Wave scattering by many small particles embedded in a medium

TL;DR

This paper develops a theoretical framework for wave scattering by many small particles embedded in a medium, deriving formulas for the effective refraction coefficient under various assumptions about particle size, spacing, and boundary conditions.

Contribution

It introduces new models for wave scattering with small particles, analyzing different regimes of particle size and spacing, and deriving formulas for the resulting effective medium properties.

Findings

Formulas for the refraction coefficient of the limiting medium.

Analysis of wave behavior as the number of small particles increases.

Conditions under which the effective medium approximation holds.

Abstract

Theory of scattering by many small bodies is developed under various assumptions concerning the ratio $\frac{a}{d}$, where $a$ is the characteristic dimension of a small body and $d$ is the distance between neighboring bodies $d = O(a^{\ka_1})$, $0 < \ka_1 < 1$. On the boundary $S_m$ of every small body an impedance-type condition is assumed $u_N = \zeta_m u$ on $S_m$, $1 \leq m \leq M$, $\zeta_m = h_m a^{-\ka}$, $0 < \ka$, $h_m$ are constants independent of $a$. The behavior of the field in the region in which $M = M(a)\gg 1$ small particles are embedded is studied as $a\ra 0$ and $m(a)\ra \ue$. Formulas for the refraction coefficient of the limiting medium are derived under the assumptions: a) $ \ka_1=(2-\ka)/3$, $0<\ka\leq 1$, and b) $\ka_1=1/3$, $\ka>1$.