378 PROPAGATION THROUGH THE STANDARD ATMOSPHERE 
mitter, whence Ey) = E,/d and, from equation (6), 
= —VGiA,- (10) 
Factors Affecting Attenuation 
and Gain 
The above definitions are quite general. In the 
absence of the earth, there remains only the free- 
space attenuation which results from the spreading 
out of the radiated energy as it moves away from 
the transmitter. At a distance which is several times 
larger than the wavelength, the field strength varies 
inversely as the distance from the antenna. 
The presence of the earth affects the field through 
two sets of quantities. One set is geometric and 
includes the heights of the antennas and their dis- 
tance apart, the curvature of the earth, and shape 
of terrain features. The other set is electromagnetic 
and depends on the dielectric constant and con- 
ductivity of the earth and of its atmosphere, the 
polarization and the wavelength of the radiation. 
Simplifying Assumptions 
The present chapter is mainly concerned with the 
computation of the field-strength distribution of a 
transmitter for certain idealized standard condi- 
tions, so chosen as to give a fair average picture 
of propagation conditions for very high-frequency 
radiation. The reasons for this limitation are stated 
in Chapter 1. In substance, the limitations are 
imposed by the great complexity of the general 
problem, which makes it necessary to proceed in 
successive steps. The first step is to consider propa- 
gation under standard conditions, which will be 
defined farther on. Successive steps take into 
account diffraction by terrain, that is, by trees, 
hills, mountain ranges, or shore lines, or by non- 
standard propagation effects in the atmosphere. 
The fundamental importance of a knowledge of 
propagation under standard conditions is first of all 
due to the fact that in a large number of cases condi- 
tions do not differ significantly from standard. On the 
other hand, when they do deviate significantly, the 
standard solution sets up a criterion for the discov- 
ery of deviations and the evaluation of the influence 
of the nonstandard conditions upon propagation. 
The basic assumptions which define what we have 
been calling standard propagation conditions will 
now be given. 
1. Standard atmosphere. It is assumed that the 
index of refraction of the atmosphere has a uniform 
negative gradient with increasing elevation. As has 
been pointed out in Chapter 4, the influence of such 
an atmosphere upon propagation is equivalent to 
that of a homogeneous atmosphere over an earth of 
radius ka, where k is a constant that usually is taken 
equal to 4/3. 
2. Smooth earth. The earth is assumed to be 
perfectly smooth. It ean be considered sufficiently 
smooth if Rayleigh’s criterion is satisfied, that is 
when the height of surface irregularities times the 
grazing angle (in radians) is less than 4/16 (see 
page 375). 
3. Ground constants. The dielectric constant and 
conductivity of the earth are assumed uniform. For 
wavelengths less than one meter this assumption is 
particularly valid since in this case propagation is 
largely independent of the ground constants. In 
the VHF (1 to 10 m) range, the same is true with the 
important exception of vertically polarized radiation 
over sea water. For the VHF range, the assumption 
of uniform earth constants is unsatisfactory for 
paths partly over land and partly over sea water, or 
over sea water with large land masses near-by (see 
Chapters 8 and 10). 
4. Doublet antenna and antenna gain. For the 
formulas of this chapter, the radiating system is 
assumed to be a doublet antenna (ie., a straight 
wire, short compared to the wavelength). Actual 
antennas have radiation patterns different from 
that of a doublet, usually having greater directivity. 
The antenna gain of a half-wave dipole is 1.09 times 
(or 0.4 db greater than) that of a doublet, the field 
maximum being the same in the two cases. This 
gain is insignificant in practice. For other types of 
antenna systems and for microwave frequencies, the 
gain may be many times larger. 
The propagation problem, thus limited, has been 
solved mathematically ; but the explicit mathematical 
formulas are far too complicated to be of much use to 
the practical computer. Much additional work has 
been done, however, to bring the solution into a 
form suitable for practical use. This involves 
reducing the computations to the use of graphs, 
nomograms, and tables, and it is this final stage of 
the problem which is the subject of subsequent 
parts of this chapter as well as of Chapter 6. 
Curved-Earth Geometrical 
Relationships 
Let fi; and he denote the heights of transmitter 
and receiver above the earth’s surface, respectively, 
and let d denote the distance from the base of the 
transmitter. to the base of the receiver, measured 
along the earth’s surface. For a number of cases 
concerned with high-frequency radiation over the 
earth’s surface, it is sufficient to identify the straight- 
line distance from transmitter to receiver with the 
distance d between the bases measured along the 
curved earth. But when path differences are of 
importance, as they are in interference problems of 
reflection and in diffraction, it is necessary to com- 
pute distances to a higher order of accuracy. 
Throughout this chapter, the earth will be assumed 
