Nonlinear gravity-capillary water waves

Two-dimensional gravity-capillary water waves are analyzed using a fully-nonlinear Cauchy-integral method with spectral accuracy. Standing waves are generated in experiments by vertical oscillation and measured by a non-intrusive optical system along with a wave probe. Nonlinear resonance of standing waves with non-wetting contact line effects are discussed in detail. Amplitude-dependent wave frequency and damping in a glass rectangular tank suggest a new contact-line model. A new type of sideband resonance due to modulated forcing is discovered and explained by weakly-nonlinear analysis. This analytical solution is verified by our numerical simulations and physical experiments.; New standing waveforms with dimpled or sharp crests are observed in experiments and computations. These new waveforms have strong symmetry breaking in time as a result of nonlinear harmonic interaction. With increasing wave steepness, steep standing waves experience period-tripling with three distinct forms: sharp crest, dimpled or flat crest, and round crest. Significant breaking occurs in the sharp-crest mode and the dimpled-crest mode. Using a complex-demodulation technique, I find that these breaking waves are related to the same 1:2 internal resonance (harmonic interaction) that causes the new steep waveforms. Novel approaches are used to estimate the (breaking and non-breaking) wave dissipation in steep and breaking standing waves. The breaking events (spray, air entrainment, and plunging) approximately double the wave dissipation.; Weak capillarity significantly affects the limiting wave height and the form of standing waves, as demonstrated by both computations and small-scale Faraday-wave experiments. Capillary ripple generation on traveling waves is shown to be significant even at moderate wave steepness. The ubiquitous horizontal asymmetry of traveling waves is shown to be critical to capillary ripple generation. Two new asymmetric modes are identified and are shown to have an important effect on the unsteady generation and modulation of capillary ripples. For standing waves at critical capillary numbers, new waveform with multiple dimples is found at finite amplitude with complex dynamical behavior. These discoveries are shown to bear important implications to remote sensing of the ocean surfaces.