38 
If #(g, 9) = 1, then integrating over a hemisphere, the 
directivity function becomes 
T 
2 
poe | 
@-08=0 
2 TT 
3 
expjzasing /(cosa- cosa )* +(cos8- cosBo )* cos(8- vy | 
. ese cos @(cosy- cosyo | singdgd 8 
Integrating with respect to 4 
= 
et 
R = 2na* be (xa sing,/(cosa- cosdo )* +(cos8- cosBo ) 
(e) 
. fexeseacosstoos COSyo | sing ag 
Using the transformation & = cos@, 
1 1 
R = 2na" lhe (q /1- 2? )cospedét+j es (q./1-2° )sinp§dé 
9 oO 
where 
g = ka J(cosa-cosdo )® +(cos8- cosBo )* 
p = ka(cosy-cosyo ) 
A preliminary search of the literature has failed to produce 
an evaluation of the integral. A simple method of evalua- 
ting it is to expand the Bessel function into its series and 
then integrate term by term. (For further details see 
remarks in Appendix C.) Another method of approxima- 
tion is to place a sufficient number of discrete elements 
on the spherical surface and use the formula for the direc- 
tivity function of a discrete element array for reception 
of planar sound waves (the limit of equation (3) as D-=). 
