Body resonances for classical waves

Abstract

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