Dynamics of Nonlinear Gravity-Capillary Waves in Deep Water Near Resonance

Abstract

The minimum phase speed of linear gravity-capillary waves in deep water ($c_\mathrm{min}$) is known to be the bifurcation point of three-dimensional solitary waves (``lumps"). In the present thesis, various aspects of unsteady gravity-capillary lumps are investigated in the context of three sets of experiments. In the first set, cinematic shadowgraph and refraction-based techniques are utilized to measure the temporal evolution of the free surface deformation pattern downstream of a surface pressure source as it moves along a towing tank, while numerical simulations using a model equation are used to extend the experimental results. The focus of this study is on exploring the characteristics of the observed periodic shedding of lump-like depressions for towing speeds close to $c_\mathrm{min}$. From the experiments, it is found that the speed-amplitude characteristics and the shape of the depressions are nearly the same as those of the freely propagating gravity-capillary lumps of inviscid potential theory. The periodic behavior is found to be analogous to the periodic generation of two-dimensional solitary waves in shallow water by a source moving at trans-critical speeds of pure gravity waves. In the second set of experiments, the effect of viscous dissipation on freely propagating lumps is examined. A steady forced lump is first generated by applying appropriate forcing and towing speed. The forcing is then removed suddenly and the change in shape and speed of the lump is measured as it propagates freely under the action of viscosity. It is found that the localized structure of the lump is maintained during the decay and the first measurement of the decay rate of gravity-capillary lumps is reported. In the third set of experiments, the interactions of state III lumps generated by two pressure sources moving in parallel straight lines are investigated. The sources are adjusted to produce nearly identical periodic responses. The first lump generated by each source, collides with the lump from the other source in the center-plane of the two sources. It was observed that a steep depression is formed during the collision but breaks up soon after and radiates energy away in the form of small-amplitude radial waves. After the collision, a quasi-steady pattern is formed with several rows of localized depressions that are similar to lumps but exhibit periodic oscillations in depth.