[45] In this paper we present the quite explicit formulas for full asymptotical description of solutions of the Cauchy problem with a general localized initial disturbance (source) for the wave equation with slow varying wave velocity. In the case of the Gaussian type disturbance the answer is expressed via the special (hypergeometric) functions. The given description includes: the special trajectories of the simple Hamiltonian system, their first derivatives and the function, implied by the initial disturbance, and the integer numbers (the Maslov or Morse indices) when focal points appear. All objects are well know in geometrical optics and semiclassical approximation, and our main pragmatic result is that only they are needed to construct the final quite explicit formulas for solution to the problem (1, 2) presented in Propositions 1-3. Let us emphasize again that the derivation and proof of these formulas is not simple and use fundamental mathematical theories.
[46] One of the basic conclusions, demonstrated for the source of the Gaussian type (when the answer is expressed via the hypergeometric functions) is that the wave profile crucially depend on the form of initial disturbance of the bottom. As in the real conditions it is very problematic to obtain any detailed information of this disturbance not only at the instant when it happens but and at later times, we propose to develop a more active the researches for application to tsunami warning systems, which used simplified seismic sources. The ones considered in this paper source of the Gaussian type, could be the first ones. We hope also that the given asymptotic formulas can be useful in this application because the visualization of these formulas on a PC is easy and takes not too much time.
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