TIIK lil.KCTKOMAGNETIC FIELD 



39 



lobes. The problem is primarily one of geometry, 

 taking into account the modification of the direc- 

 tion, phase, and intensity of the reflected ray caused 

 by the earth's curvature. It can lie solved by suit- 

 able numerical and graphical methods such as are 

 given in V)lume 3 where the details are extensively 

 treated. It may suffice here to enumerate the main 

 modifying factors. 



If a tangent to the earth is drawn at the point of 

 reflection (Figure 11), the distances h\ and /i' 2 of 

 transmitter and receiver from this line are the equiv- 



Whcn /,■ = ;',,, //, is in meters and d T in kilometers, 

 this becomes 



d T = 4.12 V/ii 



(26) 



The diffraction region actually extends at least from 

 the lower surface of the first lobe downward to the 

 earth's surface. In the diffraction region well below 

 the line of sight, the field strength decreases very 

 rapidly and very nearly exponentially with the 

 distance. 



Figure 12 shows a typical example for the ground 



Figure 11. Geometry over spherica earth. 



alent heights in terms of which the problem is a 

 plane-earth problem for that particular ray. They are 

 smaller than the heights above the ground hi and 

 h.2, but clearly they are functions of the angle of 

 elevation. Thus a set of implicit equations has to be 

 solved for each angle of elevation giving h\ and h\ 

 as functions of hi, h 2 , and d, whereupon the inter- 

 ference between the direct and reflected rays is 

 computed as in the case of a plane earth. 



In addition to the modification of direction and 

 phase at reflection, there is also a change in intensity 

 of the reflected ray caused by the fact that the 

 reflecting surface is curved. This modification is 

 taken into account by the divergence factor, a purely 

 geometrical quantity which is part of the reflection 

 coefficient, reducing the intensity of the reflected 

 ray. 



The behavior of the field below the line of sight 

 requires a more powerful line of attack. The line of 

 sight itself is given by a tangent to the earth's surface 

 passing through the transmitter. The distance from 

 the transmitter to the horizon, when a modified 

 earth's radius ka is used is 



0.5 



OJ 

 0.0 5 



0.01 

 0.0 OS 



0.00 1 

 04)005 



d T = s/2kahi 



(25) 



20 30 40 SO 60 70 



_ d IN KILOMETERS 



Figure 12. Field strength versus distance for fixed 

 height, vertical polarization. 



constants indicated. The ordinate is the ratio of 

 field strength to the free space field ; the transmitter 

 and receiver heights are fixed and d is plotted as 

 abscissa. Above the line of sight the typical lobe 

 pattern is exhibited. The decrease of the field in 

 the diffraction region is the more rapid the shorter 

 the wavelength. In the centimeter band this decrease 

 is so rapid that for most practical purposes the field 

 is nonexistent near the ground at distances exceeding 

 the horizon distance by more than a few kilometers. 

 Figure 13 shows a similar diagram for fixed distance 

 and variable receiver height. 



Modes 



The description of the electromagnetic field above 

 the line of sight is adequately given by means of rays 

 and their phases as used in optics. This method ob- 

 viously breaks down in the diffraction region into 

 which the rays do not penetrate. For this region a 

 solution of the wave equation is required. Many 

 distinguished mathematicians have contributed vary- 

 ing techniques for solving the wave equation. The 



