1908-9.] On Group- Velocity and Propagation of Waves. 
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XXIX. — On Group- Velocity and on the Propagation of Waves in a 
Dispersive Medium. By George Green, M.A., B.Sc., Assistant to 
the Professor of Natural Philosophy in the University of Glasgow. 
Communicated by Professor A. Gray, F.R.S. 
(MS. received March 16, 1909. Read June 7, 1909.) 
§ 1. The theory of group- velocity has been developed by Stokes, Osborne 
Reynolds, Lord Rayleigh, and later by Professor Lamb. The application of 
the theory to light- waves was made by Gouy and Lord Rayleigh, and its 
importance in this connection was emphasised by Professor Schuster in his 
paper “ On Interference Phenomena,” Phil. May., vol. xxxvii., 1894. 
The question of the velocity of a group of waves, as distinct from the 
velocity of a wave, arose from the well-known observation that when a 
large group of waves advances into smooth water, each separate crest 
travels through the group towards the front, where it gradually dies down 
and disappears, while similar crests meantime have in turn taken its place 
in the group, so that the main group moves forward at a velocity less than 
that of the separate waves. 
§ 2. It seems that Stokes was the first to prove that the phenomenon 
was capable of being dealt with analytically, by showing that when 
two infinite trains of waves, of equal amplitudes and nearly equal wave- 
lengths are superposed, we obtain an infinite succession of wave-groups, 
each of which advances with the half -wave- velocity. None of the groups 
maintains its outline constant for any interval of time, however short ; but 
the whole disturbance periodically returns to the configuration it had at 
any particular instant if the periods of the superposed trains be com- 
mensurable, and the effect is the same as if each group had moved forward 
in the interval, without change of shape, at the half-wave-velocity. 
A most important contribution to the theory of group-velocity was 
made by Osborne Reynolds in Nature, Aug. 23, 1877, where he gave 
a dynamical explanation of the fact that the regular part of a large group 
of waves of equal wave-lengths advances with only half the velocity of the 
separate waves. He proved that the energy propagated across a plane, 
when a regular train of waves is passing, is just sufficient to feed a regular 
procession of waves, travelling with half the corresponding wave- velocity. 
§ 3. Lord Rayleigh pointed out in his article “ On Progressive Waves ” 
( Theory of Sound, vol. i., Appendix) that the theorem given by Osborne 
