Asymptotic solutions of 2D wave equations with variable velocity and localized right-hand side

In the paper, we consider the Cauchy problem for the inhomogeneous wave equation with variable velocity and with a perturbation in the form of a right-hand side localized in space (near the origin) and in time. In particular, this problem is connected with the question about the creation of tsunami and Rayleigh waves. Using abstract operator theory and in particular Maslov's noncommutative analysis, we show that the solution is separated into two parts: the transient one, which is localized in a neighborhood of the origin and decreases in time and the propagating one, which propagates in space like the wave created by the momentary “equivalent source.” We present several examples covering a wide range of perturbation resulting in rather explicit formulas expressing the solutions it terms of the error function of complex argument.