Traveling Waves in Shallow Seas of Variable Depths

Abstract

The problem of the existence of traveling waves in inhomogeneous fluid is very important for enabling an explanation of long-distance wave propagations such as tsunamis and storm waves. The present paper discusses new solutions to the variable-coefficient wave equations describing traveling waves in fluid layers of variable depths (1D shallow-water theory). Such solutions are obtained by using the transformation methods when variable-coefficient equations can be reduced to the constant coefficient equation when the existence of traveling waves is evident. It is shown that there is a wide class of monotonic bottom profiles (discrete set) that allow the existence of traveling waves that are not reflected in a strongly inhomogeneous water medium. Their temporal shape changes with distance, mainly near the water–land boundary (shoreline). Traveling waves can transfer the wave energy over a long distance that is often observed at the transoceanic propagation of tsunami waves.

Euler–Darboux–Poisson Equation in Context of the Traveling Waves in a Strongly Inhomogeneous Media

“…From Equation (17), we can immediately say that all such non-reflective profiles h(x) (up to multiplication by a constant) lie between x 4 3 and x 2 . Thus, we reproduced the calculated equation of the bottom profiles obtained in [6], at which the non-reflective propagation of long waves is possible.…”

“…In this section, we briefly present the method for obtaining a family of non-reflective bottom profiles and, accordingly, solutions to the η offset obtained with such profiles h(x), following [6].…”

“…As an example, we give here the classic case of the bottom profile shape, well-described in the paper [10]. However, it is worth mentioning that by using slightly different techniques in [6,10], the same non-reflective profile was obtained, which is given by the equation…”

“…The linear system of shallow water equations is chosen as the mathematical model of this paper and described in Section 2. Furthermore, in Section 3, the principle of reducing the wave equation with variable coefficients to the Euler-Poisson-Darboux equation, described in [6], is touched upon. Section 4 contains the second component of the wave field-the velocity averaged over the current depth.…”