392 Prof. H. Lamb on Waves due 



The continuity of pressure at the interface between the 

 two liquids requires 



I -y cosh kh + sinh kh 1 pA — (— r sinh kh + cosh kh \ pB 



=S- i )^ va • • (40) 



The condition that the vertical displacement at the inter- 

 face must be the same for the two liquids gives 



AsinhM-BcoshM=-(V-^ . . . (41) 



This system of equations has two solutions, which it is 

 convenient to distinguish by suffixes. In the first of these 

 we have 



°i*=gk (42) 



as in the case of a single fluid, with 



B^C^A (43) 



In the second solution we have 



a 2 = gk(p'-p) sinh k h 

 2 p' cosh kh 4- p sinh kh ' ^ ' 



with 



B2 = ^! C_ 2 _ pf^crl 



A 2 gk 9 A 2 p'-pgk' ' ' ' l J 



The wave-velocities corresponding to these two modes of 

 oscillation are given by 



Cl= T' c *=f (46) 



As the wave-length (2?r/£) increases from to co, 

 c 2 increases from to an upper limit c , given by 



c *=(l- £)<,/, (47) 



If U 1? U 2 be the corresponding group- velocities, we have 



Ci-Ui=ic l5 (48) 



9 



kh 



<-2- 



u = 1 C ) i p' 



2 2 2 \ p' cosh kh-\-p sinh kh ' sinh kh J 



As the wave-length increases from to go , U 2 increases 

 from \c 2 to c . 



