新型高阶Boussinesq水波方程

New high-order Boussinesq equations for water wave

从经典的Boussinesq方程出发,引入两个参数并对方程中的部分项进行替换,通过严格的数学推导给出量级为O(εμ2)的高阶非线性项,得到一种新型的高阶Boussinesq方程。该方程的色散关系比经典Boussinesq方程提高了一阶,变浅作用性能也得到了改善,方程的适用范围由浅水达到中等水深。利用Crank Nicloson格式的有限差分法对方程进行数值模型在一维方向上进行离散计算,建立了高阶Boussinesq方程的数值模型。为验证数值模型的正确性,将数值计算结果与Zou等(2001)的物模试验结果以及Beji与Nadaoka方程的数值结果进行对比,本文的数值结果与试验结果吻合程度较好,表明本文方程可适于模拟变水深下的波浪场数值模拟。

Two parameters for improving the dispersion precision are introduced into the classical Boussinesq equations and to strictly derive the high-order nonlinear terms by mathematical method. On this basis a new type high-order Boussinesq equations is deduced. Comparing with the classical Boussinesq equations the new equations possesses better linear dispersion characteristic and the linear shoal characteristic is improved, so that the Boussinesq equations can be applied to describe the wave action in shallow water and moderate depth water. Furthermore,the finite differential method with Crank-Nicloson format is applied to numerically calculate the 1-D mathematical model and to establish the numerical model for high-order Boussinesq equations. The comparison of the calculation results shows that the new method gives better agreement with experimental data than other methods.