1908. | Groups of Waves in Dispersive Media, etc. 422 
any point P the lines of constant phase in the two wave patterns cross at an 
angle ¢, which is easily seen to be given by 
tan @ = 4 cosec a (9 cos? «—8)}, (92) 
As P approaches either radial boundary the two waves ultimately have the 
same direction, and they will also have the same phase when they meet; 
consequently an abnormal elevation is to be expected along the two outer 
boundaries, where the two systems unite in lines of cusps. As we see from 
(75), the two points A, B coincide for a point P on the line of cusps ; and it 
is on account of this fact that the previous approximations fail for both 
systems. We have in fact a double root of the equation for finding the 
chief groups of the integral (72). 
Consider the integral 
y= | d(o)sin (/@)} du, (93) 
when wp is such that 
F (uo) = 0; J” (to) = 0. 
Following the previous method, we have 
FU) = fu) +5 (U= Uo) fF (to) 5 
(uw) is not small, we can write the value of the group for 
AG 
and provided f 
the double root as 
Y= 1 pray} | $ewsin (fet) der (94) 
= {om | $ (UW) sin f (2) [ cos a do. 
Now at the line of cusps the integral (72) becomes 
Rae whe NA ur du ce gu? 95 
f 23arp |, (w?—4cum,/24 u?)? sae (w?—4cum,/24 u?)2 ) 
And we find that Cy = w/2 
makes J’ (w) = 0; JF (Mm) = 0; 
Fo) = 9ar/3/22; Fu) = Bgex/6/ 200% 
Also we have | cos a? do = 27/1 (2). 
Hence, substituting these values, we have 
x9 : GB / 3 
S = 96 
°=— Tanja Be ) 
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