Cross-waves excited by distributed forcing in the gravity-capillary regime

Cross-wave descriptions based on a boundary forced nonlinear Schrodinger equation, which have been widely used since Jones [J. Fluid Mech. 138, 53–74 (1984)], rely on the assumption that modulations occur on a slow lengthscale compared with the extent of the forcing. This assumption does not hold for recent higher frequency (large aspect ratio) experiments. We extend the established theory of modulated cross-waves in horizontally vibrated containers by including surface tension and, most importantly, a spatially extended forcing term. The resulting amplitude equations provide predictions for onset values, spatial profiles, and temporal modulations that are compared with previous theory and with experimental measurements. The appearance of temporally modulated solutions, confirmed experimentally, is interpreted as the result of weak symmetry-breaking related to the interaction of waves generated at opposite ends