1908-9.] On Group- Velocity and Propagation of AVaves. 463 
wave-length 2 and period J ir. Figs. 9 and 10 show the wave-system at 
time t = 2bjir. Fig. 9 shows the front of the wave-system which has 
formed while the original group of wave-length 2 has travelled from the 
origin to point 25 on the diagram, which is exactly the distance required 
by group-velocity theory, g being taken equal to 4 in Lord Kelvin’s 
calculations. As an example of the application of group-velocity theory to 
the front of the wave-system, we may take the case of wave-length 6. The 
place in the wave-system where this wave-length, initially near the origin, 
should be observable at time 25 J 7r is given by x = 25 J ir x J\/\ X ^ =. 43*3, 
which agrees well with fig. 9. The individual waves forming the front initi- 
ally belonged to the main group, and their places in the group have been 
taken by other waves, so that somewhere in the rear of the whole system, 
not indicated in the diagrams, fresh waves must be continually forming and 
then advancing with increasing length and speed towards the extreme front 
of the system. The point F on the diagram marks the place where the regu- 
larity of the main group perceptibly begins to fail. As we should expect 
from the continual advance of the individual waves through the group, the 
perceptible front is much more extensive than the perceptible rear, which 
is shown in fig. 10. The irregularity of the main group in the rear con- 
sists in the main in a variation in amplitude, without any falling off in the 
wave-length till we reach the extreme perceptible rear at R, beyond which 
we have a large number of imperceptible waves of continually diminishing 
wave-length. From fig. 10 we see that the rearmost wave of wave-length 
2 at the time of the diagram has just reached the point 25, starting from 
point 0, which is in accordance with group-velocity theory. The perceptible 
rear never extends far beyond the last wave-length of the main group of 
wave-length 2, but the number of the perceptible waves in the front 
increases with the time, and the increase in wave-length from the regular 
waves forward becomes more and more gradual owing to the gradual 
increase in the group-velocity as we go towards the extreme front of 
the wave-system. We hope to give the law of falling off from regularity, 
as time goes on, at the front of a large group of sinusoidal waves, in a later 
paper dealing with the wave-system arising from a given distribution of 
pressure moving steadily over the surface of infinitely deep water. 
§ 24. The diagrams of figs. 34 and 35 are taken from Lord Kelvin’s 
last Waves paper, referred to in § 4 above. Fig. 34 deals with the case of 
a disturbance mainly confined to the neighbourhood of the origin ; and all 
the diagrams are outlines of the water surface at the times indicated below 
each, J 7 r being the period of an infinite train of waves of wave-length 2 
