86 BELL SYSTEM TECHNICAL JOURNAL 



the other array is given by 



V$ = 1 1 + z + • • • + s"-i |2 



= I 1 + 2z + 3z2 + \- nz^'~' + {n - l)z" 



+ • • • + 2z-"-3 + z2"- 



(10) 



Thus the amplitudes of the individual sources are proportional to 1, 2, 3, 

 • ■ • n — I, n, n — I, ■ ■ • 3, 2, 1. Figure 5 depicts the effect of such 

 "triangular" amplitude distribution. 



A UNIFORM AMPLITUDE DISTRIBUTION 

 COS (^ COS e) 



{^' 



K? 



7 cosl-y- COS e 



B^ TRIANGULAR AMPLITUDE DISTRIBUTION 



j-T^ [ SIN can COS e)1 ^ 

 '^ [4COs(5cose)J 



C- I-3-6-7-6-3-I AMPLITUDE DISTRIBUTION 

 C05^(^ COS e) 



\ry= 



27 cos^(5 COS e) 



D: BINOMIAL AMPLITUDE DISTRIBUTION 



^1"= sin6(^ cos e) 



\d 



\ 'n .'"N 



Fig. 5 — Space Factors — (A) is for a uniform array and (B) for an array with "triangular" 

 amplitude distribution. 



Evidently we could raise (9) to any given power 



Vi = 1 1 + z + z- + • • • + z»-i I' 



(11) 



This process does not change the number of separate radiation lobes. The 

 so-called "binomial" distribution of amplitudes was first suggested by 

 John Stone Stone.^ His scheme is a special case of (11) if we let n = 2. 

 For the effect of the binomial amplitude distribution see Fig. 5. 



The relative merits of two forms for the radiation intensity as given by 

 (1) and (3) can now be appraised in the light of the foregoing examples. 

 Using (1), we have for the instantaneous radiation intensity of the uni- 

 directional couplet 



6U. S. Patents 1,643,323 and 1,715,433. 



