390 SURFACE WAVES. [CHAP. IX 



These are both real, provided 



&amp;lt;^.Co (10), 



and they have, moreover, opposite signs, if 



c (ID- 



In this latter case waves of the prescribed length (2-7T/&) may 

 travel with or against the wind, but the velocity is greater with 

 the wind than against it. If u lie between the limits (10) and (11), 

 waves of the given length cannot travel against the wind. Finally 

 when u exceeds the limit (10), the values of c are imaginary. This 

 indicates that the plane form of the common surface is now un 

 stable. Any disturbance whose wave-length is less than 



(12) 



tends to increase indefinitely. 



Hence, if there were no modifying circumstances, the slightest 

 breath of wind would suffice to ruffle the surface of water. We 

 shall give, later, a more complete investigation of the present 

 problem, taking account of capillary forces, which act in the 

 direction of stability. 



It appears from (6) that if p = p , or if g = 0, the plane form of 

 the surface is unstable for all wave-lengths. 



These results illustrate the statement, as to the instability of 

 surfaces of discontinuity in a liquid, made in Art. 80*. 



When the currents are confined by fixed horizontal planes y= -h, y = h , 

 we assume 



I S _inlO^+A) I .................. 



|f smh kh j 



sn 7 



(n). 



The condition for stationary waves on the common surface is then found 

 to be 



(iii)f. 



* This instability was first remarked by von Helmholtz, I.e. ante, p. 24. 

 t Greenhill, Lc. ante, p. 388. 



