A Two-Scale Multiple Scattering Problem

Abstract

Consider the scattering problem of a time-harmonic plane wave incident on a heterogeneous medium consisting of isotropic point (small scale) scatterers and an extended (wavelength comparable) obstacle scatterer in three-dimensional space. To compute the scattered field from the interaction between the incident wave and the point scatterers only, the Foldy–Lax method provides an effective approach, while boundary integral equation methods play an important role for solving the scattering problem solely involving an extended obstacle scatterer. It is a challenging two-scale multiple scattering problem when both the point scatterers and the extended obstacle are present. In this paper, a generalized Foldy–Lax method is developed to fully take account of the multiple scattering in the heterogenous medium. The method is viewed from two different formulations: the series solution and the integral equation. The series solution formulation is shown as an efficient iterative scheme to the integral equation formulation. The convergence of the scattered fields and the far-field patterns from the series solution formulation are characterized in terms of scattering coefficients. Numerical experiments are presented to show the agreement and the effectiveness of the proposed two approaches.