Propagation of Groups of Waves in Dispersive Media, etc. 400 
The range of values of « being infinitesimal, the group as a whole may be 
written, as in the previous case, in the form 
¥ = $(a—Upt) cos {xy (e—Vot) +8}, (4) 
where ¢ is a slowly varying function ; and the group-velocity Uy is given by 
Uo = = (eV). (5) 
Ko 
The group, to an observer travelling with velocity Up, appears as consisting 
of approximately simple waves of length 27/x. The simple group is, in fact, 
propagated as an approximately homogeneous simple wave-train ; the impor- 
tance of the group-velocity lies in the fact that any slight departure from 
homogeneity on a simple wave-train, due to local variation of amplitude or 
phase, is propagated with the velocity U. 
§3. The Fourier Integral regarded as a Collection of Groups. 
An arbitrary disturbance can, in general, be analysed by Fourier’s method 
into a collection of simple wave-trains ranging over all possible values of «; 
thus after a time ¢ the disturbance will be given by an expression of the type 
[ $0 cos « (vx—Vi+a) de, (6) 
where V is a given function of x. 
The method adopted with these integrals is based on Lord Kelvin’s* treat- 
ment of the case, in which the amplitude factor ¢ («) is a constant, so that 
WY -& | cos K (w— Vt) dk. 
0 
An integral solution of this kind is constructed to represent the subsequent 
effect of an initial disturbance which is infinitely intense, and concentrated in 
a line through the origin; Lord Kelvin’s process gives an approximate 
evaluation suitable for times and places such that —Vé is large, and the 
argument may be stated in the following manner :— 
In the dispersive medium the wave-trains included in each differential 
element of the varying period are mutually destructive, except when they are 
in the same phase and so cumulative for the time under consideration, this 
being when the argument of the undulation is stationary in value. Thus 
each differential element as regards period, in the Fourier integral, represents 
a disturbance which is very slight except around a certain point which itself 
changes with the time. 
Now if we apply this method to the more general integral (6), we obtain an 
* Sir W. Thomson, ‘ Roy. Soc. Proc.,’ vol. 42, p. 80 (1887). 
