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

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.