776 On Water-Waves as Asymmetric Oscillations. 



3. Since the free wave of finite amplitude is appreciably 

 asymmetric, it would seem possible from the foregoing that 

 the free train might tend to magnify other relatively stationary 

 trains of nearly the same wave-lengths and relatively small 

 amplitudes. This question is a partial test of the stability of 

 a free train of waves. To determine the free wave of finite 

 amplitude *, putting 



7] = a cos kx + ft cos 2l\v, 

 where a and j3 are functions of y, we find 



j Pace' 2 -Waft - 6/t V/9 - - c'a! = 0, 

 4 c 



I I 7 9 , X 7 9 9 1 1 



a — k ct -\- — Ir a. a. 



/3"-4£ 2 /3=0, 



c 

 subject to the boundary conditions a = ft — when y— — co , and 



9 



,a-2ct' + 3* / /3' -3a /z + 2Paj3--} 2 lr* 2 *' = y 

 -/3-^ + |a' 2 -i^ 2 = 0, 



when 2/ — 0. 



Hence a = ae k 'J + % a z k 2 e^, 



/3 = } 2 a 2 ke 2k *, 



The equation for the disturbance, f, is 

 ,4 + ^ + W <| cos te (^ _ , _ij + sm /,, (__ + , _i) j 



+ sm2/,r( 3 ^-+4i-^)"|=0. 



* The method is evidently applicable when the depth is finite. In the 

 case of long waves the process is necessarily different. We have then 



*=*.+* 5.1 + 



<°^\dy)j2\W)o 3!UWo 



when small quantities of order higher than the second are neglected. 

 Hence, putting y=— h> and substituting for the y derivatives from (1) 

 and the surface condition we obtain 



£(l)+M(SMl^GrrH- 



the well-known equation for the contours. 



