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We investigate second order linear wave equations in periodic media, aiming at the derivation of effective equations in $\mathbb{R}^n$, $n\in\{1,2,3\}$. Standard homogenization theory provides, for the limit of a small periodicity length $\varepsilon>0$, an effective second order wave equation that describes solutions on time intervals $[0,T]$. In order to approximate solutions on large time intervals $[0,T\varepsilon^{-2}]$, 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.

Bloch-Wave Homogenization on Large Time Scales and Dispersive Effective Wave Equations