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Euler–Darboux–Poisson Equation in Context of the Traveling Waves in a Strongly Inhomogeneous Media
The existence of traveling waves in an inhomogeneous medium is a vital problem, the
solution of which can help in modeling the wave propagation over long distances. Such waves can
be storm waves or tsunami waves in the seas and oceans. The presence of solutions in the form of
traveling waves indicates that the wave propagates without reflection and, therefore, can transfer
energy over long distances. Traveling waves within the framework of the 1D variable-coefficient
wave equation exist only for certain configurations of an inhomogeneous medium, some of which can
be found by transforming the original equation to the Euler–Darboux–Poisson equation. The solution
of the last equation for certain parameter values is expressed in elementary functions, which are the
sum of waves running in opposite directions. The mathematical features of such a transformation are
discussed in this paper.