CALCULATION OF RADIO GAIN 427 
DECIBELS 
EHH 
FIGURE 58. Shadow factor F, versus 7 = {f(6) = sdf(6)._ 
See Figures 32, 33, and 57. Curve +-10 corresponds to a 
perfect conductor. 
function f, can be represenied by 
2 81 exp {455 = 5 [(R) — 21] *?} 
V 2a (2eh)'/4 J13(@) + J-13(@) 
(197) 
where, from equation (159), 
1/3 —2/3 1/3 
7 (Gane © 60 \3k 
and where the argument of the two Bessel functions 
is 
z= 1% (— 27.) 7" 
For the nth mode, if (eh) >> 27,, the magnitude of 
fn can be written 
3 I exp(jtn V2eh) 
Vox (2eh)!/4 Jy /3(@) + J (2)! 
(199) 
For large 6, using the first two terms of equation 
(195), and writing x, for x when 7, is replaced by 
Tn, » 
— 27, \'? 
sn -(=B2) 
6 
Substituting this in J;/3(x) + J_1/3(x), writing down 
the first two terms of the Taylor expansion, making 
Ne of the fact that the 7,,, are roots of Jy/3(x) + 
J_1/3(x) = 0 and of the relation given bye a prop- 
eri of the Bessel function, 
Jy )3'(x) + J-1/3/(z) = — 1/82) [Js /3(x) + J-1/3(2)] 
+ J_2/3(2) aa J2)3(2), 
we have 
Jy /3(x) + J-1/3(x) 
1/2 
~ (- =) eed 
(200) 
If these results are substituted into equation (192) 
for both antennas, the factor (6 + 27,) becomes 
1 + 2r7,,,/65, which approaches unity for large 6. 
This means that 2f both antennas are sufficiently 
elevated, short-wave propagation is practically inde- 
pendent of ground constants. 
The value of f given by equation (199) can be 
written as glh so that g represents the gain over lh, 
the value approached by H;, when lh > 4. Thevalue 
of g for 6 —> = is represented in Figure 36. 
If 6 is not very great, as in the case of vertically 
polarized VHF over sea water, the effect of 6 can be 
taken into acocunt by changing e to eg(5) and g to 
gg’. The functions g(6) and g’ are given by Figure 55. 
5. Plane earth gain factor and shadow factor. The 
field near the ground over a plane earth with infinite 
conductivity is equal to 2Ho, twice the free-space 
field. For an imperfectly conducting ground, the 
field for antennas at zero height over a plane earth 
may be written 
E = 2EvAi. (201) 
A, is represented in Figure 56 as a function of p’d, 
where 
, 24 |eg—1| _ 2x V(e, — 1)? + (600d)? (202) 
62 + hee oh)? 
es = 
pollo V(e, — 1)?+ (600d)?. (208) 
The curve parameterisQ = ¢,/60o). 
Comparing equations (202) and (203) with 
equations (193) and (194), we find that 
= |s|¢. 
Hence, equation (192) may be written as 
= 2Ea(2n) San | >~— 
(204) 
ise! fala (205) 
If p’d is large, we see from Figure 56 that 
gal 
A= re (206) 
so that the physical significance of 1/p’d in equation 
(205) becomes apparent. 
