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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Allaire, G., Palombaro, M. & Rauch, J. Diffractive behavior of the wave equation in periodic media: weak convergence analysis. Annali di Matematica 188, 561–589 (2009). https://doi.org/10.1007/s10231-008-0089-y
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DOI: https://doi.org/10.1007/s10231-008-0089-y
Keywords
- Homogenization
- Bloch waves
- Diffractive geometric optics
Mathematics Subject Classification (2000)
- 35B27
- 35J10