Theoretical Analysis And Analytical Calculation Of Some Steady-state Nonlinear Resonant Waves Problems

Nonlinear water waves have always been a research hotspot in naval architecture and ocean engineering,and the study of nonlinear water waves has important theoretical and practical significance.There has never been a lack of research on nonlinear water wave problems.The predecessors have done a lot of basic and in-depth researches on the nonlinear water wave problems.However,due to the complexity and difficulty,many nonlinear water wave problems still need to be studied.Regarding to the resonance problem of waves,the resonant wave components will lead to the existence of singularities in the calculation process.It is difficult to obtain a steady-state resonant wave system which wave spectrum does not change with time based on traditional analytical methods.In this dissertation,the homotopy analysis method(HAM)is applied to solve the following three problems of steadystate resonant waves:(1)The steady-state resonant acoustic-gravity waves problem in consideration of compressibility under deep sea conditions;(2)The collinear steady-state resonant waves problem with infinite singularities in deep water;(3)The steady-state second harmonic resonance problem in a circular basin.The main results are shown as following:(I)Based on HAM,for the first time,we successfully obtain the convergent series solutions of the steady-state acoustic-gravity waves in cases of both non-resonance and exact resonance.Considering the compressibility of seawater,a hydro-acoustic wave can be produced by the interaction of two progressive gravity waves with the nearly same wave length traveling in the opposite directions.It is found that most of the wave energy is provided by the two primary wave components and the resonant hydro-acoustic wave component in the considered cases.The dynamic pressure on the sea bottom caused by the resonant hydro-acoustic wave component is much larger than that in the cases of non-resonance,which might even trigger microseisms of the ocean floor.(II)Based on HAM,when a complete nonlinear governing equation is used to solve the steady-state resonant acoustic-gravity waves problem,the results are found to be similar to those obtained by using a linear governing equation.It is found that for the cases of solving the linear governing equation in this paper,the nonlinear part of the governing equations can be indeed ignored.This provides a theoretical basis for the calculation of seafloor pressure values under the steady-state resonance conditions.(III)Based on HAM,we successfully gain the steady-state wave systems with an infinite number of resonant wave components,consisting of the nonlinear interaction of the two primary waves traveling in the same/opposite direction.It indicates the general existence of the so-called steady-state resonant waves,even in case of an infinite number of resonant wave components.The results show that most of the energy of the entire wave system is provided by the two fundamental wave components,but as the nonlinearity increases,the wave energy is slowly transferred from the two fundamental wave components to the higher order wave components.(IV)Based on HAM and Galerkin method,we successfully obtain the convergent series solutions of the steady-state second harmonic resonances in a circular basin.Two different steady-state second harmonic resonant wave groups are found,which indicates that the steady-state resonance wave systems also widely exist in the restricted waters.