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  • Осреднение и дисперсионные эффекты в задаче о распространении волн, порожденных локализованным источником

Article

Осреднение и дисперсионные эффекты в задаче о распространении волн, порожденных локализованным источником

Грушин В. В., Доброхотов С. Ю., Сергеев С. А.

We construct asymptotic solutions to the wave equation with velocity rapidly oscillating 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.