陡重力波上稳态寄生毛细波纹的指数渐近性
Exponential asymptotics for steady parasitic capillary ripples on steep gravity waves
摘要：
本文发展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。在伴随的工作中[Shelton等人（2021），J. Fluid Mech.，vol 922]证明了与关于前导阶重力波（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，单纯的庞加莱展开式不足以描述它们。在这里，我们将开发专门的方法，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。分析依赖于至关重要的推导和分析的奇异性的经典斯托克斯波的解析延拓。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。渐近结果进行比较，充分的数值解，并显示出良好的协议。这项工作提供了对由Longuet-Higgins首先提出的寄生毛细波的开创性理论的修正和见解[J. Fluid Mech.，第16（1）卷，1963年，第16页。138-159]。
英文摘要：
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. (2021), J. Fluid Mech., vol 922] 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 shall 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 analysis relies crucially upon the derivation and analysis of singularities in the analytic continuation of the classic 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 to 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 (1), 1963, pp. 138-159].
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