DISINTEGRATION OF WAVE TRAINS ON DEEP WATER .1. THEORY
DISINTEGRATION OF WAVE TRAINS ON DEEP WATER .1. THEORY
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DOI：
10.1017/s002211206700045x
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发表时间：
1967-01-01
影响因子：
3.7
通讯作者：
FEIR, JE
中科院分区：
文献类型：
--
作者：
BENJAMIN, TB;FEIR, JE
当一个基频为ω的周期性前进波列在深水中形成时，比如说，通过振荡桨的辐射，并且在相邻的边带频率ω（1 ± δ）处也存在残余波运动，例如，如果桨的运动在低频下受到轻微调制，就会产生这种现象。由于通过自由表面处的非线性边界条件的耦合，能量然后以如下速率从主运动传递到边带，如本文将示出的，该速率可以随着相互作用的进行而指数地增加。其结果是，波列在远离原点的地方变得非常不规则，即使偏离周期性在开始时几乎检测不到。本文从理论上研究了周期波列对一对边带模形式的小扰动的稳定性，第二部分将是与本预言雅阁的一些实验观测。该理论的主要结论是：如果k和a是扰动波列的基波数和振幅，则所考虑的这类无穷小扰动将经历无限放大。当δ = ka时，渐近增长率是最大值。
The phenomenon in question arises when a periodic progressive wave train with fundamental frequency ω is formed on deep water—say by radiation from an oscillating paddle—and there are also present residual wave motions at adjacent side-band frequencies ω(1 ± δ), such as would be generated if the movement of the paddle suffered a slight modulation at low frequency. In consequence of coupling through the non-linear boundary conditions at the free surface, energy is then transferred from the primary motion to the side bands at a rate that, as will be shown herein, can increase exponentially as the interaction proceeds. The result is that the wave train becomes highly irregular far from its origin, even when the departures from periodicity are scarcely detectable at the start.In this paper a theoretical investigation is made into the stability of periodic wave trains to small disturbances in the form of a pair of side-band modes, and Part 2 which will follow is an account of some experimental observations in accord with the present predictions. The main conclusion of the theory is that infinitesimal disturbances of the type considered will undergo unbounded magnification ifwhere k and a are the fundamental wave-number and amplitude of the perturbed wave train. The asymptotic rate of growth is a maximum for δ = ka.