451 
1908-9.] On Group-Velocity and Propagation of Waves. 
coincident-phase-velocity or the stationary-phase-velocity for that wave- 
length. 
§ 11. At present we have established an equation relating to the Fourier 
trains effective at each point of the medium, but nothing definite regarding 
the resultant wave-form of £. Indeed, it is evident from the above discussion 
that the resultant wave-form may be something differing considerably in 
each part from the constituent wave-trains which predominate at different 
points along it. It is to the constituent Fourier trains that the idea of 
group-velocity primarily applies, and we cannot speak of a group-velocity 
with reference to an endless succession of regular periodic waves ; that is, to 
a single train. We may remark that, so far as the group-velocity relates to 
the resultant wave-system it refers essentially to each single point of the 
system and not to an extended succession of waves. Before we can arrive 
at the conditions under which several consecutive wave-lengths of the 
resultant disturbance have the same group-velocity, it is necessary to find 
the relation between the predominating wave-period and the displacement £ 
at each point of the medium. 
§ 12. The above presentation of group- velocity is effective in showing 
the intimate relation existing between group-velocity and dispersion. Dis- 
persion is in fact the result of the gradual separation of the points of 
predominance of trains of nearly equal wave-length and wave-velocity. 
This is clearly illustrated in our problem of § 7. Initially, the points of 
predominance of all the trains coincide at the origin ; immediately after, the 
points of predominance of very quickly moving trains occupy the most 
distant points of the medium, each having travelled out at its own coincident- 
phase-velocity. The separation out of the closely packed predominant 
points from the neighbourhood of the origin becomes continuously more 
and more complete for the less quickly moving trains, and the same process 
of separation goes on in the front and rear of each predominant point as it 
moves uniformly forward at its coincident-phase-velocity. Returning to 
equation (4) above, we see that the points of predominance corresponding to 
the two wave-periods k and k + 8k are continually separating from each 
other at the rate v given by the equation 
V — OK 
dk 
( 6 ), 
which shows that the points of predominance of wave-periods intermediate 
to k and k + Sk are being spread over an ever-increasing length of the medium. 
The extent of the medium occupied by these points at time t is given by 
vt = /~Sk . 
dk 
( 7 ): 
