104 BELL SYSTEM TECHNICAL JOURNAL 



. (n — 1)4/ . in — k)^ 

 sm i-, — ,, • • • sin ^, -^ 



pQ ^ I, pn~l-k = pk , pk = 



sin —. TV • . . sin —, rr (^2^ 



2(w — 1) 2{n — 1) ^^^J 



2 2{n — 1) 



i ( n— 1) < 



divide the last expression in (50) by e ^ and combine the terms equi- 

 distant from the ends. Thus we obtain 



/— r. (n — 'i.)<p , r, (n — 3)(p 

 V* = 2 cos ^ ~--^ + 2pi cos ^ 2"^ 



(53) 



(w - 5)^ , 

 + 2p2 cos ^ — ~ — — + • • • , 



where the last term is 2pn_ , cos - if n is even and p "-i if n is odd. 



2 2^ 2 



Let D be the maximum value of \/$; then the gain of the array over a 

 single source is given by 



iirD^ 21)' 

 G = 10 Logio = 10 Logio -^ decibels, (54) 



Ih'" [ 



$ sin 9 dd 



where 12 is the solid angle and the integration is extended over a unit sphere- 

 For an end-on array with nulls equispaced in the range of z, the maximum 

 radiation is in the direction ^ — 0. Thus we shall have 



(w — l)(po . r,. (n — 3)(po 

 D = 2 cos ^ + 2pi cos ^ 



+ 2p2 cos —^— + 



(55) 



where 



^0 = iS^ + -^ + X. (56) 



w — 1 



A convenient expression for the radiation intensity can be obtained from 

 (50) by taking its norm 



^ = [po + pie^^ + ^e'^^ + • • • + pn-ie'^"-^^^] 



(57) 

 •[/>o + pie-''^ + • • • + />.-ie-'^"-^'1. 



