SOME VIEWS ON LORD KELVIN'S WORK 431 



duced by an Impulse delivered at the origin is given for place 

 X and time t by £ where 



S=i/*°°dkcos[k{x-tV}] .... (1) 



o 



where V = f (k), V being the velocity of the Fourier train of 

 wave-length \, and k = 2irJ\. When t is large, the effect at any 

 point is due to the trains whose phases agree or nearly agree at 

 the point chosen for observation. The remaining trains being 

 infinite in number and differing in phase can be assumed to 

 produce zero effect. Thus the predominant trains at point x 

 are determined by 



8 [k { x -t V}] - 0, or x = t (f(k) + kf (k)} - t U 



where U is called the group-velocity of the trains which produce 

 the maximum effect at place % and time t. In this the idea of 

 group-velocity is restricted simply to mean the principle of 

 stationary-phase as employed by Prof. Lamb in his investigation 

 of Ship Waves (Hydrodynamics, § 253), but applied to the Fourier 

 trains which constitute any wave disturbance. When this view 

 is accepted, the difficulties referred to by Lord Kelvin in the 

 passage quoted above are removed ; and the results to which it 

 leads are consistent with the dynamical theory of group-velocity 

 given by Osborne Reynolds and Lord Rayleigh. 



Strangely enough, this is the meaning attached to group- 

 velocity in Lord Kelvin's paper of 1887, and the ke}' to the 

 explanation of the problems regarding groups of light waves in 

 glass, and indeed of any problem involving dispersion, lay 

 unnoticed in his earlier work. The development of the funda- 

 mental process of dispersion along the lines laid down in Lord 

 Kelvin's original paper was completed by Dr. T. H. Havelock 

 in 1908, in his paper on " The Propagation of Groups of Waves 

 in Dispersive Media," Proc. R.S., vol. lxxxi, and by G. Green in 

 Proc. R.S.E., 1909. Lord Rayleigh, however, has pointed out 

 that the principle of stationary phase applied to the fundamental 

 Fourier trains, as indicated above, does not account for an 

 instantaneous propagation of any disturbance which occurs in any- 

 dispersive medium, thus calling attention to a gap between 

 initial actions and those determined by group-velocity theory, 

 which would call for some new method of determining the 

 value of the above integral. 



