Zakharov Equation For Mild Slope And Its Application

Nonlinear evolution of the wave group is one important research topic of the wave theory. Our understanding of the nonlinear dynamics of water waves grew substantially only in recent years. On one hand, this progress can be attributed to the fundamental research work of Phillips (1960). Here Phillips describes the resonance mechanism, which explains that four wave trains can slowly exchange energy among one another if their wave numbers and wave frequencies are suitably matched. This resonance mechanism is now recognized as an important factor in energy exchange among surface wave components in the ocean and waves, in general. On the other hand, this progress can be attributed to the analytical investigation of Benjamin&Feir (1967) and their decisive demonstration that the Stokes’wave is unstable to infinitesimal perturbations. The perturbations, which are of the modulational nature, manifest themselves in the form of a pair of side band frequency modes around the fundamental mode (the carrier wave). From then on, extensive efforts, both in theory and experiment, have been investigated in the field of the stability analysis at the short time scale and the long time evolution of the nonlinear wave train on deep water.Based on the complete water wave equations, an important theoretical model for studying the long time behavior of the nonlinear wave was developed by Zakharov (1968). The Zakharov integral equation describes the resonant interaction between waves and was originally derived for infinitely deep water. Further, Stiassnie&Shemer (1984) re-derived and modified the Zakharov equation on finite water depth. The applicability of this modified Zakharov equation was studied by Stiassnie&Shemer (1984), in which the linear stability analysis of both class I and class II based on the modified Zakharov equation was compared with that of McLean (1982). Good agreement was obtained. The numerical study is based on the Zakharov nonlinear equation,which is modied to describe slow spatial evolution of unidirectional waves as they move along the tank, groups with various initial shapes, amplitudes and spectral contents are studied by Lev Shemer et al.(2001). It is demonstrated that the applied theoretical model, which does not impose any constraints on the spectral width, is capable of describing accurately, both qualitatively and quantitatively, the slow spatial variation of the group envelopes. However, the model just considers constant water depth. Here we improve the model by adding the terms proportional to (?)h, while higher order terms such as ((?）h)2and (？)2h are ignored. This improved model can accurately simulate general variable water depth cases.Th-e new model is of third order nonlinearity, and suitable for waves with any spectrum width.Zakharov equation is applied to study the two problems, one is considering the effect of the water depth and wave steepness on class I instabilities of a Stokes wave, the second is to study the evolution of wave group under different slopes, periods and amplitudes. The characters of the evolution of wave group under these conditions are attained.