434 PROPAGATION THROUGH THE STANDARD ATMOSPHERE 
TABLE 1 
Antenna Y mr 
height, h; » Minimum Minimum Minimum n at 
(meters) (degrees) (meters) \ = 3 meters 
120 1.44 12.9 4.3 
60 1.1 4.5 1.5 
30 0.78 1.6 0.53 
15 0.56 0.6 0.2 
9 0.4 0.25 0.08 
which the approximation ~ = y holds depend upon 
the wavelength and transmitter height. The follow- 
ing table shows the minimum angle for various 
transmitter heights and wavelengths at which the 
error in the path difference A introduced by this 
- assumption is less than 1 per cent. To satisfy equa- 
tion (2), and have an error less than 1 per cent, 
y should lie between the minimum value and 0.2 
radians. Table 1 shows, for increasing transmitter 
antenna height, how the angle for which equation (2) 
is valid within 1 per cent also increases. 
If n is set equal to 2m — 1, integral values of m 
correspond to lobe maxima and half-odd integers to 
lobe minima. The advantage of this notation is that 
the value of m is the number of the maxima or lobe 
number. Thus for the fifth lobe, m = 5. The gen- 
eral expression for the grazing angles corresponding 
to lobe maxima, for a plane earth and for horizontal 
polarization, is 
(2m — 1) 
Y th r, (3) 
where integral values of m give maxima and half- 
odd integers give minima. 
Angles of Lobe Maxima 
(Vertical Polarization) 
With vertically polarized radiation the reflection 
phase shift @ [equation (27) in Chapter 5] is less 
than z radians (i.e., @’ = @ — 7 is negative). 
It follows that the path difference for lobe maxima 
must be greater than \/2 and greater than \ for the 
nulls. In other words, the value of 7 in equation (2) 
must be increased by (A + 7) to compensate for the 
decreased phase shift of reflection, so that 
(An) le) “Geoch © 
Hence 
(an) = 2 =1= 42. (6) 
Tv us 
Yor vertical polarization, equation (2) becomes 
= a Oy) 6) 
Ah, 
Lobe Equation 
When p = 1, and ¢=7, ¢’ =0, and Q=54, 
equation (46) in Chapter 5 may be written as 
wes Q cae 5 
d = 2VGid sin 3 2VGidp sin Re (7) 
Substituting 
Ss haha (2) 
d \x 
gives 
aes Qrhyhe 
a I Eah cin ( <= ); (8) 
where do is the free-space range which may be com- 
puted from the gain corresponding to the given 
coverage diagram by use of the nomogram given in 
Figure 3 in Chapter 2. Equation (8) may be written 
es 2VGidy sin (a - sin 7) (9) 
Equation (9) shows that for fixed values of free- 
space range do, transmitter height hf, and wave- 
length, the coverage lobe may be represented by a 
polar sine function of the angle 7 at the base of the 
antenna. This assumes that the slant range measured 
to any point on the lobe may be considered equal 
to the distance d measured along the surface of the 
earth. 
If n in equation (2) is allowed to assume all frac- 
tional and integral values from 0 to 2, sin y— 7 
may be expressed as 
CON 
fio, ea q7 a 2, (10 
sin y = 7 an ) 
Substituting this value into equation (9) gives 
— n — 
d = 2VGido sin (2) = 2VGido sin (90°n). (11) 
Equation (11) is useful in sketching the lobe contour. 
It holds only when the reflection coefficient equals 
—1 (i.e., p = +1) and the angle y is small enough 
so that equation (10) is valid. 
SPHERICAL EARTH 
Lobe Characteristics 
Figures 2 and 3 are typical vertical coverage 
diagrams for a smooth spherical earth. They illus- 
trate two important points. 
The first is the dependence of the number of lobes 
on the ratio of transmitter height to wavelength. 
Figures 2 and 3 show that for h; equal to 75.4 wave- 
lengths, the lobes are much more closely spaced 
than for a transmitter height of 32.3 wavelengths. 
When the number of lobes is large, there is little 
possibility for a target to escape detection in the 
small null areas. The shape of the contour is, there- 
fore, less important and it is sufficient to find the 
maximum and minimum ranges and then to sketch 
