or, expanding the dot product: 
D=y.co0OSaty,cosB8ts.cosyto.cosy. 
a8, eg ae y ae Ve 
ee 
= = D-9. = x.cosaty,cosBts.cosytp.(cosy,-1) 
aD D p, = %, cosa ty, cosB Za y p.,( v, 
Equation (1) follows on multiplication by x. 
The response of the see element is by definition 7,exp(jwt) - 
exp( jo ,). * (See list of references at end of report. ) 
Letting the time variation factor of the response equal 1, 
the response becomes 
PR. = Hexp jo, 
If a direction defined by the unit vector (cosq,, COS85, COSy7o ) 
should be compensated for, the response of the 7t4 element 
becomes 
R, = H,exp je, expr 8, = £,exp j(6,-6, ,) 
1K 
The directivity function is by definition” 
Vv Vv 
R= ye = \ Eexp j( 8% ;) 
oil 
BA Al 
that is, the sum of the responses of the elements. 
Hence, compensating for a spherical wave, the directivity 
function for a finite number of discrete elements is 
*Page 15 of reference 2. 
**This function is also known as the directional characteristic 
or the beam pattern of the array. 
