The fourth-order evolution equation for deep-water gravity-capillary waves
The fourth-order evolution equation for deep-water gravity-capillary waves
复制标题
深水重力毛细波四阶演化方程
DOI：
--
复制
发表时间：
1985
期刊：
影响因子：
--
通讯作者：
S. J. Hogan
中科院分区：
文献类型：
--
作者：
S. J. Hogan
考虑了无限深度的理想流体表面上一系列非线性重力毛细管波的稳定性。导出了波包络的演化方程，该方程的波陡度精确到四阶。该推导是在窄波带假设下根据 Zakharov 方程进行的，并且包括重力-毛细波相互作用系数的完整形式。假设条件远离次谐波谐振波长。正如 K. B. Dysthe (Proc. R. Soc. Lond. A 369 (1979)) 对于纯重力波的发现一样，与三阶演化方程的主要区别在于，就稳定性而言，引入了平均流响应。有一个波带保持稳定至四阶。一般来说，纯毛细波的平均流量效应与纯重力波的平均流量效应的符号相反。对一阶稳定性特性的二阶修正显示取决于控制方程中平均流量和包络频散项之间的相互作用。结果表明，对于波陡度值足够小的情况，结果与整个问题的一些最新计算结果一致。
The stability of a train of nonlinear gravity-capillary waves on the surface of an ideal fluid of infinite depth is considered. An evolution equation is derived for the wave envelope, which is correct to fourth order in the wave steepness. The derivation is made from the Zakharov equation under the assumption of a narrow band of waves, and including the full form of the interaction coefficient for gravity-capillary waves. It is assumed that conditions are away from subharmonic resonant wavelengths. Just as was found by K. B. Dysthe (Proc. R. Soc. Lond. A 369 (1979)) for pure gravity waves, the main difference from the third-order evolution equation is, as far as stability is concerned, the introduction of a mean flow response. There is a band of waves that remains stable to fourth order. In general the mean flow effects for pure capillary waves are of opposite sign to those of pure gravity waves. The second-order corrections to first-order stability properties are shown to depend on the interaction between the mean flow and the envelope frequency-dispersion term in the governing equation. The results are shown to be in agreement with some recent computations of the full problem for sufficiently small values of the wave steepness.