FINITENESS RESULTS CONCERNING NONSCATTERING WAVE NUMBERS FOR INCIDENT PLANE AND HERGLOTZ WAVES

Abstract

In this paper we introduce an approach to establish finiteness results for the set of wave numbers that may lead to vanishing scattering effects. We use this approach to establish two results concerning the two dimensional Helmholtz equation in the context of a penetrable obstacle and (1) incident plane waves as well as (2) incident Herglotz waves. For a smooth, strictly convex, bounded domain, we show that there are at most finitely many positive wave numbers at which a plane wave with a fixed incident direction is nonscattering. For a disk there exist densities such that the corresponding incident Herglotz waves are nonscattering for infinitely many positive wave numbers. Here we show that any small perturbation of the disk to a proper ellipse will lead to at most finitely many such wave numbers.