Time-dependent analytic solutions for water waves above sea of varying depths

TL;DR

This paper derives analytic solutions for water wave propagation, including tsunamis, over varying seabed depths using a time-dependent self-similar approach, covering constant, linear, and oblique seabed profiles.

Contribution

It introduces new analytic solutions for water wave behavior over different seabed geometries, enhancing understanding of tsunami dynamics in open ocean and near-shore conditions.

Findings

Analytic solutions for wave height and velocity over constant seabed

Solutions for linear and oblique seabed profiles

Application of traveling wave Ansatz with simple solutions

Abstract

We investigate a hydrodynamic equation system which - with some approximation - is capable to describe the tsunami propagation in the open ocean with the time-dependent self-similar Ansatz. We found analytic solutions how the wave height and velocity behave in time and space for constant and linear seabed functions. First we study waves on open water, where the seabed can be considered relatively constant, sufficiently far from the shore. In the second part of the study we also consider a seabed which is oblique. Finally, we apply the most common traveling wave Ansatz and present almost trivial solutions as well.