具有精确色散性的非线性波浪数学模型

以完全非线性的自由表面边界条件为基础，以波面升高\eta和自由表面速度势\phi _\eta为待求变量，建立了新的波浪方程．方程在色散性上是完全精确的，非线性近似至三阶．与缓坡方程相比较，两者都具有精确的色散性，但该方程属于非线性模型，可模拟波浪的非线性效应，且适用于不规则波．方程的特点是属于微分-积分方程，对如何处理方程中积分项进行了讨论，并数值模拟了不同周期的线性波和二阶Stokes波，也模拟了波群的非线性演化，以对模型进行验证． 

THE NONLINEAR WATER WAVE EQUATIONS WITH FULL DISPERSION

A 2D nonlinear water wave model with full dispersion is developed. The model is based on the nonlinear kinematic and dynamic free surface boundary conditions and is expressed in terms of free surface elevation $\eta $ and the velocity potential $\phi _\eta $ at the free surface. The derivation of the equations is accurate to third order in nonlinearity and keeps exact dispersion. The mild slope assumption is adopted and the derived equations can be seen as the extention of the mild slope equation of Berkhoff (1972) to the nonlinear and irregular wave case. The corresponding numerical scheme is presented, and the special attention is paid on the treatment of the integration terms in the equations. The validation of the model is made by simulating the first and second order Stokes waves and the nonlinear evolution of wave groups, the advantage of the model is shown by the good prediction of amplitude dispersion and four-wave resonant interaction for the wave group evolution.