390 PROPAGATION THROUGH THE STANDARD ATMOSPHERE 
The dependence of path difference upon distance 
and height may be seen by considering the path 
difference parameter 
_ kaa 
Indr 
Since he’ = hy'd2/d,, it follows from equations (78) 
and (59) that 
(83) 
d? \2 
i Qhy'ha! _ 2(Ii')?dz _ 2de he ( a) 
d dd, d d, 
and 
{1—(di/dr)?? 
di/dr 
Hence, using dy? = 2kahy, 
p=-@ 
d 
x 
_ 2h? (d\ [1 — (hh/dz)’? ot 
age ee 
= iS _ o/ 2) [1 = (di/dr)??P (85) 
d/dy di/dp 
The form of this expression suggests the introduction 
of two new dimensionless parameters 
d 
=—andign ee 86 
aun v ‘a (86) 
In terms of these parameters, equation (85) for R 
assumes the form 
z= (1-2)S—PY, (87) 
v Pp 
and in terms of s and v 
R= (ey mca) (88) 
sv 
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 preceding text. 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 
Vi + 2hi'he!/kad tan? y 
Using equation (60), D becomes 
1 
Ss 10 
V1 + 2d,d2/kah'd « CY) 
If &— d, 
A <3°) (91) 
7 Via oh! /ka tan? y 
and if y is small, so that tan y —>y, 
1 
7 Vion /kay? 
(92) 
Parameters p and q 
Useful expressions for the divergence factor, path 
difference, and receiver height may be obtained by 
use of the dimensionless parameters, 
dy dy 
= =2 93 
sisi, | Gp cy) 
and 
7-5 (94) 
or 
hs @= deed, (95) 
The divergence factor may be expressed directly 
in terms of p and g by modifying equation (90) as 
follows: 
jah aE 
V1 + 2d,2d,/kahy'd 
1 
ali jy ae as) Nee eines) 4(d2/d) (d2/2kah,) 
1— 1 = (@2/2kah,) 
D= 
where h;’ has been replaced by its equivalent ex- 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: 
1 
V1-+ 4p?q/(1— p*) 
Figure 17 shows 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): 
(96) 
ae 2hy! he! i: 2(hi’)? do Ns 2de hy? (i— dy?/2kahy)? 
d dd, d d; 
Hence, on using equations (93), (94), and (95), 
2 — »y2)2 
iyo 2e Cae (97) 
dp Pp 
Figure 18 shows (1 — p?)?/p asa function of p. 
The receiver height he may also be expressed in 
terms of i, p, and g. This will be found useful in 
drawing coverage diagrams in which both hz and d 
are unknowns. From equation (60) 
or 
