Traveling Waves in Deep Water with Gravity and Surface Tension

This paper is concerned with the simulation of periodic traveling deep-water freesurface water waves under the influence of gravity and surface tension in two and three dimensions. A variety of techniques is utilized, including the numerical simulation of a weakly nonlinear model, explicit solutions of low-order perturbation theories, and the direct numerical simulation of the full water wave equations. The weakly nonlinear models which we present are new and extend the work of Akers and Milewski [SIAM J. Appl. Math., 70 (2010), pp. 2390–2408] to arbitrary Bond number and fluid depth. The numerical scheme for the full water wave problem features a novel extension of the “Transformed Field Expansions” method of Nicholls and Reitich [Euro. J. Mech. B Fluids, 25 (2006), pp. 406–424] to accommodate capillary effects in a stable and rapid fashion. The purpose of this paper is apply the new numerical method, then compare small amplitude solutions of potential flow with those of the approximate model. Particular attention is paid to the behavior near quadratic resonances, an example of which is the Wilton ripple.