The point scatterer approximation for wave dynamics

Given an open, bounded and connected set $\mathrm{\Omega }\subset {\mathbb{R}}^3$ and its rescaling ${\mathrm{\Omega }}_{\epsilon }$ of size $\epsilon \ll 1$, we consider the solutions of the Cauchy problem for the inhomogeneous wave equation $$({\epsilon }^{-2}{\chi }_{{\mathrm{\Omega }}_{\epsilon }}+{\chi }_{{\mathbb{R}}^3\mathrm{\backslash }{\mathrm{\Omega }}_{\epsilon }}){\mathrm{\partial }}_{tt}u=\mathrm{\Delta }u+f$$ with initial data and source supported outside ${\mathrm{\Omega }}_{\epsilon }$; here, ${\chi }_S$ denotes the characteristic function of a set $S$. We provide the first-order $\epsilon$-corrections with respect to the solutions of the inhomogeneous free wave equation and give space-time estimates on the remainders in the $L^{\mathrm{\infty }}((0,1\mathrm{/}{\epsilon }^{\tau }),L^2({\mathbb{R}}^3))$-norm. Such corrections are explicitly expressed in terms of the eigenvalues and eigenfunctions of the Newton potential operator in $L^2(\mathrm{\Omega })$ and provide an effective dynamics describing a legitimate point scatterer approximation in the time domain.