CALCULATION OF RADIO GAIN 395 
p y (DEGREES) hy, (METERS) 
10 
03 
04 75 
05 
60 
45 10> 
1 30 
20 
10 
.2 
i 10° 
3 = 
lo 
- 2 
4 ~_ 
Cat 
=1N 
s lels 
Pomel 10? 
6 < 
u 
S} a eo 
el 
8 i 
or} 10' 
.02 
I 
01 
10 
005 
002 
001 
98) ye 
Figure 24. yas a function of h, and p=sv=d,/d7. [See 
equation (107).] (See Figure 14 for definition of lengths.) 
and v = d/dr by the equation 
e-(-Domm 
(112) 
Resolved with respect to 1/v, this equation assumes 
the form 
ie =( a, MED ) 8 
D0 Qin) Dien 
Another convenient expression for R is obtained by 
replacing, in equation (83), the path difference A 
by nd/2, where n (see below) may assume any posi- 
tive value. Substituting V2kah, for dy, equation (83) 
assumes the form 
(113) 
R=nr, (114) 
where 
1 [ka X 
pa aa|eee et 115 
2° 2h??? Go) 
ner 
or r = 1030 " 
hi? 
(fork = 4/3). 
A graphical representation of r is given in Figure 15 
in Chapter 6. 
Then for a reflection coefficient of p = 1,@ = 180 
degrees (i.e., 6’ = 0), equation (29) gives 
On IN ee 
r 
(116) 
If r is fixed, a complete pattern of contour lines 
(along which A is constant) is determined. Take as 
independent variables p and n (rather than u and v). 
A given choice of p and n determines & by equation 
(114), 2 by equation (116), v by equation (113), 
s by equation (118), w by equation (121), D by 
equation (117), and finally 20 log A by equation 
(110). By varying r, new patterns are obtained. 
Accordingly, r may be called a pattern or chart 
parameter (see page 446)..3). 
The lobes on the charts depend on n, in accordance 
with equation (116). Accordingly, for p=1, 
@ = 180 degrees, n is the lobe variable. For the 
first (lowest) lobe, n = 0 gives the first null, n = 1 
gives the first maximum and n = 2 the second null. 
For the second lobe, m varies from 2 to 4, with a 
maximum at n = 8, and so on. It should be remem- 
bered that if n < 1, corresponding to the lower side 
of the lowest lobe, the value of the field (or of A) given 
by the optical formula is too low. A more accurate 
value can be obtained by joining the curves found 
in the optical and diffraction regions into a smooth 
overall curve. 
Combining equations (102), (103), and (113) gives: 
the divergence factor D, 
4 4) =P -1/2 
belie | -[(+—22 
@ = 7) TP a 
(117) 
The variable s = d;/d is 
s = p/2, (118) 
and, repeating equation (101), 
2. sfiltaw ) 1 
si fg —= -i)+—=0. (1 
2 2 ( ve i Qu? ee) 
In terms of p, equation (119) becomes 
2p* — 3p*v + pv? —u—1)+v=0. (120) 
Equation (120), resolved for v and wu, gives 
1 1 z 
“sot 
2 Pp p 
(121) 
wu = 2p? — 3pv + — —1-+ 2% 
Pp 
Some useful approximations : 
For v large,q— land R = £ (1 — p?)? ap- 
Pp 
proaches the value 
Ree a 
P 
(122) 
