CALCULATION OF RADIO GAIN 379 
to have the equivalent radius ka, and the atmosphere 
to be homogeneous, and radiation to travel along 
straight lines. 
The straight line from the transmitting antenna 
and tangent to the earth’s surface (the so-called 
line of sight) touches the earth along a circle which 
constitutes the radio horizon of the transmitter. 
The distance measured along the earth from the 
transmitter to the radio horizon will be denoted by 
dy; and the horizon distance of the receiver by dp. 
These geometrical relations are illustrated in 
Figure 1. 
From this figure, it follows that 
ka 
k i ———— , 
apie cos (dz/ka) 
Inasmuch as 
eAAHE a pth ul BE a uo ~kat+ L dr’ 
cos (d7/ka) 
1 (dp/ka)? cs 
equation (11) assumes the form 
1 dr’ 
——— 12 
Seren (12) 
or 
dp = V2 kahy . (13) 
Similarly, the horizon distance of the receiver is 
dp = V2 kahs. (14) 
The sum of the two horizon distances is given by 
dz, where 
dy = dr + dp. (15) 
Optical and Diffraction Regions 
The points visible from the transmitting antenna 
(on an earth of equivalent radius ka), i.e., the points 
above the line of sight, constitute the optical region 
(Figure 1). The rest of space lies beyond the trans- 
mitter horizon and below the line of sight and is 
called the shadow or diffraction region. 
OPTICAL REGION 
OPTICAL HORIZON 
LINE OF SIGHT 
NC AAUY oS 
TRAN SMITTER 
Figure 1. Geometry for radio wave propagation over 
curved earth. 
It is frequently necessary to know whether a 
receiving antenna lies in tne optical region or the 
diffraction region of a given transmitter. This 
evidently is equivalent to knowing whether the 
distance d of the receiver from the transmitter is 
smaller or larger than the combined horizon distance 
d,;. By equations (13), (14), and (15), it follows 
that in the optical region 
d <V2ka (Vin + Vin), (16) 
and in the shadow region 
d > V2ka (Why + Vio). (17) 
A graphical representation ot the equation 
dy, = V2ka (Vi, + Vin) (18) 
is given in Figure 2. For k = 4/3, a = 6.37 X 10°m, 
this takes the form 
dy, = 4120 (Vn, + Vi») meters, (19) 
where /y, he are given in meters and dz, in meters. 
h, METERS d. km ha METERS 
Ce) lo) Ce) 
25 25) 
100 100) 
208 i 333 
388 400 
600 200 600 
800] 800 
1000 1000 
300 
2000 2000) 
400 
3000 3000 
500 
4000 4000 
5000 Ge 5000) 
6000 6000 
7000 700 7000 
8000} 8000 
9000 800 9000 
10000) 10000 
900 
15000) 1000 15000 
1100 
20000 20000 
1200) 
d=4.i2 (fh YRa)= d+ de 
Figure 2. Sum of transmitter and receiver horizon 
distances for standard refraction. To change scale: 
Divide h; and hz by 100, and divide dz, by 10. 
