On the direct method in problems of diffraction by smooth infinite wavy surfaces in inhomogeneous media

Direct method in problems of diffraction

159

4.

BARANOV, V. and KUNETZ, G. Fllm synthetique avec reflexions multiples theorie et calcul practique. Geophys. Prospecting, 8,2,315-325, 1960.

5.

ALEKSEEV, A. S. Inverse dynamic problems of seismics. In: Some methods and algorithms of interpretation of geophysical data (Nekotorye metody i algorithmy interpretatsii geofii. dannykh), 9-34, Nauka, Moscow, 1967.

6.

KABANIKHIN, S. 1. The finite-difference regularimtion of the inverse problem for the equation of oscillations. In: Problems of the correctness of problems of mathematical physics(Vopr. korrektnosti radach matem. fm.). 57-70, VTs SO Akad, Nauk SSSR, Novosibirsk, 1977.

7.

LADYZHENSKAYA, 0. A. Bounoky value problems of mathematical physics (Kraevye zadachi matematicheskoi fmiki), Nauka, Moscow, 1973.

8.

FADDEEV, D. K. and FADDEEVA, V. N. Computational methods of linear algebra (Vychislitel’nye metody lineinoi algebry), Fizmatgiz, Moscow-Leningrad, 1963.

9.

FILATOV, A. N. and SHAROVA, L. V. Integral inequalities and the theory of non-linear oscillations (Integral’nye neravenstva i teoriya nelineinykh kolebanii), Nauka, Moscow, 1967.

LI.S.S.R. Comput. Maths. Math. Phys. Vol. 19, pp. 159-177 0 Pergamon Press Ltd. 1980. Printed in Great Britain.

0041-5553/79/0401-0159$07.50/0

ON THE DIRECT METHOD IN PROBLEMS OF DIFFRACTION BY SMOOTH INFINITE WAVY SURFACES IN INHOMOGENEOUS MEDIA* A. S. IL’JNSKII and 0. B. POPOV

Moscow (Received 29 March 1978)

WITHIN the limits of the direct projection method a new approach is made to scalar problems of diffraction by a wide class of reflecting surfaces in media with variable refractive index. A special boundary-straightening coordinate transformation, taken as a basis, permits the partial radiation conditions to be formulated accurately in a fured plane. A proof is given of the solvability of the corresponding approximate problems and of the convergence of the approximate to the exact solutions in the energy norm.

Introduction

To obtain approximate solutions of external diffraction problems in locally inhomogeneous media in the long-wave band, commensurate with the characteristic dimensions of the obstacles, direct projection methods, being modifications of GaMrkin’s method, are most efficient [l-3] . They permit the initial external boundary value diffraction problem to be reduced to an internal problem, and then to a boundary value problem for a system of ordinary differential equations. This is achieved by formulating the radiation conditions as partial radiation conditions, which are formulated as exact integral relations on a definite surface unlike the usual asymptotic conditions of Sommerfeld. In [3,4] these methods were successfully applied to problems of diffraction by periodic structures.

*Zh. vjkhisl. Mat. mat. Fiz., 19, 2,426-443,

1979.

A. S. Il’inskii and 0. B. Popov

160

In [5,6] a different type of periodic structure, topologicahy not equivalent to the above, was considered, namely, smooth wavy inhomogeneous surfaces, and the region of inhomogeneity along the lattice had the form of a layer of finite width. The fundamental difficulty in satisfying the boundary conditions on such a surface is removed by a coordinate transformation, to which there corresponds a family of coordinate surfaces asymptotically passing into planes at infinity. The reflecting surface is one of the elements of this family. In the new system difficulties arose with the formulation of the partial radiation conditions on surfaces only approximately plane. As a result, in the calculation of the reflection coefficients, which in the majority of important practical cases are also of interest in the numerical realization, it is necessary to make additional approximations not connected with the nature of the projection method. On the other hand, with this transformation the choice of the length of the segment u 4 t Q 0 on which the system is considered, is not fured in advance and the variation of u over a wide range - Q)< u < ul is permitted. Using majorant estimates of the rate of convergence, this permitted the dependence of the approximations on the segment length I u 1and the order N of the system to be exhibited. Although it was not possible to obtain convergence estimates uniform in u, it followed from the estimates that for every N the accuracy of the approximations increased with I u I. A feature of the transformation used below (similar to that used in [2] ), which straightens the boundary of the domain, is that the originally arbitrarily chosen domain for the formulation of the radiation conditions in the initial coordinates (situated in a domain of homogeneity of the medium) is mapped onto a fmed plane in the new coordinates, in such a way as to permit the exact formulation of partial radiation conditions on it. All the subsequent discussion is conducted in a fmed domain and on the segment 0 < u < 1. The reflection coefficients are defined by simple exact conditions. Since the choice of the transformation has a considerable effect on the form of the coefficients of the fundamental equation, it becomes necessary to prove anew the solvability of the approximate problems and the convergence of the approximate to the exact solutions in the energy norm. An important advantage of the new approach is also the fact that the actual coefficients of the system of differential equations have a considerably simpler construction.

1.

We have retained the basic formulation of the problem and much of the notation of [6] . We briefly recall them. The profile of the reflecting surface (a) is defined in the plane (x, y) as follows: A=-0, y=Av, a(vf2n) -a(v), s=&(V), --m
Direct method in problems

161

of diffraction

The region of diffraction is situated on the left of (u) (see the figure). The diffraction field where in the domain JI(x, y) is sought as the solution of the equation (A., ,-I-k* (t, y) ) $=O, s 0 to the x-axis exp [ iko (z cos gP+y sin cp) ] (the variation with time is of the form e- rw t), and to one of the homogeneous boundary conditions $((I~ = 0 or (8$/r%) 1(0j=O. The third supplementary condition corresponds to the impedance boundary condition

We retain the assumptions about the nature of the smoothness of the functions k (5, y) , o (v), a(P), (9 (P), and also the conditions and notation: A*=-s, Ag ==p=Av! Xo=kd, X-kA, p=~osincp>O, 7(n)=[~~*-(n+p)*]‘~, r(n)>0 for xo >In+pl, Im 7 (n) >O, the functions k (x, y), a(P), @ (P) are periodic iny with period 27rA, Ima > 0. The last condition is used essentially within the limits of the methods of proof used below, therefore the third boundary condition does not include the first two. For the first two supplementary conditions the radiation condition is defined as the plane-wave expansion in the domainx
(x, y ) = eibg {exp(iX,i.cosq)+

E C,exp[-iy(n)f+in~]} n--m

with undefined amplitudes C, . For the third supplementary condition the first term in the curly brackets is absent. The specific coordinate transformation for this paper is defined as follows. We write xl = AZ, so that I is dimensionless and 2< min u (v). We put x-Al

Y

u=A[o(v)-Z]

U=-.

’

A

Here the domain G between the surface (u) and the plane x =x1 is mapped onto a layer with boundaries u = 0 and u = 1. Introducing the notation eiCtDf(u,V) =F (u, V) =Q (5, y) , we obtain for the function f the equation

(1) + ( kZ--g**$) g’“f=O,

where ul = U, 242= Y. The coefficients of the equation have the form gI’g’” =

$,

g’2g’” -

g,,=($,‘+ ($)’ gz2

=

_

5,

g=g’” =

$_,

g,*=:g+tg, )

(S)’+(Z,’ ,

g’”=(gIig2*-g122)"~,

162

A. S. Il’inakii and 0. B. Popov

so that finally

?+u2irz(u)

g”g’”-

u(u)

_

z

g’2g”‘-g2’g’“=-uir(u)

’

(

g’“=A”[a(u)-Z].

g=g”==a (u) -I,

We note immediately a simpler construction of the coefficients than in the previous case. Because of the periodicity of the function f we restrict ourselves in what follows to the rectangle D=-(OGz~l, O,(uG?n), requiring the satisfaction of the following conditions of adjunction on its upper and lower bases:

&(U?0)-- -f(% _&

24,

i=i,2.

(2)

The three types of boundary conditions assume for u = 1 the form

f(% u) -0,

r[L-‘==O,

r~flU-l==&>f(l, u)+@(u).

(3)

Here the notation used is

The radiation condition for the function f in the domain u < 0 is defined as follows:

f ,~0=exp{i~,c0scp[Z+u(a(u)-Z)]}

(4) zk-

+

c

C, exp{-Q(n) n---oo

[Z+u(o-Z)]+ inu}.

We suppose that the functions u(u), a(v), Q(v) are so smooth as to ensure in the domain a solvability of the problem (l)-(4) and expansion of the function f together with its second derivatives in uniformly convergent Fourier series. We now impose on the equation additional conditions of a form convenient for the formulation of the approximate problem. We write down on the segment 0 6; tl < 1 the expansion

f(u,u)=

‘r,A,(u)e’“‘. la--0D

Direct method in problems of

163

diffraction

Then for the functions An(U) we obtain the following conditions: an

5Mfle -ino av==o,

n&,

(9

0

for u = 1 boundary conditions of one of the following forms must be satisfied: an

5r[f]e-‘““dv=O,

&(1)=0,

0

(6)

~(~))~-Th=o.

J&f]-&jf(i,v)0

Finally, since 1

cr(v)--2-z

af I u-0

-

Uf L-0,

we obtain not as before an approximate equation, but an exact one sn

Ufl 5 0

,,Io e-‘“” dv=2ni[

-2niy

(n)&

to),

D cos cp+y (n) ] exp (ix&Z cos cp)iLo

n=Z,

giving the partial radiation conditions (for the third supplementary condition we omit the first term on the right). As before we will assume the presence of weak absorption in the medium, 80 and Im 7 (n) Xl. that Im +=e*

2. We establish some integral equalities for the function fin the domain 9 uniqueness of the solution of problem (5)-(7). We first write f.h[f]dudv-0, IS 43

-

SI(I?-pzgZZ)g~If12dudv.

and prove the

(7)

164

A. S. ll’imkii

and 0. B.

Popov

Therefore,

(Re kZ-p2gP2)g”*lf12da dv. JJ a

-

Using the condition of adjunction, and aiso the equations

we

obtain Irn JJ f’L[f]dudv=-

P

# --

_-

e” A2 e2

A2

J J [+p~gv~/~) +$qtwjj a

JJ

g”‘,fl’dndv=~~

s

dudv

(g”g”lfl:,o-g”g~~Ifl:,,)dv 0

g”‘lft’dzzdv. JJ d

UsingGreen’sformula for the transformation of the left sides in (8) and (9), we obtain f’L[f]dudv=I-K, JJ 9D

By the adjunction conditions

(9)

165

Direct method in problemsof dfffrction

We note that the quantity Re I can be written in terms of the operator I’:

We transform K:

We obtain Re JJ~L[f]d”dv=-Re~(f.r[fl)~_~dv+Rej’ 0 2a _ JJgvs(gii

lzl2

+g22 l$l’)

(f’l?[f])Y_ldv 0

(10)

dudu

a - 2 Re J J gi2g” +Lz

du dv,

&3

ImJJf’L[f]dudv

af

(g"~u+g'2~)

=-Imjg’

a

fLOdo

0

+ Im rgs

(11)

af

(g” du + gS22)

f,,_, dv.

0

Equating the corresponding parts of Eqs. (8~(1 l), we obtain finally 2 Re J J g’2g’“~~ a + JJgs(gllI

l

%I’

dudv+2pImJJg’h(gi2$+glt~) a, +g”l

z12)

dudv

a =

JJ

(12)

(Re kZ-p2g22)g”*lf12du dv

0

-

Re

fWIf1) 0

“-0

dv

-I-

Re 7 (f’I’[fl)uli 0

f’dudv.

dv,

166

A. S. Ii’indii

and 0. B. Popov

(13)

The last equation implies the uniqueness of the solution of the problem posed. Indeed, let

be the difference of two solutions. From the radiation conditions (7) we obtain

c

J

(fwlL-o~u=-2ni

y(n) IA*(O) I’,

so that Eq. (13) assumes the form

(14)

For the first two boundary conditions the right side is zero. For the third boundary condition the right side is

Writing al = Im &> 0, we obtain the equation

-e2

JJg”l~l’du &I+2n c

-0D

A2 sa

+ p,(u) 0 Since gs > 0,

in all three

Re 7(n) IA, (0) I’

If&

v) I’dv4J.

cases f(u, v) z 0 in 2J.

3. On the segment 0 < u 6 1 we introduce the system of functions

Direct method in problems of diffraction

For every N we require that the functions Q’(u)

167

satisfy conditions (S)-(7), that is, the equations

2%

s

cl

h[f&

-inud~=O,

n=-N,

. . . ,O,.. . , N,

(15)

the boundary conditions for u = 1 of one of the three forms:

J‘*F[f,]e-‘“mdv=O,

B,“(l)=O,

0

(16) ~(rlf.l-a(~)f~-i(D))e-‘““dv=O, 0 and also the partial conditions for u = 0 2n

J. r[fN]e-‘“”

dv=&ti[ x0 co9 cp+y (n) 1exp (ixoI COBcp)L(17)

-2~ciy(n)B,~(O),

w-N,.

. . ,O,. . . , N.

Our purpose is to prove the unique solvability of problem (15)-( 17) and the convergence of the approximate to the exact solution in the energy norm in the domain a. In view of the importance of the problem posed we write it in the explicit form of an equation and boundary conditions. In matrix form system (5) is written as follows:

d2BN dBN A(u)~+uB~+C(U)B~=O where @‘(u) is the vector {B-MN(u) , . . . , BON(u) , . . . , BNN(u) } , A(u), B and c(u) have the following construction:

and the matrices

A (u) =A+u2A, an _if,,=

e, J a(v)--1 f(rn--n)o

dv,

a&,,,=

0

B mI=2zwn 4 (m+n+2p)

zn6(v)

e

i(m--n)o

J o(v)--I

0

&,,

76 (v) ei(m-n)odv, 0

C’..(U)=~ [U(U)- Z] [$(u,

~)-((m+~)2]e*(m--n)odu,

0

n,

m--N,. ..,0, ...,N.

The form of the matrix c(u) is simplified if it is assumed that will hold if x in the coordinates (x, y) is of the form

x (u, V) =X (u) .

This relation

168

A. S. ll’imkii and 0. B. Popov

Then

Conditions I and II on the boundary u = 1 are as follows: BN(l)==O,

A(l)i3N(I)+HBN(I)=0,

where

In these two casesthe following unction

muat be satisfied at the left end of the segment:

tiN (0) +EBn (0) =-F,

E,,=GTi~

(4 6,m

F,--2ni(~~ 009cp+r (4 > exp (W co8 cp>L For a supplementary condition of the third kind we have on the right the equation A(l)jBN(l)+PBN(l)-Q, where aal

P=H+ji,

Ja^(u) e’(m-n)”du,

A_,--

0

Qn

- ji (Y)t+=

du,

0

and on the left the equation aN

(0) +EBN (0) -0.

But if O(V)is an even function an additional simplification of the form of the matrix results.

4. We consider for the function f&d, v) the equation

(18)

Direct method in problems of diffwtion

169

similar to Eq, (13). For the fast two forms of supplementary conditions the integral on the right is zero and the equation assumes the form

$SSg~lfNlzd~du+~N(0)=~XoCos’p+2n(Rey(O)+Imy(0)), 9 where

~,(O)-22n~Rey(n)IB,N(O)I2+~l~oN(O)12(Rey(O)-~oeoscp) iMo ~~~ocos~lexp(~~oZc~~)-~~N(0)l*~~Rey(O)~~o~(O) -2exp(iXJcoscp) ++(Re

lz+$

Imy(0) IB,N(0)-2ie~p(L~OEwsq)) I”

y(O)- Imy(O)f tB,N(O) I”.

Since Re y (0) N, cos Al, Re y (0) >Im y (0), then &v(O) ;B 0. Considering that g”==AL(a(v)-Z)&A%in

(u(u)-Z)>O,

we arriveat the condition of boundedness of I &N(O) 1with respect to N and at the inequality

with constant & independent of N. Writing down Eq. (18) for the difference of two solutions, we obtain a proof of the unique ~lvab~~ of problem (lS)-(17) for the fust two types of supplementary conditions. For the third supplementary condition, using the fact that

(20)

we rewrite Eq. (18) as follows:

(21)

A. S. LVnskii and 0. B. Popov

170

We again obtain the bo~dedne~ condition (19) and a proof of the unique solvable of the problem (to establish the latter it is necessary in the preceding equation to replace formally f~ by the difference TN of two solutions, and e(v) by zero). Another important consequence of Eq. (21) is the boundedness with respect to N of the quantity as

JIfN(1,v) I” du,

0

used below. Since

spI

ai b-9 fr?($ v) +

a

1 as i6(U)12 .I

6(v) adu< 2ia, (v)

dv

4 ID at(v)

therefore

0

We take any number n 4 0, provided that n2 < 2 mm of (v), and write

%, Jl i--qrlfrf

aa

+i

2

p-&~

1?

0

l~(v)I~dv. 0

From this we obtain the ~equ~ty

We return to the proof of the boundedness with respect to N of the quantities

JJ I~~‘dzsdu H JJ 1~1’ a

&3

For this we consider the equation, similar to Eq. (12):

dudv.

’

Direct method in problems of diffraction

171

Proceeding just as in [7], we transform this equation into the inequality

E, JJ 1% 9,

fkv+e,

- Re jlWh1)

JJ l+l’ &s

dadv< JJ q(u,v)~j~~~dudv a

.-,du+Rej(f,‘rlfnl)r-ldo,

0

0

where ~1 and e2 are positive constants, and the function Q(U,V) is bounded in a. two types of supplementary conditions this inequality assumes the form

For the first

(23)

By (19) and the boundedness with respect to N of I &N(O) I proved above, we obtain the inequalities

(24)

with constants rr and S2 independent of A! For the third supplementary condition the second expression on the right in (23) is replaced by

Using (22), we again arrive at conditions (24).

172

A. S. Ihskii

and 0. B. Popov

5. We will now prove that fN converges to f: As usual we introduce the function

*n-f-fN-

)‘.A,(u)e*“‘-

)‘.dnN(u)e’““-0D

wetransfom

the equation

)1.efY3~N(~). -N

Y

Y

Y

-0D

fN]

h[ qN] = -h[

for the function $N to the fom

JJ a3

JJ 2a

$v*h[$N]dUdv=-

RN(u, V) -

c

. .

&‘h[f&Udv,

A,(u)

(25)

e’““.

IYIl>N

Using the definition of the operators h and L, we write

0

On the other hand

We finally obtain

sJJ -hIi Im

g,‘h[g,]dudV=

f

Jj g”i$rri’dUdV

a

(26)

(~N'r[~N]),-Odv+Im~(b'~~~Nl)u-ldv.

0

0

We now transform the right side in (25): - JJRN*h[fN]dudv=43

-iv i -@is

23

(gi2g’lrfNRN*) ,,,dv--ip g”g’” (2

s

JJR,‘L[ f,]dUdV+ip

RN’-fN s)

7 (g”g’“f,R,‘).,,dv~ 0

JJ gl*g’” (%N,N’-fN au dUdv+ JJ (pzg22-~~~~~fNRN*dudv. 53

Direct method in problemsofdiffaction

173

Direct calculation gives -

f.]dudv=l(g’lg+P %) 1 ,_,dv

Jj R..L[

R,'g'h

0

g” g

)I

+ gf2 g

dv

u-1

0

For the imaginary part we obtain the following representation:

-Im JJ RN%[fN]du dv==Im

&3

J- VW%1 1u-odv

0

(27)

-

Im JJ (k2-p2gZa) g”f,R,’

du dv.

PD

The radiation condition for the function $N at u = 0 assumes the form N

1%

c

J

kkN(0) 1’

+I

(t,$r[$N.]),~lOdv=-hi

0

(28)

-N

- 2ni

c

r(n) IA,(O) I’.

Illl>N

Equating the right sides of (26) and (27) taking into account (28), we obtain the fundamental equation for the proof of convergence: EZ -7 A +

+

N

JJ ZB

2n

Im

g”l$,l’du

dv + 23-t c

Rey(n) Inl>N zn

IA,(O) I2

J (qN*r[qNI

)u-1 dv=-

(29) an Im

J (RN’r[fN])u_i

dv

0

0

(RN’r[fNl)u-o 0

Id,,N(O) I”

-N

c

+ Irni

Re r(n)

dv +

ImJJ t 83 1,1-i

afx aR*’ t?“ih xaa’

du dv

174

A. S. Ii’imkii and 0. B. Popov

(cont’d)

+‘Rejjg.h[g”(f,~-~R,‘) a

+-ti’ -

Im

(

aRN’

fN.av-a,.R,’

afN )I

ss

da dv

(k2-p*g2) gsfNRN' da

&I.

P,

We now use the boundary conditions at u = 1. For the first boundary condition $N( 1, v) = 0 and RN( 1, v) = 0, therefore the last integral on the left and the first integral on the right in (29) equal zero. For the second boundary condition we consequently obtain 2s InldV,

0, lTgN]u_ie-if~dv

-

s 0

Inl>N.

-rF[t,l.-,a.'".&, I

0

that is,

Ir

au

u+~tPnlybr*L-r dv=- I w~firlR~*)~~dv, 05 0

that the last integral on the left and the f3rst on the right cancel each other. For the third boundary condition we write so

’

au

J G(v)

InlcN,

(f-fN)u,le-'n*dv,

0

wt 2# (G(v)f(i,v)+@(v))e-*"* I

dv _i

\O

r[fN]ul,e-iwdv,

8

Multiplying by d,,N*( 1) and summing over n, we obtain

@

0

2n P

~~(v)ih(l,v),'du-ji(v)$hV(*,v)R,*(i,v)~v 0

+T

0

[;(v)f(i,v)+@(v)]RN*(i,v)dv 0

lni>N,

Direct method in problems of diffiction

175 (cont’d)

+ j

b,(v)R,‘(f,

u)du

t-j &)&(I,

u)RN*(l, u)du

0

0

-7

(RN*r[fNI)U-i

du.

0

In this representation it is important that the imaginary part of the first integral is non-negative, the imaginary part of the latter cancels the first integral on the right in (29), and the remaining terms are simply estimated by using (22). Finally, we show that the second integral on the right in (29) equals zero. Indeed,

c f, 7so(v:_l B,N(0)An’(O)ei(m-‘)odv

-

InllNm--N

0

InbNm--NO

Therefore, taking into account all the conditions, Eq. (29) assumes the form

JJg” 1$lyI2du du+2,‘ c

AB

+ 2n

Re y(n) i&“(O) I’.

Re y (4 IA, (0) 1’

c InI>N

l+N(l,

+ror,(~)

P

U)

@(u)RN*(~, u)du

I’& --hi 0

an

- h

s0

G(u)f~(i,

u)RN*(~, U)dU

2 +

-

Im

g'Jg'"

23i.j-i

Im s

afN

URN'

ad ad

du dv

( k2-p’g”) g”fN RN*du dv.

(30)

176

A. S. ll’inakii and 0. B. Popov

For supplementary conditions of the first and second kind we omit the last integral on the left and the first on the right. We note that all the expressions on the left are non-negative. We introduce the notation

M rj

max 9)

=

I g’jg”* I ,

M = m~~lk”(u, ~)-$g~)g’~[.

We obtain

I’ J CnU,(u)

2x -&+YF 0

-0

Similarly,

Considering this, and also (19), (22), (24), we estimate successively all the expressions on the right in (30):

I 2aJJ(I?Im

1p.e

I

$gzz) gv*fNRN’du dv G + M (6062)IA,

JJgitpA(

f$$-$tN*)

audv

1

23

<

+pM.,[ @,A,)"

1Imjii(v)i,(l, 0

<~maxl~(v)lA”( 288 Im h(v)RN*(l,

I s 0

+(6&)“21,

v)&*(I,v)dv

1

ylgll_idv)“, 0 v)dv I+=$maxli(v)I <

(~l~l~_,dv)“. 0

Direct method in problemsof diffraction

Therefore, the right side in (30) is 0(1/N). Since means that in 9)

g’&~d2~~in(~(~)

177

4) 30,

this

where limd,N(0)==lim (A,(0)-B,N(O))=O, N-c_

n--N,...,N,

Nlcra

that is, the reflection coefficients tend to the exact vaiues, since

T~n~fed

by J. Berry.

REFERENCES 1.

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2.

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