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

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