Far-Field Patterns for Acoustic Waves in an Inhomogeneous Medium

Abstract

This paper is concerned with the class of far-field patterns corresponding to the scattering of time-harmonic acoustic plane waves by an inhomogeneous medium of compact support. This class is shown to be complete in $L^2 (\partial \Omega )$ (where $\partial \Omega $ is the unit sphere) for any positive value of the wave number, with the possible exception of a discrete set of wave numbers called transmission eigenvalues. The existence of a unique weak solution to the interior transmission problem (which plays a basic role in a new method for solving the inverse scattering problem) is also established for any positive value of the wave number provided the wave number is not a transmission eigenvalue.