On The Study Of Steady-state Resonant Waves And Wave-current Interaction

Nonlinear resonant wave interaction and wave-current interaction play an important role in the wave energy spectrum, which involved in a wide range of engineering applications. In a majority of previous studies on these topics, most research works focused on investigating the effects of these two factors on the time evolving of wave spectrum from the point of view of initial-boundary value problems. This dissertation is devoted to investigate the related steady-state phenomenon from the point of view of boundary value problem, and it includes the existence of steady-state resonant waves and the effect of vorticity on wave interactions. Six typical problems are investigated in detail and described briefly as follows:1. The steady-state resonance of multiple surface gravity waves in deep water was investigated in details to extend the existing special quartet to more general and coupled resonant quartets, together with higher-order resonant interactions. The exact, nonlinear wave equations are solved without assumptions of the existence of small physical parameters. Multiple steady-state resonant waves are obtained for all the considered cases, and it is found that the number of multiple solutions tends to increase when more wave components are involved in the resonance sets.2. Taking the special quartet and sextet as examples, the topology of wave energy distribution in the parameter space is analyzed. It is found that the steady-state resonant waves indeed form a continuum in the parameter space. Besides, the significant role of the near resonance was revealed. It is found that all of nearresonant components as a whole contain more and more wave energy, as the wave patterns tend from two-dimension to one-dimension, or as the nonlinearity of steady-state resonant wave system increases.3. The linear stability of the steady-state resonant waves is analyzed. It is found that the steady-state resonant waves are stable, as long as the disturbance does not resonate with any components of the basic wave. Considering the infinitesimal amplitude of perturbations, there might be a considerable time lag until the generic perturbations become important and destabilize the steady-state wave pattern. Thus the steady-state resonance waves will be, at least in principle,metastable three-dimensional waves, which might be observed in experiments.4. Based on the freedom of choosing linear operator in Homotopy analysis method,a piecewise linear operator is proposed such that steady-state resonant waves based only on nonlinear resonance condition was obtained. Compared with the case when both Phillips’ linear resonance condition and the nonlinear one are satisfied, the dimension of the parameter space increases. And as the nonlinearity of whole wave system increases, the domain of steady-state resonant waves increases, too.5. The steady-state wave spectra are quantitatively observed within the inherent system error of the basin and identified by means of a contrasting experiment. Both symmetrical and anti-symmetrical steady-state resonant waves are observed and the experimental and theoretical results show excellent agreement.6. The nonlinear interaction between unidirectional bichromatic wave-train and exponentially sheared currents in water of infinite depth is investigated. The effect of nonlinearity on the phase velocity of primary waves in a bichromatic wavetrain is considered. Accurate high-order approximations of phase velocity are obtained under consideration of both the nonlinear wave self-self and mutual interaction. Then the combined effect of vorticity and nonlinearity on the phase velocity is investigated. It is found that the characteristic current slope determines the effect of vorticity on the phase velocity caused by nonlinear wave self-self and mutual interaction, and the surface current strength may amplify/reduce this effect.