460 Proceedings of the Royal Society of Edinburgh. [Sess. 
to which the disturbance is propagated throughout the medium is the 
process o£ separation of points of predominance of the constituent wave- 
trains, so that we can at once say that the disturbance which we are 
considering gives rise to a wave-system consisting of a succession of waves 
of all possible wave-lengths, each wave-length appearing in the succession 
according to the order of its group-velocity. The energy of the initial 
disturbance is ultimately diffused throughout the entire medium. 
The “ group,” defined as the part of the whole disturbance which has the 
same wave-length as the original group, moves along at the group-velocity 
corresponding to its wave-length, being distinct for a time owing to the 
energy it retains. But the regularity of its shape cannot be maintained as 
it proceeds, for it must supply the energy necessary to feed the ultimately 
infinite succession of waves of greater and less wave-lengths which 
constitute its front and rear. As time goes on, therefore, a falling off from 
sinusoidality, which proceeds inwards from the front and rear of the group, 
must become more and more evident as the front or rear increases in 
importance ; and the amplitude of the sensibly regular central part must in 
time diminish. Thus it would seem that we cannot expect perfect regularity 
to be maintained in any part of a finite group for any time, however short. 
It is to the part of the group which remains sensibly regular that the 
dynamical theorem given by Lord Rayleigh applies. The law of diffusion of 
energy towards the front or rear of the wave-system essentially involved in 
the process of dispersion described in § 12 does not seem to be easily 
derived from the dynamical theory of group-velocity as hitherto developed. 
LTnless we can arrive at the law of falling off from regularity of the main 
group in the front and rear, it seems impossible to follow the distribution of 
energy throughout the entire system. The kinematical group-velocity 
theory, however, accounts satisfactorily for the speed at which a group of 
waves of sensibly the same length advances, and explains in a general w T ay 
the process by which any initial disturbance is modified in invading 
undisturbed space. 
§ 23. The formation and development of the front and rear of a large 
initially regular procession of waves in deep water, in accordance with the 
theory of group-velocity given above, is illustrated by the diagrams of 
Lord Kelvin’s paper, Proc. Roy. Soc. Edin., vol. xxxv., 1904, figs. 9 and 10. 
For water-waves, the group-velocity ~ x / increases with increasing 
wave-length, so that at any time long wave-lengths appear in the front 
and short wave-lengths in the rear of the main group, which, in the case 
treated by Lord Kelvin, consists initially of a large procession of waves of 
