On the justification of the Foldy-Lax approximation for the acoustic scattering by small rigid bodies of arbitrary shapes

TL;DR

This paper establishes conditions under which the Foldy-Lax approximation accurately models acoustic scattering by many small rigid bodies of arbitrary shapes, providing explicit error estimates and analyzing different potential representations.

Contribution

It offers new sufficient conditions for the validity of the Foldy-Lax approximation and derives explicit error bounds based on obstacle parameters and potential types.

Findings

Foldy-Lax approximation validity depends on obstacle parameters and potential representation.

Explicit error estimates are derived in terms of number, size, and spacing of obstacles.

We analyze the impact of multiple scattering in inverse acoustic problems.

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 it 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.