Chapter 8 
NOMOGRAPHIC SOLUTIONS FOR THE STANDARD CASE* 
| EQUATIONS GIVEN in the preceding chapter 
have now been thrown into nomographic form. 
When these nomograms are employed a rapid method 
for constructing coverage diagrams results. 
Let hi denote the height of the transmitter in 
feet, fn. be the frequency in megacycles, n be an 
integer (1, 2, 3, ---) specifying the number of the 
lobe, (0 S b < 1) a “phase” factor specifying the 
position on the lobe, and r the radius of the earth. 
Introduce the quantity B defined as follows. 
150 (n — b) y2N (3.281)! 
hilfine 
108 (n — 6) 
hifme 
where we have taken r = 8.50 X 10° m, as the 
approximate % earth value. We have to decide on 
the interval for b. By taking b = 0, %, %, %, %, % 
we actually obtain seven points on each lobe, 
which should be sufficient for the purpose of drawing 
a coverage diagram. Hence, n — b = 0, %, %, 4, 
%, %, 1, %, ---, etc., spaced at intervals of %. 
Equation (1) is represented in the nomogram of 
Figure 1. We are given h; and f,,, the height and 
frequency of the transmitter. Connect the appro- 
priate values on the scales by a straight line and 
mark the point of intersection on the central vertical 
line. 
Define a quantity k by the equation 
k 
6 ? 
B= 
= 3.676 X (1) 
n—b= 
so that k = 3 corresponds to the maximum of the 
first lobe, k = 6 to the minimum, k = 9 to the next 
maximum, k = 12 to the minimum, etc. k = 15, 21, 
and 27 correspond to the third, fourth, and fifth 
maxima, respectively. Other values of k determine 
intermediate points on the lobe. 
Now draw a straight line from & = 1 through the 
point previously determined on the central vertical 
line until it intersects the left-hand axis of B. Read 
off B or 1/B, whichever is given. Repeat the process 
for k = 2,3, ---, etc., until a value of B is obtained 
that exceeds 10; in other words, continue until the 
straight line runs off the lower edge of the left-hand 
scale. 
There will be cases, however, usually involving 
large values of fi or fme, Where B will still be small 
(1/B large) even for k = 27. When this condition 
8By Lt. Comdr. D. H. Menzel and Lt. A. L. Whiteman ,Office 
of the Chief of Naval Operations. 
59 
exists, the lobes tend to be so closely spaced that 
the individual maxima are difficult to define and 
even more difficult to draw on a coverage chart. 
For such conditions an alternative procedure is 
recommended, which will be given later. 
If no difficulty is encountered, however, enter the 
values of B or 1/B (designate the latter with an 
asterisk) in a table such as Table 1. 
TABLE 1 
fmc = Frequency in me 
hi = Height of antenna in ft 
k= Bt n b 
1 1 4 
2 1 3 
3 il } max. 
4 1 3 
5 1 $ 
6 2 0 min 
7 2 4 
8 2 3 
9 2 % max. 
10 2 3 
11 om A 
12 3 0 min. 
15 3 4 max 
21 4 } max 
27 5 $4 max 
*Put an asterisk after an entry if the value read off is equal to 1/B. The 
corresponding values of n and b are entered in columns 3 and 4 of the form 
sheet. 
It should be noted that equation (1) is easy to 
solve, and the operator familiar with mathematical 
procedures may prefer to use direct calculation, by 
slide rule or logarithm tables, as much more accu- 
rate. In general, however, the nomogram ‘values are 
sufficiently accurate for the work. 
Next, for the five or six assumed values of decibels 
for which contours are desired, we solve a subsidiary 
equation for Y by means of a nomogram (not repro- 
duced here). We note that 
Y = db + 60 — 10 log (2rrhi) , 
and slide-rule calculation is extremely convenient. 
For each of the selected values of b, we have pre- 
pared a nomogram connecting Y, B, and a. Although 
there are six adopted values of b, the expressions for 
b = %, %, %, % coincide, so that four charts 
suffice. A representative sample of these charts, for 
b = 0, is given in Figures 2 and 3. Connect each 
value of Y, for which a contour is desired, with the 
value of B on the appropriate chart, according to 
the value of 6 (or k). Read off the corresponding 
value of a. ; 
Having determined @ for a given point on the 
coverage chart, we now calculate S from the nomo- 
