457 
1908-9.] On Group! Velocity and Propagation of Waves. 
follows easily from the case of two trains ; for the coincident-phase-velocity 
differs for each pair of trains, and, except for occasional agreement of phase 
of three or more trains, no new features occur. When the number of trains 
is exceedingly large, and their wave-lengths and wave-periods vary very 
slightly, as in the explanation of group-velocity given by Gouy and Lord 
Rayleigh,* each wave-train has its own particular point of predominance 
in the wave-system ; which point moves at its own coincident-phase- velocity. 
The argument is, in fact, exactly the same for a disturbance represented by 
£= 2C cos k{x- tf(k) + *} .... (25) 
as that given in §§ 8-10 above; the point of the medium at which the 
wave-period k predominates being again found from equation (4). Here 
the number of terms is exceedingly large, and C and k vary continuously. 
The general features of propagation of the disturbance are the same as in 
the case of a single initial impulse. The wave-system is continually being 
modified by the process of § 12, and the resultant curve ultimately takes a 
shape in each neighbourhood corresponding in wave-length to the component 
Fourier trains which predominate there. As time goes on, the rate of change 
of the wave-length as we pass along the resultant curve becomes more and 
more gradual, and the whole disturbance occupies an ever-increasing extent 
of the medium on account of the difference in the coincident-phase-velocities 
maintaining throughout the wave-system. 
§ 20. The argument regarding group-velocity contained in §§ 7-14 
above has been confined to the particular case of the wave-system arising 
from an infinitely intense disturbance at a single point of a dispersive 
medium, but it is evident throughout the later parts of the discussion that 
the results arrived at are of much more general application. Group-velocity 
has been shown to depend upon the principle of “ stationary phase,” which, 
it seems reasonable to assume (see § 9), can be applied to any infinite com- 
bination of Fourier trains. It is practically Huygens’ principle in optics. The 
agreement in wave-length between the effective Fourier trains and the result- 
ant displacement curve at any point of the medium, on which the applica- 
tion of the theory of group- velocity to the resultant curve depends (§§ 14, 
15), requires only that the predominant wave-period should vary continuously 
in the neighbourhood of the point considered. The condition that t be very 
great is required in § 14 to enable us to completely evaluate the integral in 
equation (1), but is unnecessary to prove the correspondence of wave-lengths 
referred to, which follows at once from considerations of continuity by 
Taylor’s theorem. In general, at only a short interval after a disturbance the 
* Lamb’s Hydrodynamics , 3rd ed., § 234. 
