# The Foldy-Lax Approximation for the Elastic Scattering by Many Small Obstacles with Arbitrary Shapes

Durga Prasad Challa^{*} and Mourad Sini

We are concerned with the $3$ dimensional scattering by obstacles for the linearized, homogeneous and isotropic elastic model at a fixed frequency, i.e. the LamÃ© model. We assume the obstacles to have sizes small compared to the wavelength. The main result of this talk is the following. We give a sufficient condition linking the number $M$ of obstacles, their diameters and the minimum distance between them, under which the Foldy-Lax approximation is valid. In particular, using this approximation, one can capture the scattered fields due to the first order interactions between the obstacles. As an application, we study the inverse problem of locating (and estimating the sizes of) the obstacles using far field data. It is known that any wave propagating in such elastic media is a combination of two fundamental body waves, called the pressure waves (P-waves) and the shear waves (S-waves). We show that the 'pressure' parts of the far field patterns corresponding to 'pressure' (or 'shear') incident plane waves are enough to solve this inverse problem. Similarly, 'shear' parts of the far field patterns corresponding to 'pressure' (or 'shear') incident plane waves are also enough.

Mathematics Subject Classification: 35J47 45Q05

Keywords: Scattering by small obstacles, Foldy-Lax approximation, inverse scattering, elastic waves.

Minisymposion: Inverse Problems in Elasticity