Highest waves, phase speeds and particle trajectories of nonlinear capillary waves on sheets of fluid

We derive exact criteria for the highest capillary waves on sheets of fluid and present the profiles for both symmetric and antisymmetric waves. In addition we provide correct definitions of the phase speed of these waves. Previous essentially nonlinear solutions to the problem are shown to have been misinterpreted. Trajectory properties of particles in both types of wave are presented. We also give results for the drift velocity ratios as a function of the mean displacement of a streamline from the free surface. Many of the results are exact, being given in terms of elliptic functions and elliptic integrals. An application to wave motion in thin films on pipe walls is considered.