408 PROPAGATION THROUGH THE STANDARD ATMOSPHERE 
20. LOG F, 
20L0oc fe —--— 
(sd)? 
20L0G Fy 
FictreE 33. Shadow factor for dielectric earth. 
made, it is found that 
= [2A0AiF si] f(Mi) - (he). (160) 
For the dielectric case equation (160), using equa- 
tions (140), (147), (151), and (156), becomes 
-; E “2 (gh) (gh)e (161) 
The same formula holds for sea water, vertical 
polarization, wavelengths in VHF range, d > 50/p’, 
and h > 4/lI, as described above, except that F, or 
F.,; is represented by various curves in Figure 32, 
60 
ae, 2 345 
mem 
PEGE mc ea 
il Sec 
20 itd ie 100 500 
according to the value of \, and g is modified slightly 
by a correction factor g’. 
FoRMULA FOR SHA Water. VERTICAL POLARIZA- 
TION. VHF. 
Equation (160), the general formula for the first 
mode, in the VHF (1 to 10 meters) range, becomes 
A = [2A Ai Fa |(H 199’): H199’)2, (162) 
where H; is the low antenna height-gain function 
whose formula is given by equation (152), and 
gg’ = 1 for low antennas. As pointed out above, 
if h > 4/l and d > 50/p’, equation (162) reduces to 
equation (161). g’, the correction factor for g, is 
given in Figure 36. For a more extended discussion, 
see page 416. 
Effect of Changing the Value of k 
In the optical region, the effect of a linear gradient 
of refractive index has been shown to be equivalent 
to replacing the radius of the earth a by an effective 
radius ka and then treating the atmosphere as 
homogeneous (see Chapter 4). 
In view of the equivalence of the sum of modes to 
the optical formula in the optical region, it follows 
that the modes should be changed in the same way 
in the optical region, i.e., a should be replaced by kd. 
These same modes supply the solution of the wave 
equation in the diffraction region, so that in both 
regions the substitution of ka for a will take care of 
an atmosphere with a linear variation of refractive 
index with height for all values of k. 
For given transmitter and recewer antenna heighis, 
hy and he, the first maximum of the field-strength 
versus distance curve (see Figure 4) will frequently 
represent the limit of detection. The first maximum 
occurs not far from the line of sight which, for a 
Pa ee 
eal a EEG 
Figure 34. 20 log z versus z. 
