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BELL SYSTEM TECHNICAL JOURNAL 



where the values of the a's are given in Table I. A similar expression 

 results for horizontal polarization so that the a's are also evaluated 

 for this case. 



With the aid of equation (9) this may be expressed as a power ratio 

 between short doublets : 



1 +ai 



1 + 1 

 hi hi 



+ "'^1 + 1 



+ (Ja. 



+ a4 



1 



hih 



1 + 1 



hi hi 



1 



hihi 



1/2 



(31) 



For antennas at the heights above the ground that are usual in the 

 ultra-short-wave range, the bracket in equation (31) is unity so we 

 have the useful result that within certain limitations the ratio of 

 received to transmitted power with simple antennas is independent of 

 the wave-length. 



When the Q of the ground is large in comparison with unity, equation 

 (30) reduces to the somewhat simpler expression 



E_ 



4iTrhih2 



1 + 



hi' 



1+^ 



(32) 



where ao' = e''K'/4:T'(e—l) for vertical polarization and ao^ — 'K'^l4:Tr'^{e—l) 

 for horizontal polarization. Likewise when the Q of the ground is 

 small in comparison with unity, equation (30) reduces to 



E^ 



E, 



4:irhih', 

 \d 



^ hi 2/2i2 



^ hi Ihi' 



(33) 



where bo = — V2gX/27r for vertical polarization and bo = V2/gX/2x for 

 horizontal polarization. 



Equations (28), (30), (31), (32) and (33) are valid for all distances 

 beyond those for which the received field strength begins to vary 

 inversely with the square of the distance provided the antennas are 

 not too high. This range of validity contains all practical distances 

 for ultra-short-wave propagation over land and fresh water and the 

 longer distances for ultra-short-wave propagation over sea water. 

 For antennas at greater heights above the ground, equation (27) is 

 required; but usually the range of antenna heights between those for 

 which equation (21) and those for which equation (30) are valid, is 

 small. 



The applicability of the approximate equations to the problem in 

 hand may be ascertained as follows. First calculate the parameter x 

 of Fig. 2 to determine if the distance is sufficient for the field strength 



