波浪绕射非线性数学模型

Nonlinear mathematical model of wave diffraction

本文用波数矢量无旋波能守恒方程和非线性弥散关系一起建立了一个浅水波浪绕射的非线性数学模型。在此模型中以非线性弥散关系 (经验非线性弥散关系和理论非线性弥散关系 )引入浅水波浪绕射理论 ,组成浅水波浪非线性数学模型 ,并且在波能守恒方程中考虑了底摩擦的影响。模型中通过对一个斜坡浅滩水域的波浪绕射现象 ,对使用经验非线性弥散关系和理论非线性弥散关系以及线性弥散关系的计算结果进行了比较。结果表明 :使用非线性弥散关系比使用线性弥散关系的计算结果与试验结果更加吻合 ;水波在浅水传播时使用经验非线性弥散关系和使用理论非线性弥散关系两者相差甚微 ,但采用经验弥散关系近似波浪在浅水传播时的非线性的计算量远小于理论非线性弥散关系的计算量 。

In the present study, a nonlinear mathematical model was developed to describe wave diffraction in areas of shallow water using the irrotationality of the wave number vector and the energy conservative equation along with nolinear dispersion relations. The nonlinear effect was included by employing an nonlinear empirical dispersion relation and a nonlinear theoretical one, compared with linear dispersion relation. The model was tested against laboratory measurements for the case of submerged elliptical shoal on a slope, where both refraction and diffraction were equally significant. The computed results show that it is better to use the nonlinear dispersion relation than linear one. And the discrepencies are small between using empirical dispersion relation and theoretical dispersion relation. But the computational workload using the empirical relation is estimated to be much smaller than using theoretical one, so it is advantageous and feasible to use nonlinear empirical dispersion relation instead of theoretical dispersion relation for modelling the wave diffraction in shallow water. The nonlinearity in diffraction of the shallow water wave is important and can not be ignored.