FACTORS INFLUENCING TRANSMISSION 375 
Fiaure 18. Geometry for divergence factor. 
sphere like the earth, the divergence of a bundle of 
rays is increased when it suffers reflection, and the 
plane earth reflection coefficient, R, must be multi- 
plied by a divergence factor denoted by D, which 
accounts for the earth curvature. This factor ranges 
from unity at close range where the earth can be 
considered plane to zero at points just above the 
tangent line. [Note: When the divergence factor 
approaches zero at grazing angles less than the last 
minimum, other components of the wave must be 
considered.] To a sufficient approximation D is 
given by the expression 
b=|1+ 
where (see Figure 18), 
yr 4-} 
2hy he | (24) 
dka tan? y 
hy’, ho’ = heights of transmitter and receiver above 
tangent plane at reflection point. 
d = distance between transmitter and re- 
ceiver measured along the surface of 
the earth. 
y = grazing angle above tangent plane. 
ka = equivalent earth radius. 
Irregularity of Ground 
The formulas for reflection from a plane or a 
spherical earth can only be applied with confidence 
granting a certain smoothness of the reflecting 
surface, depending on the wavelength. A rule of 
thumb for the applicability of the reflection formulas 
is that the vertical height of the irregularities should 
not exceed 4/16, where \ is the wavelength and y 
the grazing angle in radians. Suppose, for instance, 
that the wavelength is 1 meter and the grazing angle, 
1 degree. The limit of tolerance is then 56/16 = 3.5 
meters. Hence, on this wavelength one may expect 
specular reflection over sea in most cases. For 
d = 10cm, on the other hand, the limit is only 35 cm, 
ana for \ = 3 cm it is 11 em. For larger grazing 
angles, the limit of tolerance will be correspondingly 
smaller. 
DIFFRACTION (GENERAL SURVEY) 
Definition 
The term diffraction will be understood to apply 
to those modifications of the field produced by 
material bodies outside the transmitter that cannot 
be described by the ray methods of geometrical 
optics. 
With this limitation of the term diffraction, there 
are three main topics to be considered: 
1. Diffraction by the earth’s curvature. 
2. Diffraction by irregular features of the terrain, 
such as hills, houses, ete. 
3. Diffraction by objects, primarily metallic 
(targets) in two-way transmission (radar echoes). 
Also scattering by raindrops. 
Diffraction by Earth’s Curvature 
The diffraction field in this case is the field appear- 
ing below the line of sight determined by use of the 
equivalent value of the earth’s radius ka. The case 
of an idealized earth with smooth surface and given 
electrical properties can be treated mathematically, 
and the field obtained is often designated as the 
standard field (see Chapter 5). If one moves away 
from the transmitter horizontally, at a fixed height 
above the earth, the field strength decreases expo- 
nentially with distance once the line of sight is 
passed. Similarly the field strength decreases expo- 
nentially with height above the ground on going 
vertically downwards from the line of sight. In 
many instances, the variation in the field strength, 
in the diffraction region is independent of the electric 
properties of the ground. The main exception occurs 
in a comparatively shallow layer near the ground. 
Only for the important case of propagation over sea 
water and for frequencies below 100 megacycles 
does this layer become high enough to cover an 
appreciable part of the whole diffraction region. 
Diffraction by Terrain 
The problem of diffraction by terrain features 
requires special treatment. Frequently a field of 
appreciable magnitude is found behind hills, houses, 
etc. Diffraction is also important when there is a 
sudden change in ground properties, as for instance 
in a transition from land to sea. In this case the 
shore line acts as a diffracting edge. Only a limited 
number of cases lend themselves to evaluation by 
simple formulas. The cases aré those which can be 
treated by the Fresnel-Kirchhoff method of optics 
which leads to a somewhat intricate but straight- 
forward mathematical formula for the diffracted 
field strength. In spite of its apparent limitations, 
