TABLE OF CONTENTS Modified Rayleigh Conjecture (MRC) in scattering theory is proposed and justified. MRC allows one to develop numerical algorithms for solving direct scattering problems related to acoustic wave scattering by soft and hard obstacles of arbitrary shapes. It gives an error estimate for solving the direct scattering problem. It suggests a numerical method for finding the shape of a star-shaped obstacle from the scattering data. Section 12.1 is based on [116]. A numerical implementation of MRC method is given in Section 12.2 and is based on the paper [119]. Section 12.2.1 is based on the paper [117].
Consider a bounded domain D ? ? n, n = 3 with a boundary S. The exterior domain is D' = ? 3\ D. Assume that S is smooth and star-shaped, that is, its equation can be written as
| (12.1) |  |
where ? ? S 2 is a unit vector and S 2 denotes the unit sphere in R 3. Smoothness of S is used in (12.18) below. For solving the direct scattering problem by the method described in the beginning of Section 12.2, the boundary S can be Lipschitz. The acoustic wave scattering problem by a soft obstacle D consists in finding the (unique) solution to the problem (12.2)-(12.3):
| (12.2) |  |
| (12.3) |  |
Here u 0 := e ik? x is the incident field, v is the scattered field, A( ?
