Lord Rayleigh : Hydrodynamieal Notes. 179 



since the periodic time (r) is constant, as X. Conservation 

 of energy then requires that 



(height) 2 xV= constant; . . . . (1) 



or since V varies as fc\ height varies as Z~ ? *. 



But for a dispersive medium corresponding parts are not 

 proportional to V, and the argument requires modification. 

 A uniform regime being established, what we are to equate 

 at two separated places where the waves are of different 

 character is the rate of propagation of energy through these 

 places. It is a general proposition that in any kind of waves 

 the ratio of the energy propagated past a fixed point in unit 

 time to that resident in unit length is U, where U is the 

 group-velocity, equal to dcr/'dk, where cr = 27r/r, k = 2irj\^. 

 Hence in our problem we must take 



height varies as U~S (2) 



which includes the former result, since in a non-dispersive 

 medium U = V. 



E^or waves in water of depth /, 



(T 2 -=gh tanh kl, (3) 



whence 



2(7TJIg = t&nhkl + kl(l-tnnh 2 kl). ... (4) 



As the wave progresses, a remains constant, (3) determines 

 k in terms of I, and U follows from (4). If we write 



•%=''. (5) 



(3) becomes 



kl. tanh kl = V, (6) 



and (4) may be written 



2aV/g = kl + (l'-l /2 )/kl (7) 



By (6), (7) U is determined as a function of /' or by (5) 

 of l. 



If kl, and therefore V, is very great, kl = l\ and then by 

 (7) if U be the corresponding value of U, 



2*U /g=l, (S) 



and in general 



TJ/Uo=kl + (l / -l' 2 )/kl (9) 



* Loc. cit. p. 25o. 



t Proc. Lond. Math. Soc. ix. 1877; ' Scientific Papers/ i. p. 326. 



JS 2 



