On the Limiting Stokes' Wave of Extreme Height in Arbitrary Water Depth

TL;DR

This paper introduces the homotopy analysis method (HAM) to analytically approximate limiting Stokes' waves with sharp crests across all water depths, overcoming previous convergence issues in shallow water.

Contribution

It demonstrates the first successful, convergent analytic approximation of limiting Stokes' waves with sharp crests in arbitrary water depths using HAM, unifying wave models across different water regimes.

Findings

Achieved convergent wave profiles for limiting Stokes' waves in all water depths.

Successfully modeled solitary waves of extreme form in shallow water.

Provided a unified analytical framework for various wave types.

Abstract

As mentioned by Schwartz (1974) and Cokelet (1977), it was failed to gain convergent results of limiting Stokes' waves in extremely shallow water by means of perturbation methods even with the aid of extrapolation techniques such as Pad\'{e} approximant. Especially, it is extremely difficult for traditional analytic/numerical approaches to present the wave profile of limiting waves with a sharp crest of $120^\circ$ included angle first mentioned by Stokes in 1880s. Thus, traditionally, different wave models are used for waves in different water depths. In this paper, by means of the homotopy analysis method (HAM), an analytic approximation method for highly nonlinear equations, we successfully gain convergent results (and especially the wave profiles) of the limiting Stokes' waves with this kind of sharp crest in arbitrary water depth, even including solitary waves of extreme form in extremely shallow water, without using any extrapolation techniques. Therefore, in the frame of the HAM, the Stokes' wave can be used as a unified theory for all kinds of waves, including periodic waves in deep and intermediate depth, cnoidal waves in shallow water and solitary waves in extremely shallow water.