224-225] SURFACES OF DISCONTINUITY. . 391 



It appears on examination that the undisturbed motion is stable or 

 unstable, according as 



^ p coth kk+p' coth kh' 



u < " 3-*- T.CO (iv), 



> (pp f coth kh coth M')* 



where u is the velocity of the upper current relative to the lower, and c is 

 the wave- velocity when there are no currents (-Art. 223 (ii)). When h and h' 

 both exceed half the wave-length, this reduces practically to the former 

 result (10). 



225. These questions of stability are so important that it is 

 worth while to give the more direct method of treatment*. 



If </> be the velocity-potential of a slightly disturbed stream 

 flowing with the general velocity U parallel to x t we may write 



= -Cfo + <fc ........................... (1), 



where fa is small. The pressure-formula is, to the first order, 



and the condition to be satisfied at a bounding surface y = 77, where 

 rj is small, is 



...................... . 



dt dx dy 



To apply this to the problem stated at the beginning of Art. 

 224, we assume, for the lower fluid, 



^ = Cfe*iH-*ta+rf) ........................ (4); 



for the upper fluid 



</ = O'tf-fclH-itte+irf) ..................... (5); 



with, as the equation of the common surface, 



The continuity of the pressure at this surface requires, by (2), 

 p{i(<r + kU)C-ga}=p'{i(<r + kU')C'-ga} ...... (7); 



whilst the surface-condition (3) gives 



a = - kC, 



* Sir W. Thomson, " Hydrokinetic Solutions and Observations," Phil. Mag., 

 Nov. 1871; Lord Bayleigh, "On the Instability of Jets," Proc. Lond. Math. Soc., 

 t. x., p. 4 (1878). 



