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

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 $a\le a_0$ and $\sqrt{M-1}\frac{a}{d}\le c_0$ 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).