1908. ] Groups of Waves in Dispersive Media, etc. 418 
that is, from cu? — 3cum cos a+ 2n7 = 0, (74) 
or cu = 40 {3 cosa+(9 cos? a—8)*}, (75) 
We have then different cases to consider according to the nature of these 
values for cw, remembering that cw gives a position of the moving impulse, at 
time w previously, for which the waves sent out reinforce each other at the 
point (w, «) at time ?¢. 
(i) In the region where 9 cos?« < 8, both roots are imaginary; thus the 
previous position is non-existent, and there is no principal group in the 
integral (72). Hence all the regular wave pattern is contained within two 
straight lines radiating from the point impulse, each making with the line of 
motion an angle cos~! 2,/2/3, or approximately 19° 28’. 
(ii) When 9 cos?«<8, there are two different real roots for cu. Thus we 
have two chief groups in the integral (72), corresponding to two regular 
wave systems superposed on each other. 
At any point P within the two bounding radii the disturbance consists of 
two parts: one part sent out from A at time w previously, where 
OA = 4a {3 cos 2+(9 cos? a—8)*} and uw = OA/c; (76) 
and another part sent out from B at time wz before, where 
OB = 4m {3 cos 2—(9 cos?#—8)*} and uw, = OB/c. (77) 
Fig. 4. 
We have then two wave systems, which may be called the transverse 
waves and the diverging waves; we shall examine them separately. 
(a) The transverse wave system—Taking the larger value of cw in (76) 
we find 
w?— 2cum cos a+¢cu? = 30? {3 cos?«—2+cos a (9 cos? «—8)}}, 
f(u) = uw — 98/2 18 cos? a—8 + 6 cos « (9 cos? a—8)t 
4 (w?—2cus cos a+ c?u*)t 16c? {3 cos?a—2+ cos @ (9 cos? a—8)}t 
(78) 
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