PART IT 
CONFERENCE REPORTS ON STANDARD 
PROPAGATION 
Chapter 7 
A GRAPHICAL METHOD FOR THE DETERMINATION 
OF STANDARD COVERAGE CHARTS: 
Ae POWER DENSITY at distance S from a, trans- 
mitter of unit power depends upon h; and he, 
the heights of the transmitting and receiving anten- 
nae, and upon X, the wavelength of the radiation. 
For the high frequencies under discussion, we assume 
the earth to be a perfectly conducting sphere, of 
effective radius r, equal to % that of the earth. We 
are to take inte account the so-called divergence 
factor D resulting from the earth’s curvature. 
Even with the simplifying assumptions above, one 
cannot express the power as a simple function of S, 
hy, he, and d in a single equation. Accordingly, most 
workers on this problem have introduced various 
arbitrary parameters, as intermediate steps. Differ- 
ences in procedure lie primarily in the choice of 
parameters. Whether a method is simple or difficult 
depends upon the character of the parameters. 
Certain procedures suggested are satisfactory for 
determining the number of decibels by which the 
signal is below the adopted standard. of 1 uw per 
square meter, designated here by A; but if we are 
given A, hf, and f and then are asked to compute he 
as a function of S, as for a coverage diagram, some 
of the methods become very unwieldy. The present 
method works satisfactorily for either case. 
Ficurer 1. Geometry for determination of standard cov- 
erage. 
In selecting a parameter we have been guided by 
the following conditions. The number of parameters 
should be kept to a minimum; the remaining vari- 
ables hi, he, and S should appear in the final equa- 
tions, if possible. Also it should be unnecessary to 
interchange transmitter and receiver according to 
“By Lt. Comdr. D. H. Menzel, USNR, Office of the Chief 
of Naval Operations. 
53 
the condition that he is or is not greater than h; 
The arbitrary parameter a is defined as follows. 
Let d, be the distance from the transmitter to the 
point at which the ray is reflected and dy the distance 
to the point where a ray is tangent. Then 
at = dg(1 — a) = 2hir(1'— a) (1) 
a, therefore, is constant along a reflected ray; a = 0 
corresponds to the continuation of the tangent ray; 
a = % corresponds to a reflected ray perpendicular 
to the mast of the transmitting antenna; a = 1 is 
the vertical ray. Thus 
0<a<1, 
with a > 24 over a large portion of the range of 
interest for the frequencies involved. 
Equation (1) leads to the following relationship 
(=2 + 3a) 
Spaz 
SV 2rhi + 2rhi - (1 — 2a) 
— 2rh, = 0, (2) 
an irreducible cubic in a. It is this fact that makes 
the problem mathematically difficult and makes 
impossible the explicit elimination of a. 
Additional equations are 
a a 
DP = —_ eee 
4 — 3a —40/2ih; S(1—a)i 4 — 3a 
(3) 
an approximation holding well over the region of 
interest since a > 24. The phase difference @, result- 
ing from the difference in optical path between the 
reflected and direct rays, is 
aoe site| 1 ae ;| 
x V2rhy(l—a) SJ’ 
and for the transmitted power 
10° 1 | (1 — D)? _ =, ® 
LMP. a SS = Se Pps 
10- = ral q + D sin s|. (5) 
Here we have four equations. If h;, \, and A are 
specified, there remain five unknowns: D, 2, a, he, 
and S. Thus we should be able theoretically to elimi- 
