S02 Mr Priestley, On'the Oscillations of Superposed Fluids. 



Equation (1) gives us the transcendental equation of the free 

 surfoce in the form 



X = -%- A^ sin ^-^1 - A. sin 2Jc% - A., sin SA-^i, 

 y — A,, + ^1 cos A-'^i 4- A ., cos 2A-^i + A^ cos SA-^i. 

 From these we find 



^j = _ X + ^1 sin hv + (^o - ^Auli^) sin Ihx, 

 and, using values of coefficients already found, 

 y = [A-, + 1^-/33 (4/)i TJ;^K-^ - 1)] cos hx 



+ \h^- (pi U;' - p, U^ K-' cos 2kx 



+ ^Jc'/S'il - 6p, C/i" . p.,U.^ . Z-") cos Skx. 



Writing b for the amplitude of the principal harmonic term 

 we have, to order Jc"^'^, 



y=h cos kx + ^Jcb- (pi f/j- — p.. U.f) K~^ cos 2k-x 



+ |M)S [1 - 6pi U^' . p., CTo^ . K-^] cos 3Aw (6). 



Both this equation of the wave profile and the period equation 

 reduce to Stokes' equations for waves on a free surface on putting 

 P. = 0. 



In discussing these results we shall denote the wave velocity 

 by c and the stream velocity of the upper fluid by u, so that 



U^ = — c; U.2= u — c. 



Writing the period equation in the form 



gk-' {p, -p,)-K = - Jc^^-^ [K - 2p, ^r . p. U^ • K-'l 



and remembering that K may be put equal to gk~'^ (pi — p.j) in the 

 small terms, we notice that, corresponding to a given wave length, 

 the small terms have their greatest importance when U2 = or 

 c = w and their least importance when piC/i'' = poC/o'^. 

 The first case gives 



c- = gk~^ (1 — s) [1 + k^fi'] , where s = ps/pi. 



Turning to the equation of the wave profile we see that the 

 small terms have their greatest importance for this value of c. 



The equation takes the form 



y = h cos kiV + h kb- cos 2kx + ^k^b^ cos Skx, 

 which is the form Stokes found for waves on a free surface. The 

 expression for the velocity only differs from Stokes' free surface 

 velocity by the factor (1 — s). 



The second case gives 



2pxC^=/C 

 whence c" = ^ gk-' {1 - s) (I + ^k''^) (7), 



and u=±c(l±s^)ls^ (8). 



