Analytic and Numerical Methods in the Study of the Inverse Scattering Problem for Acoustic Waves

Abstract

In this paper we wish to briefly survey some recent results we have obtained in studying the inverse scattering problem for acoustic waves. Since the literature on inverse scattering proplems is enormous and considers a wide variety of problems, it is first necessary to be more explicit as to what inverse scattering problem we are talking about. Here we are concerned with the problem of determining the shape of a two dimensional acoustically soft obstacle from a knowledge of the far field pattern corresponding to an incoming time harmonic wave at frequencies in the resonant region, i.e. it is not possible to use either high or low frequency approximation methods (In those cases where such asymptotic methods are valid, our methods are still applicable, and enjoy the corresponding simplicity afforded by such an approach). Although this problem is quite special, our methods can be generalized to treat a variety of related problems, including different boundary conditions, higher dimensional problems, the determination of the surface impedance of an obstacle of known shape, and corresponding problems in time harmonic electromagnetic wave propagation. For details of both the results outlined here, as well as the above mentioned generalizations, we refer the reader to the forthcoming monograph by the first author in collaboration with Rainer Kress of the University of Göttingen ([4]) as well as the papers[1],[2],[3] and [5].

The research of this author was partially supported by AFOSR Grant 81-0103