fe SC 
FACTORS INFLUENCING TRANSMISSION 365 
TO CENTER OF EARTH 
Figure 3. Refraction over curved earth. 
T = aandr=a-+h. This is the practical form of 
Snell’s law for the atmosphere above a curved earth. 
The reference level (see Figure 3) is here taken at 
the surface of the earth where m is the index of 
refraction. 
Modified Refractive Index 
In place of the sum(n — np + h/a)appearing in 
equation (2), it is customary to define and use a 
quantity M given by 
i= G Zit | 10°. (3) 
M is called the modified refractive index. It gives a 
unit that is convenient for practical use. The modi- 
fied index is then said to be expressed in M units, 
values of which commonly lie in the range of 300 to 
500. Using this definition, equation (2) becomes 
5 @ ea (Me fey 10; Se, 14) 
An important special case is that in which the 
refractive index decreases linearly with height, 
nN — mo = constant X h. Then equation (2) may be 
written in the form 
By ex, te. 
a (5) 
where & is the factor mentioned in text on p.364 
which determines the modified earth’s radius ka. 
Comparing the above expression with equation (2), 
and differentiating, it follows that 
liam, 
ka dh a 
or (6) 
ee a Buk a 10°. 
nN a 
1 = 
ae 
Proof of the fact that refraction is negligible unless 
the angle is very small may readily be deduced from 
the preceding formulas. Thus, on differentiating 
equation (4) 
-6 
de Mis oe 
Qa 
and in the standard linear case, by equation (5), 
da = dh/kaaw ~ 1.2- 10°’ dh/a 
for k = 4/3. Taking a = 0.05 radians (3°) and 
dh = 100 meters, one finds da = 0.00024 radians 
(50 seconds of arc), a very small change in angle. 
This is the standard deflection which is accounted 
for by replacing a by ka. The deviations from this 
value experienced with nonstandard refraction are 
even smaller. The larger the angle a with the hori- 
zontal at which a ray issues from the transmitter, the 
less the angular deviation. In communication work 
and for certain radar problems, however, angles of 
less than one degree are of importance, and da 
may then become comparable to a. 
Graphical Representation 
Figures 4 to 6 show three different ways of repre- 
senting rays subject to refraction. Figure 4 gives a 
true picture apart from the exaggeration of heights. 
In the case of standard refraction, the curvature 
of the rays is always concave downwards, the center 
of curvature being below the surface of the earth. 
The middle ray shown is the horizon ray and to. the 
lower right is the diffraction region into which rays 
do not penetrate. Figure 5 shows a diagram with 
modified earth’s radius, ka, in which the rays are 
straight lines. Figure 6, finally, is a plane earth 
diagram; the rays are here curved upwards. 
These diagrams may be considered as resulting 
FiaurE 5. Rays in a homogeneous atmosphere 
(equivalent radius ka). 
