to a Travelling Disturbance. 393 



From this point we may without confusion use the symbols 

 r), t) to denote the elevations at the upper surface and at the 

 interface, respectively. Hence, putting y = in (36), and 

 y= —li in (37), and combining the solutions, we have 



r) = (o"iA x sin crxt + cr 2 A 2 sin a- 2 t) cos kx, .... (50) 



7)' = — - ( o-^-^Ai sin a x t f^- cr 2 e kh A 2 sin a 2 t J cos kx. (51) 



Use has here been made of the relations (39), (43), (45). 

 These formulae correspond to the following initial distribution 

 of </>, cf>', viz. 



cj>= | A 1 e^ + A 2 (cos\ikt/+^-smhky\ 1 cos foe, (52) 



a/ = (Ai-^-^LAji^cos^ (53) 



I p-pgk J 



The coefficients A l5 A 2 are to be regarded as functions 

 of k } depending on the particular manner in which the 

 oscillation (of prescribed wave-length) is started. When it 

 is necessary to call attention to this, they may be written as 

 &i{k), A s (i). 



The wave-trains due to a travelling disturbance are 

 accordingly given, on the analogy of (21), by 



V= - U .^! TJ ) Aifa) sin k y x- /'rjj ^M sin« s .r, (54) 

 v ' = - aie ' A x fa) sin g 3 a ^ °" 2fc A 2 (/c 2 )sin/g 2 ^, (55) 



#\c— Uxj p—pg\ c —u 2 ) 



where 



_ 



1 C 



(56) 

 (57) 



and a: 2 is the positive root of the equation 

 c >2 = g jp' - p) s'mh kh 



k ' p' cosh kh + p sinh kh ' 



It is also understood that in the preceding formulae 



a 1 = k 1 c, cr 2 = K 2 c. . . . . . (08) 



If c> c , k 2 is imaginary, and the second terms in (54) 

 and (55) disappear. 



The periods ^irja^ 27r/<r 2 being different, the mean energies, 

 and the mean rates of transmission of energy, in the two 

 systems of waves may be calculated independently. 



