1908. ] Groups of Waves in Dispersive Media, ete. 410 
an initial disturbance consisting of five crests and four hollows of approxi- 
mately sinusoidal shape ; the following remarks are made :—* “ 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 leftwards from the end of the initial statie group far beyond the ‘ hypo- 
thetical 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 7f the free groups remained uniform. 
How far this 7/ 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 coexistence 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”. . . . . Thus the per- 
ceptible 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 2 
This extract will serve to emphasise the importance of strict definition and 
use of the word “group.” A simple group, of whatever structure, has asso- 
ciated with it a definite velocity depending only on the wave-length, but not 
so an arbitrary limited displacement. In various cases we have found it 
convenient to analyse such into its important elementary groups, each with 
definite velocity ; in special cases the disturbance may be equivalent practi- 
cally to one simple group. 
§8. Lnitial Impulse on Deen Water. 
Suppose that initially the surface is horizontal, but that given impulses 
are applied to it. Then for any given symmetrical distribution of impulse 
7 (x), suitable for Fourier analysis, with no initial elevation, the surface 
elevation at any subsequent time is given by 
moon BSE ( «Vd (x) sin x (2—Vt) de—} | KV p(x) sin« (a+ Vt)de, (40) 
0 
0 
) 
where o (x)= | 7 (@) cos kw dw. 
* Lord Kelvin, ‘ Phil. Mag., vol. 13, p. 11 (1907). 
. 
13 
