8 TECHNICAL SURVEY 
Peles = and (23) 
with n=1, 3, 5- - -for the maxima, 
n=0, 2, 4- - -for the minima. 
If ho > In, the angle of elevation is ¥ = h2/d and 
the formula for the maxima and minima can be 
written 
eee (24) 
If the earth curvature is taken into account the 
pattern remains essentially the same above the line 
of sight, but a’number of corrections enter which 
change somewhat the position and strength of the 
lobes. The problem is primarily one of geometry, 
taking into account the modification of the direc- 
tion, phase, and intensity of the reflected ray caused 
by the earth’s curvature. It can be solved by suit- 
able numerical and graphical methods such as are 
given in Volume 3 where the details are extensively 
treated. It may suffice here to enumerate the main 
modifying factors. 
If a tangent to the earth is drawn at the point of 
reflection (Figure 11), the distances h’; and h’s of 
transmitter and receiver from this line are the equiv- 
RECEIVER 
Ficurt 11. Geometry over spherica earth. 
alent heights in terms of which the problem is a 
plane-earth problem for that particular ray. They are 
- smaller than the heights above the ground h; and 
he, but clearly they are functions of the angle of 
elevation. Thus a set of implicit equations has to be 
solved fer each angle of elevation giving h’; and h’s 
as functions of hi, he, and d, whereupon the inter- 
ference between the direct and reflected rays is 
computed as in the case of a plane earth. 
In addition to the modification of direction and 
phase at reflection, there is also a change in intensity 
of the reflected ray caused by the fact that the 
reflecting surface is curved. This modification is 
taken into account by the divergence factor, a purely 
geometrical quantity which is part of the reflection 
coefficient, reducing the intensity of the reflected 
ray. 
The behavior of the field below the line of sight 
requires a more powerful line of attack. The line of 
sight itself is given by a tangent to the earth’s surface 
passing through the transmitter. The distance from 
the transmitter to the horizon, when a modified 
earth’s radius ka is used is 
dp = V2kah, (25) 
When k = %, hi is in meters and dz in kilometers, 
this becomes 
dp = 4.12 Sh, . (26) | 
The diffraction region actually extends at least from 
the lower surface of the first lobe downward to the 
earth’s surface. In the diffraction region well below 
the line of sight, the field strength decreases very 
rapidly and very nearly exponentially with the 
distance. 
Figure 12 shows a typical example for the ground 
mim 
° 
0.0005 
00015 10 20 30 40 50 60 70 80 90 ic 
> d IN KILOMETERS 
Ficure 12. Field strength versus distance for fixed 
height, vertical polarization. 
constants indicated. The ordinate is the ratio of 
field strength to the free space field; the transmitter - 
and receiver heights are fixed and d is plotted as 
abscissa. Above the line of sight the typical lobe 
pattern is exhibited. The decrease of the field in 
the diffraction region is the more rapid the shorter 
the wavelength. In the centimeter band this decrease 
is so rapid that for most practical purposes the field 
is nonexistent near the ground at distances exceeding 
the horizon distance by more than a few kilometers. 
Figure 13 shows a similar diagram for fixed distance 
and variable receiver height. 
Mopers 
The description of the electromagnetic field above 
the line of sight !s adequately given by means of rays 
and their phases as used in optics. This method ob- 
viously breaks down in the diffraction region into 
which the rays do not penetrate. For this region a ~ 
solution of the wave equation is required. Many 
distinguished mathematicians have contributed vary- 
ing techniques for solving the wave equation. The 
theory in its present form as applied to short and 
microwave propagation has been worked out by 
van der Pol and Bremmer?? for vertical polarization 
