Chapter 4 
FACTORS INFLUENCING TRANSMISSION 
REFRACTION 
Survey 
EFRACTION is caused by the variation of the 
dielectric constant (square of refractive index) 
of the atmosphere. Although the atmosphere is 
tenuous and the variations of refractive index 
are small, the effect of refraction upon the field- 
strength distribution of waves is considerable. As 
will be shown, refraction under average conditions 
may be taken into account by using an earth with a 
modified radius. A representative average value of 
modified earth radius commonly used is ka with 
= 4/3. Under certain conditions, especially in 
warmer climates, a slightly higher value of k might 
be preferable. 
The case where a is replaced by 4a/3 is referred to 
as standard refraction. It corresponds to a linear 
variation of refractive index with height in the 
atmosphere. In recent years, more complicated 
variations of refractive index in the atmosphere 
have received considerable attention and have 
proved to be of great operational interest. This 
volume, however, is restricted to consideration of 
standard atmosphere propagation. 
Snell’s Law 
Let mo and 7; denote the refractive indices of two 
media separated by a plane boundary. The ordinary 
law of refraction known as Snell’s law is then usually 
stated (see Figure 1), as 
Mm Sin Bo = m Sin fi, 
where Bo and #; are the angles which the ray makes 
with the perpendicular to the boundary. It is con- 
venient to use the complementary angle a, so that 
NM COS ao = NM COS ay. 
Ficure 1. 
y Refraction at boundary between two 
media. 
364 
For several plane-parallel boundaries, Snell’s law 
generalizes to 
NM COS Ao = Ny COS A] = Ny COS QA. =—-ee. 
In the atmosphere, the refractive index is a con- 
tinuous function of the height. Again, it is usually 
legitimate to consider the atmosphere as horizon- 
tally stratified, so that the refractive index is a 
function of height only. The case of a continuously 
variable refractive index is readily obtained by 
passing to the limit of an infinity of parallel bound- 
aries infinitely close together, Snell’s law remaining 
the same; thus 
n(h) + cos a = No COS ao, 
where now n and @ are continuous functions of the 
height. In place of a discontinuous change in direc- 
tion, there will now occur a bending of the rays 
(Figure 2). 
HEIGHT 
DISTANCE ALONG EARTH 
as 2. Refraction in the atmosphere with variable 
n . 
If the boundaries are not plane but spherical, 
Snell’s law must be modified. Analysis shows that 
over a spherical earth surrounded by an atmosphere 
in which the refractive index n is a function of the 
distance r from the earth’s center, the law of re- 
fraction becomes 
() 
where a is the angle between a ray and the horizontal 
(see Figure 3). 
Refraction is of practical importance only when 
the angle between the rays and the horizontal is 
small. In the determination of gain as given in later 
chapters, the effect of refraction becomes com- 
pletely negligible when a is more than a few degrees. 
For small angles, cosa may be replaced by 
1 — a?/2. In this case equation (1) is well approx- 
imated by 
n(r) + T COS @ = No" COS ao, 
(2) 
1 
Loot) saoms 2, 
2 a 
where h is the height above the ground, so that 
