455 
1908-9.] On Group- Velocity and Propagation of Waves. 
and when we eliminate k 0 from the argument of the cosine in equation (14) 
by means of this equation, the argument 0 may be written in the form 
«+i 
6 = B : 
x 
+ - : B = 
n 
n+1 1_ 
( 1 + n) A n 
(17), 
where the negative sign is taken when n lies between 0 and —1. If we 
follow the crest of a wave whose equation is given by 
6 = '2rTT ...... (18), 
we can find the velocity of this crest by differentiating (17). Thus we have 
y _clx _ 1 x \ 
dt n+1 t > . . . . . (19) ; 
= by (16)) 
Since the phase varies by 2 tt as we advance one wave-length along the £ 
curve, the length A of the wave whose crest we are following is given 
approximately at any time t by the equation 
n+ 1 n + 1 
= 2rr 
■ ( 20 ). 
With A small compared with x, this gives finally by means of equation (18) 
n x 
n+\ (r + \) 
■ ( 21 )- 
These equations show that each wave lengthens as it proceeds, and that its 
velocity alters accordingly. If the wave-velocity increases with increasing 
wave-length, each wave is continually accelerated; if the wave-velocity 
diminishes with increasing wave-length, each wave-crest moves with con- 
stantly diminishing speed. The individual waves which at any time 
constitute the part of the disturbance which has a certain wave-length A 
immediately pass ahead of this part, if the medium is one in which the 
wave-velocity exceeds the group-velocity, or fall behind it if the group- 
velocity exceeds the wave-velocity for that wave-length. In the former 
case succeeding waves in turn become of length A ; in the latter case, pre- 
ceding waves in turn become of length A. A point which moves so as to 
coincide at each instant with the point of the wave-system at which the 
wave-length is A travels at the coincident-phase-velocity corresponding to 
that wave-length. 
§ 17. At this stage it is interesting to examine the coincident-phase- 
velocity with reference to a disturbance which can be analysed into a 
limited number of wave-trains. The most interesting case is that of two 
