[Paper Review] On a new type of solitary surface waves in finite water depth

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

Ocean Waves and Remote Sensing34 references2 citations

TL;DR

This paper presents a new type of solitary surface wave in finite water depth, derived analytically using exponentially decaying base functions in both linear and fully nonlinear wave equations. The waves exhibit a peaked crest, can be depression waves, are non-dispersive (phase speed independent of amplitude), and have higher bottom velocity than surface velocity—distinguishing them fundamentally from classical solitary waves and offering new insights into rogue wave phenomena.

ABSTRACT

In this paper, a new type of solitary surface waves in a finite water depth is found by analytically solving the fully nonlinear wave equations. Using a new type of base functions which decays exponentially in the horizontal direction, this new type of solitary surface waves is gained first by means of linear wave equations, and then confirmed by the fully nonlinear wave equations. The new type of solitary surface waves have many unusual characteristics. First, it has a peaked crest. Secondly, it may be in the form of depression, which has been often reported for internal solitary waves but never for free-surface solitary ones, to the best of author’s knowledge. Third, its phase speed has nothing to do with wave height, say, the peaked solitary waves are non-dispersive. Finally, its horizontal velocity at bottom is always larger than that on surface. All of these are so different from the traditional periodic and solitary waves that they clearly indicate the novelty of the peaked solitary waves. Based on the new peaked solitary surface waves, a new explanation to the so-called rogue waves and some theoretical predictions are given. All of these are helpful to deepen our understandings and enrich our knowledge about solitary waves.

Motivation & Objective

• To identify and analytically describe a novel type of solitary surface wave in finite water depth that differs fundamentally from classical solitary and periodic waves.
• To resolve the long-standing absence of depression solitary surface waves in free-surface flows, which are common in internal waves but not in surface waves.
• To investigate the non-dispersive nature of these waves, where phase speed is independent of wave height.
• To explore the unusual velocity structure, particularly the condition where bottom horizontal velocity exceeds surface velocity.
• To provide a theoretical basis for understanding rogue waves through the lens of these newly discovered wave characteristics.

Proposed method

• Employing a new set of base functions that decay exponentially in the horizontal direction to model wave profiles in both linear and fully nonlinear formulations.
• Solving the fully nonlinear wave equations using the new base functions to confirm the existence of the novel wave solution.
• Applying analytical techniques to derive wave solutions under the constraint of finite water depth and surface boundary conditions.
• Verifying the wave’s non-dispersive behavior by showing phase speed remains constant regardless of wave amplitude.
• Analyzing the horizontal velocity distribution across the water column to demonstrate that bottom velocity exceeds surface velocity.
• Using the derived wave solution to explore implications for rogue wave formation and theoretical predictions.

Experimental results

Research questions

• RQ1 Can a new type of solitary surface wave with a peaked crest exist in finite water depth, distinct from classical solitary waves?
• RQ2 Is it possible for a free-surface solitary wave to exhibit a depression form, similar to internal solitary waves but previously unobserved in surface flows?
• RQ3 Does the phase speed of such a wave remain independent of wave height, indicating non-dispersive behavior?
• RQ4 How does the horizontal velocity profile of this wave compare across the water column, particularly at the surface versus the bottom?
• RQ5 Can this new wave type provide a plausible theoretical explanation for the occurrence of rogue waves?

Key findings

• A new type of solitary surface wave with a peaked crest is analytically confirmed in finite water depth using fully nonlinear wave equations.
• The wave can exist in a depression form—previously unreported for free-surface solitary waves—indicating a significant departure from classical wave behavior.
• The phase speed of the peaked solitary wave is independent of wave height, confirming its non-dispersive nature.
• The horizontal velocity at the bottom of the water column is always greater than that at the surface, a unique and counterintuitive feature.
• The wave solution is derived using a novel set of exponentially decaying base functions, enabling accurate modeling of the wave’s localized structure.
• The findings offer a new theoretical framework for understanding rogue waves, suggesting such extreme events may be linked to these non-classical solitary wave dynamics.