The Foldy-Lax approximation of the scattered waves by many small bodies for the Lamé 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 a0 and c0, depending only on the Lipschitz character of the obstacles, such that under the conditions a ≤ a0 and≤ c0 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).

Multiscale analysis of the acoustic scattering by many scatterers of impedance type

“…The accuracy of the approximation of the scattered field by the Foldy-Lax field depends of course on the error estimates. In its generality, this issue is still largely open but there is an increase of interest to understand it, see for instance [11,12,[14][15][16][24][25][26].…”

“…It is important to remark that, if we do not distinguish the near by and far obstacles, as it is discussed in the beginning of section 3.2.2, and by following the way it was done in [15,16] we can get the estimate DK * + λK ≤ M −1 (3.46) and hence the condition (3.52) will be replaced by (M − 1) a 2−β d 2 < c 0 , for some suitable constant c 0 . However, this condition is too strong to enable us to apply our asymptotic expansion to the effective medium theory where we need to choose M ∼ a −s with s = 2 − β and d ∼ a t with t ≥ s 3 .…”

Location and size estimation of small rigid bodies using elastic far-fields

“…The forward problem is to compute the P-part, U ∞ p (x, θ), and the S-part, U ∞ s (x, θ), of the far-field pattern associated with the Lamé system (1.1-1.3) for various incident and the observational directions. The main result is the following theorem, see [7,Theorem 1.2], which justifies the Foldy-Lax approximation, in order to represent the scattering by small scatterers taking into account the three parameters M , a and d. then the P-part, U ∞ p (x, θ), and the S-part, U ∞ s (x, θ), of the far-field pattern have the following asymptotic expressions…”

“…Regarding the first step, we provide the asymptotic expansion in terms of the three parameters M , a and d, while in the previous literature the two parameters M and d are assumed to be fixed, [3,4]. This expansion is justified for the Lamé model, under consideration here, in our previous work [7]. Regarding the second step, which is the object of this paper, we apply the MUSIC algorithm, see [5,9], to the P-parts (respectively the S-parts) of the elastic far-fields to localize the centers of the scatterers.…”

“…The rest of paper is organized as follows. In section 2, we recall, from [7], the Foldy-Lax approximation of the elastic fields. In section 3, we use these approximations to justify Theorem 1.2.…”