Bloch-wave homogenization on large time scales and dispersive effective wave equations

TL;DR

This paper derives a dispersive effective wave equation for periodic media that accurately models wave behavior over large time scales, extending classical homogenization with Bloch-wave analysis and numerical validation.

Contribution

It introduces a well-posed, weakly dispersive effective wave equation for large time scales in periodic media, using Bloch-wave analysis and symmetry assumptions.

Findings

Derived a dispersive effective wave equation valid on large time intervals.

Provided error estimates between original and effective solutions.

Validated results through numerical experiments.

Abstract

We investigate second order linear wave equations in periodic media, aiming at the derivation of effective equations in $\R^n$, $n \in \{1, 2, 3\}$. 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 under symmetry assumptions on the periodic structure. The analytical results are confirmed and illustrated by numerical tests.