BELOW THE INTERFERENCE REGION 



119 



From these limiting values of r the value of t in 

 general for any gi\'en 5 can be found from the follow- 

 ing two series, of which only the first is of interest 

 in short-wave work. 



5 large, 



T,i = ni., 



+ • 



S-l/2 - 



— - T-„ 



;-3/2 



+ z5" 



"t 2 s -3 ,'2 



o 



(195) 



5 vimtll, 



Tn.n 



Tn.O 







4. The height-gain functions fQi). 



(a) For low antennas {h < SOX"''^), the height-gain 

 functions, to the first approximation, are inde- 

 pendent of n. 



f(h) = 1+i 



/(/0 = i+i 



'2irh V e, - 

 '2Trh 



_ for vertical polarization, 



(196) 



e.-l] 



Ve^— 1 for horizontal polarization 



Note that the magnitudes of the bracketed quantities 

 are equal to Ih. The magnitude of / has been denoted 

 b.v Hl and is represented in Figure 47 as a function 

 of Ih. The phase of the bracketed quantities in 

 equation (196) is taken into account by using differ- 

 ent curves with the parameter Q = €r/60o-X. For 

 large values of Ih, H^ — > Ih. 



(b) For elevated antennas (h > > 30X^^^). The 

 function /„ can be represented by 



/« = 



1 exp{ + n~Jl[{eh)-2r„]''"} 



V27r(2e/i)^ 



/i 



Jl/six) + J -I /six) 



(197) 



ii-f 



(198) 



where, from equation (159), 



(X^fca)'/^ 60 



and where the argument of the two Bessel fimctions 

 is 



X^H-^r^f'-e-'" 

 For the «th mode, if {eh) > > 2t„, the magnitude of 

 /„ can be written 



3 1 exp(jr„ V2e/i) 



^27r{2eh)'^*J^/,{x) + J.„s(x) 



(199) 



For large 5, using the first two terms of equation 

 (195), and writing x^ for .r when r„ is replaced by 





Substituting this in Ji/six) + J.^/^ix), writing down 

 the first two terms of the Taylor expansion, making 

 use of the fact that the r„, „ are roots of Ji/z{x) -\- 

 J-i/six) = and of the relation given by a prop- 

 erty of the Bessel function. 



we have 



Ji,,(x) + J.,,,(x) 



1/(30)) [Ji/sW + ^-i/sW] 



+ J -2 /six) - J 2 /six), 



-(-'t)"l^- 



ixj 



J2,,sixj]. 



(200) 



If these results are substituted into equation (192) 

 for both antennas, the factor (5 + 2t„) becomes 

 1 + 2r„,^/S, W'hich approaches unity for large 8. 

 This means that if both antennas are sufficiently 

 elevated, short-ivave propagation is practically inde- 

 pendent of ground constants. 



The value of / given by equation (199) can be 

 written as glh so that g represents the gain over Ih, 

 the value approached by Hi, when ?/) > 4. The value 

 of g for 5 — > oo is represented in Figure 36. 



If 5 is not very great, as in the case of verticalh* 

 polarized VHF over sea water, the effect of 5 can be 

 taken into acocunt by changing e to eg{S) and g to 

 gg'. The functions g{8) and g' are given by Figure 55. 



5. Platie earth gain factor and .shadow factor. The 

 field near the ground over a plane earth with infinite 

 conductivity is equal to 2Eo, twice the free-space 

 field. For an imperfectly conducting ground, the 

 field for antennas at zero height over a plane earth 

 may be written 



E = 2£;o-4i. (201) 



Ai is represented in Figure 56 as a function of p'd, 

 where 



P' 



27r 



- 1 



27r V(6, - 1)2+ (60o-X)- 



(60(rX)-2 



(202) 



for A-ertical polarization. For horizontal polarization. 



P = 



= yV,., 



ly -\- (60(rX)=. (203) 



The curve parameter is Q --= tJGOvX. 



