392 SURFACE WAVES. [CHAP. IX 



Eliminating a, G, C&quot;, we get 



p(&amp;lt;r + kU)* + p (o- + kUy = gk(p-p ) ............ (9). 



It is obvious that whatever the values of U, V, other than 

 zero, the values of a will become imaginary when k is sufficiently 

 great. 



Nothing essential is altered in the problem if we impress on 

 both fluids an arbitrary velocity in the direction of x. Hence, 

 putting U = 0, U = u, we get 



p&amp;lt;T* + P (&amp;lt;r + ku)* = gk(p-p ) ............... (10), 



which is equivalent to Art. 224 (7). 



If p = p, it is evident from (9) that a will be imaginary for all 

 values of k. Putting U = U } we get 



o-=ikU ........................... (11). 



Hence, taking the real part of (6), we find 



&amp;lt; n = ae kut coskx ........................ (12). 



The upper sign gives a system of standing waves whose height 

 continually increases with the time, the rate of increase being 

 greater, the shorter the wave-length. 



The case of p~p, with U=U , is of some interest, as illustrating the 

 flapping of sails and flags*. We may conveniently simplify the question by 

 putting U=U =0 , any common velocity may be superposed afterwards if 

 desired. 



On the present suppositions, the equation (9) reduces to o- 2 =0. On 

 account of the double root the solution has to be completed by the method 

 explained in books on Differential Equations. In this way we obtain the two 

 independent solutions 



and 



The former solution represents a state of equilibrium ; the latter gives a 

 system of stationary waves with amplitude increasing proportionally to the 

 time. 



In this problem there is no physical surface of separation to begin with ; 

 but if a slight discontinuity of motion be artificially produced, e.g. by impulses 



* Lord Kayleigh, I.e. 



