Cyclic gravity waves in deep water

Numerical evidence is presented for the existence of unsteady periodic gravity waves of large height in deep water whose shape changes cyclically as they propagate. It is found that, for a given wavelength and maximum wave height, cyclic waves with a range of cyclic periods exist , with a steady wave of permanent shape being an extreme member of the range. The method of solution, using Fourier transforms of the nonlinear surface boundary conditions, determines the irrotational velocity field in the water and the water surface displacement as functions of space and time, from which properties of the waves are demonstrated. In particular, it is shown that cyclic waves are closer to the point of wave breaking than are steady permanent waves of the same wave height and wavelength