223] OSCILLATIONS OF SUPERPOSED FLUIDS. 387 



The case where the two fluids are bounded by rigid horizontal planes 

 y=h, y = h , is almost equally simple. We have, in place of (1), 



sinh k (y + A) 



^--y-g^&quot;^.**** 

 smh kK 



leading to c 2 = ?. ., .f ^ ,. 77&amp;gt;t . ...(ii). 



p coth M -Hp coth kh 



When M and M are both very great, this reduces to the form (2). When kh 

 is large, and kh small, we have 



(iii), 



the main effect of the presence of the upper fluid being now the change in 

 the potential energy of a given deformation. 



When the upper surface of the upper fluid is free, we may assume 



A Binh(v+A) 7 \ 



bc= -y+p -- . .y.T cos&g, I 



(iv), 

 !// /&amp;lt;? = y + (/3 cosh ky + y sinh &y) cos &*? / 



and the conditions that ^ = \^ , ^&amp;gt;=j0 at the free surface then lead to the 

 equation 



c 4 (p coth kh coth M + p ) - c 2 P (coth kh + coth M) | + (p - p ) ^ = ...... (v). 



Since this is a quadratic in c 2 , there are two possible systems of waves of any 

 given length (2?r/^). This is as we should expect, since when the wave-length 

 is prescribed the system has virtually two degrees of freedom, so that there 

 are two independent modes of oscillation about the state of equilibrium. For 

 example, in the extreme case where p /p is small, one mode consists mainly in 

 an oscillation of the upper fluid which is almost the same as if the lower fluid 

 were solidified, whilst the other mode may be described as an oscillation of the 

 lower fluid which is almost the same as if its upper surface were free. 



The ratio of the amplitudes at the upper and lower surfaces is found 

 to be 



kc* 

 kc* cosh kh -g sinh kh 



Of the various special cases that may be considered, the most interesting 

 is that in which kh is large ; i. e. the depth of the lower fluid is great compared 

 with the wave-length. Putting coth kh = 1, we see that one root of (v) is now 



c 2 =#/ ..................................... (vii), 



exactly as in the case of a single fluid of infinite depth, and that the ratio of 

 the amplitudes is e* A . This is merely a particular case of the general result 

 stated at the end of Art. 222 ; it will in fact be found on examination that 



252 



