Different forms for nonlinear standing waves in deep water
Different forms for nonlinear standing waves in deep water
复制标题
深水中非线性驻波的不同形式
DOI：
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发表时间：
1994
影响因子：
3.7
通讯作者：
M. Stiassnie
中科院分区：
文献类型：
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作者：
P. J. Bryant;M. Stiassnie
本文以Zakharov方程及其修正为解析解，得到了深水周期驻波在空间和时间上的多种形式，并以全非线性边值问题的二维无旋解进行了计算研究。不同的形式是基于2、3、…倍频率和4、9、…倍波数的基频和共振谐波之间的弱非线性相互作用。新形式的驻波的振幅在共振谐波处有局部最大值，而不像经典（斯托克斯）驻波是由基谐波主导的。从初始扰动为数值误差的驻波解出发，通过数值计算研究了小到中等波能下新驻波的稳定性。斯托克斯驻波对边带扰动的不稳定性首先得到了证明，当一组9个相等的斯托克斯驻波被一个长度等于9个驻波总长度的驻波扰动时，就会发生循环复发的演变。这种循环递推与斯托克斯渐进波的众所周知的线性不稳定性和边带调制中观察到的循环递推相似，也与一、九次谐波占主导地位的新驻波演变所产生的循环递推相似。新的驻波只有在小到中等波能量时才略微不稳定，谐波在演变成缓慢调制的振荡或发散之前，通常在100-1000波周期内保持在其初始振幅和相位附近。
Multiple forms for standing waves in deep water periodic in both space and time are obtained analytically as solutions of Zakharov's equation and its modification, and investigated computationally as irrotational two-dimensional solutions of the full nonlinear boundary value problem. The different forms are based on weak nonlinear interactions between the fundamental harmonic and the resonating harmonics of 2, 3,…times the frequency and 4, 9,…respectively times the wavenumber. The new forms of standing waves have amplitudes with local maxima at the resonating harmonics, unlike the classical (Stokes) standing wave which is dominated by the fundamental harmonic. The stability of the new standing waves is investigated for small to moderate wave energies by numerical computation of their evolution, starting from the standing wave solution whose only initial disturbance is the numerical error. The instability of the Stokes standing wave to sideband disturbances is demonstrated first, by showing the evolution into cyclic recurrence that occurs when a set of nine equal Stokes standing waves is perturbed by a standing wave of a length equal to the total length of the nine waves. The cyclic recurrence is similar to that observed in the well-known linear instability and sideband modulation of Stokes progressive waves, and is also similar to that resulting from the evolution of the new standing waves in which the first and ninth harmonics are dominant. The new standing waves are only marginally unstable at small to moderate wave energies, with harmonics which remain near their initial amplitudes and phases for typically 100–1000 wave periods before evolving into slowly modulated oscillations or diverging.