which becomes 
R = 2ad\sinka* 
Thus f# is a constant. This is to be expected since an 
uncompensated spherical array would not favor any parti- 
cular direction, because of its symmetry. This result 
corresponds to the fact that a circular array, if uncom- 
pensated, has a constant response for a plane wave arriv- 
ing in the plane of the array. 
For a compensated hemispherical array the directivity 
function is 
R= [ at, Ys B) 
S! 
exp Jk ((cosa- COSM )+y(cosB- cosBo ) 
+2(cOs y- COSYo )) | as 
where the subscript zero indicates direction of compensa- 
tion. If the radius of the hemisphere is a, and expressing 
x“,Y,#, in spherical coordinates as before, R becomes 
R= \ x0, §),expjzal singcos§ (cosa- cosdo ) 
+singsiné (cos8- cos8> )+cos@(cosy- cosyo) |} dS 
_ cos8- cos8o : 
Let tanvo i eanGCGaGs. Then the integral becomes 
[at 6){ exp jxasing,/(cosa- cosas )* +(cos8- cosBo )* 
S 
cos(e- V5) esp ocossteos y- COSyo )} dS 
*Reference 7, page 378-9 
