392 SURFACE WAVES. [CHAP. IX 



Eliminating a, C, C', we get 



p(<T + kU)* + p'(<T + kUy = gk(p-p') ............ (9). 



It is obvious that whatever the values of U y U', 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<T* + p'(<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 



<r = ikU ........................... (11). 



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



7j = ae kut coska} ........................ (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 Eayleigh, I.e. 



