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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。
In this paper, we use the approximate solution iterative method to deduce the trajectory equation of the Stokes wave water mass without approximating the potential function parameter β of the Stokes wave to the wave amplitude a. In the specific derivation, it is found that the horizontal and vertical displacements converge to exact solutions at different speeds of approximation, the vertical displacements are very similar to the exact solutions after one iteration, and the horizontal displacements must be done in two The second iteration can achieve better accuracy. A new version of the Stokes drift formula is obtained from the quadratic iteration of the horizontal displacement equation, which shows that Stokes drift is proportional to the square of the amplitude A of the water particle at the depth at the third-order accuracy. Using the relation between the amplitude obtained by the vertical displacement equation and the parameter β, we can prove that the classical Stokes drift equation can be regarded as the result of the new drift equation at a specific vertical section under a specific coordinate system.