INTRODUCTION 
The work described here was done as part of the general 
program to improve long-range sonar receiving arrays. It 
had a twofold purpose: first, to show that previous work 
on array directivity functions can be subsumed under a 
more general method, and, second, to bring together 
results that are scattered throughout the literature. The 
advantages of the general results of this report are that 
the directivity pattern can be determined for any distance 
and direction of the sound source from the array, and for 
any geometrical configuration of the array. 
PART I - GENERAL THEORY 
1. Discrete Distribution of Elements 
If the source of the sound wave is at a distance D from the 
origin of the array, then the sound wave will be spherical 
in shape when arriving at the array. Furthermore, the 
sound wave will expand radially from its source. Let 
(x0, Yo. Zo) be the coordinates of the source with respect 
to the origin of the array. Let (x;,y;,2;) be the coordi- 
nates of the 7'% element of the array. A unit vector in the 
direction of the line joining (0, 0, 0) and (xo, Yo» 20) will be 
(cosa, cosb, cosy), the direction cosines of the line. Then, 
from figure 1, the phase difference with respect to the 
origin from the 7°" element is 
ae + + + = 
a, # [ x, cosa y, Coss Zz, Cosy p, (cosy, 1)] (1) 
where & = Zing and } is the wavelength of the incoming 
sound wave. Equation (1) is obtained as follows: 
D=(x.,y.,2.) ° (cosa, cos8, cosy)+p, cosy, 
U U U U U 
