RADIO PROPAGATION 



63 



added to the primary disturbance gives the following expression. 

 Since it is now known ^ that no exponential term must be added to this 

 expression, it may be used to calculate the received field. 



4t 



' g—2iTiRil\ g— 2)ri/22/X 



where the geometry and nomenclature are given in Fig. 4. 

 gi — R = — Ke^* is the reflection coefficient and 

 n - 1 ^ sin ^2 ^ , , cos- ^2 ^ „ 



(24) 



(25) 



where 7r/2 — ^2 is the angle of incidence and the primes denote differ- 

 entiation with respect to sin ^2- Performing the operation indicated 

 in (11) on (24) the complete expression for the received field strength 

 on vertical polarization is found to be , 



E= ^(-^^cos- ^r + -^^gr cos^ ^.+^7— 



g-2-iJ2/x^^(l _3 sin^ ^2) +2^1' sin ^2 cos^ ^2-^2 cos^ ^2 



— 3 sin^ ^i 



2wiRil\ 



R2 



iTiRi/X 



g-2TtiJi/X 1—3 sin^ ^1 g-2a-i«2/X 



Ri {iTriRi/Xy- ' Ro 



5'i(l-v3sin2 ^2) +55^1' sin ^2 cos^ ^2 



{2TiR2i\y 



-^/'cos" ^2-5'2(l-5 sin^ ^2)-2fl'2'sin ^2 cos^ ^2+^30032 ^2" 



+ L 



{iTriRi/Xy 



e~2-iR^i^ V g„_,{n- I) (l-[_n + 1'] sin^ ^2) 

 ; Ri i-l-jriRilX)" 



(2n-{-l)gn-i sin ^2 cos'' ^^-gn-i" cos-^ ^2 



(-27rii?2/X)" 

 -yn(l-[2w+l]sin ^2) -2^/ sin ^2 cos'' ^2+ gn+i cos'' I2] 1 



{-liriRilxy 



\ 



(26) 



The first term on the right of equation (26) is the vertical com- 

 ponent of the electric field radiated by a vertical electric doublet in free 

 space. The second term is the corresponding component reflected 

 from the earth. The third and fifth terms are sometimes referred to as 

 the induction and electrostatic components respectively. The re- 

 maining terms complete the effect of the ground. When the antennas 

 approach the ground Ri -^ R2, cos ^1 -^ cos ^2-^1 and gi^ — 1 so 



L 



