142 

 follows : 



-— - = when 

 8A 



BELL SYSTEM TECHNICAL JOURNAL 



XT- • . / 2TrH . . 



II = Sin • tan A • tan ( ^ — sin A 



/ 



27r(l — sin 0-cos A) 



Substituting (3) into (8) gives 



tan 



-- (1 — sin 0-cos A) 



tan 



= 2 



(1 — sin 0-cos A) 



\i 



— (1 — sin 0-cos A) 

 X 



(8) 



(9) 



This is a transcendental equation of the form tan x = 2x, whose roots, 

 to four significant figures, are 



X = 0, 



0.37l07r, 



1.4667r, 



2.4807r, 



3.4867r, 



etc. 



Alignment of the major ear of the directive pattern with the wave- 

 angle occurs only for the first solution greater than zero. Therefore we 

 have for this condition. 



/ 



0.371X 



1 — sin (j) ■ cos A 



(10) 



To obtain the dimensions for greatest output in the alignment design 

 case, / and in equation (2) are no longer independent v^ariables, but 

 are related by equation (10). We can, therefore, eliminate either one 

 we choose from equation (2) and maximize this new equation for the 

 required dimensions. 



Substituting (10) into (2) gives: 



/// 



sin 



/27r// 



I 



sin A 



cos 4) 



1 — sin cf) cos A 



and 



when 



5V 

 sin (j) — cos A. 



(Hi 

 (12) 



Since equation (12) is identical with equation (6), the recommended 

 design for directive pattern alignment with the wave-angle is obtained 

 by changing only the antenna length which was given for the maximum 



