介质中许多小颗粒对波的散射
Wave scattering by many small particles embedded in a medium
摘要：
在关于$\frac{a}{d}$比的各种假设下发展了许多小物体的散射理论，其中$a$是小物体的特征尺寸，$d$是相邻物体之间的距离$d = O(a^{\ka_1})$，$0 < \ka_1 < 1$。在每个小天体的边界$S_m$上，假定一个阻抗型条件$u_N = ζ_m u$在$S_m$上，$1 \leq m \leq M$，$ζ_m = h_m a^{-\ka}$，$0 < \ka$，$h_m$是与$a$无关的常数.研究了$M = M(a)\gg 1$小颗粒嵌入区域中的场行为，作为$a\ra 0$和$m(a)\ra \ue$。在下列假设下导出了极限介质折射系数的计算公式：a）$ \ka_1=(2-\ka)/3$，$0<\ka\leq 1$; b）$\ka_1=1/3$，$\ka>1$。
英文摘要：
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 $u_N = ζ_m u$ on $S_m$, $1 \leq m \leq M$, $ζ_m = h_m a^{-\ka}$, $0 < \ka$, $h_m$ 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) $\ka_1=1/3$, $\ka>1$.
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