(b) Three-Dimensional Array 
By reasoning analogous to that used in deriving equation (5), 
it can be shown that, if 7 is a region in 3-space, then the 
directivity function is 
(® 
R= | 1a ($6. vafemp [xcosatycosstzcosy 
dE 
+o(x, y, z(cosv(x, y, z)-1)] bday 
Or, if compensated for a spherical wave front 
r 
R= / E(x, y, 2) {exp jk [x(cosa- cosa, )ty(cos8- cosB, ) 
df 
+ z(cosy-cosy,)+ (x, y,z)(cos (x, y, z)-1) 
-0o (x, y, z)(cosl> (x, y, z)-1) 5 | QV) 
In Appendix A, it is shown that 
Li P 
D ae | IB Ges Ws &)) jo [xcosatycos8tzcosy 
Te 
to(x, y, z)(cosh(x, y, z)-1)] | dy 
Li 
i | Cae E(x, unferos [xcosatycos8tzcosy 
oF 
t+o(x, y, z)(cosv(x, y, z)-1)] jer 
which is 
R= { E(x, y, 2) | exp jk [xcosatycos8tzcos 7] | ay 
Op 
11 
