GENERAL SOLUTION 



69 



In calculating the field intensity reflected from a 

 spherical earth, the inverse distance attenuation 

 factor l/r^ used for the direct wave must be multi- 

 plied by the divergence factor D, which is always 

 less than unity. 



As a result of this divergence the reflection 

 coefficient for a spherical surface is less than that 

 for a plane surface as given by 



pD = p' (38) 



^\here p' is the spherical earth reflection coefficient. 

 Equations (30) and (36) may then be written as 



E 



= Eoyj{l - p'Y+ip' sin- 



(39) 



and 



5.2.5 



= d„^ 



(1 



p')^ + 4p'sin^^- 



(40) 



Antenna Gain and Directivity 



The effects of antenna gain and directivity are 

 expressed by means of the gain factor G, defined in 

 Section 2.2.2, and the antenna pattern factors Fi 

 and Fi, which are the fractions of the maximum radi- 

 ation amplitude in the direction of the direct and 

 reflected rays respectively. The maximum ampli- 

 tude for a transmitting antenna mth gain Gi is 

 Eo = VGi Eq, where £"0 is the free-space field 

 strength radiated by a doublet. 



Wlien the antenna pattern factors are taken into 

 account, the resultant field intensity, following 

 equation (30), is 



E = ylCh E, V Fi^ + F-^p-D- - 2F1F2PD cos 9.. (41) 



III 1 1 III III II II in 1 11 1 n I II 1 1; I' 1 1 III 1 1 n 1 1 1 1 II n 1 1 

 Figure 11. Antenna pattern factors Fi and F2. 



The factors Fi and F-, are functions of the angles 

 7i — a and V + a which the direct and reflected 

 rays make with the axis of the beam. Figure 11 is a 

 typical antenna pattern. 



53 GENERAL SOLUTION 



^ * ' Generalized Reflection Coefficient 



Equation (41) may be simplified by introducing a 

 generalized coefficient which includes the effects of 

 reflection, divergence, and directivity. The ampli- 

 tude of this coefficient will be denoted by K and is 

 given by 



(42) 



Fo 

 A' =^pD. 



F^ 



Substituting F^ = FJi/pD in equation (41) gives 



E = <(hFiE^ V 1 -t- K" - 27v cos Q (43) 



or 



E = ^GiFiEoyj (1 - K)" + 4K 



sm- 



(44) 



Figure 12. 

 K and fi. 



^(1 - A', 



-H 4iv sill- as a function of 



2 



If the transmitting antenna is pointed so that the 

 direct ray Ues in the direction of maximum gain, 

 Fi = 1, and 



and 



- = VG'iFo V (1 - T^Y + 4K 

 d = Vc^rfo V (1 - Ky- -I- 4A' 



sin' 



(45) 



2 



sin^ - . (46) 

 2 



