Diffractive behavior of the wave equation in periodic media: weak convergence analysis |
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Authors: | Grégoire Allaire Mariapia Palombaro Jeffrey Rauch |
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Affiliation: | (1) Centre de Mathématiques Appliquées, école Polytechnique, 91128 Palaiseau, France;(2) Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22-26, 04103 Leipzig, Germany;(3) Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA |
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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 initial 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 Schrödinger 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 Schrödinger equation. |
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Keywords: | Homogenization Bloch waves Diffractive geometric optics |
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