246-247] WAVES ON THE BOUNDARY OF TWO CURRENTS. 449 



If we apply the method of Art. 224, we find without difficulty 

 that the condition for stationary waves is now 



the last term being due to the altered form of the pressure 

 condition which has to be satisfied at the surface. Putting 



U c, U = - c + u, p /p = s, 

 we get 



where 



i.e. c is the velocity of waves of the given length (Zir/k) when 

 there are no currents. 



The various inferences to be drawn from (2) are much as in 

 Art. 224, with the important qualification that, since c has now a 

 minimum value, viz. the c m of Art. 246 (5), the equilibrium of 

 the surface when plane is stable for disturbances of all wave 

 lengths so long as 



u&amp;lt; L J f .c 1 ........................ (4). 



When the relative velocity u of the two currents exceeds this 

 value, c becomes imaginary for wave-lengths lying between certain 

 limits. It is evident that in the alternative method of Art. 225 

 the time-factor e i&amp;lt;Tt will now take the form e*=*t+W t where 



& &quot;\ T~: rr U CA f K 3 /3 z /t U ( O ). 



J/llo\2 1lr \/ 



The real part of the exponential indicates the possibility of a 

 disturbance of continually increasing amplitude. 



For the case of air over water we have s = -00129, c m = 23 2 (c.s.), whence 

 the maximum value of u consistent with stability is about 646 centimetres 

 per second, or (roughly) 12*5 sea-miles per hour*. For slightly greater values 

 of u the instability will manifest itself by the formation, in the first in- 



* The wind-velocity at which the surface of water actually begins to be ruffled 

 by the formation of capillary waves, so as to lose the power of distinct reflection, is 

 much less than this, and is determined by other causes. We shall revert to this 

 point later (Art. 302). 



L. 29 



