INTRODUCTION 



63 



A graphical representation of the equation 



(18) 



is given in Figure 2. Fork = 4/3, a = 6.37 X 10*^ m, 

 this takes the form 



(II = 4120 (yllh + V]Q meters, (19) 



where hi, /lo are given in meters and d^ in meters. 



h, METERS 

 0|— 



25- 

 100 — 



200 

 300 

 400 



600 

 800 

 1000 



2000 

 3000 



4000 

 5000 

 6000 

 7000 

 8000 

 9000 

 10000 



15000 



20000 



d^ km 



Oi— 



100 



200 



300 



500 



600 



700 



800 



900 



1 000 



1100 



1200 



hj METERS 

 0(- 

 25 — 

 100 



200^ 



300- 



400: 



600 — 

 800- 

 1000 



2000 



3000 — 



4000 



5000 



6000 



7000 



8000 



9000 



10000 



15000 



20000 



dL = 4.l2 (/^T +/*^) =. dT + dR 



Figure 2. Sum of transmitter and receiver horizon 

 distances for standard refraction. To change scale: 

 Divide Ai and h« by 100, and divide d^ by 10. 



' '' Nature of the Radiation Field 

 in the Standard Atmosphere 



The mathematical solution for the radiation field 

 takes various forms for particular cases. The treat- 

 ment for low antennas, for instance, differs from 

 that for high antennas, and similarly the equations 



must be handled differently for the two types of 

 polarization. These and related problems are dis- 

 cussed in general in the following. 



1. General form of field variation. The mathemat- 

 ical expression for the radio gain of the radiation 

 field of a doulslet under standard conditions is given 

 as the sum of an infinite number of complex terms or 

 modes. (See Section 5.7.6.) Disregarding the phase 

 factor, a representative term (mode) of this series- 

 has the form 



F{d) . fiih) . f,ih) 



(20) 



These modes are attenuated unequally. Well within 

 the diffraction region, the first mode contributes 

 practically all of the field so that the effects of dis- 

 tance and height are separable. In this region, the 

 problem of numerical computation is simplified, 

 since it is possible to use separate graphs for the 

 dependence on height and distance. As the receiver 

 is moved toward the transmitter, the number of 

 modes required for a good approximation increases. 

 For low antennas, the addition of the modes is 

 practicable and the graphical aids are useful for 

 short distances. These conditions are illustrated in 

 Figiu'e 3 for horizontal polarization or ultra-short 

 waves. 



In the optical i-egion, the methods of geometrical! 

 optics give a result equivalent to that of the rigor- 

 ous solution at points which are not close to the line 

 of sight. The field is then the sum of a direct and a 

 reflected wave, resulting in an interference pattern. 



The preceding discussion is illustrated by Figure 4 

 which shows the variation of field strength with 

 distance for fixed antenna heights, for propagation 

 over dry soO with a wavelength of 0.7 meter on 

 vertical polarization. The numbers refer to the 

 number of modes required for a better tlian 99 per 

 cent approximation. The interference pattern is 

 illustrated by the oscillatory nature of the curve. It 

 will be observed that beyond the first maximum, 

 the points found by geometric optics gi-\'e a value 

 of the field which is slightly too low. (See dots in 

 Figure 4.) hi fact, as the line of sight is approached, 

 the optical formula approaches zero whereas the exact 

 solution does not. The geometric-optical method 

 breaks down in the optical region as the line of 

 sight is approached. It may be noted that Figure 4 

 has been drawn for fc = 1 rather than for the cus- 

 tomary value of fc = 4/3 corresponding to standard 

 atmosphere conditions and is for a hypothetical' 

 isotropic radiator. 



