The highest standing surface wave at infinite depth is a classical hydrodynamic problem, illuminated by Taylor's excellent experiments [G. I. Taylor, “An experimental study of standing waves,” Proc. R. Soc. London, Ser. A 218, 44–59 (1953)]. Based on length scale arguments, we present a compact analytical approach to the highest standing wave. Our physical postulate is that the highest deep-water wave has a single length scale, i.e., its wavelength. The single-scale postulate for standing periodic deep-water waves is confronted with two distinctly different cases where zero and two length scales are postulated as follows: (i) No physical length scale for an isolated rogue-wave peak at deep water suggests a similarity solution. (ii) Two length scales for the periodic peaked surface at constant depth suggest a one-parameter family of standing waves. Moreover, the two length scales are the wavelength and average fluid depth. The deep-water limit with its single-length scale postulate confirms Grant's theory [M. A. Grant, “Standing Stokes waves of maximum height,” J. Fluid Mech. 60, 593–604 (1973)], taking the highest standing wave as a state of zero kinetic energy. The reversible motion is irrotational according to Lord Kelvin's theorem. The acceleration field for the highest deep-water wave has a single Fourier component according to our single length scale postulate. The resulting free-surface shape follows from the exact nonlinear dynamic condition. Our analytical theory confirms the ratio 0.203 for maximal wave height to wavelength, found by Grant. We test its robustness by extending the theory to a moderate spatial quasi-periodicity. Appendix A provides a simple proof for the right-angle peak, representing a regular extremal point of a locally quadratic complex function. Appendix B presents a quadrupole solution for an isolated peak of stagnant deep-water rogue waves.

中文翻译：

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最高驻波水波的长度尺度方法

无限深度处的最高驻波是一个经典的流体动力学问题，泰勒出色的实验阐明了这一点[GI Taylor，“驻波的实验研究”，Proc。R. Soc。伦敦，爵士 一个218, 44–59 (1953)]。基于长度尺度参数，我们提出了一种针对最高驻波的紧凑分析方法。我们的物理假设是最高的深水波有一个单一的长度尺度，即它的波长。常设周期性深水波的单尺度假设面临两种截然不同的情况，其中假设零和两个长度尺度如下： (i) 深水孤立无赖波峰的物理长度尺度没有表明相似性解决方案。(ii) 恒定深度处周期性尖峰表面的两个长度尺度表明驻波的单参数族。此外，两个长度尺度是波长和平均流体深度。具有单一长度尺度假设的深水极限证实了格兰特的理论 [MA Grant，“最大高度的立斯托克斯波”，J.60 , 593–604 (1973)]，将最高驻波视为零动能状态。根据开尔文勋爵定理，可逆运动是无旋的。根据我们的单一长度尺度假设，最高深水波的加速度场具有单一的傅立叶分量。由此产生的自由表面形状遵循精确的非线性动态条件。我们的分析理论证实了格兰特发现的最大波高与波长之比为 0.203。我们通过将理论扩展到中等空间准周期性来测试其稳健性。附录 A 提供了直角峰的简单证明，表示局部二次复函数的规则极值点。附录 B 给出了停滞深水流氓波孤立波峰的四极解。