COVERAGE DIAGRAMS 445 
Since the angle y is of the order of a few degrees 
only, it is permissible to write A = d; in the above 
equation. 
Neglecting further the term n\/4 in comparison 
with d,, which is permissible for short waves and 
small values of n, the above expression for B reduces 
to equation (44g). 
The method of determining the locus of points 
having a path difference of d/2 (ie., n = 1) is as 
follows. Assume a value of d; and calculate the 
corresponding values of ‘ 
hy’ by equation (44d), 
y by equation (44a), 
B by equation (44g), 
rq by equation (44e), 
Wa by equation (44f), 
6 by equation (44b), 
1 by equation (44c). 
The assumed values of d; are limited to those which 
will give positive values of B in equation (44g). 
Select as many values of d, as are necessary to plot a 
smooth path-difference locus. Repeat for n = 3, etc. 
This method of determining the angles of lobe 
maxima is of particular value in constructing the 
first few lobes, since the approximations in text on 
pp-441- 444 may cause considerable error for low 
angles. These path-difference loci will intersect a 
vertical line drawn from the antenna to the ground 
below at heights equal to n\/4. For short waves, 
this height is negligible for the lower lobes. 
LOBE-ANGLE METHOD 
(VERTICAL POLARIZATION) 
Angles of Lobe Maxima 
The values of n corresponding to the angles of 
lobe maxima are determined exactly as in text on 
p434 forthe case of a plane earth. The values of n 
in the expression y’ = n\/4h;’ are increased above 
those for horizontal polarization by an amount (An) 
to compensate for the reduced phase shift on reflec- 
tion. In other words, the path difference must be 
greater than integral multiples of \/2 to compensate 
for the reduced phase shift. The expression for this 
compensation, from equation (5), is 
An = — (45) 
Hence nN 
%= A S 
vy’ = (n+ An) ane 
[See Figure 8 and equation (32).] 
(46) 
Construction of Lobes 
As a first approximation, the angles of lobe maxima 
are calculated on the basis of horizontal polarization. 
A table is constructed giving values of n and y’ 
for maxima and minima. The next approximation 
is to let Y = y’. This assumes that d, << d. The 
values of d’ and p may then be found from reflection 
curves, and (An) calculated from equation (45). 
The corrected values of y’ may be determined from 
equation (46). It is simpler to find y’ by interpolat- 
ing between integral values of n in the n versus ’ 
table previously constructed. The values of dmax 
and dj, are 
Omax = VGido(1 af kK), (47) 
Anin = V@yds(1 — K), (48) 
where K = (F2/F\)pD. The divergence factor may 
be found directly from Figure 11 for the corrected 
values of n and y’. It will be found that for the higher 
lobes, the effect of (An) upon the value of the 
divergence factor is negligible. 
Table 5 shows calculations of the corrected values 
of m and y when the radiation from antenna A of 
p-441 and Table 4 is vertically polarized. “Trans- 
mission over sea water is assumed. Tables 6 and 7 
illustrate the effects of vertical polarization on reduc- 
ing the maxima and increasing the minima. 
TABLE 5 
o |e Io ea 
(in (in Ay! y' | 6 
n |degrees)| degrees) | An (rad) (rad) | (rad) 
0} 180.0 0 0 0 0 — 0.00788 
1| 175.0 5.0 0.0278 | 0.000064 | 0.00369) — 0.00071 
2)171.5 8.5 0.0472 |.0.000112 | 0.00602 0.00284 
3| 168.0 12.0 0.0664 | 0.000155 | 0.00843 0.00593 
4| 164.7 15.3 0.085 | 0.00204 | 0.01080 0.00878 
5 | 160.6 19.4 0.108 | 0.000250 | 0.01325 0.01153 
6 | 157.3 22.7 0.126 | 0.000290 | 0.01569 0.01423 
7|153 27.0 0.15 0.000374 | 0.01807; 0.01681 
8 | 149 31.0 0.172 |0.000413 | 0.02061 0.01947 
9)144.5 35.5 0.197 |0.000491 | 0.02309 0.02208 
10 | 140.0 40.0 0.222 | 0.000532 | 0.02563 0.02472 
11 |135.2 44.8 0.249 | 0.000623 | 0.02812 0.02735 
12 130.5 49.5 0.274 | 0.000685 | 0.03068 0.02991 
13 |125.8 54.2 0.301 | 0.000722 | 0.03312 0.03241 
14 |120.8 59.2 0.329 |0.000790 | 0.03559 0.03493 
15 |116.0 64.0 0.355 | 0.000886 | 0.03819) 0.03778 
16 |110.2 68.8 0.388 | 0.000968 | 0.04077 0.04019 
17 }105.3 T4A.7 0.415 | 0.000995 |.0.04320 0.04177 
18 |101.1 78.9 0.437 | 0.001091 | 0.04569 9.04519 
19 | 96.1 83.9 0.466 | 0.001130 | 0.04823 0.04775 
20! 91.8 88.2 0.490 |0.001224 0.05082} 0.05037 
*% corresponds to y’in Table 4. ¢/ = 1—@. 
TABLE 6 
dmax(HP) dmax(VP) @max (VP) 
n K (km) (km) dmax (HP) 
1 0.904 150.2 145.5 0.968 
3 0.765 181.7 162.5 0.895 
5 0.670 191.0 161.0 0.844 
¢ 0.585 194.3 155.2 0.80 
9 0.516 196.4 149.8 0.762 
11 0.458 197.3 144.6 0.733 
13 0.415 198.2 140.8 0.710 
15 0.385 198.9 138.0 0.695 
17 0.362 199.1 135.7 0.681 
19 0.360 199.4 135.8 0.680 
