On functional equations leading to exact solutions for standing internal   waves
  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. 
Felix Beckebanze (1) and Grant Keady (2) ((1) Institute for Marine and   Atmospheric Research Utrecht, The Netherlands, (2) ...
math.AP