302 BELL SYSTEM TECHNICAL JOURNAL 



The similarity among the diagrams for arrays of antennae within a 

 given fixed interval is illustrated by Figs. 8, 9, and 10. 



In Fig. 8 are shown diagrams for 2, 3, and 4 antennae situated 

 within a total distance of one wave-length with separations of 1/2, 

 1/3, and 1/4 wave-length, respectively, and with zero phase difference 

 between adjacent antennae. The curve (oo) gives the limit of the 

 family of directive diagrams for arrays of n antennae with a separa- 

 tion of \/n wave-length, as n becomes infinite. 



Fig. 9 gives similar curves for arrays of n antennae within an interval 



of two wave-lengths, with the parameters ( -X, 07") . Fig. 10 shows a 



\ 11 / 



similar set of diagrams for arrays within an interval of one wave- 

 length and within a total phase inter\'al of one period, that is, for 



( \ '1 \ 

 arrays of n antennae with the parameters I -X, -T\ . 



For any interval {A\, BT) a similar family of curves can be ob- 

 tained for arrays of n antennae with the parameters (—X, —T). As 



^n n ^ 



the number n is increased without limit, the directive diagram ap- 

 proaches a limiting curve. This limiting curve never has more than 

 two directions of unit amplitude. There are zero, one, or two such 

 directions, depending upon whether A is less than, equal to, or greater 

 than B. The diagrams for the infinite case are reflected about the 

 vertical axis upon changing the sign of the phase difference, but they 

 do not repeat cyclically with inci easing phase difference. 



The rapidity with which the diagrams approach this limiting curve 

 as n is increased is well illustrated by Figs. 8, 9, and 10. On the 

 scale of these drawings, the curves for 16 antennae would be indis- 

 tinguishable from the limiting curves for the infinite case. The upper 

 left-hand corner of Fig. 5 may thus serve as a chart of the directive 

 diagrams for the infinite case if the column and row headings are 

 multiplied by the factor 16 to give the total separation and phase 

 difference of the interval. For larger values of these parameters, 

 however, the curves for the infinite case depart more and more from 

 those of Fig. 5. 



The diagrams with A=B are of particular interest since these are 

 unilateral, with the main lobe growing narrower as the total separa- 

 tion and phase difference are increased. In the case of the Beverage 

 antenna, the ideal system " consists essentially of a long loop, which 

 we may think of as the limiting case of a succession of a large number 

 of narrow loops. The directive diagram of such an antenna system 



" H. H. Beverage, C. W. Rice, and E. W. Kellogg, loc. cit., pages 372, 373. 



