深水表面有限振幅周期波的稳定性

Stability of periodic waves of finite amplitude on the surface of a deep fluid

本文研究了无限深流体中非线性定常表面波的稳定性(Lamb 1964,Moiseev 1960).在第1节中,将带有自由表面的理想流体动力学方程转换为有关正则变量的方程:以正则变量来表示波形和表面速度势函数.通过引入正则变量,可以将表面波的稳定性问题视为色散介质中非线性波这一更具普遍性问题的一部分(Akhmanov 1964,Zakharov 1965).本文其余部分的结果也适用于一般情况.在第2节中,使用与van der Pohl类似的方法,得到了一个用于描述小振幅近似下的非线性波的简化方程.如果假设波包很窄,方程将特别简单.该方程具有精确解,该解近似于一个有限振幅的周期波.在第3节中,研究了有限振幅周期波的不稳定性,发现了两类不稳定性.第一类不稳定性是破坏不稳定性,类似于等离子体中波的破坏不稳定性(Oraevskii&Sagdeev 1963,Oraevskii 1964).在该类不稳定性中,一对波被同时激发,其频率之和是原始波频率的整数倍.对于毛细波,破坏不稳定产生得最快;而对于重力波,破坏不稳定产生得最慢.第二类不稳定性是负压类型的不稳定性,它是由于非线性波的波速依赖于振幅而产生的,这导致波的调制率被无限放大.当非线性波通过色散介质时,如果色散关系对波数的二阶导数的符号与因非线性效应导致频率漂移的符号不同,则会产生此类不稳定性.正如Litvak A N和Talanov V I(1967)所提到的那样,这类不稳定性已经在非线性电磁波中被独立发现.

We study the stability of steady nonlinear waves on the surface of an infinitely deep fluid(Lamb 1964,Moiseev 1960).In section 1,the equations of hydrodynamics for an ideal fluid with a free surface are transformed to canonical variables:the shape of the surfaceη(r,t)and the hydrodynamic potential at the surface are expressed in terms of these variables.By introducing canonical variables,we can consider the problem of the stability of surface waves as part of the more general problem of nonlinear waves in media with dispersion(Akhmanov 1964,Zakharov 1965).The resuits of the rest of the paper are also easily applicable to the general case.In section 2,using a method similar to van der Pohl's method,we obtain simplified equations describing nonlinear waves in the small amplitude approximation.These equations are particularly simple if we assume that the wave packet is narrow.The equations have an exact solution which approximates a periodic wave of finite amplitude.In section 3 we investigate the instability of periodic waves of finite amplitude.Instabilities of two types are found.The first type of instability is destructive instability,similar to the destructive instability of waves in a plasma(Oraevskii&Sagdeev 1963,Oraevskii 1964),In this type of instability,a pair of waves is simultaneously excited,the sum of the frequencies of which is a multiple of the frequency of the original wave.The most rapid destructive instability occurs for capillary waves and the slowest for gravitational waves.The second type of instability is the negative-pressure type,which arises because of the dependence of the nonlinear wave velocity on the amplitude;this results in an unbounded increase in the percentage modulation of the wave.This type of instability occurs for nonlinear waves through any media in which the sign of the second derivative in the dispersion law with respect to the wave number(d2ω/dk2)is different from the sign of the frequency shift due to the nonlinearity.As announced by A.N.Litvak and V.I.Talanov(1967),this type of instability was independently observed for nonlinear electromagnetic waves.