Wave scattering by many small particles embedded in a medium

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) $\ka_1=1/3$, $\ka>1$.