On solitary water-waves of finite amplitude
On solitary water-waves of finite amplitude
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DOI：
10.1007/bf00250799
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发表时间：
1981-03
影响因子：
2.5
通讯作者：
C. Amick;J. Toland
中科院分区：
文献类型：
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作者：
C. Amick;J. Toland
1.1. 在描述有限深度的水中的孤波时，瑞利勋爵写道：“这是斯科特·罗素先生在1844年英国协会报告中描述的一种特殊波浪的名字。由于它的长度大约是运河深度的六到八倍，粗略地说，这个波被包括在长波理论中；但罗素先生观察到的几种情况表明，它具有不同于其他长波的特性。其中可能会提到孤立波的不同行为，根据它们是正的还是负的，也就是说，根据它们由未受干扰的水平的上升或下降组成。在前一种情况下，波具有显著的持久性，可以传播到很远的距离而没有太多损失；但负面的浪潮很快就会被打破和消散。”他把这个问题看作是一个稳定运动的问题，用一种级数展开的方法对这种行为给出了理论解释，这种方法最终依赖于波幅很小的假设。独立地，BOUSSINESQ~ 11也得出了类似的结论。KORTEWEG和DE VRIES[21]方程是浅水长波的模型方程，它有一组精确的孤立波解，这些解是已知的封闭形式。在流体力学中，模型方程还可能具有许多其他情况：孤立波解[3]-[6]，[15],[27],[37],[38],[45],[46]，最近人们对这些解[2]，[5],[7]-[10],[16],[26],[37],[45]，[46]的存在性问题非常感兴趣。然而，有限深度水中孤立波理论的许多工作一直致力于波廓线的估计问题。最近，LONGUET-HIGGINS和他的同事，FENTON， FOX， BYATT-SMITH和COKELET[12]，[13],[30]-[32]，将物理洞察力与复杂的数值技术相结合，计算了非常高的波的剖面，包括孤立波和周期波。结果显示出很强的稳定性和一致性，解决了如何计算大振幅水波形状的问题。
1.1. BackgroundIn describing the solitary wave on water of finite depth, Lord RAYLEIGH [40] wrote:" This is the name given by Mr. SCOTT RUSSELL* to a particular wave described by him in the British Association Report of 1844. Since its length is about six or eight times the depth of the canal, this wave is, to a rough approximation, included under the theory of long waves; but there are several circumstances observed by Mr. RUSSELL which indicate that it has a character distinct from other long waves. Among these may be mentioned the very different behaviour of solitary waves according as they be positive or negative, viz. according as they consist of an, elevation or a depression from the undisturbed level. In the former case the wave has a remarkable permanence, being propagated to great distances without much loss; but a negative wave is soon broken up and dissipated." Regarding the problem as one of steady motion, he gave a theoretical explanation of this behaviour, using a series expansion method which depends ultimately on an assumption that the wave amplitude is small. Independently, BOUSSINESQ~ 11] had reached similar conclusions. The equation of KORTEWEG & DE VRIES [21], which is a model equation for long waves in shallow water, has a family of exact solitary wave solutions which are known in closed form. There are many other situations in hydrodynamics where model equations may be expected to possess: solitary wave solutions [3]-[6],[15],[27],[37],[38],[45],[46], and recently considerable interest has been focused on questions about the existence of such solutions [2],[5],[7]-[10],[16],[26],[37],[45],[46]. However, much of the work on the theory of solitary waves on water of finite depth has been devoted to the problem of estimating the wave profile. Most recently, LONGUET-HIGGINS and his colleagues, FENTON, FOX, BYATT-SMITH and COKELET [12],[13],[30]-[32], have combined physical insight with sophisticated numerical techniques to calculate the profile of very high waves, both solitary and periodic. Their results show great stability and consistency, and the problem of how to compute the shape of large amplitude water-waves appears to have been solved.