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 $\mathrm{\Omega }$ 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 $D_m,m=1,\mathrm{.}\mathrm{.}\mathrm{.},M$ is modeled by four parameters: their number $M$, their maximum radius $a$, their minimum distance $d$ and the surface impedances ${\lambda }_m,m=1,\mathrm{.}\mathrm{.}\mathrm{.},M$. 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 $a\to 0$, 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 $a\to 0$. 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.