We study the homogenization of a transmission problem arising in the scattering theory for bounded inhomogeneities with periodic coefficients in the Helmholtz equation. The coefficients are assumed to be periodic functions of the fast variable, specified over the unit cell with characteristic size $\epsilon$. We consider both boundary and bulk corrections and show convergence results, in particular we show that the boundary correction is a larger order effect in general than the mean field. For periodic squared index of refraction, we obtain convergence results that assume lower regularity than when the periodicity was also in the second order operator, and find the limiting boundary correctors for general domains at all orders. In particular we show that at first order $O(\epsilon)$ the boundary corrector is the only effect and can be seen in the far field, while at higher orders boundary, bulk and mean field effects all appear.

Joint work with Fioralba Cakoni, Bojan Guzina and Tayler Pangburn.