Mr. stokes, on THE THEORY OF OSCILLATORY WAVES. 451 



14. Let us consider now the case of waves propagated at the common surface of two liquids, 

 of which one rests on the other. Suppose as before that the motion is in two dimensions, that the 

 fluids extend indefinitely in all horizontal directions, or else that they are bounded by two vertical 

 planes parallel to the direction of propagation of the waves, that the waves are propagated with 

 a constant velocity, and without change of form, and that they are such as can be propagated into, 

 or excited in the fluids supposed to have been previously at rest. Suppose first that the fluids 

 are bounded by two horizontal rigid planes. Then taking the common surface of the fluids when 

 at rest for the plane xz, and employing the same notation as before, we have for the under fluid 



^'■d?^'' • <^«)' 



dd) 



~ = when y= h, (29), 



p = C + gpy + cp-L , 

 ax 



neglecting the squares of small quantities. Let h^ be the depth of the upper fluid when in equi- 

 librium, and let p^, p^, (p^, C^ be the quantities referring to the upper fluid which correspond to 

 p, p, (p, C referring to the under : then we have for the upper fluid 



d'(p (P(b 



-TT+ r^ = ^ (30), 



dd) 



—— = when y = - h (31), 



dy 



P,= C, + gp,y + cp^ -J-^ . 



We have also, for the condition that the two fluids shall not penetrate intoy nor separate from each 



other, 



dd) dd) 



~ = -~, when » = (32). 



dy dy " 



Lastly, the condition answering to (u) is 



-('T:-4:')--(/r?-.3)=» <"'• 



when C- C\ + g {p ~ p) y + c (p -f- - p,-p] =0 (34). 



V dx ' ' dx I 



Since C - C is evidently a small quantity of the first order at least, the condition is tliat (3,3) 

 shall be satisfied when y = 0. Equation (34) will then give the ordinate of the common surface of 

 the two liquids when y is put = in the last two terms. 



The general value of (b suitable to the present case, which is derived from (28) subject to the 

 condition (2f>), is given by (13) if we suppose that the fluid is free from a uniform liorizontal motion 

 compounded with the oscillatory motion expressed l)y (13). Since the equations of the present 

 investigation are linear, in consequence of the omission of the squares of small quantities, it will be 

 sufficient to consider one of the terms in (13)- I-et then 



0= ^(e^i'-J' + f-'-i^-y) sin m.v (S.'i). 



