Integral equations for the diffraction problems in the wedge

✍ Scribed by Ju. K. Podipenko

• Publisher: John Wiley and Sons
• Year: 1995
• Tongue: English
• Weight: 863 KB
• Volume: 18
• Category: Article
• ISSN: 0170-4214
• DOI: 10.1002/mma.1670180604

✦ Synopsis

In the present paper we investigated the boundary value problems appearing in the study of diffraction of acoustical or electromagnetic waves on arbitrary bounded body contained within the wedge. The potential theory has been developed making it possible to reduce the boundary value problems to Fredholm integral equations on the body's boundary. We prove existence and uniqueness of solutions for these integral equations and the boundary value problems.

For some other types of domains with infinite boundaries similar problems have been studied in [4, 6-8, 12-14, 19, 20, 22, 23, 25-28].

1. Formulation of problems

Introduce in R3 the spherical co-ordinate system r, 8, u, (0 < r < + co, 0 d 8 d x , 0 < v, ,< 274 and denote by R:= {(r, 8, v , ) ~R ~l r > 0,O < 8 < x, 0 < 9 < @} a wedge with the spread angle 0 , 0 < 0 < 27r, and by aR the boundary of the wedge. Let 9 be a bounded domain with the boundary 8 9 consisting of a finite number of nonintersecting closed C2 surfaces. We assume that g c R and the domain i 2 \ 8 is connected.

Definition 1. We shall denote by 92 the linear space of all complex-valued functions u E C2(R\G) n C( a\9) such that at any point N E 8 9 there exists the limit ~--lim ( v N , grad u ( N + hvN)), h -0 h > O uniformly in N. Here vN is the exterior unit normal to 9 in N . 0 < u,< 9 r = R } . Let us also set RR := { ( r , 8, v,)ERlr < R } , and SR := { ( r , 8, p ) ~ R310 d 8 d n, The letters A, C equipped with indices will denote positive constants. We investigate the following three boundary value problems. Problem I . Find a function u (r, 8, p) E C 2 ( R \ g 1 n C ( a \ 9 ) satisfying Au(r, 8, p) + k2u(r, 8, u,) = O(Im k 2 0) in R/@ (1.1) CCC 01 70-4214/95/060449-23 Q 1995 by B. G. Teubner Stuttgart-John Wiley & Sons Ltd.