GAIN OF DIRECTIVE ANTENNAS 87 



The equation for the diagram in the (XY) plane may be had by 

 placing 6 — ir/2 giving 



sin (iVx^ sin </)) tt, ^ .. ,. .. 



^ = AT • / — 1—- — ttcos- (cos 4> — 1), (14) 



iVsm (tt^I sm ^) 4^ ^ ^ 



which is the equation of the diagrams in Fig. 5 above. The corre- 

 sponding equation for the principal vertical section may be had by 

 placing = and = tt giving 



sin (NttA cos 6) tt , . . <n • „ 

 cos — (sm 6—1) sm d 



Nsin {tA cos 6) 4 

 and 



sm (NtA cos d) T . a \ A\ • a 



-rr^—. 7 -. 77 COS — (sm 0+1) SHl Q 



Nsm (ttAcos 6) 4 



(15) 



which is the equation for the diagrams of Fig. 17. 



The diagram of a single linear array of point sources is specified 

 by the first term of equation (12) where 6 = x/2 or 



sin W7r(a cos <^ + 6) ,... 



^~ = 7 , I , X • (16) 



n sm 7r(a cos <p -\- o) 



The diagrams of Figs. 3 and 4 above may be calculated from equation 

 (16; by placing w = 2 and n = 16, respectively. This also agrees with 

 Foster's equation (1), page 307.^ 



The diagram of a field of coplanar linear arrays such as depicted 

 in Fig. 16 above follows from equation (12) by placing N = 1, a = I 

 b = - I SLXid B = 0. 



If the diagram is to be restricted to the (XY) plane, 6 = t/2 and 



• /TVT A • ,\ sin I w- (cos — 1) I 

 _ sm (NtA sm 0) ^ \ 4 ^ V ^ . ^. 



^ ~ iV sin (tt^ sin 0) ' . / tt , ^ ,,\' ^'^ 



w sm I - (cos (/) — 1) I 



Calculated Gains from Arrays 



The flow of power through each unit area due to an advancing 

 electric wave is given by the Poynting vector as 



s=^EXH, (18) 



47r 



where E and H are vectors representing respectively, the electric and 

 magnetic components of the advancing wave. 

 2 Loc. cit. 



