452 Proceedings of the Royal Society of Edinburgh. [Sess. 
an amount which becomes appreciable as time goes on, no matter how small 
dV 
dll 
Sk may be, provided -P is not zero. If -p- is zero, then all wave-periods 
whose points of predominance are initially in coincidence or almost in 
coincidence maintain the same relation to each other throughout all time, 
and there is consequently no dispersion. This will be illustrated later for 
media for which V = a + b/k, where a and b are constants. 
§ 13. Now, according to the principle of stationary phase adopted in 
§§ 8-10 above, only trains whose points of predominance are in the 
immediate neighbourhood of a point x contribute effectively to the resultant 
displacement £ at x ; hence we arrive at the conclusion that when the 
process of dispersion just described is sufficiently far advanced we can 
obtain the resultant displacement at any point by considering only the 
effects of trains differing infinitely little in period from the predominant 
period at the point. In the early stages of dispersion, when points of pre- 
dominance of widely differing wave-periods are very near one another, 
the resultant wave-curve £ near any point x will in general only very 
roughly correspond in wave-length and wave-period to the wave-period 
of the Fourier trains effective in the neighbourhood. It will now be shown 
that the correspondence becomes more and more pronounced as time goes 
on ; and we shall arrive at an understanding as to how group-velocity is 
to be applied to the resultant wave-curve. We shall find that the process 
described in § 12 leads ultimately to the following result referred to by 
Professor Lamb : “In a medium such as we are considering, where the 
wave- velocity varies with the frequency, a limited initial disturbance gives 
rise in general to a wave-system in which the different wave-lengths, 
travelling with different [constant] velocities, are gradually sorted out,” * 
and arrive at any given point in the order corresponding to their group- 
velocities. Each separate crest or trough, however, moves with continually 
increasing or continually diminishing speed. 
§ 14. Returning now to equation (1), we assume that the dispersion is 
so far advanced that the phases of the effective Fourier trains in the 
neighbourhood of a point x are determined with sufficient accuracy by two 
terms of a Taylor’s series. Thus, taking k 0 as the predominant period at x 
at time t, we have by equation (4) 
« - W%) - tk 0 f(k 0 ) = 0 (8) ; 
and by Taylor’s theorem we have 
k{x - tf(k)} = k^x - tf(k Q )} + {k- k^bLk {) {x - tf(k Q )} + ^ ~ ^ -Jp k 0 {x - tf(k 0 )} (9). 
* Hydrodynamics , 3rd ed., § 234. 
