268 Lord Rayleigh on Waves. 



is given by 



- / h\ 



f=A-h!ta Rsin(a^+^J, 



r)=:fc + T{e R cos[a^+-pJ. 



It is not difficult to show that the motion represented by 

 these equations satisfies the condition of continuity, and is con- 

 sistent with the principles of fluid mechanics ; but it involves a 

 molecular rotation, whose amount is 



Ik 2k 



a«"»-f-l— e~*. 



This molecular rotation, being constant for each particle, is not 

 inconsistent with the properties of frictionless fluid when the 

 motion is once set up ; but it is known that a motion of this 

 kind could not be generated from rest in such fluid by any 

 natural force. We proceed to consider the theory of periodic 

 waves in deep water when there is no molecular rotation. 



As in the case of long waves, the problem may be reduced 

 to one of steady motion by attributing to the water a velocity 

 equal and opposite to that of the waves. If x be measured 

 horizontally and y downwards from the surface, the conditions 

 of continuity and of freedom from rotation are satisfied by 



<$>=.CX + Cte- Ky S1YIKX) \ ... 



yjr = cy — ue~ Ky cos kx ; J 



where <f> and ^ are the equipotential and stream functions, c 

 the velocity at a great depth, a a constant depending on the 

 amplitude of the waves, and /e = 27r-f-\, X being the wave- 

 length or distance from crest to crest. The motion represented 

 by (A) passes into a uniform horizontal flow at a great depth ; 

 and we have only to inquire how far the surface-condition of 

 constant pressure can be satisfied. 



If U be the resultant velocity at any point, 



U 2 = (Q\ \ f (^t V= c 2 + %cic*e-*y cos kx + «V*-*» 



= c 2 + 2c/c(cy -yjr) + /cVr 2 % 

 and therefore 



? - = const. + (g — kc 2 )ij + acty — J *•*« V 2 *y. . . (B) 

 Hence, when -ty is constant and a is so small that a 2 can be 



