Lord Rayleigh on Waves. 269 



neglected, p will also be constant, provided that 



'=v/f=\/i CO) 



If c has this value, the surface-condition is satisfied approxi- 

 mately, and (A) may be understood to represent a train of free 

 periodic stationary waves, or, if the motion relatively to deep 

 water be considered, a train of periodic waves advancing with- 

 out change of type and with a uniform velocity c. 



The profile of the wave is determined by the second of equa- 

 tions (A), in which ty is made constant. By successive ap- 

 proximation we may deduce the value of y in terms of x. If 

 yjr be taken so that the mean value of y is zero, we get 



a A , 5*V\ k* 2 . , 3*V q 



y= - I 1 + ^ — 2~ ) cos KX ~ 9~2 cos * KX + q — r cos v*®) 



which is correct as far as a 3 . Let 



then 



<f> = c# + ca ( 1 — o /c 2 a 2 J e~*tf sin *#, 



•^=cy—ca(l— q /e 2 a 2 J e"~*^cos #; 



(D) 



and for the equation of the surface, 



y — a cos kx 5- cos 2 jm? + o /c 2 a 3 cos 3/e#. . . (E) 



From (B) we may obtain a closer approximation to the 

 value of c. Expanding the exponential, we have (approximately) 



so that 



- = const. + (g—tcc 2 + K 3 a?)y + . . . ; 

 P 



c 2 = 9 -+H?u 2 = g - +K 2 a\\ 



K K 7 



or 



C 2 =«(1+«V), . (F) 



where /c=27r-t-\. 



Formulae (E) and (F) are given by Professor Stokes in a 

 memoir published in the Cambridge Philosophical Transac- 

 tions, vol. viii. 



So long as the depth is everywhere sufficiently great in com- 

 parison with the length of the waves, uniformity of depth is 



