382 SURFACE WAVES. [CHAP. IX 



and therefore, for the group-velocity, 



d(kc)_ 

 dk ~* 



The ratio which this bears to the wave-velocity c increases as kh 

 diminishes, being \ when the depth is very great, and unity when 

 it is very small, compared with the wave-length. 



The above explanation seems to have been first suggested by 

 Stokes*. The question was attacked from another point of view 

 by Prof. Osborne Reynolds f, by a calculation of the energy propa- 

 gated across a vertical plane of particles. In the case of infinite 

 depth, the velocity-potential corresponding to a simple-harmonic 

 train of waves 



v) = a sin k (x ct) ........................ (5), 



is <f> = ace ky cos k(x ct) ..................... (6), 



as may be verified by the consideration that for y = we must 

 have drj/dt = dfyjdy. The variable part of the pressure is pd$/dt, 

 if we neglect terms of the second order, so that the rate at which 

 work is being done on the fluid to the right of the plane x is 



- I p dy = ptffcc* sin 2 k (x - ct) I 



J x ^* 7oo 



(x ct) ..... . ...... (7), 



since c 2 = g/k. The mean value of this expression is %gpa?c. It 

 appears on reference to Art. 219 that this is exactly one-half of 

 the energy of the waves which cross the plane in question per 

 unit time. Hence in the case of an isolated group the supply of 

 energy is sufficient only if the group advance with half the 

 velocity of the individual waves. 



It is readily proved in the same manner that in the case 



* Smith's Prize Examination, 1876. See also Lord Eayleigh, Theory of Sound, 

 Art. 191. 



t " On the Kate of Progression of Groups of Waves, and the Kate at which 

 Energy is Transmitted by Waves," Nature, t. xvi., p. 343 (1877). Professor 

 Reynolds has also constructed a model which exhibits in a very striking manner 

 the distinction between wave-velocity and group-velocity in the case of the 

 transverse oscillations of a row of equal pendulums whose bobs are connected 

 by a string, 



