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 $\epsilon$), 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 ${\omega }_M$. Here we explain the origin of such a phenomenon: at first we show that the scattering of an incident wave of frequency $\omega$ is described by a self-adjoint $\omega$-dependent Schrödinger operator with a singular $\delta$-like potential supported at the inhomogeneity interface. Then we show that, in the low energy regime (corresponding in our setting to $\epsilon \ll 1$) such an operator has a non-trivial limit (i.e., it asymptotically differs from the Laplacian) if and only if $\omega ={\omega }_M$. 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 ${\omega }_M$, the scattering process undergoes a transition between an asymptotically trivial behaviour and a non-trivial one.

arXiv commentLatex, 31 pages, improved version

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Cited by in corpus (1)

• Localization of point scatterers via sparse optimization on measures