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

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 $\mathrm{\Omega }$, 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 $\mathrm{\Omega }$, 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 $a,a<<1$.