A Novel Method for Solving the Inverse Scattering Problem for Time-Harmonic Acoustic Waves in the Resonance Region II

Abstract

In a previous paper [SIAM J. Appl. Math. (1985), pp. 1039–1053] we presented a new method for determining 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. The method given there was based on knowing the far field pattern for an interval of values of the square of the wave number k such that this interval contained the first eigenvalue $\lambda _1 $ of the interior Dirichlet problem. The purpose of this paper is to extend the methods of our earlier paper to treat the case when the far field pattern is only known for a single value of the wave number such that $k^2 $ is not equal to $\lambda _1 $. In addition we show how to combine the results of this paper with the first to obtain a method that is valid for any fixed value of k independent of whether or not $k^2 $ is an eigenvalue. We also extend our analysis to include the case of the impedance boundary value problem. Numerical examples are given showing the practicality of our method.