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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
Inria Saclay - Ile de France, CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
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.
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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. ⟨10.1007/s10231-008-0089-y⟩. ⟨hal-00784060⟩



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