On the Origin of Minnaert Resonances

Abstract

It is well known that the presence, in a homogeneous acoustic medium, of a small inhomogeneity (of size ε), enjoying a high contrast of both its mass density and bulk modulus, amplifies the generated total fields. This amplification is more pronounced when the incident frequency is close to the Minnaert frequency ωM . Here we explain the origin of such a phenomenon: at first we show that the scattering of an incident wave of frequency ω is described by a self-adjoint ω-dependent Schrödinger operator with a singular δ-like potential supported at the inhomogeneity interface. Then we show that, in the low energy regime (corresponding in our setting to ε 1) such an operator has a non-trivial limit (i.e., it asymptotically differs from the Laplacian) if and only if ω = ωM . The limit operator describing the non-trivial scattering process is explicitly determined and belongs to the class of point perturbations of the Laplacian. When the frequency of the incident wave approaches ωM , the scattering process undergoes a transition between an asymptotically trivial behaviour and a non-trivial one. As a by-product, we get the existence of a local maximum of the scattering amplitude occurring at the Minnaert frequency. It is this last property that is experimentally observed and used in the applications.