Two Methods for Solving the Inverse Scattering Problem for Time-Harmonic Acoustic Waves

Abstract

The inverse scattering problem we are considering in this paper is to determine the shape of an acoustically soft obstacle from a knowledge of the time-harmonic incident wave and the far field pattern of the scattered wave, where the frequency is assumed to be in the “resonant” region, i.e. high frequency asymptotic methods are not available. For a survey of the research done in this area, we refer the reader to the monograph of COLTON and KRESS [4] as well as the survey paper by COLTON [2]. Our aim is to provide methods that are numerically implementable, and the basic problem in trying to do this is the fact that the inverse scattering problem is both nonlinear and improperly posed ([2], [4]). In this paper, we shall present two methods for solving the inverse scattering problem. The first of theses is based on the use of integral equations and has been numerically implemented by KIRSCH [7] (see also [2]). Our second method avoids the use of integral equations ad is based on the theory of Hergoltz wave functions (c.f. [6]). Numerical experiments using this method are presently being carried out, but as yet we have no results to report. Hence, the practicality of our second method has yet to be established.

This research was partially supported by AFOSR Grant 81-0103 and NSF Grant DMS-8320550