394. A PROPAGATION THROUGH THE STANDARD ATMOSPHERE 
and 
=1>=]=1—s (103) 
@) 5 
Q 
Qa 
Since s is a function of w and »v only, it follows that D 
may be plotted in u,v coordinates. This may be 
accomplished by solving equation (96) with respect 
to q, which gives 
_@ =p) (=D) te 
4p*D? 
Equation (103) gives s=1—q; v= p/s, and u 
may be read from Figure 19 or 20. Contours of 
constant D are shown in Figures 21, 22, and 23. 
The grazing angle y is important in calculations 
for vertical polarization since it determines the 
magnitude and phase of the reflection coefficient 
for a particular frequency and reflecting surface. 
The grazing angle y may be expressed in terms of 
s, u, and », as follows. From equation (60), 
’ 2 
tan y = hie hs ( | ‘ (105) 
1. 
dh a \ kak 
Hence 
tany = Li sy’) = == — w) 
dz(sdfdr) dr Sv 
or 
tan y 1 1 
Te valet) 009 
and for k = 4/3, 
tan y 
= = 1 — 
Ais 2.4 X 10 (Z x). (107) 
From Figure 24, y may be obtained for given values 
of hy and sv = p. 
The generalized coordinates described in this 
section will be found highly useful both in field 
strength and coverage calculations. 
ILLUSTRATIVE CALCULATIONS FOR 
THE OPTICAL-INTERFERENCE REGION 
Introduction 
The general expression for the gain factor A in the 
interference region is obtained by combining equa-_ 
tions (44) and (7). Then 
A = AF, Va — K)?+ 4K sin? g. (108) 
The value of the radio gain is then given by equa- 
tion (3) and the value of radar gain is given by equa- 
tion (5). The value of the radical which defines the. 
interference pattern has a range of values between 
0 and 2. The extreme values can occur only when 
K =1 (p = 1, D = 1, F./Fi = 1); the value 0 
(nulls) is then given by sin? (0/2) = 0 and the value 
2 (maxima) is given by sin? (Q/2) = 1. 
In general, the value of A lies between the two 
extremes 
A = AoF; (1 + Kk), 
the positive sign giving a maximum and the negative 
giving a minimum. At any other point, the value 
lies between these two extremes. For range calcula- 
tions (which involve maxima), the variation in A 
is from 1 to 2 times the free-space value, according 
to the value of K, so that in practice a quick rule of 
thumb for range may be devised. Assume (1 + K) 
equal to 1.9 for favorable conditions (sea water, 
horizontal polarization, or, in the case of vertical 
polarization, small grazing angles) down to (1 + K) 
= 1 or K = 0 for propagation over rough terrain. 
The problem of finding the range is thus reduced 
to a problem for free space. In range calculations, 
P;/P, is given by the ratio of minimum detectable 
power to power output. A is then determined by 
equations (8) or (5), and the range is given by find- 
ing d from the relation (writing Ay = 3d/87d) 
AS (2) +8). (109) 
More detailed calculations are presented in this 
section. However, the assumption is made generally 
that the reflection coefficient is equal to —1 (e., 
p = 1 and @ = 180 degrees) and that the direct and 
reflected rays do not differ appreciably on account 
of the shape of the antenna beam pattern (F2 = F). 
For large distances over sea water, these assumptions 
are approximately realized. For most of the calcula- 
tions it offers no inherent difficulty to consider the 
effect of directivity or of a reflection coefficient 
different from — 1 but may require considerable 
additional calculation. 
For convenience, the formulas required in the 
calculations are recapitulated here. Putting p = Fy 
= F, = 1, equation (108) takes the form 
N= Ao ( — D)? + 4D sin’ > (110) 
or, in decibels, 
20 log A = 20 log Ay+ 10 log [a — D)?+ 4D sin? 2 
The reflection point variable 
dy dy 
jos 111 
dp V2kahy oD 
= es alee — (whenk = 4/3) 
(4120 Vix) 
will be used extensively. It has been found that the 
interference pattern is very sensitive to slight changes 
in p, so that an accuracy to the fourth significant 
figure is generally required. 
The path difference variable Ris related to p = di/dr 
