A MATHEMATICAL THEORY OF LINEAR ARRAYS 



83 



Fig. 2 — A tj'pical direction in space is represented by a complex variable which is repre- 

 sented in a complex plane by a point lying on the circumference of a circle of unit 

 radius, having its center at the origin. As the angle 6 made by a typical direc- 

 tion with the line of sources, increases from 0° to 180°, point z moves clockwise. 



(A) 



(B) 



Fig. 3- 



-(A) The active range of z, corresponding to «? = 0f and one-quarter wave-length 

 separation between the elements. (B) The active range of s, corresponding to 

 & = pf ande = |X. 



z moves in the clockwise direction. When 6 — 0, xp = ^C — i}; and when 

 d = 180°, rp = —j3i — '&. Hence the range ^ described by z is 



^ = 2^(. 



(5) 



\Vlien the separation t between the successive elements of the array is 

 equal to one-half wavelength, the range of z = lir and as 6 varies from 0° 

 to 180°, z describes a complete cycle and returns to its original position. 



