410 SURFACE WAVES. [CHAP. IX 



a?/\ 3 , the solution of the problem in the case of infinite depth 

 is contained in the formulae 



sn 



The equation of the wave-profile ijr = is found by successive 

 approximations to be 



y = fi^ cos kx = ft (1 + ky + py + . . . ) cos kx 

 = p/3 2 + ft (1 + f & 2 /3 2 ) cos kx + %kj3 2 cos 2kx + f & 2 /3 3 cos 3A;# + ... 



............ (2); 



or, if we put 0(1+ & 2 /3 2 ) = a, 



y %ka? = a cos kx + ^ka? cos 2&# + |& 2 a 3 cos 3^ 4 ...... (3). 



So far as we have developed it, this coincides with the equation of 

 a trochoid, the circumference of the rolling circle being Zir/k, or X, 

 and the length of the arm of the tracing point being a. 



We have still to shew that the condition of uniform pressure 

 along this stream-line can be satisfied by a suitably chosen value 

 of c. We have, from (1), without approximation 



= const. - gy - \<? {1 - 2k/3e& cos kx + k^e^} ..... (4), 

 and therefore, at points of the line y = fie ky cos kx, 

 = const. + (kc 2 -g)y- -p 2 c 2 /3V^ 



= const. + (ktf-g-k 3 c*l3' 2 )y+ ............... (5). 



Hence the condition for a free surface is satisfied, to the present 

 order of approximation, provided 



(6). 



This determines the velocity of progressive waves of per- 

 manent type, and shews that it increases somewhat with the 

 amplitude a. 



For methods of proceeding to a higher approximation, and for 

 the treatment of the case of finite depth, we must refer to the 

 original investigations of Stokes. 



* Lord Eayleigh, 1. c. ante p. 279. 



