On functional equations leading to exact solutions for standing internal waves

TL;DR

This paper explores how functional equations like Abel and Schr{"o}der can be used to derive exact solutions for standing internal waves in variable-depth domains, providing a novel analytical approach.

Contribution

It introduces a method using functional equations to construct exact internal wave solutions for specific depth profiles, advancing analytical techniques in wave modeling.

Findings

Exact solutions for internal waves with simple depth functions

Use of Abel and Schr{"o}der equations to organize solutions

Potential for extending methods to more complex geometries

Abstract

The Dirichlet problem for the wave equation is a classical example of a problem which is not well-posed. Nevertheless, it has been used to model internal waves oscillating sinusoidally in time, in various situations, standing internal waves amongst them. We consider internal waves in two-dimensional domains bounded above by the plane z=0 and below by z=-d(x) for depth functions d. This paper draws attention to the Abel and Schr{\"o}der functional equations as a convenient way of organizing analytical solutions. Exact internal wave solutions are constructed for a selected number of simple depth functions d.