18 
Substituting these polar coordinates in equation (3) and 
FA 0 
t 
WV 
R= > B,expJk jr I(cosp , (cosa- costo sing, (cos8-cos8o ) | 
Gai 
+ - 
p, (7, ,\(cos¥,(r, 9 ,) 1) 
It is easy to verify the relationships 
cosa = cos@sin y 
cos8 = sin§ sin y 
obtained by introducing spherical coordinates. Substituting 
into R and simplifying: 
WV 
R= ) B,expJjh | r[(sinycos(6-9 ,)- sinyo COS(Qo - ,)1 
t=1 
+o,(8,8, (cosy, (6, 0,)-1) 
Let D-~; then, the directivity function becomes that of a 
planar wave front for a circular array: 
MV 
R= ) B,expjkr| sin y(cos(6-9 ,)- sinyo cos(8o -9,)] 
Call 
Let WV = 2m elements be placed symmetrically with respect 
to the x-axis with equal angular displacement between any 
two consecutive elements. Let this angular displacement 
be 7.5°; then 2; = (t+1/2) 7.5° for the ith element. 
Further, let 6 = 0° and y = At WOOm sethen 
W/2 
R= ) Bvexpsjer[cos(-(i41/2)7. 5°)-cos(t+1/2)7.5°]* 
i=-W/2 
*Reference 3, pages 13-15 
