444 PROPAGATION THROUGH THE STANDARD ATMOSPHERE 
Ps p| 6 (42) 
Fy 
The values of D to be used in equations (40), (41), 
and (42) may be read directly from Figure 11 or 
Figure 12, or calculated by equation (39). 
Intermediate points in the lobe may be formed by 
assigning fractional values to n. The corresponding 
path-difference angle 6 may be calculated in the 
following manner. Suppose it is desired to find 
intermediate points on the fourth lobe. The values 
of n for this lobe range from n = 6 to n = 8, with 
the maximum at n = 7. It follows that a change of 
2 in n corresponds to a change of 27 in 6. Thus if 
n = 6.5, 6 = (0.5/2)(27) = 2/2 = 90°. For hori- 
zontal polarization 2 ( = 6 + @’) reduces to 6, 
since @’ & 0. The distance from the transmitter 
base to this intermediate point in the lobe is equal to 
din = VGido E = 
Hype Gag — K) + 4K sin? (2), (43) 
as given in equation (46) in Chapter 5, in which 
K = (F2/F;)pD and p = 1. The value of D may 
be read from Figures 11 and 12, using the assigned 
value of nd and the relation y’ = n)d/4h,;’. The 
proper value of d; to be used in equation (63) in 
Chapter 5 to determine the antenna-pattern angle v 
may be read from Figure 9. 
Correction for Low Angles 
The method outlined in the last five sections 
depends upon the assumption that y’ > y or d; > 0. 
This assumption gives good results when n is a large 
number but serious errors are involved for small 
values of n. The method described in this paragraph 
is designed to avoid this difficulty. The procedure 
consists in plotting point by point the lobe-center 
lines and angles of lobe maxima. The points will be 
located by polar coordinates with the pole at the 
transmitter. The coordinates are shown in Figure 
13 as rq and 7. 
Referring to Figure 13, the following relations 
RECEIVER, 
Ficure 13. Geometry for radio propagation over 
spherical earth. See Figure 14 in Chapter 5. 
hold when the angle y is less than 10 degrees: 
oo aed 
(a) y= Te So 
AMO 8 geal 
(b) oe ka ka 
(c) 7, = va — 0 
d,* 
d le? = hp = SE 
(d) 1 ee ope (44) 
(e) Peale Ps Ghose 
0) w= (2=*)y 
Ta 
_ __(nd/2)d, 
(g) We 2QdyW? — nd/2 
Equations [(44a to e)] are obtained by inspection of 
Figure 13. Equations (44f) and (44g) are derived as 
follows: 
_sin2y _ ta_, 
sn@t+va) B wtva’ 
vivwe= 
Tq 
2) == 
Ta Ta 
The path difference A is given by 
mr 
A = Bap BS PS (@ = 180°). 
Hence 
nr 
A+B =) 42+ B? + 2AB cos y= > - 
Squaring, 
md \? 
B(2A — nd — 2A cos 2f) = nA — (2) ; 
Therefore, 
man 
B= 2 AN 
A(1 — cos 2p) _ 2 
2 
For small angles, 
cos 2 = 1 — GH" _ 1 — oy. 
Hence 
ee ees 
Re 2 2 _ -* —. 
n n 
2A — — 2Ay? — — 
v B) ¥ 2 
