The Transmission Acoustic Scattering Problem for Bi-Spheres in Low-Frequency Regime

Abstract

An acoustically soft sphere covered by a penetrable eccentric spherical shell disturbs the propagation of an incident plane wave field. It is shown that there exists exactly one bispherical coordinate system that describes the given geometry. The incident wave is assumed to have a wavelength which is much larger than the characteristic dimension of the scatterer and thus the low-frequency approximation method is applicable to the scattering problem. The incomplete R-separation of variables in bispherical coordinates and the normal differentiation involved in the transmission boundary conditions lead to a three-term recurrence relation for the series coefficients corresponding to the scattered fields. Thus, the potential boundary-value problem for the leading low-frequency approximations is reduced to infinite tridiagonal linear systems, which are solved analytically.