Calculation of resonant short-crested waves in deep water

Solving the problem of resonant short-crested waves is very challenging because the appearance of small divisors causes the classical perturbation methods to fail. From a numerical point of view, the case of resonant gravity short-crested waves has been studied, but as far as we know, there are very few results in the case of resonant capillary-gravity short-crested waves. In fact, to the best of our knowledge, the most related study which has been made is the one of Craig and Nicholls [SIAM J. Math. Anal. 32, 323 (2000)] who gave existence theorems for the case where the surface tension is supposed not to be too small. There is a need for such an investigation, and the work considered herein therefore provides a calculation technique and presents new results on resonant short-crested gravity-capillary waves. We overcome the technical problems associated with small divisors by using a method derived from Whitham's variational formulation of the classical problem of short-crested waves. Whitham's method is not modified in essence, but computations are carried out and organized to obtain a method that has been applied to series of cases demonstrating the robustness and flexibility of the approach. In particular, numerical solutions corresponding to three-dimensional Wilton ripples have been obtained. Moreover, these waves are also obtained for long wave configurations. This method is able to handle the case of small or zero surface tension, including the resonant cases, and works well very near the limiting two-dimensional cases.