BELOW THE INTERFERENCE REGION 



93 



in which p' is the distance coefficient given by 

 equations (144) and (145). Let the value of / for 

 a low antenna be denoted by Hj^. 

 The magnitude of Hj^ is given by 



HL-^\ 



1 + 



2lh 



m\ 



(152) 



4Q2+ 1 



and is plotted in Figure 47. 



For height h larger than 4//, the two first terms 

 under the square root may be neglected in comparison 

 to the third term, and 77^, becomes approximately 



Hl = Ih. (153) 



In order to show more cleai-Iy when this approxima- 

 tion is justified, Table 4 gives values of 4-/1 for differ- 

 ent wavelengths and vertical polarization. For 

 horizontal polarization, 4/Z is quite small. 



z 

 o 



t3 



z 



3 



I 



Figure 29. Height-gain as a function of height. (See 

 Figures 7, 25, and 47.) 



Inspection of Table 4 shows that, except for the 

 case of sea water at wavelengths above 1 meter, 

 the approximate equation (153) is good for heights 

 above about 50 meters. 



Table 4. Values of 4/Z for different wavelengths 

 (vertical polarization). 



To a first approximation, the value of / for low 

 antennas is the same for all modes, so that the height- 

 gain functions may be factored out, as was pointed 



out in Section 5.1.7. The gain factor for the case 

 when both antennas are low can then be represented 

 by 



A = 2AoA,F, {HlUHl),, (154) 



where F^ is sum of the shadow factors of all the 

 modes. F^ has been plotted in Figure 32. Equation 

 (154) is also valid in the optical region, provided 



2/li/l2 



d>> 



(155) 



This condition is added to insure that the point is 

 well below the center of the lowest lobe. 



2. Elevated antenna; h > 30X"'^. In this case the 

 height-gain function / increases exponentially. Rep- 

 resenting the increase over Ih by g, we have 



/ = glh. (156) 



For the dielectric case 5 > > 1, refer to equation 

 (193) to define 5. (See Sections 5.1.6 and 5.7.3.) 



g = 0.1356 (eh)- 



10' 



,0.948 Meh 



where 



\\^ka/ 



= — X 



1/3 



60 



irf 



(157) 



(158) 

 (159) 



See Figures 35 and 36. 



For the case of sea water, vertical polarization and 

 wavelength in the VHF (1 to 10 meters) range, 

 g requires a correction factor g' which depends on X. 

 A table of g' is given \^ith the graph of ^ in Figure 36. 



The change of / with height h is represented 

 schematically in Figure 29 (see also Figures 7, 25, 

 and 47). 



Formula for the Dielectric Earth. (See Sections 

 5.1.7 and 5.7.3.) 5 > > 1. 



The first mode has the form $i(d) • f(hi) • fihi). 

 If the value of the gain is given substantially by the 

 first mode, and substitution from equation (148) is 

 made, it is found that 



A = [2AoA^Fsi]f{h) ■ fih). (160) 



For the dielectric case equation (160), using equa- 

 tions (140), (147), (151), and (156), becomes 



^^Wrigh),. 

 2 a- 



(161) 



The same formula holds for sea water, vertical 

 polarization, wavelengths in VHF range, d > 50/p', 



