Chapter 11 



A GRAPHICAL METHOD FOR THE DETERMINATION 

 OF STANDARD COVERAGE CHARTS" 



Tin: power DENSITY at distance »S' from a trans- 

 mitter of unit power depends upon hi and hi, 

 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 into 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, 

 hi, hi, and X 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 juw per 

 square meter, designated here by A; but if we are 

 given A, hi, and / and then are asked to compute hi 

 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. 



TRANSMITTER S_ — -^RECEIVER 



Figure 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, h^ and S should appear in the final equa- 

 tions, if possible. Also it should be unnecessary to 

 interchange transmitter and receiver according to 



a By Lt. Comdr. D. H. Menzel, USNR, Office of the Chief 

 of Naval Operations. 



the condition that hi is or is not greater than hi 

 The arbitrary parameter a is denned as follows. 

 Let rfi be the distance from the transmitter to the 

 point at which the ray is reflected and d the distance 

 to the point where a ray is tangent. Then 



d§(l - a) = 2hir(l - a) 



(1) 



a, therefore, is constant along a reflected ray; a = 

 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 



< a < 1 , 



with a > % over a large portion of the range of 

 interest for the frequencies involved. 



Equation (1) leads to the following relationship 



S 2 + ( ~ 2 + y S V2WU + 2rhi ■ (1 - 2a) 



(1 -a)* 



- 2rh, 







(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 



D- 



4 - 3a - Wl'-lt i S~ l (I -a)' 4 - 3a 



, (3) 



an approximation holding well over the region of 

 interest since a > %. The phase difference $, result- 

 ing from the difference in optical path between the 

 reflected and direct rays, is 



* = 



4irh 



*lT 1 -11 



k lV2rhi(l-a) S_T 



* |_V2/7li(I -a) 

 and for the transmitted power 



10 6 1 Rl - DY- 



10-'" 



i f a - d)- , n ■ 2 *] 



(4) 



(5) 



Here we have four equations. If hi, X, and A are 

 specified, there remain five unknowns: D, *, a, h 2 , 

 and S. Thus we should be able theoretically to elimi- 



93 



