440 g PROPAGATION THROUGH THE STANDARD ATMOSPHERE 
TABLE 3. Values of v and R for sin? (2/2) = 0.7, do = 2. 
aa) (PS UP) 
= 2 2 
D 4/( D) + 4D sin*5 , 
(Figure 12, Chapter 5) K | n | R = nr | Lobe 
0.2 1.1 2:2) 0 |0.63) 0.756 1 
0.3 1.15 2.3 0 |1.37| 1.644 1 
0.4 1.22 2.44|| 1 |2.63} 3.16 2 
0.5 1.28 2.56 || 1 |3.37| 4.04 2) 
0.6 1.36 2.72|| 2 |4.63] 5.56 3 
0.7 1.43 2.86 || 2 |5.37) 6.44 3 
0.8 1.52 3.04|| 3 |6.63| 7.96 4 
0.9 1.6 3.2 3 |7.37| 8.84 4 
0.95 1.64 3.28 
1.0 1.68 3.36 
and 1.377 + 2k, in which k is an integer. Then 
n = 0.63 + 2k and 1.37 + 2k, and R = nr. Values 
of n and RB are listed in Table 3. The intersections 
of the R values and the locus for sin? (2/2) = 0.7 are 
plotted as circles on Figure 7. 
The entire lobe structure for one contour may be 
drawn by choosing additional values of sin?- (2/2). 
A large number of contours have been calculated 
by the Radiation Laboratory and are plotted in 
Figures 16 to 39. 
In order to construct one contour of a coverage 
diagram, it remains to find the intersection between 
the curves giving values of w for constant sin? (2/2) 
and the corresponding path-difference contours. 
The equations relating R to 0/2 are given below. 
From equation (18) 
5=2+4+20n (ob = 180°,¢’ = 0), (23) 
and from equation (19) 
=) 
From equation (83) in Chapter 5 
An assigned value of © fixes two values of 6 for each 
lobe, as explained in the previous paragraph. All 
values of sin? (2/2) other than 1 or 0 determine two 
intersections with the lobe. When sin? (2/2) = 1, 
the envelope of maxima is obtained, while sin? (0/2) 
= 0 corresponds to the envelope of minima. 
By selecting several values of sin? (2/2) in Figure ~ 
12, Chapter 5, and following the method outlined 
above, a coverage diagram may be constructed in 
generalized (u,v) coordinates. The actual values of 
he and d are 
he = hw, (24) 
d = dy. (25) 
Construction of Lobes 
(Vertical Polarization) 
Problems involving vertical polarization or cases 
where the ratio of the antenna-pattern factors 
F,/F, cannot be neglected, may be solved by suc- 
cessive approximation. 
As a first approximation the method on pp.438- 
440 is applied to determine points (hz,d) on the lobe. 
The corresponding values of w and v determine s 
in Figure 19 or Figure 20, in Chapter 5, and tany 
may be found from Figure 24 in Chapter 5 for the 
given transmitter height h;. An alternate method 
is to calculate tan y from equations (73), (58), and 
(60) in Chapter 5, which are 
d; = sd, 
d,2 
PE re) 
tany = > 
1 
The angles, and y required to calculate the antenna 
pattern factors F; and F, are found from equations 
(62) and (68) in Chapter 5, 
he — hy d 
tanyg = ———' — — 
Ma d 2ka 
BG dy 
OS Pe 
The values of p, @ (or ¢’) may now be read from 
the reflection curves in Chapter 4. 
Equation (46) in Chapter 5 may now be applied 
with K = (F:/F;)pD where D assumes the same 
values as in the approximate solution. Equation (21) 
determines the value of d from which u = d/dr 
may be calculated. This value of u = d/d7 is laid 
off on the original divergence contour in Figures 4, 
5, or 6. This determines v. The assumption under- 
lying this procedure is that the divergence factor is 
not appreciably affected by the change in coordinates 
caused by imperfect reflection and an unsymmetrical 
antenna pattern. The corrected phase difference 6’ 
is found from 
v= 2-¢' (26) 
and the path difference A’ and parameter R’ from 
aC 
Fiaure 8. Lobe angles corrected for earth curvature. 
