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 -fy = ^r = 0, say, and therefore y = /3 cos kx, ap 

 proximately. 



The pressure-equations are 



* = const. gy -~ (1 2&/3e^ cos kx), 



$ c 2 



, = const. gy -= (1 + 2kj3e~ ky cos lex), 



which give, at the common surface, 



p/p = const. -(g-kc*)y, 

 p lp = const. -(g + kc 2 ) 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 s)/(l + 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 

 ( dty/dy) changes from c(lk/3 cos Tex) to c(l + k/3 cos kx) as we 

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

 of strength kc/3 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 &amp;lt; /&amp;gt; , the value of c is imaginary. The undisturbed 

 equilibrium -arrangement is then of course unstable. 



