by means of singular integral equations

Journal of Sound and Vibration (1976) 45(l), 15-28

SURFACE

WAVE

DIFFRACTION INTEGRAL

BY MEANS OF SINGULAR

EQUATIONS

B. RULF AND N. KEDEM Department of Mathematical Sciences, Tel-Aviv University, Tel-Aviv, Israel (Received 22 March 1975, and in revisedform

4 August

1975)

Reflection, transmission and diffraction of surface waves that impinge upon a jump discontinuity in the properties of the guiding structure are investigated. A set of singular integral equations for the reflection, transmission and scattering coefficients is derived. For small jumps these equations are solved asymptotically. Several examples are shown in some detail. 1. INTRODUCTION Surface wave phenomena are encountered in the analysis of many problems in continuum mechanics and electrodynamics. The mathematical models of these problems lead to boundary value problems which exhibit wave spectra having both continuous and discrete parts. The discrete eigenfunctions correspond to the surface waves. They carry energy in directions tangent to some guiding surfaces, and are both of theoretical interest and of practical importance. The continuous eigenfunctions correspond to radiation which is not bound to the guiding surfaces. When surface waves impinge upon a discontinuity in the propagating medium or the guiding surface, reflection and transmission of the surface waves across the discontinuity as well as diffraction effects may occur. These are characterized by coupling between the various components of the spectrum. The physical meaning is that some of the energy of the surface wave is scattered away from the guiding surface. In this paper we present a method by which reflection and diffraction of surface waves at discontinuities is reduced to a system of singular integral equations for the scattering coefficients. For small discontinuities these equations can be solved asymptotically, yielding approximate expressions for these physical observables. For arbitrary discontinuities these integral equations may be solved in certain simple cases. In other cases numerical methods may be applied. We shall only give the formulation and asymptotic solutions for a few simple cases. We hope however that this method will enable us to analyse a large class of problems in acoustics and optics now under investigation. While this paper was being reviewed, another paper [l] with a similar basic approach came to our attention, and requires some comments. Equations (7) in reference [I] are analogous to our equations (2.22), (2.23) or (3.36), (3.37). In reference [l], these integral equations are converted to an infinite set of linear algebraic equations, which is then truncated and solved numerically. The expansions of the unknown functions A and B in series of orthogonal polynomials (equations (8) in reference [l]) are formal, and may not be mathematically justifiable, since it seems that these functions have singular points. The same may be said about truncating an infinite set of equations : the solution of the truncated finite set may or may not be a good approximation to the solution of the original problem. We found A and B to be singular (see equations (2.39) and (2.40) at p= 1, equations (A18) and (Al9) at p = #, and the same in equations (B6) and (B7)). Similar behavior of 15

16

B. RULF AND N. KEDEM

solutions of singular integral equations has been reported elsewhere [2,3]. These singularities may provide a starting point for numerical procedures of solving this class of problems.

2. SURFACE

WAVES ON A REACTIVE

SURFACE

The simplest model for a system which allows surface waves is a plane with a surface impedance [4, 51. For the case of a constant impedance, 2, the boundary value problem is u,, + u,,, + k2 u = 0,

iny>O,-=
(2.1)

on y=O,

(2.2)

u,+kZu=O,

asr=dm+co,

IUI < a,

(2.3)

where the wavenumber, k, and the surface impedance, look for solutions of the form

Z, are both real and positive. We may

u(x, y) = v(y) exp(+ikxm, which, upon substitution

in expressions

(2.1)-(2.3),

(2.4)

yields in y > 0,

v” + k2 /Iv = 0,

on y=O,

v’+kZv=O,

(2.7)

The eigenvalues, I, of equations (2.5)-(2.7) can be shown there is a single eigenvalue, namely & = -z2, eigenfunction,

(2.6)

asy+m.

IUI < a,

and a corresponding

(2.5)

to be real. In the range

(2.8)

which we denote by &Z, y) = eekzy.

The corresponding

solutions

of equations

(2.1)-(2.3)

ui = A,&(Z, y) exp(ikx m), Further are

(2.9)

are surface waves :

u; = B, &Z, y) exp(-ikxdm).

it is easy to verify that for 0 < 2 < 00 the eigenfunctions $(Z, y; 1.) = cos k&y

and the corresponding

solutions

,J < 0

of equations

- ZA-“2

(2.1)-(2.3)

u+ = mA(L)t,G(Z, y; 1)exp(ikx1/1 s

of equations

sin kfiy,

(2.10) (2.5)-(2.7) (2.11)

are - A)dn,

0

u- = r B(A)

$(Z, y ; 1) exp(-ikx

dm

dl,

(2.12)

0

where A(A) and B(A) are arbitrary (subject to existence of the integrals). Since 0 ,< 2 < ~0, it is obvious that u+ exists only in x > 0, while u- exists in x < 0 only (unlike equations (2.10) which exist for all x). If v and w are solutions of equations (2.5)-(2.7) which correspond to L and p, then k2(p - 2) (v, w) s k2(p - A) rvw dy = WV’- VW’(~. 0

(2.13)

17

SURFACE WAVE DIFFRACTION

When either v or w (or both) belong to the discrete spectrum, then from equations (2.6) and (2.7) one obtains (u, w) = 0, or equivalently (2.14)

(4(Z, Y),$(Z> Y ; 4) = 0.

When both u and w belong to the continuous spectrum, then by interchanging orders of integration one obtains

The inner product ($(Z, y ; A),J/(Z, y ; p)) may be calculated either classically [5] or by a generalized functions approach, yielding MZ, y; A)>, $(ZLY; P)) = (n/k) fl(l

(2.16)

+ ZZ/d) 80 - cl).

Equations (2.14) and (2.16) are the orthogonality relations for the eigenfunctions of equations (2.5)-(2.7). If a surface wave impinges from the left upon a discontinuity in Z at x = 0, one seeks a continuously differentiable function u(x,y) which satisfies u,*+u,,+k2U=O,

iny>O,--03
(2.17)

ony=O,-co
u,+kZ,u=O, u,+kZ,u=O,

ony=O,O
(2.18) to,

(2.19)

u and u, are continuous at x = 0.

(2.20)

One can represent u as an incident surface wave of unit amplitude which gives rise to reflected and transmitted surface waves, as well as scattered waves : e-‘*~X]+ r,4(l)$(Z1, y; A)e-‘8”dE., u(x, Y)

x < 0,

a

=

(2.21)

mB(A)$(Zz. y; A)eiB”dl, J

x > 0,

0

(2.21a) It is not hard to verify that equation (2.21) satisfies equations (2.17)-(2.19). The coefficients R, T, A(A) and B(A) have to be determined. The continuity condition (2.20) yields

One now can use the orthogonality relations (2.14) and (2.16): i.e., equations (2.22) and (2.23) are multiplied by Cp(Z,,y) and integrated over 0 < y < tc, and then the process is repeated with I,@~, y; 11).This yields the four equations

Il4tZAll’t1+ R) = <4tZ,),4(2,)) T+

~W.) <4tZ,),W&

dJ.9

(2.24)

0

~1ll4tZ,)ll~ (1- W = %+#@I>,KG)> T + J'atl<4cz,,, 0

$(Z,)> dA,

(2.25)

B. RULF

18

AND N. KEDEM

m SA(4(~(Z,;l.),~(Z,;~))dl=(g(Z,),i(Z,;~))T+~B~~~~jr(Z,;i),~(Z,;p))di, 0

-

jh)

Kll/(Z, ; 9,ll/(Z,

; cl)> dd A ~2
(2.26)

0

$(Z, ; 4) T + p B(k) P<$(zz; 4, @G ; 10\ a. 0 (2.27)

0

Similar to expression (2.16) one calculates ($(Z,; 4, KG ; 4) = (44

&
+ Z, z,/4

W. - ~1 i- (z, - &)/(A - P).

(2.28)

The calculations of the other inner products that appear in equations (2.24)-(2.27) are straightforward. Thus one obtains OD B(A) dl

l+R=

s

(2.29)

3TFy

0

(2.30) 0 p+zz: 7c2/r;

G-22

T+nP+Z,Z,

A(p)=-

Nd+(Zl

-Z2)

fi

(P+z)

m B(A) s 1-P 0

dR,

(2.31)

112

m

BW)

dl

(2.32)

x(~+ZlZ2)B(~)+(zl-Z2) s k(l-P) o

.

Equations (2.29)-(2.32) determine R,T,A(k) and B(1). By some simple algebraic manipulations one can obtain from equations (2.29)-(2.32) a single singular integral equation of the form (2.33) wheref,g,h and t are known functions which are integrable in [0, a). In this particular case, equation (2.33) may have a known solution, since the problem has been solved in a different way by means of the Wiener-Hopf technique [6,7] (which may be regarded as a special case of the general singular integral equation). Since we are interested in solving more involved problems, whose solutions are not known, we shall concern ourselves here with an asymptotic approach only. That is, for Z, - Z1 small, equations (2.29)-(2.32) can be solved asymptotically. The first order asymptotic approximation is particularly easy to obtain. We shall write z, =z Zz=Z(l+&) (2.34) J&J<( 1. Upon inspecting equations (2.29)-(2.32) one sees that for E = 0 the solutions are, as one may expect, R=A=R=O T= 1. (2.35) Thus one writes T=1+&TI+O(c2) R = &RI + 0(e2)

A(P) = MP)

+ 0(c2)

(2.36) B(P) = E&(P) + 0(e2).

(2.37)

19

SURFACE WAVE DIFFRACTION

Equations (2.29) and (2.30) become mB,(l) + O(s) do 1 + &RI+ O(E*)= (1 - s/2 + O(E*))(1 + ETA+ O(E*)) + 2Z2 E* A+22 ’ s0

The O(E) terms yield RI= -

Z2

T,=

2(1 +z*)

1

(2.38)

2(1 +z*)’

In a similar way equations (2.31) and (2.32) yield

1’2(1 + z*)“* + (1 - py*

&W=

g

& i

z Al(P) = -

1

(p + z*y

I

(P+z*)*

‘12 (1 +zy*

P

-

27ri 1-p

(2.39)

’

- (1 - /p*

(2.40) .

It should be noted that A1 and B1 have branch point singularities at p = 0 and p = 1, but the singularities are weak enough to be integrable. One must be careful, however, if it is desired to calculate the scattered field in the far field region, since there will be enhancement of the far field in the directions that correspond to a saddle point near a branch point [4]. In a similar way one can calculate higher order terms in the expansions for R, T,A and B. One may set (instead of expressions (2.36) and (2.37)) T= 1 + cTl+ E*T2+ O(E~), R = &RI+ E'R*+ O(e3), A(P)= &AI(P) + &*A*(P)

+

B(P)

O(E3)9

=

W,4

+

E* B2b4

+

N&31,

and expand all the E dependent quantities in expressions (2.29)-(2.32) up to 0(e3). One thus obtains a hierarchy of equations which can be solved successively. Since the integrals in equations (2.29)-(2.32) are multiplied by Z, - Z,, which is O(E), one never has to solve integral equations. Once T,,R,,A, and B1 are found, one can find T2, R2, A2 and B2 by solving algebraic equations. The calculation is not hard, but involves the evaluation of integrals of the forms m jBG)l(n - p) d& ~B&%‘(~ + Z’)dJ. 0

0

Thus it seems that for higher accuracy, rather than trying to find higher order terms in the asymptotic expansion it would be advisable to try to solve equations (2.29)-(2.32) numerically. This has to be done in any case when Eis not small (and in particular, when E 2 1). Numerical methods for solving singular integral equations similar to (2.33) have appeared in the literature [2,3], and we are trying to adapt them.

3. SURFACE WAVES OVER A LOW VELOCITY LAYER A more realistic configuration of a guiding structure for surface waves is that of a layer of finite thickness adjacent to a half space. The propagation velocity in the medium of the layer must be smaller than the propagation velocity in the half space. This model is used both in

20

B. RULF AND N. KEDEM

elasticity (Love waves) and in electrodynamics [4]. The boundary value problem is now U,, + U,, + p2 u = 0,

in 0 < y < h, 1x1< to,

(3.1)

U,, + U,, + q2 u = 0,

in h < y < a, 1x1< cc,

(3.2)

ony=O,

(3.3)

U, = 0,

asr=m+

IUI< a,

co,

(3.4)

where p and q are given real and positive wavenumbers, with i.e., A2 s pz - q2 > 0.

P ‘9;

(3.5)

At y = h one may prescribe finite jump discontinuities in u and u,. The following conditions are commonly met in problems of elastic and electromagnetic waves : lim[U(x,h-s)--&h+a)]=O E’O f@[Zl,(X,

h -

E) - bU,(X, h + E)] = 0

(i.e., continuity of u),

(3.6)

(prescribed jump in u,),

(3.7)

with b real and positive. (For Love waves, b = p2/p1.) One may look for solutions of the form u(x, y) = o(y) exp[*ixm], (3.8) thus obtaining, for u(y), v”+(A2+~)u=0, in 0 6 y < h, (3.9) vn + Ilv = 0, VI= 0,

14< “7

inh
a,

(3.10)

ony=O, asy-t

u(h - 0) = u(h + 0) v’(h - 0) = bu’(h + 0)

5,

(3.11) (3.12)

ony=h,

(3.13)

ony=h.

(3.14)

As before the spectrum has a discrete part, in the range -A* < L < 0, and a continuous part, whichconsists of the real positive axis 0 < i < co. The eigenfunctions of the continuous spectrum are cosdd’

$(y; 2) =

+ Ly,

for 0 < y < h,

D(1) cosV%(y - h) + C(1) sin fi(y = /I(J) cos [ V%
- h) = forhGy<

(3.15) co,

where (3.16a)

C(~)=-(1/b){(d2+1)/~}“2sin~h,

D(n) = cos v’mh

(3.16b)

tan a@) = C/D.

/3(n) = m The eigenfunctions of the discrete spectrum are cos&PX* 4*(Y) =

y,

i cos~‘mheexp[-&&(y

for 0 < y < h, - h)],

(3.17) forh
m.

In order for condition (3.14) to be satisfied, JVihas to be a solution of msinmh=bGcosmh.

(3.18)

Equation (3.18), which determines the eigenvalues, can be written as Q tan Q = 62/A’ h2 - Q2,

(3.18a)

21

SURFACE WAVE DIFFRACTION

4

Q=hm.

(3.19)

It is easily seen that equation (3.18) has a finite set of solutions in the range 0 Q Qt 6 Ah,

or -AZ ( Lr = -A2 + (QJh)’ G 0,

i= 1,2, . . ., M.

(3.20)

In order to get useful orthogonality relations, we define the inner product of two solutions of equations (3.9)-(3.14) as follows: (u,w)-(1,6)juwdy+jo~d~. 0

(3.21) h

As in the previous section, we deduce from equation (3.21), and equations (3.6) and (3.7), that (u, w) = (WV’ - VW’),,and MY), I(l(r; 12>> = (MY), 4,(Y)> = 0,

i, j=

MY; &$(Y; P)> = ; [o(n) D(P) + C(l) C(P)] S(h

1,2 , **., M, i#j,

- 4)

= 7~&‘(A)

(3.22) a(2 - P). (3.23)

with C, D and B defined by equation (3.16). We shall now formulate and analyze the problem of diffraction of a surface wave at a discontinuity in p: i.e., in the properties of the low velocity layer. (See Figure l(a).) Let a

(b) Figure 1. (a). A layer with discontinuous medium, adjacent to a homogeneous discontinuous thickness, adjacent to a homogeneous half-space. _

half-space. (b). A layer of

surface wave of unit amplitude impinge from the left upon a discontinuity in x = 0. The appropriate equations are u,,+u,,+p:u=o,

for 0 Q y < h, x < 0,

(3.24)

u,,+u,,+p;u=o,

for 0 < y < h, x > 0,

(3.25)

forh
ux,+uy,+q2u=o,

co,--co
co,

(3.26)

where pi, p2 > q. We shall assume the boundary conditions to be 24,= 0

at y = 0,

(3.27)

u is continuously at y = h,

(3.28)

u,(x, h - E) = b, u,,(x, h + E),

&+O,xtO

(3.29)

u,(x, h - E) = b, u,(x, h + E),

.T+O,x>O

(3.30)

24is continuous at x = 0,

Ody<

03,

(3.31)

u, is continuous at x = 0,

h
03,

(3.32)

22

B.RULFANDN.KEDEM

for 0 < y < h, E -+ 0,

d--E,y) = (h/~2)4(~9y)~

(3.33)

forr=m+w.

IUI< *

(3.34)

These boundary conditions correspond to a class of problems in elastic and electromagnetic waves. We shall assume that only one surface mode exists in both x < 0 and x > 0. The generalization to M and Nsurface modes, respectively, will be discussed later. The solution of equations (3.24)-(3.34) may be represented as follows: ~l(y)[e’““~ + Re-IXS I] + TA(,i)$,(y;

I) e-ixs(‘) dl,

x
0

K=

m @2(y)Te++

dl,

B(I)$2(y;I)ei""'") 5

1 s 1,z

=

&,

s(A)=

x > 0,

(3.35) (3.35a)

q.

In expression (3.35), and thereafter, the subscripts 1 and 2 correspond to x < 0 and x > 0, respectively. Thus $r and & are given by equation (3.17) with A (defined in equation (3.5)) replaced by A, or AZ. Similarly 1, and A2are determined by equation (3.18), with A and b replaced by Al, bl or A,, bl, respectively. The boundary conditions (3.31)-(3.33) yield now

where c(y) is defined as follows: for 0 < y < h, forh
(3.38)

co.

One now takes inner products of equations (3.36) and (3.37) with ~$iand el, according to the rule (3.21). Due to the orthogonality relations (3.22) and (3.23) one obtains the following equations for R, T, A(R) and B(A): IlM~)ll’(l + R) = (A(Y),MY))

T+ j B(~)
(3.39)

(3.40) W)

<$,(Y;P), Il/2(~;4>dk

(3.41)

The inner products that appear in expressions (3.39)-(3.42) can be calculated in a straightforward way. The equations can be reduced to a single singular integral equation for B(p) which is similar to equation (2.33). As in the previous section, one looks for approximate solutions of equations (3.39)(3.42) for the case of a small discontinuity. There are two quantities that undergo a jump discontinuity at x = 0, namely p and b, and they are independent of each other. In order to solve equations (3.39)-(3.42) asymptotically one assumes, bl =b

b, = b(1 + E)

(3.43)

23

SURFACE WAVE DIFFRACTION

Pl =p

P2

=Pu

+

(344)

74

where E is small, while y is arbitrary. Next, one expands all the quantities with subscript 2 in terms of the quantities with subscript 1. For example, equation (3.18) can be written in the form (3.45) W,P, 6) = 0; thus aF/ab

-b yp + aF/an

1

E + O(E~)= a(p, b) E + O(E’).

(3.46)

The partial derivatives are evaluated at p =pl, b = bl, 1= 1,. The calculation is straightforward but cumbersome, and is outlined in Appendix A. Similarly, one expands the modal functions $2 and 11/2in terms of 41 and $I as follows: 42(Y)

=

CM9

+

&WI

+

(3.47)

o(&2),

where E(Y) =

Edp,

O
b)y sin Ly,

[E2(p, b) + E,(P, b) (y - h)l exp[--fl

(y -

hfy<

WI,

LE%QX,.

co,

(3.48) (3.49)

Similarly (3.50)

~~(Y;~)=~~(Y;~)+F(Y;~)&++O(E~),

where O
Fl,Cp, b; 2)~ sinW)y,

F(y’ ‘) =

F,(p, 6; A) sin[fi(y

- h) + F,(p, 6; A)],

h
(3.51)

The quantities a, El, E,, E,, F,, F2 and F3 in equations (3.46), (3.48) and (3.51) are given in Appendix A, It should be noted that E(y) and F(y) in equations (3.47) and (3.50), respectively, are bounded, indicating that the asymptotic expansions are uniform in y. Finally, one substitutes 42 and tiz from equations (3.47) and (3.50) in equations (3.39)(3.42). As in the previous section, the system of equations (3.39)-(3.42) becomes a system of algebraic equations (up to O(.s’)), from which one finds that R, T, A(p) and B(p) have the same form as in equations (2.36) and (2.37). The explicit expressions for R1, T,, AI(,u) and B,(p) are given in Appendix A. Another interesting problem is that of diffraction of a surface wave at a “step”: i.e., a discontinuity in h at x = 0. (See Figure 1(b).) As before, we assume that only one surface mode exists in both x < 0 and x > 0. The appropriate equations are u,,+u,,+p2u=o,

for

u,,+u,,+q2u=o,

for 24, =

IUI< *,

0,

0 < y < hl, x < 0,

0 4 y < h2, x > 0, hl fy<

co,x
h2Gy-c

co,x>O,

at y = 0,

asr=w-+w,

u is continuous at the interface, i.e., at y = hi,

(3.52) (3.53) (3.54) (3.55) (3.56) (3.57)

B. RULF AND N. KEDEM

24

where we use hi = h, when x < 0 and hi = h, when x > 0. Also u is continuous at x = 0,

au/ax is continuous at x = 0, au/axj,=4 = ba~/axj,=+o

(3.58)

when y < h, or y > hI,

(3.59)

for h,
(3.60)

The field representation for this problem will be the same as in equation (3.35). The functions $l(y) and Il/l(y; A) will be different, however, They are given now by A(Y) =

cosdFqy,

0 < J’ < hi, h,] exp[-a(y - hi)],

cos[m[

(3.6 1)

0 < y -c hi,

cosvT-TTy, $dy; A) =

hi
Q(1) cosd(y

- h,) + C&t) sind(y

s B*(J)cos[fi(Y

- hi) + ai(A

- h,) = h,
(3.62) 00.

The eigenvalues 1, are determined by the equation Ql tan Qi = bw,

(3.63)

Q1 =him,

(3.64)

AZ =p2 - q2 > 0.

(3.65)

C,, Di, PI and @iin equation (3.62) are defined as in equations (3.15) and (3.16). Using conditions (3.58)-(3.60) and the orthogonality relations for the functions 4I and $i as before, one obtains the same four equations (3.39)-(3.42) for R, T, A and B; only c(y) has a different definition now, namely C(Y) =

1,

for y < h, or y > hI,

b,

for h,
(3.66)

The asymptotic solution of equations (3.39)-(3.42) for a small jump in h proceeds as before. One defines h, =h

h,=h(l

+E)

(3.67)

and expands all the quantities with subscript 2 in terms of quantities with subscript 1 and E. The expressions for R, T, A and B are again in the forms (2.36) and (2.37). The calculations and the expressions for RI, T,, A,(p) and B,(p) are given in Appendix B. 4. GENERALIZATIONS

The method outlined in the previous sections can be applied to other problems of greater complexity. Both problems of section 3 are special cases of the problem shown in Figure 2, where the line x = 0 separates between two different layered structures. One may assume in general that there are A4possible surface modes in x < 0 and N possible surface modes in x > 0. A surface mode that impinges on x = 0 (either from the left or from the right), will excite M + N surface modes, as well as radiation. By our method it is possible to write a set of equations for the coefficients of the scattering matrices. For the case that M = N and the discontinuities are small, these equations simplify considerably, and approximate expressions (to O(E)) for the scattering coefficients can be easily found. When M # N however, the discontinuities are not “small” (at least for some of the modes). Numerical methods for solving certain types

2.5

SURFACE WAVE DIFFRACTION

x

Figure 2. A general discontinuous

configuration

for surface wave diffraction.

of singular integral equations are available, and we are trying to apply them to some of our problems. We are also applying our methods to problems involving vector functions, and to a three dimensional cylindrical geometry. The latter generalization should be useful for the analysis of certain problems that occur in fiber optics [8], such as calculation of the reflection and radiation losses that occur when an optical fiber passes from one dielectric medium to another (Figure 3).

Figure 3. A surface wave diffraction problem with cylindrical geometry which occurs in fiber optics.

Among other applications we mention the area of microwave acoustics: the equations governing the behaviour of elastic surface waves guided by thin films can be simplified, to yield equations and boundary conditions that fall into the class of problems that we have treated [9]. ACKNOWLEDGMENT

The work of one author (B.R.) was sponsored by NSF Contract No. MPS 75-07328, while he was a visitor at the Department of Mathematical Sciences, Rensselaer Polytechnic Institute (Troy, N.Y.). This author wishes to express his gratefulness to Professors Handelman, DiPrima and Tiersten for their help. The contribution of the second author (N.K.) is included in an M.Sc. Thesis, submitted to the Department of Mathematical Sciences at Tel Aviv University (1975). REFERENCES 1. S. F. MAHMOUD and J. C. BEAL 1975Institute of Electrical and Electronic Engineers Transactions MTT-23, 193-198 Scattering of surface waves at a dielectric discontinuity on a planar waveguide.

B. RULF AND N. KEDEM

26

2. F. ERDOGAN, G. D. GUPTA and T. S. Coon 1973 in Methods of Analysis and Solutions of Crack Problems (editor: G. C. Sih). Leyden: Noordhoff International Publishing. Numerical solution of

singular integral equations. 3. S. KRENK 1975 Quarterly of Applied Mathematics 32, 479-484. On the use of interpolating polynomials for solutions of singular integral equations. 4. L. B. FELSEN and N. MARCUVITZ 1973 Radiation and Scattering of Waves. Englewood Cliffs, New Jersey: Prentice Hall. (This book contains an extensive bibliography.) 5. V. V. SHEVCHENKO 1971 Continuous Transitions in Open Waveguides. Boulder, Colorado: The Golem Press. 6. A. E. HEINS and H. FESHBACH 1954 Proceedings of the 5th Symposium in Applied Mathematics, 75-88. On the coupling of two half planes. New York: McGraw-Hill Book Company, Inc. 7. H. M. BARLOW and J. BROWN 1962 Radio Surface Waves. Oxford: Clarendon Press. See Chapter XII. 8. B. RULF 1975 Journal of the Optical Society of America. Discontinuity radiation in surface waveguides. 9. H. F. TIERSTEN 1969 Journal of AppliedPhysics 40,770-789. Elastic surface waves guided by thin films. APPENDIX One expands all the quantities with subscript 1. From equations

with subscript

A

2 in terms of the corresponding

quantities

(3.18), (3.45) and (3.46) one obtains L tan Lh - ypyp2 [(tan Lh)/L + h/cos2 Lh]

4p, b) = 2

(Al)

(tan Lh)/L + h/co? Lh + (L tan Lh)/& ’

where L = d/d: + A,. (Note that A, = A,(p,b).) Similarly, from equation (3.17) one can find &(y) approximately by considering the first order changes in A and in A: cos Ly - {y sin Ly[2p2 y + a(p, b)]/2L} E + O(E~), 42(Y)

=

exp[-G

cosLh

(y - h)] + exp[-a

This determines

(y - h)].

( a(p, 6) cos Lh (Y - h) - !$$a@, 22/rx;

O
b) + 2~’ Y)

IiE+O(E2), (A2)

h
E(y), Et, Ez and E, in equations (3.47) and (3.48). Similarly F(y; A)in equation

(3.50) is defined by for 0 < y < h,

cos L(A) y + [F,(p, Il)y sin L(I) y] E + 0(e2), VUY; 2) =

8G> cos[fly

I

+ F,(p, b; 4

WI + V’,(p,b; 4 sin[v’%r - 4 + 00, I>E+ O(E’), forh
(A3)

- h) +

where L(A) = l/A: + i, g(J) and /?(A)are defined by equation (3.16b), and FI(P, 2) = - P2 r/W)

.

(A4)

F, and F3 in expression (3.51) are defined by F2 sin[dx(y

sin L(A) h

- h) + FJ =

+ hcosL(l)h

L(A) - h) - F,(p,~)hsinL(~)hcos~(y

x sinfi(y The first approximations

to the various

inner products

(~I(Y),~AYD

-

114111~ + (A(Y),

x II

- h).

(~45)

would be now E(Y))* 6,

W)

SURFACE WAVE DIFFRACTION

om),$z(Y)

C(Y)> - 11~111’ + <41(Y), C(Y) E(Y)) *6

(O,(y),c(y)~~(1)~2(y;i.)di.)0

~.~~(~)<~,(y),c(y)F2(y;i)>dl,

27 647)

(A9)

0

(Il/l(Y;L4*42(Y)) N ~.($l(Y;P),E(Y)),

($l(Y;P),c(Y)4J2w-

&.($1(Y), &9Jw));

WO) (All)

also

(Al3 Substituting all these expressions in equations (3.39)-(3.42), and separating the O(1) and O(e) quantities, yields

(Al@ (A17)


(AW

($1, E) + ($1, cE)

6419)

The expressions ~~~1~~2, (c/J~,E), (&cE), ($l,E) and (til,cE) are rather long, even though their calculation is elementary. We shall not write them out in detail. APPENDIX B In this problem only one quantity (h) changes. For example, from equations (3.18) and (3.45) one obtains ,4l= A2- II = - [(aF/%)/(aF/&I)] hs + 0(e2) E b(h) E + O(e2).

(W

B. RULF AND N. KEDEM

28

The approximate calculation of the mode functions and the inner products proceeds as in Appendix A : MY> = 4IW + Jw *&+ W&7 * &(W Y sin Q,

NdJIw + E

O
[E,(h) + &W (Y - h)l exp[-GA

(v -

41,

h
W)

similarly

buYi PL) - $l(Yi P)+ 8’

O
- h) + F,(h ; P>I,

h
(B3)

The quantities El, E,, Es, Fz and F3 are evaluated as in Appendix A, by expanding all quantities with subscript 2 in terms of the corresponding quantities with subscript 1. The calculation is simpler here, because only one parameter changes. We omit the trivial details. Note that because h, - h, = O(E), quantities of the forms

ht

hl

hl

hl

are all O(E). Thus, the first order expansion of T, R, A and B yields T=l+e

(41, E> b(h) - + 0(&Z), %J*- 4) h#Jl12

I

R=E.

A(p) =

B(p)=-

b(h) + WE*), %I* - 4)

2;~p)[l-($y]+om

E’

E.

(B4) (BS)

(B6)

(B7)