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MiS Preprint

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

Gregoire Allaire, Mariapia Palombaro and Jeffrey Rauch


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.

Nov 26, 2007
Nov 26, 2007
MSC Codes:
35B27, 35J10
homogenization, Bloch waves, diffractive geometric optics

Related publications

2009 Journal Open Access
Grégoire Allaire, Mariapia Palombaro and Jeffrey Rauch

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

In: Annali di matematica pura ed applicata, 188 (2009) 4, pp. 561-589