410 SURFACE WAVES. [CHAP. IX 



a?l\*, the solution of the problem in the case of infinite depth 

 is contained in the formulae 



. . # 

 



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

 approximations to be 



y = fidw cos ka; = P(I+ky + |-% 2 + . . .) cos kx 

 = i&2 + (1 + f & 2 /3 2 ) cos kx + |&/3 2 cos Zkx + -f & 2 3 cos 3kx + ... 



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



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



y ^ka 2 = a cos kx + ^ka? cos 2kx + f & 2 a 3 cos 3Aj# + ...... (3). 



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

 a trochoid, the circumference of the rolling circle being 2ir/&, 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 - Jc 2 { 1 - 2kp& cos Tex + k^e 2k y} ..... (4), 

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

 = const. + (kc* -g)y- 



-const. 



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

 order of approximation, provided 



c^ + ^c^^a+^a 2 ) .................. (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 Kayleigh, I c. ante p. 279. 



