such that if properly chosen will produce a high main lobe 
and low side lobes in a given directivity function.™ If 
E(x, y, Zz) = 1 or 1g C= 1, then the equations reduce to the case 
of an unshaded array (i.e., all elements equally weighted). 
So far the directivity function has been concerned principally 
with the effect of phase differences. However, the sound 
pressure will also affect the pattern in that it decreases 
with distance from the source. This effect can be taken 
care of, in the continuous distribution case, by letting 
sles 
Pd 
hi D 
E (x,y) =4 (x, y) G5) 
Then, the uncompensated directivity function is given by 
p= | awl, Woy fexps1D-olx, we 
y) XH» | 
In the above equation, w(x, y) is a shading factor, CS 
is a weighting factor that takes account of the decrease in 
sound pressure amplitude with distance from the source, 
and D-o(x, y), the exponent of the exponential function, is 
the phase difference of the element at (x, 1) with respect 
to the origin. There is a corresponding, similar result 
for the discrete distribution case: 
N 
= D ; — 
R )) Dy exp jz[D p, | 
ae @ 
Y 
D 
In most applications a 1, so that F(x, y)~wls, y). 
Thus this correction factor for the decrease of sound 
pressure amplitude is absorbed into the factor 7x, y). 
*Reference 2, page 16, and reference 3, page 15. 
**This treatment is for the 2-dimensional array, but it 
can be extended to 3 dimensions. In either case, a spheri- 
cal wave is incident on the array. 
