Traveling Waves in Shallow Seas of Variable Depths

Symmetry. 2022. Vol. 14. No. 7. Article 1448.

Pelinovsky E., Kaptsov O.

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

Keywords: бегущие волны, travelling waves, Euler-Darboux-Poisson equation, уравнение Эйлера-Дарбу-Пуассона