Affiliations: [a] Faculty of Mathematics, Indian Institute of Technology Tirupati, Tirupati, India. E-mail: durga.challa@iittp.ac.in | [b] LMR, EA 4535 URCA, Fédération de Recherche ARC Mathématiques, FR 3399 CNRS, Université de Reims Champagne-Ardenne, France. E-mail: andrea.mantile@univ-reims.fr | [c] RICAM, Austrian Academy of Sciences, Altenbergerstrasse 69, A-4040, Linz, Austria. E-mail: mourad.sini@oeaw.ac.at
Abstract: We deal with the time-harmonic acoustic waves scattered by a large number of small holes, with radius a, a≪1, arbitrarily distributed in a bounded part of the homogeneous background R3. We assume no periodicity in distributing these holes. Using the asymptotic expansions of the scattered field by this cluster, we show that as their number M grows following the law M∼a−s, a→0, the collection of these holes has one of the following behaviors: (1) if s<1, then the scattered fields tend to vanish as a tends to zero, i.e., the cluster is a soft one. (2) if s=1, then the cluster behaves as an equivalent medium modeled by a refraction index, supported in a given bounded domain Ω, which is described by certain geometry properties of the holes and their local distribution. The cluster is a moderate (or intermediate) one. (3) if s>1, with additional conditions, then the cluster behaves as a totally reflecting extended body, modeled by a bounded domain Ω, i.e., the incident waves are totally reflected by the surface of this extended body. The cluster is a rigid one. Explicit errors estimates between the scattered fields due to the cluster of small holes and the ones due to equivalent media (i.e., the refraction index) or the extended body are provided.
Keywords: Scattering by small particles, Foldy–Lax approximation, effective medium theory
