23 



but take it decidedly smaller in the case of the finite depth, such for example as to 

 bear the same proportion to the greatest possible value in the two cases. 



But with all due allowance to this consideration, it must be confessed that the 

 approximation is slower in the case of a finite depth. That it must be so is seen 

 by considering the character of the developments, in the two cases, of the ordinate 

 of the profile in a harmonic series in terms of the abscissa, or of a quantity having 

 the same period and the same mean value as the abscissa. The flowing outline 

 of the profile in deep water lends itself readily to expansion in such a 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. In extreme cases the fact of the waves 

 being in series at all has little to do with the character of the motion in the neigh- 

 bourhood of the elevations, where alone the motion is considerable, and it is not 

 therefore to be wondered at if an analysis essentially involving the length of 

 a wave should prove inconvenient. 



The problem of the greatest possible height attainable by any given 

 oscillatory wave of permanent form has been considered by a number 

 of investigators. Stokes analyzed the problem on the basis of a 

 velocity superimposed on the water which would bring the wave to 

 rest. The flow being assumed as frictionless, Bernoulli's equation 

 applies and the greatest possible height is attained when the velocity 

 at the crest is zero. Writing a potential function for flow in the 

 immediate vicinity of the crest, we have: 



tf>=Ar'* sin nd 



The profile is symmetrical about the crest and only sine terms are 

 admissible. Since no flow occiu-s perpendicular to the surface, then: 



^„ ■0=—Ar^-ncosnd 

 od 



and 



^=nA sin n^r"'~'^=V=^J2gr cos 6 



Comparing the exponents of r 



therefore 



n— 1 = 2 and '^=2 



Q 



cos nd=0 and 7^9=^ 



^-3 



The complete angle is then 2 ir/3 or 120°. Concerning this result, 

 Stokes (2) remarks: 



This, however, leaves untouched the question whether the disturbances can 

 actually be pushed to the extent of yielding crests with sharp edges . . . After 

 careful consideration, I feel satisfied that there is no such earlier limit, but that 



