A Modified Form of Mild-Slope Equation with Weakly Nonlinear Effect

Nonlinear effect is of importance to waves propagating from deep water to shallow water. The non-linearity of waves is widely discussed due to its high precision in application. But there are still some problems in dealing with the nonlinear waves in practice. In this paper, a modified form of mild-slope equation with weakly nonlinear effect is derived by use of the nonlinear dispersion relation and the steady mild-slope equation containing energy dissipation. The modified form of mild-slope equation is convenient to solve nonlinear effect of waves. The model is tested against the laboratory measurement for the case of a submerged elliptical shoal on a slope beach given by Berkhoff et al. The present numerical results are also compared with those obtained through linear wave theory. Better agreement is obtained as the modified mild-slope equation is employed. And the modified mild-slope equation can reasonably simulate the weakly nonlinear effect of wave propagation from deep water to coast.

Nonlinear Dispersion Effect on Wave Transformation

A new nonlinear dispersion relation is given in this paper, which can overcome the limitation of the intermediate minimum value in the dispersion relation proposed by Kirby and Dalrymple (1986). and which has a better approximation to Hedges' empirical relation than the modified relations by Hedges (1987). Kirby and Dalrymple (1987) for shallow waters. The new dispersion relation is simple in form, thus it can be used easily in practice. Meanwhile, a general explicit approximation to the new dispersion and other and other nonlinear dispersion relations is given. By use of the explicit approximation to the new dispersion relation along with the mild slope equation taking into account weakly nonlinenr effect, a mathematical model is obtained, and it is applied to laboratory data. The results show that the model developed with the new dispersion relation predicts wave transformation over complicated topography quite well.

波浪的非线性频散关系

在概括和总结现有非线性频散关系的基础上,给出了非线性频散关系的通式形式及其显式表达式。通过与原频散关系的比较可知显式形式具有很高的精度,与隐式形式吻合很好。利用频散关系的显式形式,结合含非线性项的缓坡方程,得到考虑非线性弥散影响的波浪变形模型。模型中对采用非线性频散关系和修正的非线性频散关系的计算结果进行了比较。结果表明,用新的频散关系的显式表达式得到的模型更加精确、合理。

非线性弥散效应及其对波浪变形的影响

针对Hedges、Kirby和Dalrymple提出的非线性弥散关系的修正式在浅水区存在较大偏差的问题 ,给出了一个在整个水深范围内具有单值性的新的非线性弥散关系。比较可知 ,它具有在深水与中等水深逼近二阶Stokes波的弥散关系式 ,在浅水较Hedges、Kirby和Dalrymple的修正表达式与Hedges的关系式更加吻合的优点 ,且形式简练。用近似该非线性弥散关系的显式表达式 ,结合弱非线性效应的缓坡方程 ,得到考虑非线性弥散影响的波浪变形模型。数值验证结果表明 。

波浪在浅水传播中的弱非线性效应

在波浪从深水向浅水传播过程中 ,考虑弱非线性效应具有重要的实用价值 ,因此得到广泛的讨论和研究。本文根据文献 [6 ]导出的考虑能耗的定常缓坡方程 ,结合文献 [5 ]给出的显式非线性弥散关系 ,得出了含弱非线性效应的缓坡方程 ,用该方程对浅水中波浪的传播进行了计算 ,将计算结果和试验数据进行了比较 ,结果表明 ,含弱非线性效应的缓坡方程可以用于讨论浅水中波浪传播的弱非线性效应 ,所得计算结果与试验结果更为吻合。

考虑非线性弥散影响的波浪变形数学模型

提出了逼近Ｋｉｒｂｙ和Ｄａｌｒｙｍｐｌｅ的非线性弥散关系的显式非线性弥散关系的表达式，该显式表达式与他们的非线性弥散关系的精度几乎完全相同．采用显式非线性弥散关系，结合含弱非线性效应的缓坡方程，得到考虑非线性弥散影响的波浪变形数学模型，并对该数学模型进行了数值验证．结果表明，考虑非线性弥散影响的波浪变形数学模型更为精确．

显式非线性弥散关系在浅水波变形计算中的应用

本文参照Ｚｈａｏ和Ａｎａｓｔａｓｉｏｕ的方法，导出了逼近Ｂｏｏｉｊ的非线性弥散关系的近似显式表达式，该式给出的结果与Ｂｏｏｉｊ的非线性弥散关系相当吻合。用中文显式非线性弥散关系，结合会弱非线性效应的缓坡方程，构成含非线性影响项缓坡方程的一个求解浅水波变形问题的方程组。用实验数据对本文模型进行验证，结果表明，显式非线性弥散关系在求解浅水波变形问题时，给出了更符合实验数据的结果。

波浪的非线性显式弥散关系

波浪的非线性弥散关系在应用于求解波浪的变形问题时很不方便,需要与含非线性效应的缓坡方程一起进行迭代运算,往往导致数值计算的计算量太大,计算过于复杂。采用显式形式表达非线性弥散关系,可以克服上述缺点,大为简化波浪变形数值计算的计算量。本文通过将现有的非线性弥散关系进行分析比较,给出了一个更为一般的非线性弥散关系及其显式表达式,经比较可知,该显式弥散关系与相对应非线性弥散关系吻合的很好。本文最后用该显式结合含弱非线性效应的缓坡方程,对复式浅滩地形上的波浪折射绕射进行了计算。结果表明,考虑弱非线性可以得出与实验数据更为相符的结果,而采用显式弥散关系可以有效提高计算效率,在波浪的非线性计算中不失为一种切实有效的方法。

新的非线性弥散关系及其波浪变形数学模型

针对Hedges、Kirby、李瑞杰提出的修正非线性弥散关系在浅水区存在较大偏差的问题,给出了一个在整个水深范围内相对波速具有单值性的新的非线性弥散关系。它在深水区与二阶Stokes波的弥散关系相一致,在浅水区较前人的修正式与Hedges经验弥散关系更加吻合,在中等水深区域与二阶Stokes波的弥散关系及Hedges经验弥散关系的偏差也达到最小。为了避免非线性弥散关系引入缓坡方程而导致的迭代,采用显式形式近似表达该非线性弥散关系,得到与其精度几乎完全相同的显式表达式。用该显式表达式,结合弱非线性效应的缓坡方程,得到考虑非线性弥散影响的波浪变形数学模型。用该模型对复杂地形进行模拟,计算结果与实测值吻合很好。