We introduce a time-space two-scale transform designed to capture the high and low frequency waves in the asymptotics of the periodic homogenization of the wave equation. The asymptotical solution is the sum of the solution of known homogenized equations and of Bloch waves. We also derive the transport equations satisfied by the Bloch wave coefficients.
Nous introduisons une transformation à deux échelles en espace et en temps destinée à capturer à la fois les basses fréquences et les ondes de Bloch qui apparaissent lors du processus asymptotique d'homogénéisation de l'équation des ondes à coefficients périodiques. La solution du modèle qui en résulte comprend les ondes de Bloch et une contribution basse fréquence qui est la solution du modèle homogénéisé de l'équation des ondes. On établit aussi les équations de transport vérifiées par les coefficients des ondes de Bloch.
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@article{CRMATH_2009__347_23-24_1439_0, author = {Brassart, Matthieu and Lenczner, Michel}, title = {A two-scale model for the wave equation with oscillating coefficients and data}, journal = {Comptes Rendus. Math\'ematique}, pages = {1439--1442}, publisher = {Elsevier}, volume = {347}, number = {23-24}, year = {2009}, doi = {10.1016/j.crma.2009.10.017}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.crma.2009.10.017/} }
TY - JOUR AU - Brassart, Matthieu AU - Lenczner, Michel TI - A two-scale model for the wave equation with oscillating coefficients and data JO - Comptes Rendus. Mathématique PY - 2009 SP - 1439 EP - 1442 VL - 347 IS - 23-24 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2009.10.017/ DO - 10.1016/j.crma.2009.10.017 LA - en ID - CRMATH_2009__347_23-24_1439_0 ER -
%0 Journal Article %A Brassart, Matthieu %A Lenczner, Michel %T A two-scale model for the wave equation with oscillating coefficients and data %J Comptes Rendus. Mathématique %D 2009 %P 1439-1442 %V 347 %N 23-24 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2009.10.017/ %R 10.1016/j.crma.2009.10.017 %G en %F CRMATH_2009__347_23-24_1439_0
Brassart, Matthieu; Lenczner, Michel. A two-scale model for the wave equation with oscillating coefficients and data. Comptes Rendus. Mathématique, Volume 347 (2009) no. 23-24, pp. 1439-1442. doi : 10.1016/j.crma.2009.10.017. http://www.numdam.org/articles/10.1016/j.crma.2009.10.017/
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