On the Origin of Minnaert Resonances
| Autor: | Mantile, Andrea, Posilicano, Andrea, Sini, Mourad |
|---|---|
| Rok vydání: | 2021 |
| Předmět: | |
| Zdroj: | J. Math. Pures Appl. 165, 106-147, 2022 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1016/j.matpur.2022.07.005 |
| Popis: | It is well known that the presence, in a homogeneous acoustic medium, of a small inhomogeneity (of size $\varepsilon$), 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 Schr\"{o}dinger 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 $\varepsilon\ll1$) 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. Comment: Latex, 31 pages, improved version |
| Databáze: | arXiv |
| Externí odkaz: |
|