### 4. Conclusion

[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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