24 



we may actually approach as near as we please to the form in which the curva- 

 ture at the vertex becomes infinite, and the vertex becomes a multiple point where 

 the two branches, with which alone we are concerned, enclose an angle of 120°. 



Michell (11) found theoretically that the limiting height from trough 

 to crest is 0.14i, Havelock (12) obtained as the maximum value 

 0.1418i. Quoting Stokes further on the subject: 



There can be little doubt that in both cases (deep water and shallow water) 

 alike the series cease to be convergent when the limiting wave, presenting an 

 edge of 120°, is reached. 



In fact, high oscillatory waves in shallow water tend to assume the character 

 of a series of disconnected solitary waves, and the greatest possible height depends 

 mainly on the depth of the fluid, being but little influenced by the length of the 

 waves, that is, the distance from crest to crest. 



The flowing outline of the profile in deep water lends itself readily to expansion 

 in series. But the approximately isolated and widely separated elevations that 

 represent the profile in very shallow water would require a comparatively large 

 number of terms in their expression in harmonic series in order that the form should 

 be represented with sufficient accuracy. 



Rayleigh (13) compared the potential and kinetic energies of waves 

 which are not infinitely small and found that the kinetic energy exceeds 

 the potential but the error is small. 



These equations for a stable wave form were obtained for conditions 

 of constant atmospheric pressure on the free surface. Waves moving 

 under the influence of strong winds may well exhibit different forms; 

 but laboratory experiments on water waves under still air confirm the 

 theoretical equations. 



B. Trochoidal theory. — Classical theory concludes that the surface 

 profile of a wave of finite height is approximately trochoidal, and 

 measurements of natural and model waves conform approximately to 

 this type of curve in which the crests become sharper and the troughs 

 flattened as the height increases in proportion to the length. Experi- 

 ment has not yet conclusively shown whether or not surface water 

 waves are exactly trochoidal but the deviations therefrom are small. 

 This fact was made the basis for the trochoidal theory of deep-water 

 waves first proposed by Franz Gerstner (14) and developed independ- 

 ently much later by Rankine (15). The outstanding features of this 

 theory are that: the flow is rotational and therefore cannot be gener^ 

 ated from rest by conservative forces, and the wave movement is not 

 accompanied by a mass transport in the direction of wave travel. 



Stokes' (2) comments on this theory are of particular interest and 

 will be quoted in some detail: 



This case of motion has latterly attracted a good deal of attention, partly no 

 doubt from the facility of dealing with it, but partly it would seem, from miscon- 

 ceptions as to its intrinsic importance. 



Expressing the position of a particle at time t in terms of its initial 



