On functional equations leading to exact solutions for standing internal waves

• Standing internal waves below a horizontal plane are described by Schröder functional equations.
• We give a unified approach to many exact solutions for standing internal waves below a horizontal plane.
• Relevant results on Schröder and Abel functional equations are presented and used.

The Dirichlet problem for the wave equation is a classical example of a problem which is ill-posed. Nevertheless, it has been used to model internal waves oscillating harmonically in time, in various situations, standing internal waves amongst them. We consider internal waves in two-dimensional domains bounded above by the plane z=0z=0 and below by z=−d(x)z=−d(x) for depth functions dd. This paper draws attention to the Abel and Schröder functional equations which arise in this problem and use them as a convenient way of organising analytical solutions. Exact internal wave solutions are constructed for a selected number of simple depth functions dd.