The point scatterer approximation for wave dynamics

Given an open, bounded and connected set Ω ⊂ R³ Ω ⊂ R 3 and its rescaling Ω ε Ω ε of size ε 1 ε ≪ 1 , we consider the solutions of the Cauchy problem for the inhomogeneous wave equation aligned (ε ⁻²χ _Ω ε+χ _R³ Ω ε)∂ ₜₜu=Δ u+f aligned ( ε - 2 χ Ω ε + χ R 3 \ Ω ε ) ∂ tt u = Δ u + f with initial data and source supported outside Ω ε Ω ε ; here, χ S χ S denotes the characteristic function of a set S . We provide the first-order ε ε -corrections with respect to the solutions of the inhomogeneous free wave equation and give space-time estimates on the remainders in the L∞((0,1/ε τ),L²(R³)) L ∞ ( ( 0 , 1 / ε τ ) , L 2 ( R 3 ) ) -norm. Such corrections are explicitly expressed in terms of the eigenvalues and eigenfunctions of the Newton potential operator in L²(Ω ) L 2 ( Ω ) and provide an effective dynamics describing a legitimate point scatterer approximation in the time domain.