421 Dr. T. H. Havelock. The Propagation of [Aug. 26, 
The law of amplitude along the same crest is given by 
__const. {3 cos a—(9 cos? «—8)'}# 
(2n + 5)? (9 cos? «—8)* {3 cos? a —2—cos a (9 cos? «—8)'}? 
(89) 
In this case, and for the same reason as for the transverse waves, the 
expression for the amplitude tends to infinity at the outer end of each 
diverging crest ; we shall find an approximation in the next section. But 
(89) becomes infinitely large for small values of « From (88) we see that 
w also becomes small, so that the approximation fails; further, we should 
expect the expression to become infinite near the impulse on account of its 
special character. We can show how the infinity disappears if we remove 
this cause. Consider, as an example, a finite impulse, of constant intensity 
over a circular area of radius d round the origin, and of zero value outside 
this circle. Then, as we see from (63), we shall have the same expressions as 
before, with a new factor given by 
g(x) = [r@ Jo (Ka) a da 
ZG [ IG ade = Od WGA) 
Now in the final group for the diverging system we have 
ae] {3 cos a—(9 cos? «—8)?}? 
8c? 3 cos? «—2—cos « (9 cos? «—8)? 
Hence the additional factor due to ¢(«) is proportional to 
3 cos? a—2—cos a (9 cos? a—8) 5 { gd {3 cos a—(9 cos? —8)?} } 
{3 cos «—(9 cos? «—8)?}? * 82 {3 cos? a—2—cos a (9 cos? a—8)!} J* 
(90) 
When « approaches zero, the argument of the Bessel’s function increases 
indefinitely and we may use the asymptotic expansion; then (90) is 
proportional to 
{3 cos? «a—2—cos «(9 cos? «—8)* }# 
91 
{3 cosa—(9 cos? a —8)*}4 (91) 
If now we multiply (89) by (91) we obtain a limiting value of the 
amplitude of the diverging system near the axis; it is proportional to 
(2n+4)-= {3 cos a—,/(9 cos? «—8)}#, 
and the infinity near the axis has disappeared. 
(c) The line of cusps—We shall consider now the infinity which occurs 
at the outer boundary of the two wave systems, when « is cos”! 2,/2/3. At 
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