62 



CALCULATION OF RADIO GAIN 



gain is insignificant in practice. For other types of 

 antenna systems and for microwave frequencies, the 

 gain may be many times larger. 



The propagation problem, thus limited, has been 

 solved mathematically^; but the explicit mathematical 

 formulas are far too complicated to be of much use to 

 the practical computer. Much additional work has 

 been done, however, to bring the solution into a 

 form suitable for practical use. This involves 

 reducing the computations to the use of graphs, 

 nomograms, and tables, and it is this final stage of 

 the problem which is the subject of subsequent 

 parts of this chapter as well as of Chapter 6. 



equation (11) assumes the form 

 2 ka 



(12) 



dj. = V2 kohl . (13) 



Similarly, the horizon distance of the receiver is 



da = V2 kalh . (14) 



The sum of the two horizon distances is given by 

 f/i,, where 



dL = (It + (/fl. • (15) 



^■'■^ Curved-Earth Geometrical 

 Relationships 



Let hi and h-i denote the heights of transmitter 

 and receiver above the earth's surface, respectively, 

 and let d denote the distance from the base of the 

 transmitter to the base of the receiver, measured 

 along the earth's surface. For a number of cases 

 concerned with high-frequency radiation OATr the 

 earth's surface, it is sufficient to identify the straight- 

 line distance from transmitter to receiver with the 

 distance d between the bases measured along the 

 curved earth. But when path differences are of 

 importance, as they are in interference problems of 

 reflection and in diffraction, it is necessary to com- 

 pute distances to a higher order of accuracy. 



Throughout this chapter, the earth will be assumed 

 to have the equivalent radius ka, and the atmosphere 

 to be homogeneous, and radiation to travel along 

 straight lines. 



The straight line from the transmitting antenna 

 and tangent to the earth's surface (the so-called 

 line of sight) touches the earth along a circle which 

 constitutes the radio horizon of the transmitter. 

 The distance measured along the earth from the 

 transmitter to the radio horizon will be denoted by 

 dx; and the horizon distance of the receiver by dj^. 

 These geometrical relations are illustrated in 

 Figure 1. 



it follows that 

 ka 



From this figure, 

 ka 



/h = 



cos (dx/ka) 



(11) 



Inasmuch as 

 ka 



ka 



cos idir/ka) 



- (rfr/A-a)- 



ka + - 

 2 



ka 



' * * Optical and Diffraction Regions 



The points \dsible from the transmitting antenna 

 (on an earth of equivalent radius ka), i.e., the points 

 above the line of sight, constitute the optical region 

 (Figure 1). The rest of space lies beyond the trans- 

 mitter horizon and below the line of sight and is 

 called the shadow or diffraction region. 



OPTICAL REGION 

 OPTICAL HORIZON 



TRANSMITTER 



LINE OF SIGHT 



Figure 1. Geometry for radio wave propagation over 

 curved earth. 



It is frequently necessary to know whether a 

 recei\'ing antenna lies in the optical region or the 

 diffraction region of a given transmitter. This 

 evidently is equivalent to knowing whether the 

 distance d of the recei^'er from the transmitter is 

 smaller or larger than the combined horizon distance 

 dx,. By equations (13), (14), and (15), it follows 

 that in the optical region 



d < V2A-fl. (V/(, + V/!.,), 

 and in the shadow region 



(7 > V2fc^(V/^-(- V/i^). 



(16) 



(17) 



