群孔反射声波的时域分析及等效质量密度
Analysis of the acoustic waves reflected by a cluster of small holes in the time-domain and the equivalent mass density
摘要：
研究了多个小孔（即声软障碍物）的时域声散射问题。基于延迟边界积分方程方法，导出了散射场随孔洞尺寸趋于零的渐近展开式。在一定的几何约束的大小和最小距离的孔，我们表明，散射场近似的线性组合的点源的权重是由每个孔的电容和因果信号（这些点源）可以计算通过解决一个，延迟的时间，线性代数系统。在自然条件下证明了该线性代数系统的渐近展开和唯一可解性。作为渐近展开的应用，我们推导出，在极限情况下，当孔密集分布，并占据一个有界域，等效的有效声学介质（等效的质量密度，其特征在于由孔的电容），产生近似相同的散射场的孔群。相反，给定满足一定次谐波条件的局部可变、平滑和正质量密度，我们可以设计具有适当电容的具有孔的穿孔材料，其产生与由给定质量密度（和恒定传播速度）建模的声学介质近似相同的声场。最后，我们数值验证的渐近展开通过比较的渐近近似与数值解的散射场通过有限元法。
英文摘要：
We study the time-domain acoustic scattering problem by a cluster of small holes (i.e. sound-soft obstacles). Based on the retarded boundary integral equation method, we derive the asymptotic expansion of the scattered field as the size of the holes goes to zero. Under certain geometrical constraints on the size and the minimum distance of the holes, we show that the scattered field is approximated by a linear combination of point-sources where the weights are given by the capacitance of each hole and the causal signals (of these point-sources) can be computed by solving a, retarded in time, linear algebraic system. A rigorous justification of the asymptotic expansion and the unique solvability of the linear algebraic system are shown under natural conditions on the cluster of holes. As an application of the asymptotic expansion, we derive, in the limit case when the holes are densely distributed and occupy a bounded domain, the equivalent effective acoustic medium (an equivalent mass density characterized by the capacitance of the holes) that generates, approximately, the same scattered field as the cluster of holes. Conversely, given a locally variable, smooth and positive mass density, satisfying a certain subharmonicity condition, we can design a perforated material with holes, having appropriate capacitances, that generates approximately the same acoustic field as the acoustic medium modelled by the given mass density (and constant speed of propagation). Finally, we numerically verify the asymptotic expansions by comparing the asymptotic approximations with the numerical solutions of the scattered fields via the finite element method.
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