Convergence of the Stokes expansion was first proved by  for the case of small-amplitude waves – on the free surface of a fluid of infinite depth. This was extended shortly afterwards by  for the case of finite depth and small-amplitude waves.
Near the end of the 20th century, it was shown that for finite-amplitude waves the convergence of the Stokes expansion depends strongly on the formulation of the periodic wave problem. For instance, an inverse formulation of the periodic wave problem as used by Stokes – with the spatial coordinates as a function of velocity potential and stream function – does not converge for high-amplitude waves. While other formulations converge much more rapidly, e.g. in the Eulerian frame of reference (with the velocity potential or stream function as a function of the spatial coordinates).