  We consider the interior transmission problem associated with the scattering
by an inhomogeneous (possibly anisotropic) highly oscillating periodic media.
We show that, under appropriate assumptions, the solution of the interior
transmission problem converges to the solution of a homogenized problem as the
period goes to zero. Furthermore, we prove that the associated real
transmission eigenvalues converge to transmission eigenvalues of the
homogenized problem. Finally we show how to use the first transmission
eigenvalue of the period media, which is measurable from the scattering data,
to obtain information about constant effective material properties of the
periodic media. The convergence results presented here are not optimal. Such
results with rate of convergence involve the analysis of the boundary
correction and will be subject of a forthcoming paper.
