1908. | Groups of Waves in Dispersive Media, ete. 428 
Further, since equal values of cw are given by 
m (3—n) = 2/p, 
we see that there are no cusps, for the left-hand side cannot be greater than 2. 
The values of cw given in (115) and (116) correspond to the transverse and 
diverging waves respectively. If we substitute (116) in equations (111) 
and (112) we find that they are satisfied identically ; hence there is always a 
diverging wave system. On the other hand, if we substitute (115) we find we 
must have 
l—pm 
1—}pm (1+n) (3—n)’ 
But the greatest possible value of the left-hand side is unity. 
Hence there can be a transverse wave system only so long as p is greater 
than 1; when ¢ exceeds ,/(gh), the transverse waves disappear. 
At the outer line given by 
sin?a = p, =o = il, 
1 
pm = or m(2—n) ==. 
: ip 
we have, for the diverging waves, 
uUu=s (l—p)? = @ SEC a. 
Hence the outer line forms a wave front of the diverging wave system. 
We see also that the other wave fronts (lines of equal phase) are now concavet 
to the axis, instead of being convex as when p>. There is no definite inner 
limit to the system; as the axis is approached, the wave fronts become more 
nearly parallel to the axis, and the wave-length diminishes indefinitely. 
Finally, as the velocity c is increased, the angle « diminishes, and the regular 
waves are contained within a narrower angle radiating from the centre of 
disturbance. 
The following tables (III) and (IV) and the curve in fig. 8 show how the 
angle # varies as the velocity cis increased up to and beyond the critical 
velocity. maa 
able III. 
ee ee 
kh at cusps. Pp. a. e/ (gh). 
10 7 19 28 0°38 
8 5-4 19 28 0:42 
6 4 19 29 0°5 
5 3°38 19 30 0°55 
4 PACH f 1@) By/ 06 
3 2 20 18 0-7 
2 15 23 42 0°82 
1 1-18 39 19 0-92 
0-5 1-08 59 27 0-96 
0-2 1-01 78 0:99 
0 1 90 1 
a = cos-1/8 (1—n)/(3—x). 
TSee Editorial Note on page 33. 
31 
