This project aims at improving the fundamental understanding of the generation of waves on the surface of deep water by an external disturbance. The physical regime of interest is when the waves have wavelength of a few cm; both gravity and capillary effects then become important, and the wave speed attains a minimum value which defines a critical forcing speed: the linear response to external forcing traveling with speed equal to this minimum grows unbounded with time and, apart from damping, nonlinear effects can become important near this resonance. Theoretical models will be used to study the interplay of forcing, nonlinear and damping effects on the wave response under resonant conditions. The theoretical predictions will be compared with laboratory experimental observations reported in the literature.
Resonant forcing of gravity-capillary waves is relevant to the generation of ripples by wind, that appear as small-scale roughness on the ocean surface, and to the wave drag associated with the motion of small bodies on a free surface. More generally, this problem is prototypical of resonantly forced wave systems with a phase-speed minimum at finite wavelength, and other potential applications include the response of floating ice sheets to surface vehicles.
A mathematical model was developed for the generation of water waves by a localized pressure source moving on the surface of deep water with speed close to the minimum gravity-capillary phase speed; at this speed, classical linear inviscid theory predicts a singular response owing to a resonance phenomenon. The new model: (i) has brought out the delicate interplay between nonlinear and viscous effects that takes place near resonant conditions; (ii) has resolved the singular behavior of linear inviscid thory; (iii) has enabled the first ever experimental verification of fully localized solitary nonlinear waves ("lumps") in fluid flows. These results should prove useful in understanding the generation of ripples by wind on the ocean surface as well as the response of ice sheets to moving loads due to vehicles. A theoretical study was also made of: (i) the reflection of internal gravity wave beams from a sloping boundary in a stratifuid fluid, when the incident beam hits the slope obliquely to the isobaths; (ii) oblique beam collisions, involving beams that propagate in different verical planes. The main objective was to understand the effect of obliqueness on secondary beams induced due to nonlinear interactions of the incident/reflected or colliding primary beams. These analyses have revealed that: (i) in comparison to plane wave beam reflections from a slope, oblique reflections generally give rise to weaker secondary reflected beams; (ii) oblique collisions can be resonant, resulting in secondary beams of large steepness. These results add to the understanding of tidal conversion -- the transfer of energy from the barotropic tide to internal gravity waves, a process that is believed to be important in deep-ocean mixing.
