On the structure of parasitic gravity-capillary standing waves in the small surface tension limit

We present new numerical solutions for nonlinear standing water waves when the effects of both gravity and surface tension are considered. For small values of the surface tension parameter, solutions are shown to exhibit highly oscillatory capillary waves (parasitic ripples), which are both time- and space-periodic, and which lie on the surface of an underlying gravity-driven standing wave. Our numerical scheme combines a time-dependent conformal mapping together with a shooting method, for which the residual is minimised by Newton iteration. Previous numerical investigations typically clustered gridpoints near the wave crest, and thus lacked the fine detail across the domain required to capture this phenomenon of small-scale parasitic ripples. The amplitude of these ripples is shown to be exponentially small in the zero surface tension limit, and their behaviour is linked to (or explains) the generation of an elaborate bifurcation structure.