223-224] EFFECT OF WIND ON WAVE- VELOCITY. 389 



the origin being at the mean level of the common surface, which 

 is assumed to be stationary, and to have the form 



y = ft cos kx ........................... (3). 



The pressure-equations give 



= const. - gy - J U 2 (1 - 2A?e** cos lex), 



p , 



= const. -gy-\ W* (1 + Zkpe-* 1 * cos kx) 

 P 



whence, at the common surface, 



= const. 

 P 



= const. - (k U' 2 +g)y 

 P 



Since we must have p=p' over this surface, we get 



pU' + pW-fa-p 1 ) ..................... (6). 



This is the condition for stationary waves on the common 

 surface of the two currents U, V. 



If we put U' = U, we fall back on the case of Art. 223. Again 

 if we put 



we get the case where the upper fluid has a velocity u relative to 

 the lower ; c then denotes the velocity (relative to the lower fluid) 

 of waves on the common surface. An interesting application of 

 this is to the effect of wind on the velocity of water-waves. 



The equation (6) now takes the form 



or, if we write s for p'/p, and put c for the wave-velocity in the 

 absence of wind, as given by Art. 223 (2), 



2su su 2 



TTt'^-r 



The roots of this quadratic in c are 



su ( su 2 



c ~~ 



