Abstract
We consider the two scale asymptotic expansion for a transmission problem modeling scattering by a bounded inhomogeneity with a periodic coefficient in the lower order term of the Helmholtz equation. The squared index of refraction is assumed to be a periodic function of the fast variable, specified over the unit cell with characteristic size epsilon. Since the convergence of the boundary correctors to their limits is in general slow, we explore in detail their use in a second order approximation and show a new convergence estimate for the second order boundary corrector on a square. We show numerical examples of the higher order forward approximation in one and two dimensions. We then use the first order boundary correction as an asymptotic model for inversion and show numerical examples of inversion in the two dimensional case.