Diffractive behavior of the wave equation in periodic media: weak convergence analysis
Gregoire Allaire, Mariapia Palombaro, and Jeffrey Rauch
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Submission date: 26. Nov. 2007
published in: Annali di matematica pura ed applicata, 188 (2009) 4, p. 561-589
DOI number (of the published article): 10.1007/s10231-008-0089-y
MSC-Numbers: 35B27, 35J10
Keywords and phrases: homogenization, Bloch waves, diffractive geometric optics
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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 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.