缓坡型Zakharov方程及其应用

发布时间：2018-03-22 22:34

本文选题：Zakharov方程　切入点：波群　出处：《大连理工大学》2014年硕士论文　论文类型：学位论文

【摘要】：波群的非线性演化研究是波浪理论中重要的研究课题之一。近年来,水波的非线性理论发展很快。一方面,Phillips (1960)[1]对此做出了奠基性的工作,提出了共振机制理论。该理论认为当四个波的波数与波频满足一定条件(即共振)时,它们之间就会缓慢互相传递能量。目前普遍认为共振机理是海洋中波浪的波能传递的重要原因。另一方面,BenjaminFeir (1967)[2]提出只要有微小的边带扰动,Stokes波将是不稳定的。该扰动就是波浪缓慢调制的最根本的原因。从那时起,波群在短时间和长时间范围内非线性稳定问题的扩展性研究工作(包括理论和实验工作)得到了大量的开展。基于水波控制方程,Zakharov (1968)[3]推导了适用于深水情况下的波浪非线性演化研究的积分方程。该方程即是著名的Zakharov方程。随后StiassnieShemer (1984)[4]推导了在有限水深情况下的Zakharov方程,并用该方程研究了波浪的线性稳定性,所得结果与McLean (1982)[5]吻合。Shemer et al.(2001)[6]将Zakharov方程由时域转化为空间域,从而可模拟单向波群沿水槽演化。他们将模拟结果与实验结果定性与定量对比,吻合良好。但是,该空间域方程只适用于常水深情况。本文针对该方程的不足,添加了与水深梯度成正比的项,从而使改进后的方程能适用于一般变水深的情况。改进的方程非线性近似到三阶,适用于任意谱宽的波群,是Zakharov方程的扩展。应用Zakharov方程研究了两类问题,一是考虑了水深与波陡对波浪第一类不稳定性的影响,二是讨论了坡度、周期与波幅对波群的演化的影响。
[Abstract]:The study of nonlinear evolution of wave groups is one of the most important research topics in wave theory. In recent years, the nonlinear theory of water waves has developed rapidly. The theory of resonance mechanism is put forward, which holds that when the wave number and frequency of four waves satisfy certain conditions (i.e. resonance), At present, it is generally believed that resonance mechanism is an important reason for wave energy transfer in the ocean. On the other hand, Benjamin Feir (1967) [2] suggests that the Stokes wave will be unstable as long as there is a small side-band disturbance. Disturbance is the fundamental reason for the slow modulation of waves. From then on, The research on the expansibility of wave groups in the short and long time range of nonlinear stability problems has been carried out extensively, both in theory and in experiment. Based on the water wave control equation Zakharov (1968) [3], an integral equation for the study of the nonlinear evolution of waves in deep water is derived. This equation is known as the Zakharov equation. [4] subsequently, the Zakharov equation in the case of limited water depth is derived. The linear stability of waves is studied by using this equation. The results are in good agreement with that of McLean's 1982) [5] .Shemer et al. [6] [6] the Zakharov equation is transformed from time domain to spatial domain. Thus, the evolution of unidirectional wave group along the flume can be simulated, and the qualitative and quantitative comparison between the simulation results and the experimental results shows that the simulation results are in good agreement with each other. However, the space domain equation is only suitable for the case of constant water depth. In this paper, a term proportional to the gradient of water depth is added to the equation. The improved equation is an extension of the Zakharov equation, which is approximate to the third order and suitable for wave groups of arbitrary spectral width. Two kinds of problems are studied by using Zakharov equation. One is to consider the influence of water depth and wave steepness on the first kind of wave instability, the other is to discuss the influence of slope, period and amplitude on the evolution of wave group.
【学位授予单位】：大连理工大学
【学位级别】：硕士
【学位授予年份】：2014
【分类号】：TV139.2

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