CALCULATION OF RADIO GAIN 405 
n = $f(5) = 0.4 from Figure 58; f(6) in Figure 57 
is approximately unity, so that ¢ = sd=0.4. 
Since s, from equation (150) for k = 4/3, is given by 
4.43 x 107° it follows that d, for the plane-earth 
approximation to hold, must be less than 10*d!/%, 
d < 10*x'” (plane earth). (147a) 
3. Curved earth. The screening effect of the earth 
curvature results in a further decrease in gain. Well 
within the diffraction region, the shadow effect pro- 
duces an exponential drop in field strength with dis- 
tance, which is much greater for higher modes than 
it is for the first mode. 
Denoting the screening or shadow factor by F, 
and the distance gain factor by #:, we have 
@, = 2AcAiF,. (148) 
For the dielectric case 5 > > 1 and for distances 
greater than 1.5/s, the shadow factor is 
F, = 2.507(sd)9/? ¢ 1-87 (e4)_ (149) 
| 1 ie 
SLX Gayl 
=: Leis rota) 
3k 
Equation (149) gives the value of the shadow 
factor for the first mode only. In Figure 32, the 
curve marked “dielectric earth” is a plot of the 
shadow factor evaluated by using all terms or modes. 
However for sd > 1.5 only the first mode is impor- 
tant. Consequently, equation (149) represents this 
curve accurately for all values of sd larger than 1.5. 
where 
(150) 
Hetcut-GaIn 
For antennas at zero height, the height-gain 
functions f are equal to unity, so that equation (148) 
represents the actual value of the first mode -for 
both antennas at zero height. 
When the antennas are raised above the ground, 
it is convenient to distinguish between low and high 
antennas, the division between the two cases being 
given by the critical height h, = 307°. 
1. Low antenna; h <h, = 3007". For a low 
antenna, f is a function of Jh and of Q, where lis a 
quantity that depends on the complex dielectric 
constant and is given by 
amp’ 
l= V2, (151) 
in which p’ is the distance coefficient given by 
equations (144) and (145). Let the value of f for 
a low antenna be denoted by H;. 
The magnitude of H; is given by 
h 
His of RES ee ee wt + be (152) 
and is plotted in Figure 47. 
For height h larger than 4/1, the two first terms 
under the square root may be neglected in comparison 
to the third term, and H;, becomes approximately 
Hy, =lh. (153) 
In order to show more clearly when this approxima- 
tion is justified, Table 4 gives values of 4/1 for differ- 
ent wavelengths and vertical polarization. For 
horizontal polarization, 4/1 is quite small. 
z Zi 
° Za 
S 7 
5 7 
Zan 
4 
: 
i 
Fo he =30 5S 
= 
2 4/2 Ne 
h ——e 
Ficure 29. Height-gain as a function of height. (See 
Figures 7, 25, and 47.) 
Inspection of Table 4 shows that, except for the 
case of sea water at wavelengths above 1 meter, 
the approximate equation (153) is good for heights 
above about 50 meters. 
Tasie 4. Values of 4/I for different wavelengths 
(vertical polarization). 
Nin meters |01 123 4 5 6 7 8 9 10 
Sea water 0.06 10 28 50 80 115 133 160 235 285 300 
Fresh water 61117 22 30 33 40 44 53 57 
Moist soil 3 71115 18 21 25 28 33 36 
Fertile ground 3 5 810 13 16 17 20 25 27 
Very dry ground 23 46 8 9 10 12 15 16 
To a first approximation, the value of f for low 
antennas is the same for all modes, so that the height- 
gain functions may be factored out, as was pointed 
out in text on p.380- The gain factor for the case 
when both antennas are low can then be represented 
by 
A = 2A,AiF, (Hz)i(Hz)2, (154) 
where F, is sum of the shadow factors of all the 
modes. F’, has been plotted in Figure 32. Equation 
(154) is also valid in the optical region, provided 
haha 
d>> (155) 
This condition is added to insure that the point is 
well below the center of the lowest lobe. 
2. Elevated antenna; h > 30/°. In this case the 
height-gain function f increases exponentially. Rep- 
