  Given an open, bounded and connected set $\Omega\subset\mathbb{R}^{3}$ and
its rescaling $\Omega_{\varepsilon}$ of size $\varepsilon\ll 1$, we consider
the solutions of the Cauchy problem for the inhomogeneous wave equation $$
(\varepsilon^{-2}\chi_{\Omega_{\varepsilon}}+\chi_{\mathbb{R}^{3}\backslash\Omega_{\varepsilon}})\partial_{tt}u=\Delta
u+f $$ with initial data and source supported outside $\Omega_{\varepsilon}$;
here, $\chi_{S}$ denotes the characteristic function of a set $S$. We provide
the first-order $\varepsilon$-corrections with respect to the solutions of the
inhomogeneous free wave equation and give space-time estimates on the
remainders in the $L^{\infty}((0,1/\varepsilon^{\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 $L^{2}(\Omega)$ and
provide an effective dynamics describing a legitimate point scatterer
approximation in the time domain.
