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

TL;DR

This paper develops a multiscale asymptotic analysis for acoustic scattering by many small impedance obstacles, deriving explicit error estimates and identifying the dominant Foldy-Lax field in the small obstacle limit.

Contribution

It provides a novel asymptotic expansion for the farfield pattern of multiple impedance scatterers with explicit error bounds, considering complex scaling of parameters as obstacle size diminishes.

Findings

Explicit asymptotic expansion of farfields as obstacle size tends to zero

Identification of the dominant Foldy-Lax scattering field

Error estimates in terms of obstacle size and parameters

Abstract

We are concerned with the acoustic scattering problem, at a frequency $\kappa$, by many small obstacles of arbitrary shapes with impedance boundary condition. These scatterers are assumed to be included in a bounded domain $\Omega$ in $\mathbb{R}^3$ which is embedded in an acoustic background characterized by an eventually locally varying index of refraction. The collection of the scatterers $D_m, \; m=1,...,M$ is modeled by four parameters: their number $M$, their maximum radius $a$, their minimum distance $d$ and the surface impedances $\lambda_m, \; m=1,...,M$. We consider the parameters $M, d$ and $\lambda_m$'s having the following scaling properties: $M:=M(a)=O(a^{-s})$, $d:=d(a)\approx a^t$ and $\lambda_m:=\lambda_m(a)=\lambda_{m,0}a^{-\beta}$, as $a \rightarrow 0$, with non negative constants $s, t$ and $\beta$ and complex numbers $\lambda_{m, 0}$'s with eventually negative imaginary parts. We derive the asymptotic expansion of the farfields with explicit error estimate in terms of $a$, as $a\rightarrow 0$. The dominant term is the Foldy-Lax field corresponding to the scattering by the point-like scatterers located at the centers $z_m$'s of the scatterers $D_m$'s with $\lambda_m \vert \partial D_m\vert$ as the related scattering coefficients.