436 PROPAGATION THROUGH THE STANDARD ATMOSPHERE 
plane earth may be used. 
There exists also the intermediate case where the 
effects of divergence may not be neglected and an 
accurate knowledge of the lobe shape is required. 
Three different solutions of this problem are given 
in following sections. They are 
1. The p-qg method for horizontal polarization only. 
2. The u-v method, which may be used for both 
horizontal and vertical polarization. 
3. Lobe-angle method which has the advantage of 
determining the lobe angles directly and is used for 
either polarization. 
THE p-qg METHOD 
(HORIZONTAL POLARIZATION) 
Outline of Method 
This method consists in plotting the locus of 
points having a constant range d and locating those 
points on this curve which are at such a distance 
from the transmitter that the phase shift caused by 
path difference corresponds to the required range. 
The range corresponding to a total phase shift 
is given by 
des WGGh Va = PPL AD sin? (12) 
where D replaces K in equation (46) in Chapter 5, 
since for horizontal polarization p = 1 and since 
we are neglecting any effect due to the antenna 
radiation pattern. For this expression, D and Q are 
certain functions of the antenna heights h: and hy as 
well as d which were considered earlier in the text 
of the last chapter). For a given transmitter, hi, 
Gi, do, and X are given, so that for a given gain 
contour the only variables in equation (12) ared and 
hz. The difficulty of the problem consists in the fact 
that equation (12) provides an extremely complicated 
relation between hf, and d which cannot be solved 
explicitly for either coordinate. 
Under such circumstances, the natural procedure 
is to introduce new coordinates which make the 
handling of equation (12) easier, The method de- 
scribed in the following makes use of the variables 
dy dz 
=— and q=— 
Pp ie q d 
discussed on pages 389 and 39(), and the pro- 
cedure will be called the p-g method. 
It may be recalled that expressed in coordinates 
p and q 
d=—" «dy, (13) 
iG) 
4p’q ie 
D= [1 + 5 (14) 
(1 — p’) 
and the total phase shift, by equations (97) and 
(29) in Chapter 5, is 5 
g = 4turad — PP 
a ae oe (15) 
For horizontal polarization, @ — 0, so that for this 
case (which is the one under consideration) all 
variable quantities in equation (12) have been 
expressed in terms of p and q. 
Construction of Range Loci 
Suppose to start with that we want to compute 
the position of the extreme range of a lobe. We may 
then proceed in the following manner. To a fair 
approximation, we may assume the extreme range 
of the lobes to correspond to sin?(Q/2) = 1, so that 
by equation (12) the corresponding distance daz 18 
given by 
dmax = VG; do(1 + D). (16) 
Expressing d,,,,and D by p and q, the above equation 
determines the envelope of all lobe maxima. The 
practical way of doing this is to use a graphical 
representation of D in p and q coordinates (Figure 17 
in Chapter 5) and to start by selecting a particular 
value of D, say D = D,. Inserting this in equation 
(16) gives a corresponding value of d,,,,, and insert- 
ing this value of dx for d in equation (13) determines 
a straight line in the p,q plane, since dr = V2kah 
is known. Whatever is the value of djyax, this line 
passes through the point g = 1, p = 0. In order 
to determine the position of the line, only one more 
point is needed. A convenient point to choose is to 
take g = 0.9 and compute the corresponding p from 
p = 0.1d,,.,x/dr. The point of intersection of this line 
with the selected D,-contour then gives the desired 
p,q combination that corresponds to the given 
values of d,,,, and D). 
From this pair of values (p,q), the corresponding 
receiver height hz may be calculated by the relation 
[equation (98) in Chapter 5] 
2, 
a [1-2 + 2 iF 
1—q l1—q 
Now both coordinates of the desired point are known, 
and the point may be plotted. Plotting a series of 
different points by the same method yields a smooth 
curve, which is the envelope of all lobe maxima. 
The locus of minima may similarly be plotted by 
using sin' 5 = 0 and 
Amin = VGido(1 — D) (17) 
instead of equation (16). Intermediate range curves 
are found by assigning nonintegral valuesto m in the 
equation @ = mm and substituting in equation (12). 
