Diffractive behavior of the wave equation in periodic media: weak convergence analysis

Grégoire Allaire 1, 2 M. Palombaro J. Rauch
2 DeFI - Shape reconstruction and identification
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France, Polytechnique - X, CNRS - Centre National de la Recherche Scientifique : UMR7641
Abstract : We study the homogenization and singular perturbation of the wave equation in a periodic media for long times of the order of the inverse of the period. We consider inital data that are Bloch wave packets, i.e., that are the product of a fast oscillating Bloch wave and of a smooth envelope function. We prove that the solution is approximately equal to two waves propagating in opposite directions at a high group velocity with envelope functions which obey a Schr\"{o}dinger type equation. Our analysis extends the usual WKB approximation by adding a dispersive, or diffractive, effect due to the non uniformity of the group velocity which yields the dispersion tensor of the homogenized Schr\"{o}dinger equation.
Type de document :
Article dans une revue
Annali di Matematica Pura ed Applicata, Springer Verlag, 2009, 188, pp.561-590
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Contributeur : Houssem Haddar <>
Soumis le : dimanche 3 février 2013 - 13:56:20
Dernière modification le : jeudi 11 janvier 2018 - 06:22:34

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  • HAL Id : hal-00784060, version 1

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Grégoire Allaire, M. Palombaro, J. Rauch. Diffractive behavior of the wave equation in periodic media: weak convergence analysis. Annali di Matematica Pura ed Applicata, Springer Verlag, 2009, 188, pp.561-590. 〈hal-00784060〉

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