Characterization of the equivalent acoustic scattering for a cluster of an extremely large number of small holes

TL;DR

This paper analyzes how large clusters of tiny holes scatter acoustic waves, showing different behaviors depending on the number of holes relative to their size, and models these clusters as soft, moderate, or rigid bodies.

Contribution

It provides a detailed characterization of the acoustic scattering behavior of clusters of small holes, including explicit error estimates, for different scaling regimes of the number of holes.

Findings

For s<1, the scattered field vanishes as holes shrink.

For s=1, the cluster acts as an equivalent medium with a specific refraction index.

For s>1, the cluster behaves as a totally reflecting rigid body.

Abstract

We deal with the time-harmonic acoustic waves scattered by a large number of small holes, of maximal radius $a, a<<1$, arbitrary (i.e. not necessarily periodically) distributed in a bounded part of a homogeneous background. We show that as their number $M$ grows following the law $M:=M(a):=O(a^{-s}), \; a<<1$, 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 $\Omega$, which is described by certain geometry properties of the holes and their local distribution. The cluster is a moderate (or intermediate) one. 3. if $s>1$, then the cluster behaves as a totally reflecting extended body, modeled by a bounded and smooth domain $\Omega$, i.e. the incident waves are totally reflected by the surface of this extended body. The cluster is a rigid one. These approximations are provided with explicit error estimates in terms of $a,\; a<<1$.