Generation of upstream advancing solitons by moving disturbances
Generation of upstream advancing solitons by moving disturbances
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DOI：
10.1017/s0022112087002817
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发表时间：
1987-11
影响因子：
3.7
通讯作者：
WU T.YAO-TSU
中科院分区：
文献类型：
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作者：
WU T.YAO-TSU
本研究研究了最近发现的一种现象，即在浅水中以跨临界速度稳定移动的强迫扰动可以周期性地在扰动的上游产生一系列孤立波，而在扰动后面的一个凹陷水面区域下游发展出一系列弱非线性和弱色散波。这种现象是由Wu & Wu（1982）基于广义Boussinesq模型在数值上发现的，该模型用于描述由移动的表面压力或地形产生的二维长波。在一项理论和实验联合研究中，Lee（1985）发现实验与两个理论模型，即广义Boussinesq和强制Korteweg-de Vries （fKdV）方程，两者都包含强迫函数之间存在广泛的一致性。本研究采用fKdV模型来探讨这一现象的基本机制。为了便于分析fKdV方程初边值问题解的稳定性，建立了一类强迫稳定孤立波。根据这里的唯一性定理，任何这样的解一旦成立，将在形式上保持不变。选择最简单的平稳解之一进行稳定性计算，它是一个单参数族，可以缩放成通用的相似形式。作为计算机代码的测试，发现最初建立的平稳解在波浪在强迫下穿过600水深的距离后，在数值形式上是永久的，分数不确定性小于2%。另一数值结果表明，当波的初始扰动大到不得不从静止状态上升时，也会出现同样的上游推进孤子的产生现象，并且产生周期是确定的。该相似族的结果表明，由此产生的孤子的产生周期、Ts和标度振幅α由公式Ts = const α−3/2相关。我们进一步发现，这种关系与这里基于fKdV方程的质量、动量和能量考虑而得出的第一性原理预测很好地吻合。
This study investigates the recently identified phenomenon whereby a forcing disturbance moving steadily with a transcritical velocity in shallow water can generate, periodically, a succession of solitary waves, advancing upstream of the disturbance in procession, while a train of weakly nonlinear and weakly dispersive waves develops downstream of a region of depressed water surface trailing just behind the disturbance. This phenomenon was numerically discovered by Wu & Wu (1982) based on the generalized Boussinesq model for describing two-dimensional long waves generated by moving surface pressure or topography. In a joint theoretical and experimental study, Lee (1985) found a broad agreement between the experiment and two theoretical models, the generalized Boussinesq and the forced Korteweg-de Vries (fKdV) equations, both containing forcing functions. The fKdV model is applied in the present study to explore the basic mechanism underlying the phenomenon. To facilitate the analysis of the stability of solutions of the initial-boundary-value problem of the fKdV equation, a family of forced steady solitary waves is found. Any such solution, if once established, will remain permanent in form in accordance with the uniqueness theorem shown here. One of the simplest of the stationary solutions, which is a one-parameter family and can be scaled into a universal similarity form, is chosen for stability calculations. As a test of the computer code, the initially established stationary solution is found to be numerically permanent in form with fractional uncertainties of less than 2% after the wave has traversed, under forcing, the distance of 600 water depths. The other numerical results show that when the wave is initially so disturbed as to have to rise from the rest state, which is taken as the initial value, the same phenomenon of the generation of upstream-advancing solitons is found to appear, with a definite time period of generation. The result for this similarity family shows that the period of generation, Ts, and the scaled amplitude α of the solitons so generated are related by the formula Ts = const α−3/2. This relation is further found to be in good agreement with the first-principle prediction derived here based on mass, momentum and energy considerations of the fKdV equation.