CALCULATION OF RADIO GAIN 403 
where doy is the free-space distance corresponding to 
the given value of A. pis found from the value of , 
the grazing angle at the reflection point given by p’ 
found above in the calculation for p=1 and 
@ = 180°. For the new contour tip p changes, but 
the angle y does not change sharply, so that p and @ 
as found for the p obtained by the simplified calcula- 
tion is a close approximation. 
For p #1, d ~ 180°, 2 = 6+ ¢ — z= [see equa- 
tion (29)] consists of two parts. 6 = rR/r = (A/d) 2r 
= nrand d’ = d — x. Writing Q = Nz, N the lobe 
number for the case p ~ 1, @ ¥ 180°, the require- 
ment for the new lobe tip is N = 1; for the first 
maximum Q = 7 = Rr/r + ¢ — zw where R is the 
path difference variable at the new lobe tip. Hence 
the value of R at the new tip is given by 
R=r(2—¢/nz). (139) 
Using the results in (2), 
p = 0.10385, 
D = 0.9788, 
r= 9.4, 
dy = 566, 
we find from Figure 24 or equation (107), = 0.725°; 
from Figures 14 and 15 in Chapter 4, p = 0.76, 
¢@ = 170°. Substituting in equation (138), 
d = 566 (1.744) = 987 km. 
From equation (139), 
R = (9.4) (1.056) = 9.92. 
Also 
The above value of p can now be improved by 
substitution in equation (137) with D replaced by 
pD which gives 
p = 0.0985. 
In the calculation just made, p has been assumed 
constant. This can be checked by finding y as de- 
termined by the new value of p. The result is 
Ww = 0.766° and the corresponding values are p = 0.74 
(as against 0.76 previously) and @ = 169° (as against 
170°), which is good enough. We now find 
— P _ 0.0985 _ 9 ogg04 
v 43. 
Ig 2,359 
hy 
h, = 70,770 meters. 
Hence the new maximum point of the lobe is at a 
distance of 987 km and at an elevation of 70,700 
Meters, aS compared with a distance of 1,110 km 
and height 86,900 meters for perfect reflection. 
Maximum Range Versus Receiver 
(or Target) Height 
If the value of A has been determined by using 
the minimum detectable power in equation (38) 
or (5), the corresponding contour is a curve of 
maximum range versus receiver height for com- 
munication or maximum range versus target height 
for radar. Generally, the lower part of the lowest 
curve (n < 1) is of greatest interest. If A is suffi- 
ciently small (i.e., 20 log A numerically large and 
negative), the complete contour has points below 
the line of sight. If the transmitter antenna is low 
(li < 30° meters), the lower points are likewise 
given by the diffraction formula, discussed in text 
on pages 380-413. Several such curves, the lower 
part of the lowest lobe corresponding to various 
transmitter heights for \ = 1 meter and 20 log A 
= —130, are given in Figure 27. 
Consider, for example, the curve for h; = 2 meters. 
The uppermost points of the curve correspond to 
the tip of the lowest lobe and were found by the 
procedure used on pp.400-402, putting n = 1. 
The lowest points were found from the diffraction 
formula by the method presented before, with the 
aid of Figures 31 to 36. It has been pointed out 
that for n < 1, the optical interference formula is 
inadequate. ' 
To locate a point between the upper extreme 
(n = 1) and the diffraction points, a curve of A 
versus hp for some distance is constructed. In 
Figure 28, a set of such curves is given for the dis- 
tance 100 km and for various transmitter heights. 
By taking the intersection of 20 log A equal to —130 
with the curve h; = 2, a value of he is obtained. 
This value he = 3,300 and d = 100 km represent 
the coordinates of a point on the contour h, = 2 in 
Figure 27. 
It is of some interest to observe the shortening 
of the lobe for h: = 100 meters on account of the 
divergence which is close to unity at the tips of the 
lobes corresponding to the low transmitter heights 
but drops to 0.65 at the tip of the lobe for h; = 100. 
Since for h; = 100, the formula for both antennas 
low (h < 30X7’%) is not applicable, the lowest points 
on the curve h; = 100 in Figure 27 were obtained 
by applying the reciprocity principle to the curves 
32 
1 
-201 
UIT 
100 1000 19000 100009 
ha IN METERS 
FiGurE 28. Radio gain versus receiver height hs, for 
given values of transmitter height h,. 
