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We study the time-domain acoustic scattering problem by a cluster of smallholes (i.e. sound-soft obstacles). Based on the retarded boundary integralequation method, we derive the asymptotic expansion of the scattered field asthe size of the holes goes to zero. Under certain geometrical constraints onthe size and the minimum distance of the holes, we show that the scatteredfield is approximated by a linear combination of point-sources where theweights are given by the capacitance of each hole and the causal signals (ofthese point-sources) can be computed by solving a, retarded in time, linearalgebraic system. A rigorous justification of the asymptotic expansion and theunique solvability of the linear algebraic system are shown under naturalconditions on the cluster of holes. As an application of the asymptoticexpansion, we derive, in the limit case when the holes are densely distributedand occupy a bounded domain, the equivalent effective acoustic medium (anequivalent mass density characterized by the capacitance of the holes) thatgenerates, 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 perforatedmaterial with holes, having appropriate capacitances, that generatesapproximately the same acoustic field as the acoustic medium modelled by thegiven mass density (and constant speed of propagation). Finally, we numericallyverify the asymptotic expansions by comparing the asymptotic approximationswith the numerical solutions of the scattered fields via the finite elementmethod.

Analysis of the acoustic waves reflected by a cluster of small holes in the time-domain and the equivalent mass density