[sess. 1905 - 6 .] Lord Kelvin on Groups of Deep-Sea Waves. 411 
the curves of diagrams 5 and 6, of fig. 34. Thus what, according 
to the known law of deep-sea periodic waves (§ 19 above), would 
be definitely and precisely the wave-length, if the numbers of 
crests and hollows were infinitely great, would he 2 ; and as we 
are taking g — 4, the period would be Jtt, and the propagational 
velocity would he 2/^/7 r. 
§ 115. Immediately after the water is left free, the disturbance 
begins analysing itself into two groups of waves, seen travelling 
in contrary directions from the middle line of the diagram. The 
perceptible fronts of these two groups extend rightwards and left- 
wards from the end of the initial single static group, far beyond 
the “hypothetical fronts,” supposed to travel at half the wave- 
velocity, which (according to the dynamics of Osborne Reynolds 
and Rayleigh, in their important and interesting consideration of 
the work required to feed a uniform procession of water-waves) 
would be the actual fronts if the free groups remained uniform. 
How far this if is from being realised is illustrated by the 
diagrams of fig. 35, which show a great extension outwards in 
each direction far beyond distances travelled at half the “ wave- 
velocity.” While there is this great extension of the fronts 
outward from the middle, we see that the two groups, after 
emergence from co-existence 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. 
The extreme perceptible rear travels at a speed closely correspond- 
ing to the “halfwave-velocity,” found by Stokes as exactly the 
group-velocity of his uniform succession of groups, produced by 
the interference of two co-existent infinite processions of 
sinusoidal waves, having slightly different wave-lengths. 
§ 1 16. Our fairly uniform rear velocity is illustrated in diagrams 
1 and 3, of fig. 35. In diagram 1, R indicates the perceptible 
rear of the component group commencing its rightward progress 
at t = 0. In diagram 3, R shows the position reached at time 
8 7r (eight periods) by an ideal point travelling rightwards from 
the R of diagram 1 at a speed of half the wave-velocity. This 
R of diagram 3 corresponds to a fairly well-marked perceptible 
rear of the rightward travelling group. 
