Exponential asymptotics for steady parasitic capillary ripples on steep gravity waves
Exponential asymptotics for steady parasitic capillary ripples on steep gravity waves
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DOI：
10.1017/jfm.2022.114
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发表时间：
2021-06
影响因子：
3.7
通讯作者：
Josh Shelton;Philippe H. Trinh
中科院分区：
文献类型：
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作者：
Josh Shelton;Philippe H. Trinh
本文建立了在小表面张力极限下稳定传播的重力毛细波的渐近理论。在一项附带的工作中（谢尔顿等人，流体力学杂志，Vol.922，2021），证明了与关于首阶重力波（Stokes波）的扰动相关联的解包含具有指数小振幅的表面张力驱动的寄生波纹。因此，朴素的庞加莱展开式不足以描述它们。在这里，我们开发了专门的方法，在指数渐近周期域上的寄生纹波的推导。的波纹所示，出现在与斯托克斯线和斯托克斯现象。由此产生的分析相关联的生产寄生波纹的复值奇点与陡峭的斯托克斯波的波峰。一个可解性条件的推导，表明这种类型的解决方案不存在在某些值的邦德数。渐近结果与完整的数值解进行了比较，并显示出良好的协议。这项工作提供了对由Longuet-Higgins（J. Fluid Mech.，第16卷，第1期，1963年，pp. 138-159）。
Abstract In this paper, we develop an asymptotic theory for steadily travelling gravity–capillary waves under the small-surface tension limit. In an accompanying work (Shelton et al., J. Fluid Mech., vol. 922, 2021) it was demonstrated that solutions associated with a perturbation about a leading-order gravity wave (a Stokes wave) contain surface-tension-driven parasitic ripples with an exponentially small amplitude. Thus, a naive Poincaré expansion is insufficient for their description. Here, we develop specialised methodologies in exponential asymptotics for derivation of the parasitic ripples on periodic domains. The ripples are shown to arise in conjunction with Stokes lines and the Stokes phenomenon. The resultant analysis associates the production of parasitic ripples to the complex-valued singularities associated with the crest of a steep Stokes wave. A solvability condition is derived, showing that solutions of this type do not exist at certain values of the Bond number. The asymptotic results are compared with full numerical solutions and show excellent agreement. The work provides corrections and insight of a seminal theory on parasitic capillary waves first proposed by Longuet-Higgins (J. Fluid Mech., vol. 16, issue 1, 1963, pp. 138–159).