The Foldy-Lax approximation of the scattered waves by many small bodies for the Lame system

TL;DR

This paper establishes conditions under which the Foldy-Lax approximation accurately models elastic wave scattering by many small obstacles, providing explicit error bounds useful for inverse and design problems.

Contribution

It proves the validity of the Foldy-Lax approximation for elastic scattering by multiple small obstacles with explicit error estimates, depending on obstacle size and spacing.

Findings

Validates Foldy-Lax approximation under specific size and spacing conditions.

Provides explicit error bounds in terms of obstacle parameters.

Applicable to inverse and effective medium design problems.

Abstract

We are concerned with the linearized, isotropic and homogeneous elastic scattering problem by many small rigid obstacles of arbitrary, Lipschitz regular, shapes in 3D case. We prove that there exists two constant $a_0$ and $c_0$, depending only on the Lipschitz character of the obstacles, such that under the conditions $a\leq a_0$ and $\sqrt{M-1}\frac{a}{d} \leq c_0$ on the number $M$ of the obstacles, their maximum diameter $a$ and the minimum distance between them $d$, the corresponding Foldy-Lax approximation of the farfields is valid. In addition, we provide the error of this approximation explicitly in terms of the three parameters $M, a$ and $d$. These approximations can be used, in particular, in the identification problems (i.e. inverse problems) and in the design problems (i.e. effective medium theory).