BLOCH-WAVE HOMOGENIZATION ON LARGE TIME SCALES AND DISPERSIVE EFFECTIVE WAVE EQUATIONS

We investigate second order linear wave equations in periodic media, aiming at the derivation of effective equations in R-n, n is an element of {1, 2, 3}. Standard homogenization theory provides, for the limit of a small periodicity length epsilon > 0, an effective second order wave equation that describes solutions on time intervals [0, T]. In order to approximate solutions on large time intervals [0, T epsilon(-2)], one has to use a dispersive, higher order wave equation. In this work, we provide a well-posed, weakly dispersive effective equation and an estimate for errors between the solution of the original heterogeneous problem and the solution of the dispersive wave equation. We use Bloch-wave analysis to identify a family of relevant limit models and introduce an approach to select a well-posed effective model under symmetry assumptions on the periodic structure. The analytical results are confirmed and illustrated by numerical tests.