Title: On the Limiting Stokes' Wave of Extreme Height in Arbitrary Water Depth,
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 Padé 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.