The acoustic cavity containing small scatterers as a singular perturbation problem

Abstract

The problem of finding the perturbed eigenvalues and eigenfunctions of a two-dimensional hard walled container, containing a hard scatterer of length scale small compared to wave-length, is treated by the method of matched asymptotic expansions. The perturbing body is of arbitrary shape and results are given in general form. Notable features of the treatment are that the perturbation expansion valid near the body is used to find an equivalent scattering operator valid in a region bounded away from the body and equivalent to representing the scatterer as a dipole, and that “composite” expansions which are uniformly valid in the entire cavity are presented. The special case of two concentric circles is used to verify the expansions. Another special case treated is a rectangular cavity perturbed by a sharp edge on one wall. As a major purpose of the work is to show how matched asymptotic expansion techniques can be applied to this class of linear acoustics problems, the manipulative techniques of “matching” are shown in detail. Also, the problem chosen is picked somewhat for its illustrative character, and the principal results could most likely be obtained by more classical procedures.