深水Stokes漂流公式的一种新形式

A NEW FORM OF STOKES DRIFT FORMULA IN DEEP WATER

本文在未将Stokes波的势函数参数β近似为波面振幅a的前提下使用近似解迭代方法重新推导了Stokes波的水质点的轨迹方程。在具体推导中发现水平位移与垂向位移在作近似解迭代时,收敛于精确解的速度是不相同的,垂向位移在一次迭代后即与精确解非常近似,而对于水平位移必须做二次迭代才能达到较好的精度。由二次迭代的水平位移方程即可获得Stokes漂流公式的一种新形式,该形式表明,在三阶精度下Stokes漂流与所在深度的水质点的振幅A的平方成正比。运用由垂向位移方程解得的振幅与参数β的关系式可证明经典Stokes漂流公式可以看作是新漂流公式在特定坐标系下的特定垂直断面处的结果。

Using the iterative approximation method, the motion equation of water particles in Stokes waves are rededuced without the assumption that the coefficient β in the potential function approximates to the amplitude a. In the course of deduction, the converging speeds of iterative solutions of horizontal and vertical displacement to the exact solutions are found to be different: while the vertical displacement is accurate enough after its first iteration, the horizontal one need to be iterated for the second time. A new form of Stokes drift formula can be obtaincd from the new horizontal displacement equation, which shows that under the third order approximation, the Stokes drift is proportional to the square of the amplitude of water particle. It is proved that the classical Stokes drift formula can be regarded as a special case of the new drift formula at a special vertical section in a special coordinate system.