458 Proceedings of the Royal Society of Edinburgh. [Sess. 
separation of predominant points will be sufficiently advanced to allow us to 
recognise a distinct group-velocity in the various parts of the wave-system. 
This makes it possible to apply the theory of group-velocity to any 
wave-system, provided the word “ group ” is understood to mean any part 
of the whole disturbance which has a specified wave-length. In general, 
the “ group,” as here defined, is confined to the immediate neighbourhood 
of a single point, which moves uniformly forward ; but when the wave- 
length varies very slowly from point to point, the group may be taken in 
a modified sense to refer to an extended portion of the wave-system where 
the wave-length is nearly constant, provided we remember that such a 
group would be continually increasing in length as it proceeds, and that its 
recognition soon ceases to be useful. 
§ 21. We can now obtain from the theory of group- velocity a useful 
general understanding of the way in which any disturbance initially con- 
fined to a small portion of a dispersive medium is propagated into regions 
initially undisturbed. We may arrive at the effect of any initial disturbance 
by summing the effects due to point displacements applied at each point of 
the disturbed region ; but it is more convenient from our present standpoint 
to regard any initial displacement of the medium as due to a certain distri- 
bution of predominant points. Since the initial disturbance is limited in 
extent, it is clear that the Fourier trains into which it can be analysed must 
be infinite in number and of all possible wave periods, and we suppose that 
their points of predominance are arranged irregularly but continuously 
within the disturbed part of the medium. At any very short time after 
motion has commenced, an effect will have been produced in the most 
distant parts of the medium by the very quickly moving trains whose points 
of predominance have moved out in the interval. Near the place of the 
original disturbance the slowly moving trains still agree in phase, but 
irregularities in amplitude and wave-length must occur owing to the pre- 
dominant points of the quicker trains overtaking and out-stripping those 
of the slower. Indeed, it seems certain that irregularities in amplitude will 
persist, according to the manner in which the energy is distributed initially 
among the effective Fourier trains.f But, as time advances, the various 
trains sort themselves out according to their coincident-phase-velocities, 
and the wave-length of the resultant curve ultimately varies regularly and 
continuously as we pass outwards on either side from the place of the initial 
disturbance. At this stage the continuity of the disturbance allows us 
to regard the distance between corresponding points on two consecutive 
half-waves as equal to a half-wave-length of the train which has its point 
t Burnside, Proc. Lond. Math. Soc., t. xx. p. 22, 1888. 
