Dispersive effective equations for waves in heterogeneous media on large time scales
Tomáš Dohnal, Agnes Lamacz, Ben Schweizer · arXiv (Cornell University) · 2013
We investigate second order linear wave equations in periodic media, aiming at the derivation of effective equations in $\R^n$, $n \ge 1$. Standard homogenization theory provides, for the limit of a small periodicity length $\eps>0$, an effective second order wave equation that describes solutions on time intervals $[0,T]$. In order to approximate solutions on large time intervals $[0,T\eps^{-2}]$, 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. The analytical results are confirmed and illustrated by numerical tests.