1908. | Groups of Waves in Dispersive Media, etc. 408 
only the integral necessary for propagation in the direction of a positive, 
that is, 
y= EAL cos Gn a) tele cos x (e— Vi) die (36) 
T Jo UG hol Ke 
If 1 is very large, the main feature consists of the component waves round 
the value «’; but in general it is to be noticed that a series of subsidiary 
components appears whose effects may be of sufficient magnitude to be 
appreciable. But the component waves are cumulative only for values of « 
and ¢ such that 
K = gt? /42?, 
which is the value corresponding to a stationary argument of the cosine; thus 
the prominent effect at time ¢, of any group of parameter «, will be at localities 
where « has the value «’, or, else, a value belonging to one of the subsidiary 
maxima. The result may be evaluated in the same manner as before; 
we find 
Re zit Be ge iE ‘ 
7 = 16e'A (2 \ Tear ap oO (204+) 7 Sc0s(S°— yr). (37) 
We can obtain the prominent travelling groups above referred to, which 
this involves, by evaluating the maxima of the amplitude function 
t 
The form of this function is shown by fig. 1; it is obtained by plotting the 
COS (2n +3) 5 ze (38) 
curve 
cos $7ra?, (39) 
a 
YS Tae 
where @ is proportional to ¢, and, further, « equal to 1 corresponds to « equal 
to x’. 
The curve represents the variation of the disturbance at a given point with 
the time, neglecting the local variations of the last cosine factor in (37) ; it 
shows the grouped propagation of an initial displacement consisting of 
4} complete wave-lengths of a simple cosine wave of wave-length 27/«’, 
or 2X’. 
The main undulatory disturbance appears as a simple group around the 
predominant wave-length X’, moving forward with the corresponding group- 
velocity 3\/(g/«’) or 4V. But ia advance of this main group of undulations 
there are two or three subsidiary groups of sensible magnitude with wave- 
lengths in the neighbourhood of 9A/2, 92/4, 9/6, moving with corresponding 
group-velocities of 3V/2,/2, 3V/4, 3V/2,/6. Thus in advance of the main 
group we have slighter groups of larger wave-lengths moving with group- 
velocities which may be larger than the wave-velocity of the original dis- 
11 
