404 PROPAGATION THROUGH THE STANDARD ATMOSPHERE 
already obtained. The distance at which h, = 160 
on the hi = 2 curve is the same as that at which 
ho = 2 on the h; = 100 curve. 
BELOW THE INTERFERENCE REGION 
Analysis of the First Mode 
Except for the numerical constants involved, the 
discussion for one mode applies to all the other modes. 
Each mode is of the form ®(d) - f(hx) - f(he), i-e., the 
product of a distance function ® by two antenna 
height-gain functions f. 
DISTANCE 
&,(d), the distance function of the first mode, can 
be represented as the product of two physically 
significant factors, one for free space and one for 
the earth effect. The latter may be divided into a 
plane earth factor and a shadow factor. 
1. Free space. It has been shown in equation (18) 
in Chapter 2 that for doublet antennas, with matched 
load at the receiver and adjusted for maximum 
power transfer, the free-space gain factor Ao is 
given by 
For other types of antennas in free space, this takes 
the form 
ARGRTS Nee = AGRE GD) 
1 8nd 
Unuer actual conditions when earth and atmos- 
phe ©eets are of importance, each mode will be 
con’: .cred to have Ao as one factor. 
It may also be recalled that for given power P2 
delivered to the load, the corresponding electric- 
field strength, under matched conditions, is given by 
8nV5 P. 
== ye (141) 
Combining equations (140) and (141) gives the 
free-space value of the electric field strength E, in 
terms of transmitted power 
E= Be VPWVGh , (142) 
which is the same as equation (7) in Chapter 2 with 
the addition of the transmitter gain G}. 
2. Plane earth. The earth modifies the field by 
absorbing and reflecting radiation. If the earth 
were plane and perfectly conducting, the value of 
the gain factor would be 2Ao and the electric field 
2E» for vertical antennas several wavelengths above 
the ground and at distances sufficiently large. The 
imperfect conductivity of the earth produces a 
change in the gain. Representing this effect by the 
factor A, (where A; < 1), 
A = 2A,A;. (143) 
The plane earth factor Ai depends on distance and 
on the electrical properties of the earth. Fortunately 
the earth constants enter the problem in an intrinsi- 
cally simple way, as the main effect is taken into 
account by multiplying the distance d by a certain 
factor which we shall denote by p’, so that A; is. 
mainly a function of p’d. The new parameter p’ is 
different for the two states of polarization. For 
vertically polarized radiation, 
’ _ 2n| &o 1| 
» l,l? 
where e, is the complex dielectric constant ¢, —j60a, 
P 
? 
1 _ an Ve = 1)? + (600d? 
rN 62+ (600A)? 
For horizontal polarization, 
or p (144) 
p= =| eé,—-1| = “EV, — 1)?+ (600d)2. (145) 
A, depends also on the phase of the complex di- 
electric constant. The phase is determined by the 
parameter 
€ 
: 600A oe) 
For ultra-short waves, with the exception of 
vertically polarized waves over sea water at distances 
less than 50/p’ (See text p- 416 and Figure 45) and 
a wavelength greater than 1 meter, A; is inde- 
pendent of Q and, to a sufficient approximation, is 
given by 
1 
Ay mie (147) 
A, as a function of p’d is plotted in Figure 56. 
In the case excepted above, A: deviates substan- 
tially from 1/p’d (see Figure 45) for distances less 
than 50/p’. Table 3 gives 50/p’ as a function of X. 
It appears that the deviations are immaterial for 
practical purposes as long as the wavelength is 
smaller than 3 meters, since we are usually con- 
cerned with ranges larger than 7 km. It should 
further be mentioned that, in the above case, A; 
depends to a small degree on @. However, the 
variations are less than 1 db and may be neglected 
for wavelengths less than 10 meters. 
The condition that the earth may be considered 
plane is that the shadow factor F [see equations 
(149) and (207)] shall be approximately unity. For 
F, = 0.9, which is approximately 1 db below unity, 
Taste 3. Sea water (vertical polarization). 
WS eh By Ge Dw meters 
r 
50 
reed aioe EN) Virgie 8 km 
