2(3 1) Lord Rayleigh on Waves. 



we substitute these values of y and eliminate C, we find 



<■=& (H) 



as the relation between u and V . The constants C and u Q 

 being now determined so as to make y' vanish when y = and 

 when y—V, it will be found that the third root of the cubic is 

 also Z, so that our equation may be put into the form 



y' 2 +iG/-0 2 G/-O=0 (i) 



From this result it appears that there is only one maximum 

 or minimum value of y (besides I) ; and since y — V is neces- 

 sarily negative, it follows that the surface-condition cannot be 

 satisfied to this order of approximation by a solitary wave of 

 depression. Differentiating (I), we get 



which shows that the points of zero curvature occur when y = I 



21' + 1 2 



and when y= — x— =1+ r (V — I). Thus the curvature changes 



sign at two thirds of the height f the wave above the undis- 

 turbed level, and at these points only. The nature of the wave 

 is sufficiently defined by (I) ; but we may readily integrate 

 again, so as to obtain the relation between x and y. Thus, if 

 l'-l=j3,y-l= V ^ 



the constant being taken so that #=0 when rj=/3. This equa- 

 tion gives the height rj at any point x in terms of one constant, 

 viz. the maximum height of the wave. There is therefore (in 

 a given canal) only one form of solitary wave of given maxi- 

 mum height. On either side the height diminishes without 

 interruption, but does not (according to (J)) absolutely vanish 

 at any finite distance. Accordingly there is no definite wave- 

 length ; but if we inquire what value of x corresponds to a 

 given ratio of t) : 0, we get 



- -WW- 



being greatest for the smallest waves. 



Suppose, for example, that we regard the wave as ending 

 where the height is one tenth of the maximum. Then 



.i-*V*+!r 



