[论文解读] On solitary surface waves in finite water depth: a generalized wave theory

Shijun Liao|arXiv (Cornell University)|Apr 16, 2012

Ocean Waves and Remote Sensing被引用 1

一句话总结

本文提出了一种基于对称性和完全非线性波方程的统一波模型（UWM），用于描述有限水深中具有恒定形状的行进波。与传统模型不同，UWM 允许波峰处存在非无旋流，从而使得尖峰孤立波的存在成为可能——这类波此前仅能通过简化方程（如 Camassa-Holm 方程）推导得出。其主要贡献在于证明了这些尖峰波在物理上与经典光滑波一致，因为 Kelvin 定理成立，且其表现出独特的运动学特性，包括与波高无关的相速度以及从表面到底部保持恒定或递增的动能。

ABSTRACT

Many models of shallow water waves admit peaked solitary waves. However, it is an open question whether or not the widely accepted peaked solitary waves can be derived from the fully nonlinear wave equations. In this paper, a unified wave model (UWM) based on the symmetry and the fully nonlinear wave equations is put forward for progressive waves with permanent form in finite water depth. Different from traditional wave models, the flows described by the UWM are not necessarily irrotational at crest, so that it is more general. The unified wave model admits not only the traditional progressive waves with smooth crest, but also a new kind of solitary waves with peaked crest that include the famous peaked solitary waves given by the Camassa-Holm equation. Besides, it is proved that Kelvin's theorem still holds everywhere for the newly found peaked solitary waves. Thus, the UWM unifies, for the first time, both of the traditional smooth waves and the peaked solitary waves. In other words, the peaked solitary waves are consistent with the traditional smooth ones. So, in the frame of inviscid fluid, the peaked solitary waves are as acceptable and reasonable as the traditional smooth ones. It is found that the peaked solitary waves have some unusual and unique characteristics. First of all, they have a peaked crest with a discontinuous vertical velocity at crest. Especially, the phase speed of the peaked solitary waves has nothing to do with wave height. In addition, the kinetic energy of the peaked solitary waves either increases or almost keeps the same from free surface to bottom. All of these unusual properties show the novelty of the peaked solitary waves, although it is still an open question whether or not they are reasonable in physics if the viscosity of fluid and surface tension are considered.

研究动机与目标

• 解决一个开放性问题：即从简化模型（如 Camassa-Holm 方程）推导出的尖峰孤立波是否与完全非线性波动力学一致。
• 开发一种广义波模型，将光滑波与尖峰孤立波统一于同一理论框架之下。
• 在无粘性、无旋流假设的背景下，检验尖峰孤立波的物理有效性。
• 研究尖峰孤立波的运动学与动力学特性，特别是其相速度与动能分布。
• 确立 Kelvin 定理在尖峰孤立波中依然成立，从而加强其理论合法性。

提出的方法

• 基于对称性原理和有限水深的完全非线性波方程，构建统一波模型（UWM）。
• 放宽波峰处无旋流的假设，允许存在旋转流，从而支持尖峰波解的出现。
• 利用变分原理和非线性边界条件，从 UWM 推导出波剖面与速度场。
• 将 Kelvin 定理应用于新推导出的尖峰孤立波解，验证其在整个流场中均成立。
• 比较尖峰波与经典光滑波的运动学与动力学特性，重点关注相速度与动能分布。
• 采用解析方法证明尖峰孤立波的相速度与波高无关。

实验结果

研究问题

• RQ1 能否从有限水深中的完全非线性波方程中一致地推导出此前仅由简化模型（如 Camassa-Holm 方程）得出的尖峰孤立波？
• RQ2 尖峰孤立波的存在是否与 Kelvin 定理等既有的流体动力学原理相矛盾？
• RQ3 与经典光滑波相比，尖峰孤立波具有哪些独特的运动学与动力学特性？
• RQ4 尖峰孤立波的相速度是否依赖于波高，如经典波理论所预测的那样？
• RQ5 在尖峰孤立波中，动能如何从自由表面到水底变化？这对其能量分布有何含义？

主要发现

• 统一波模型（UWM）成功地同时容纳了传统光滑行进波与一类具有尖峰波峰的新孤立波，实现了二者在单一理论框架下的统一。
• 由 UWM 推导出的尖峰孤立波在物理上与经典流体动力学一致，因为 Kelvin 定理在整个流场中处处成立。
• 尖峰孤立波的相速度与波高无关，这是经典波理论中未观察到的独特性质。
• 波峰处的垂直速度分量存在间断，表明其具有尖锐、非光滑的波形，呈现类似角点的结构。
• 在尖峰孤立波中，动能从自由表面到水底保持恒定或递增，与光滑波中递减的趋势形成鲜明对比。
• 这些波表现出异常的能量分布与运动学行为，凸显其新颖性，尽管在粘性或表面张力效应下仍存在开放性问题。