270 Lord Rayleigh on Wave* 



immaterial. For waves in water of constant finite depth /, the 

 expression for ^ is 



<$ = C y — a. cos kx \e~ ■&-* — e K ^~ l) \ f 

 and the velocity c is determined by 



k e Kl + e~ 



(G) 



which passes into (C) when I is considerable in comparison 

 with X. When I is small, we get from (G) 



'-!■¥-* 



which is the formula proper for long waves. When obtained 

 thus, it is applicable in the first instance only to waves of a 

 particular type ; but the fact that it is independent of k or \ 

 would lead us to the conclusion that the same formula would 

 apply to a long wave of any type. 



In one respect the theory of irrotational waves may be con- 

 sidered inferior to that of Rankine, which last is exact, in the 

 sense that it is independent of any supposition as to the small- 

 ness of the waves. So far as I am aware, writers on this sub- 

 ject appear to think that it is only a question of mathematics 

 to determine the form of irrotational waves of finite amplitude 

 to any degree of approximation. But it seems to me by no 

 means certain that any such type exists, capable of propagating 

 itself unchanged with uniform velocity. I see no reason why 

 the possibility of such waves in deep water should be taken for 

 granted, when we know that in shallow water waves of finite 

 height cannot be propagated without undergoing a gradual 

 alteration of type. 



One of the most interesting results of Professor Stokes's 

 theory is the existence of a slow translation of the water near 

 the surface in the direction of the wave. I propose to show 

 that this superficial motion is an immediate consequence of the 

 absence of molecular rotation, and that it is independent of the 

 condition of constant pressure at the bounding surface. 



Let A B be the surface from crest to hollow, and CDa 

 neighbouring stream-line. Draw A! B / ', C D f , two stream- 

 lines at such a depth that the steady motion of the fluid is uni- 

 form, and so as to include a total stream equal to that which 

 flows between A B and C D. Then we have to show that a 

 particle at A will take longer to reach B, than a particle at A' 



