459 
1908-9.] On Group- Velocity and Propagation of Waves. 
of predominance somewhere between them. Thus if we call twice the 
distance between any two consecutive zeros of the resultant wave-system 
the perceptible wave-length in their neighbourhood, the perceptible wave- 
length will agree very closely with that of the wave-train which predomin- 
ates at the point of maximum displacement between the zeros observed ; 
and the further progress of the dispersion may be described as a gradual 
increase in the extent of the medium sensibly disturbed, accompanied by a 
continual diminution in the rate of change of the perceptible wave-length from 
point to point of the system, owing to the process indicated in § 12. This 
is exactly the result described by Professor Lamb, quoted in § 13 above. 
More detailed description would require a knowledge of the initial distribu- 
tion of energy and of the law of diffusion of energy throughout the medium. 
§ 22. The case of an initial disturbance consisting of a large succession 
of equal, regular waves of wave-length X, with undisturbed space in front 
and rear, is interesting in connection with the dynamical interpretation of 
group-velocity given by Osborne Reynolds and Lord Rayleigh, and on 
account of its importance in relation to sequences of light-waves in 
dispersive media. The mathematical expression representing such a group 
of waves would again involve all possible wave-periods, and we may state 
our initial conditions by saying that the wave-length X predominates in the 
part of the medium initially occupied by the group. According to the 
theory of group- velocity presented above, at any time t after the 
commencement of motion there will be a succession of waves of wave- 
length X displaced on either side from the place of the original disturbance 
a distance U t, where U is the group- velocity corresponding to wave-length 
X ; but in front of the waves of length X there will have appeared a 
continuous wave-disturbance, of which the wave-length increases con- 
tinuously as we pass farther and farther ahead if the medium is such that 
the coincident-phase-velocity of the wave-trains increases with increasing 
wave-length, and of which the wave-length diminishes continuously if the 
coincident-phase-velocity diminishes with increasing wave-length. Behind 
the main group of waves of length X there will also have appeared a 
continuous wave-disturbance of which the wave-length either decreases or 
increases continuously as we pass farther and farther behind the group of 
waves of wave-length X, according as the wave-length in front increases or 
diminishes. Each perceptible wave-length in the front or rear of the main 
group is always to be found in the wave-system at any time in advance of 
the place where it was observed at an earlier time by a distance corre- 
sponding to the group-velocity for that wave-length. The history of the 
individual waves is again that given in § 16. The process according 
