1908-9.] On Group-Velocity and Propagation of Waves. 467 
waves at each time, the perceptible wave-length predominates at a point 
slightly beyond the maximum, which is what is to be expected. The whole 
progress of the disturbance is in entire agreement with the process 
described in § 21. 
§ 25. Fig. 35 shows the manner in which a finite group of equal waves in 
deep water is modified as it proceeds, owing to the different coincident-phase- 
velocities of the component Fourier trains. The group of waves of wave- 
length 2 comprising the main “ group ” in diagram 1 can be observed in the 
two later diagrams displaced from the central position by an amount corre- 
sponding exactly to its group-velocity ; while it is clear that the individual 
waves forming this group are continually passing forward to form the front. 
Each wave originates near the origin and moves through the whole system, 
lengthening as it proceeds, and carrying some of the energy of the original 
group forward to the front, which continually increases in importance 
relatively to the main group, and becomes perceptible at greater and greater 
distances beyond it. “ While there is this great extension of the fronts 
outward from the middle, we see that the two groups, after emergence from 
coincidence in the middle, travel with their rears leaving a widening space 
between them of water not perceptibly disturbed, but with very minute 
wavelets in ever augmenting number following slower and slower in the 
rear of each group.” * It is easy to verify from the diagrams that each per- 
ceptible wave-length appears in the wave-system at any time, displaced from 
the position at which it is observable at an earlier time by the distance corre- 
sponding to the group-velocity for that wave-length. As an example we may 
take the wave-length at the maximum point of the wave marked/ in diagram 
2. The particular wave at f in diagram 2 has disappeared beyond the 
diagram limits at the time of diagram 3, but the new wave at /in diagram 3 
is of only slightly shorter wave-length and will suit for comparison. From 
the diagrams, the distance between the maximum points is about 8’4 units 
of the scale shown. The value of U for the perceptible wave-length at / is 
given by U = = = which g ives U(8 ^/tt-4 n /tt) = 8'7. 
Thus the wave-length at the maximum near / in diagram 2 corresponds to 
the wave-length appearing slightly beyond the maximum in diagram 3, as 
is to be expected from its slightly greater wave-length. 
§ 26. The general features of all these water-wave disturbances are 
the gradual increase in wave-length of the disturbance as we pass outward 
from the place of the original disturbance, and the continually increasing 
importance of the front as compared with the rear. It seems certain that 
* Lord Kelvin, Proc. Roy. Soc. Edin ., 1906. 
