STANDARD PROPAGATION 9 
and Marian C. Gray for horizontal polarization. 
The results of the theory may be summarized as 
follows. The electromagnetic field can be represented 
as an infinite series of the form 
E = Ey Vd >, ene—Snd Um(ti) Un(hs) (27) 
where E) is the free space field and c,, and [, are 
complex constants depending upon |the wavelength 
and the electromagnetic ground constants. h; and he 
are again the heights of the transmitter and receiver 
above the ground, d is the distance between the two; 
U,, are the height-gain functions, and e is the base 
of natural logarithms. The formula is symmetrical 
with respect to the interchange of transmitter and re- 
ceiver, in agreement, with the principle of reciprocity. 
19,090 
FREQUENGIES 
3000 MG 
100,200,500 MC 
HORIZONTAL POLARIZATION 
1000 
7) 
m 
> 
HEIGHT IN METERS 
10 
-200 -180 -160 
DB 
Fach of the terms which compose the sum (27) 
is called a mode. The coefficients ¢,, are complex con- 
stants with their real parts positive. They represent 
therefore an exponential decrease of the field strength 
with distance. The real part of ¢, is the attenuation 
factor of the mth mode expressed in nepers per unit 
distance. The height-gain functions U,, are found to 
increase with height above the ground. The increase 
is first slow but eventually becomes exponential and 
remains that way for large heights. 
The real part of fm, the attenuation factor, in- 
creases with increasing mode number; hence, if the 
receiver is far enough from the transmitter, all 
modes except the first one become very small and 
the sum in equation (27) reduces to its first term 
which can be computed without much difficulty. 
Z| 
eT 
SS 
q 
Ns 
\ 
AT SSS 
CATLIN 
-140 -120 -100 -80 
Figure 13. Field strength versus height of receiver for fixed distance relative to radiation field at one meter from 
transmitter. 
