Lord Kelvin on Groups of 'Deep-Sea Waves. 13 



outward from the middle, we see that the two groups, after 

 emergence from eo- 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 corresponding to the " half wave-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. 



§ 116. 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 £ = 0. In diagram 3, R shows the 

 position reached at time SVtt (eight periods) by an ideal 

 point travelling rightwards from the R of diagram 1 at a 

 speed of half the wave-velocity. This 11 of diagram 3 corre- 

 sponds to a fairly well-marked perceptible rear of the right- 

 ward travelling group. 



Look now to F, F, F, in the three diagrams of fig. 35, and 

 f, f, in diagrams 2 and 3. In diagram 1, F marks a per- 

 ceptible front for the rightward travelling component group. 

 In diagrams 2 and 3, F, / show ideal points travelling right- 

 wards from it at speeds respectively, the half wave-velocitv, 

 and the wave-velocity. We see a manifest wave-disturbance 

 far in advance of F, F ; and very small but still perceptible 

 wave-disturbance in front of f\ f. Thus the perceptible 

 front travels at speed actually higher than the wave- 

 velocity, and this perceptible front becomes more and more 

 important relatively to the whole group with the advance of 

 time, as we may judge from fig. 9 of § 20 above. 



§§117. It is interesting to see by these diagrams how 

 nearly the hypothetical group-velocity is found in the rears : 

 while the fronts advance with much greater and with ever- 

 increasing velocity. The more elaborate calculations and 

 graphical constructions of §§ 20-29 above led to corresponding 

 conclusions in respect to the front and rear of a procession, 

 given initially as an infinitely great number of regular 

 sinusoidal waves travelling in one direction. The diagrams, 

 figs. 9 and 10, showed respectively, at twenty-five periods 

 after a sinusoidal commencement, a front extending forward 

 indefinitely, and a perceptible rear lagging scarcely two 

 wave-lengths behind a point, travelling from the initial 

 position of the rear at a speed of half the wave-velocity. 



