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Proceedings of the Royal Society of Edinburgh. [Sess. 
interfering trains, which was dealt with first by Stokes, and afterwards by 
Lord Rayleigh in § 191 of his Theory of Sound, vol. i. In Lord Rayleigh’s 
notation the two interfering trains are represented by 
t x 
COS H77-I 
and 
COS 2 7 
A/ \r 
where r, t are the wave-periods, and A, A' the wave-lengths of the trains. 
At the origin, which may be chosen at the point of maximum displacement 
of any one of the wave-groups which constitute the initial disturbance, the 
phases of the two trains are initially zero ; and the point at which agree- 
ment of phase occurs at any later time is given by the equation 
t db t 'JCf 
T A T X 
( 22 ), 
which corresponds to equation (4) above. It may be written in the form 
(23), 
Krr 
from which we obtain the coincident-phase-velocity U, as 
1 1 
l 
A' 
(24). 
When the trains are of nearly the same wave-period and wave-length, this 
becomes which is the form given by Lord Rayleigh. 
§ 18. With reference to this case, the view of the matter here adopted 
frees us from what is in general a misleading conception in connection with 
group- velocity, namely, the reappearance of the entire group of waves at 
regular intervals. Thus Professor Schuster, referring to an extended suc- 
cession of waves, in his article “ On Interference Phenomena,” remarks : “ We 
can speak of definite rate of propagation U of the group, because at definite 
intervals t the group takes up the same shape, displaced through a distance 
Ut.” Our investigation proves beyond doubt that the reappearance of the 
same shape at regular intervals in a dispersive medium is confined to the 
particular case of a disturbance represented by two interfering trains. The 
original shape of any disturbance may maintain itself unchanged, or may 
reappear at regular intervals, in a medium for which the coincident-phase- 
velocity is constant or zero for all wave-periods ; but, as we have found in 
§12, in such media there is no dispersion. This will be illustrated in a 
later paper. The process of separation of predominant points referred to 
12 above essentially involves continuous non-periodic change in the 
in 
shape of any initially given group of waves. 
§ 19. The case of three, four, or any finite number of interfering trains 
