386 SURFACE WAVES. [CHAP. IX 



condition of irrotational motion, V 2 \|r = ; and they give a uniform 

 velocity c at a great distance above and below the common surface, 

 at which we have ty = ifr = 0, say, and therefore y /3 cos kx, ap- 

 proximately. 



The pressure-equations are 



v c 2 



*- = const. gy -= (1 Zkfie^ cos kx), 



P 



f)' C 2 



, = const. gy -= (1 + VkfftT** cos kx), 



which give, at the common surface, 



pfp = const. -(g-kc*)y y 

 p'lp = const. -(g + kc*) y, 



the usual approximations being made. The condition p=p' thus 

 leads to 



a result first obtained by Stokes. 



The presence of the upper fluid has therefore the effect of 

 diminishing the velocity of propagation of waves of any given 

 length in the ratio {(1 - )/(! -f s)}i, where s is the ratio of the 

 density of the upper to that of the lower fluid. This diminution 

 has a two-fold cause ; the potential energy of a given deformation 

 of the common surface is diminished, whilst the inertia is in- 

 creased. As a numerical example, in the case of water over 

 mercury (s~ 1 = 13'6) the above ratio comes out equal to '929. 



It is to be noticed, in this problem, that there is a disconti- 

 nuity of motion at the common surface. The normal velocity 

 (d^rjdx) is of course continuous, but the tangential velocity 

 (- d^jr/dy) changes from c(lkft cos kx) to c (1 + kj3 cos kx) as we 

 cross the surface ; in other words we have (Art. 149) a vortex-sheet 

 of strength - kcft cos kx. This is an extreme illustration of the 

 remark, made in Art. 18, that the free oscillations of a liquid of 

 variable density are not necessarily irrotational. 



If p < p, the value of c is imaginary. The undisturbed 

 equilibrium -arrangement is then of course unstable. 



