380 PROPAGATION THROUGH THE STANDARD ATMOSPHERE 
Nature of the Radiation Field 
in the Standard Atmosphere 
The mathematical solution for the radiation field 
takes various forms for particular cases. The treat- 
ment for low antennas, for instance, differs from 
that for high antennas, and similarly the equations 
must be handled differently for the two types of 
polarization. These and related problems are dis- 
cussed in general in the following. 
1. General form of ficld variation. The mathemat- 
ical expression for the radio gain of the radiation 
field of a doublet under standard conditions is given 
as the sum of an infinite number of complex terms or 
modes. (See pp.4'21-428.) Disregarding the phase 
factor, a representative term (mode) of this series 
has the form 
Fd) « filhi) + fo(he) « (20) 
These modes are attenuated unequally. Well within 
the diffraction region, the first mode contributes 
practically all of the field so that the effects of dis- 
tance and height are separable. In this region, the 
problem of numerical computation is simplified, 
since it is possible to use separate graphs for the 
dependence on height and distance. As the receiver 
is moved toward the transmitter, the number of 
modes required for a good approximation increases. 
For low antennas, the addition of the modes is 
practicable and the graphical aids are useful for 
short distances. These conditions are illustrated in 
Figure 3 for horizontal polarization or ultra-short 
waves. 
In the optical region, the methods of geometrical 
optics give a result equivalent to that of the rigor- 
ous solution at points which are not close to the line 
of sight. The field is then the sum of a direct and a 
reflected wave, resulting in an interference pattern. 
The preceding discussion is illustrated by Figure 4 
which shows the variation of field strength with 
distance for fixed antenna heights, for propagation 
over dry soil with a wavelength of 0.7 meter on 
vertical polarization. The numbers refer to the 
number of modes required for a better than 99 per 
cent approximation. The interference pattern is 
illustrated by the oscillatory nature of the curve. It 
will be observed that beyond the first maximum, 
the points found by geometric optics give a value 
of the field which is slightly too low. (See dots in 
Figure 4.) In fact, as the line of sight is approached, 
the optical formula approaches, zero whereas the exact 
solution does not. The geometric-optical method 
breaks down in the optical region as the line of 
sight is approached. It may be noted that Figure 4 
has been drawn for k = 1 rather than for the cus- 
tomary value of k = 4/3 corresponding to standard 
atmosphere conditions and is for a hypothetical 
isotropic radiator. 
If the earth were flat and perfectly reflecting, the 
envelope of the maxima of the curve in Figure 4 
would coincide witk the line 2H, twice the free- 
space field, corresponding to the in-phase addition 
of the direct and reflected waves. An envelope of the 
minimum points would be HE = 0, corresponding 
to the destructive interference of the direct and 
reflected waves. The curvature of the earth, resulting 
in increased divergence of the waves (see text on 
P.384),and the lack of perfect reflection (see text on 
p.383) cause the maximal and minimal envelopes to 
differ from 2p and 0, respectively. In the neighbor- 
hood of the first maximum in Figure 4 (i.e., when the 
direct ray makes small angles with the earth), the 
reflection coefficient tends to be unity in magnitude 
for both polarizations except for the increase in diver- 
gence which results in the deviation of the maxi- 
mal and minimal lines from 2H and 0, respectively 
At a smaller distance, for vertical polarization, as 
shown in Figure 4, the deviation is caused principally 
by the smaller magnitude of the reflection coeffi- 
cient. The virtual meeting of the maximal and 
minimal lines corresponds to the minimum value 
of the reflection coefficient at the pseudo-Brewster 
angle. (For horizontal polarization at small dis- 
tances, the envelope of maxima would virtually 
coincide with 2 and the minima would be closer to 
zero. As the distance is increased, the difference 
between the envelopes for vertical and horizontal 
polarization gradually decreases.) 
2. Both antennas low; h < h,. In a discussion of 
the height function, it is convenient to distinguish 
between high and low antennas. The critical height 
separating the two cases for horizontal polarization 
or ultra-short waves is given by 
he = 300 meters (21) 
where ) is expressed in meters. For \ = 0.1 meter, 
h, = 6.46 meters, and for \ = 10 meters, h, = 139.5 
meters. If both antennas are at elevations less than 
h,, the height-gain functions f(h), to a first approx- 
imation, are the same for all the modes, so that the 
complete solution 
filha) « fhe) - Fi(d) + fa(ha) « folhe) - Fad) ++ + + 
can be written in the form 
Sha) + fhe): (Fi Fa ++--), (22) 
f(x) - fe) - Fd), (23) 
where f(ii) replaces fi(fi), f(he) replaces fi(he), etc., 
while F(d) stands for the sum F', + Fo+---. 
The distance function F(d) can be calculated for 
particular cases. This has been done for high fre- 
quencies and is represented graphically on pp.403 - 
432 ,the results being valid for low antennas for all 
distances in the optical as well as in the diffraction 
region such that 2hih, << dd (see Figure 3). The 
condition 2hih, < < dd assures that the antennas are 
below the interference pattern. 
or 
