Euler-Darboux-Poisson Equation and Travelling Waves

Methods for finding exact solutions to wave equations with variable coefficients describing wave propagation in highly inhomogeneous media are presented, which are based on various transformations that lead the studied equations to the Euler-Darboux-Poisson equation. Shallow water equations of variable depth are chosen as a model for finding exact solutions. The first approach to finding accurate solutions is to represent an unknown function of water displacement (or velocity) as a product of amplitude and a new unknown function depending on time and phase (mapping technique). It is shown that for certain medium configurations, the shallow water equations are reduced to the Euler-Darboux-Poisson equation. Due to this, accurate solutions have been obtained for certain types of bottom profiles (variable coefficients of the wave equation), which are represented in the form of two generalized traveling waves propagating in different directions (non-reflective propagation). A detailed analysis of the exact solution obtained is given. Another method of transformation of linear shallow water equations is also presented, based on a Carrier-Greenspan transform generalization, well known in the theory of run-up nonlinear waves on a flat slope. Thanks to it, the initial equations for waves in a basin with arbitrary bathymetry are reduced to a wave equation, from which both the displacement of the free surface and the flow velocity can be obtained simultaneously. Exact solutions in elementary functions for seamounts of a certain shape are obtained. It is shown that when moving from a more gentle slope to a sharper one, the waveform will integrate a certain number of times, and the maximum amplitude is not reached at the very top of the mountain.