On the homogenization of a transmission problem in scattering theory for highly oscillating media

Here we study homogenization of a transmission problem for bounded scatterers with periodic coecients modeled by the anisotropic Helmholtz equation. The material property coefficients are assumed to be periodic functions over the unit cell for the fast variable. By way of multiple scales expansion, we focus on the $O(k)$, $k = 1, 2$ bulk and boundary corrections of the leading-order ($O(1)$) homogenized transmission problem. The analysis in particular provides the $H^1$ and $L^2$ estimates of the error committed by the first-order-corrected solution considering i) bulk correction only, and ii) boundary and bulk correction. We treat explicitly the $O(\epsilon)$ boundary correction for the transmission problem when the scatterer is a unit square. We also establish the $O(\epsilon^2)$-bulk correction describing the mean wave motion inside the scatterer. The analysis here also highlights a previously established, yet scarcely recognized fact that the $O(\epsilon)$ bulk correction of the mean motion vanishes identically.