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PROPAGATION OF RADIO WAVES 399 
0.039 < 10-* units per meter. 
There are several reasons why this book concerns 
propagation in the moist standard atmosphere. 
1. The atmosphere in certain places (particularly 
the temperate zones) and over considerable periods 
of time is substantially standard in character. 
2. Calculations based on the standard atmosphere 
serve as a standard against which propagation in 
nonstandard atmospheres may be compared. 
3. A great deal of propagation information now 
available in the field is based on propagation cal- 
culated for standard conditions. 
Propagation in the Moist 
Standard Atmosphere 
The radiation energy emitted by a transmitter is 
@ wave spreading out in three dimensions, which 
may be represented by a series of concentric spherical 
wave fronts or by a system of lines called rays. The 
velocity at any point on the wave front is given by 
c 3X 108 
n n 
meters per second. (2) 
Since 7 decreases with height, the upper portions of 
the wave front move with higher velocities than the 
lower portions, and the wave paths as represented 
by the rays are curved slightly downward, as shown 
in Figure 2. 
The radius of curvature of the rays p is given by 
i =— = = + 0.039 X 10-® units per meter (8) 
Pp 
and p, therefore, is equal to 25.5 X 10° meters, which 
is approximately four times the radius of the earth 
(p = 4a). As a result the distance to the radio horizon 
is some 15 per cent greater than the geometrical line- 
of-sight distance from the transmitter to the horizon. 
This curvature of the rays by the atmosphere is 
called refraction. 
Ficurse 2. Curvature of rays in the standard atmos- 
phere. 
For the purpose of calculating wave propagation, 
only relative curvature of the earth and the rays is 
of interest. Compensation is made for the effects 
of refraction by replacing the actual earth with a 
radius a by an equivalent earth with a radius ka 
and replacing the actual atmosphere (in which the 
index of refraction n decreases with height) by a 
homogeneous atmosphere (constant n) in which the 
rays are straight lines. Since 1/a is the curvature 
of the earth and 1/p the curvature of the rays, we 
may set their difference equal to 1/ka, the curvature 
of the equivalent earth. Thus 
and 7 (4) 
Since p = 4a, k = 4/8, and ka, the radius of the 
equivalent earth, equals 8.49 X 10° meters. See 
Figures 4 and 5 in Chapter 4. 
Propagation in Nonstandard 
Atmospheres 
Though this subject is beyond the scope of this 
volume it is desirable to present a brief discussion 
of the salient features. 
Not infrequently the lower atmosphere is stratified 
in horizontal layers in which the variations with 
height of the temperature and moisture content 
are nonstandard. Of particular interest is a sharp 
rise in temperature with increasing height (tempera- 
ture inversion), or a sharp decrease in water vapor 
content, or a combination of the two. If these varia- 
tions from the standard distribution are sufficiently 
great, horizontal radio ducts may be formed in the 
atmosphere. In this event an appreciable fraction 
of the wave energy (only that fraction moving in the 
nearly horizontal direction) may be constrained to 
propagate along the duct to distances far beyond 
the horizon and the field strength may be large 
compared with that obtainable under standard 
conditions. This phenomenon produces a marked 
bending of the wave paths or rays and is known as 
super-refraction. To take fullest advantage of this 
phenomenon the antennas should be located in the 
duct. 
Ducts are of various types: 
1. Overland. These are surface ducts formed at 
night by the radiation cooling of the earth. 
2. Oversea. In the trade-wind belt there appears 
to be a continuous duct of the order of 50 ft thick 
starting at sea level. 
3. Land to sea. Warm dry air flowing from land 
out over cooler water often yields surface ducts 
100 or more feet thick. 
4. Elevated. Caused by subsidence of large air 
masses, these ducts may be found at elevations of 
perhaps 1,000 to 5,000 ft and may vary in thickness 
from a few feet to 1,000 ft. They are common in 
Southern California and certain areas in the Pacific. 
