SPHERICAL EARTH 



75 



The dependence of path difference upon distance 

 and height may be seen by considering the path 

 difference parameter 



R = 



kaA 



(83) 



Since hi' = Ih'd^/di, it follows from equations (78) 

 and (59) that 



. _ 2h'h2' _ 2 (fe/)% _ ?^u 2 

 d ddj, d 



\ 2kahJ 



di 



Hence, using dj-^ = 2kahi, 



, [1 - (di/drY]' 



dr \d/ 



di/df 



(84) 



and 



^ ^ rf2 ^ [l-jdr/drn 

 d di/di' 



= f 1 - '^ll^') [1 - W'^T 



\ d/dr / di/dr 



■y? 



(85) 



The form of this expression suggests the introduction 

 of two new dimensionless parameters 



di , d 



p = — and V = — 



df dy 



(86) 



In terms of these parameters, equation (85) for R 

 assumes the lurm 



R 



= {l-P\ (1 - P')" 



vJ 



and in terms of s and v 

 R= (l-s) 



(1 - svy 



sv 



(87) 



(88) 



'•*•* Divergence Factor 



The reflection of a beam of radiation from the 

 spherical earth increases the divergence of the beam 

 and reduces the intensity of the reflected wave by 

 spreading, as explained in Section 5.2.5. This is 

 taken into account by introducing a divergence 

 factor D, less than unity, which appears in the formu- 

 las as a multiplier of the plane earth reflection coeffi- 

 cient. Expressions for D are 



1 



D = , . (89) 



Vl + 2Wh2'/kad tan' ^ 



Using equation (60), D becomes 



1 



If do 



D = 



D = 



V 1 + 2dM.2/kaWd 



(90) 



Vl + 2hi' /ka tan- ^|^ 



(y < 3°) (91) 



and if 4' is small, so that tan \}/ — > i/-, 



1 

 D = 



V 1 + 2hi'/ka^p- 



(92) 



5.5.7 



Parameters p and q 



Useful expressions for the divergence factor, path 

 difference, and receiver height may be obtained by 

 use of the dimensionless parameters, 



di di 



V = 



'\2kahi df 



and 



or 



d-i 



di = (1 -q)d = sd. 



(93) 



(94) 



(95) 



The divergence factor may be expressed directly 

 in terms of p and q by modifying equation (90) as 

 follows: 



D = 



1 



Vl + 2diM2/kahi'd 

 1 



V 



1 + 



4(d2/d) {d{-/2kah) 

 1 - (diV2fca/ii) 



where W has been replaced by its equivalent ex- 

 pression, given in equation (58). The above form 

 of D shows that it can be expressed in terms of 

 p and q only: 



D = ^ = . (96) 



Vl+4p-g/(l-p^) 



Figure 17 show's contours of constant D as a function 

 of p and q. 



The path difference A may be written in terms of 

 p and q by substituting into equation (78) : 



2h'h' 2{W)-d2 _ 2d2 , 2 {\-d,y2kahy 



A = 



d 



ddi 



Jh' 



di 



