Homogenization and dispersion effects in the problem of propagation of waves generated by a~localized source

V. V. Grushin ; S. Yu. Dobrokhotov ; S. A. Sergeev

Informatics and Automation, Modern problems of mechanics, Tome 281 (2013), pp. 170-187.

Résumé

We construct asymptotic solutions to the wave equation with velocity rapidly oscillating against a smoothly varying background and with localized initial perturbations. First, using adiabatic approximation in the operator form, we perform homogenization that leads to a linearized Boussinesq-type equation with smooth coefficients and weak “anomalous” dispersion. Then, asymptotic solutions to this and, as a consequence, to the original equations are constructed by means of a modified Maslov canonical operator; for initial perturbations of special form, these solutions are expressed in terms of combinations of products of the Airy functions of a complex argument. On the basis of explicit formulas obtained, we analyze the effect of fast oscillations of the velocity on the solution fronts and solution profiles near the front.

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