Diffractive behavior of the wave equation in periodic media
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Mon, 19/10/2009 17:00 |
Grégoire Allaire (Ecole Polytechnique) |
Partial Differential Equations Seminar  |
Gibson 1st Floor SR |
| 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. This is a joint work with M. Palombaro and J. Rauch. | |||