Body resonances for classical waves

TL;DR

This paper analyzes the spectral properties and resonance localization of a high-contrast acoustic wave operator with small inhomogeneities, providing explicit first-order expansions in terms of eigenvalues and eigenvectors.

Contribution

It offers a detailed spectral analysis of the operator modeling acoustic waves with small high-contrast inhomogeneities, including explicit resonance expansions.

Findings

Resonance localization results for the operator

Explicit first-order epsilon-expansions of resonances

Connections between resonances and eigenvalues of the Newton potential

Abstract

We provide a detailed study of the spectral properties of the linear operator $H(\varepsilon)=-(\varepsilon^{2}\chi_{\Omega_{\varepsilon}}+\chi_{\Omega^{c}_{\varepsilon}})\Delta$ modeling, through the wave equation $(\partial_{tt}+H(\varepsilon))u=0$, the dynamics of acoustic waves in the presence of a small inhomogeneity of size $\varepsilon$ having high contrast $\varepsilon^{-2}$. In particular, we give precise results on the localization of the resonances of $H(\varepsilon)$ and their first-order $\varepsilon$-expansions; the latter are explicitly expressed in terms of the eigenvalues and eigenvectors of the Newton potential operator of the set $\Omega$ whose rescaling of size $\varepsilon$ defines $\Omega_{\varepsilon}$.