Geodesic
Homogenization and dispersion effects in the problem of propagation of waves generated by a~localized source
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems of mechanics, Tome 281 (2013), pp. 170-187

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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. 
@article{TM_2013_281_a13,
     author = {V. V. Grushin and S. Yu. Dobrokhotov and S. A. Sergeev},
     title = {Homogenization and dispersion effects in the problem of propagation of waves generated by a~localized source},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {170--187},
     publisher = {mathdoc},
     volume = {281},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2013_281_a13/}
}
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V. V. Grushin; S. Yu. Dobrokhotov; S. A. Sergeev. Homogenization and dispersion effects in the problem of propagation of waves generated by a~localized source. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems of mechanics, Tome 281 (2013), pp. 170-187. http://geodesic.mathdoc.fr/item/TM_2013_281_a13/