  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.
