On the Justification of the Foldy--Lax Approximation for the Acoustic Scattering by Small Rigid Bodies of Arbitrary Shapes

Abstract

We are concerned with the acoustic scattering problem by many small rigid obstacles of arbitrary shapes. We give a sufficient condition on the number $M$ and the diameter $a$ of the obstacles as well as the minimum distance $d$ between them under which the Foldy--Lax approximation is valid. Precisely, if we use single layer potentials for the representation of the scattered fields, as is done sometimes in the literature, then this condition is $(M-1)\frac{a}{d^2} <c$, with an appropriate constant $c$, while if we use double layer potentials, then a weaker condition of the form $\sqrt{M-1}\frac{a}{d} <c$ is enough. In addition, we derive the error in this approximation explicitly in terms of the parameters $M, a$, and $d$. The analysis is based, in particular, on the precise scalings of the boundary integral operators between the corresponding Sobolev spaces. As an application, we study the inverse scattering by the small obstacles in the presence of multiple scattering.