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BELL SYSTEM TECHNICAL JOURNAL 



that the first two terms tend to cancel. Under these conditions the 

 sum of the first four terms of equation (26) may be written * 



[« 



a—iTik\h2l\d 



(27) 



^TVlQli -f hi) _ 



At the greater antenna heights this expression also gives the correct 

 result provided the distance, d, is large compared with the sum of 

 the antenna heights, h\ + Jh. For smaller antenna heights this ex- 

 pression is limited to distances for which the magnitude of the second 

 term within the bracket is small (say less than 0.1). If this term is 

 not small more terms of equation (26) must be taken into consideration. 

 While equation (26) applies to vertical polarization only, equation 

 (27) applies to both polarizations within the region for which it is 

 valid provided the appropriate reflection coefficient is employed. 



For antenna heights sufficiently small that the exponential factor 

 of equation (27) may be replaced by the first two terms of its series 

 expansion, 



E_ 



iirihihi 

 \d 



1 + 



(a - ib)\ 



1 + 



(a — ib)\ 



(28) 



where a and h are given in Table III of Appendix II (page 72). The 

 first factor gives the well known expression for ultra-short-wave 

 propagation over level land. The second two factors are important 

 for antennas near the ground. When h\ -^ h^-^ this becomes 



£ _ {a - ibY\ 

 Eo ~ 4:Tid 



which is equivalent to the first term of the asymptotic expansion of 

 the attenuation factor given in Part I. 

 A more useful form of equation (28) is 



(29) 



Eo 



Aivhihi 

 \d 



'+'"'F, + i 



+ OJ 



+ ^4 



hih'. 



i + l 



hi hi 



-^ + i 



hi hi 



+ 03 



1 



hihi 



+ fls 



hi'h, 



1/2 



(30) 



* Equation (27) differs from equation (26) only in the dropping of terms in Ifd^. 

 By leaving the exponential factor and the coefficient of reflection unexpanded the 

 useful range of this formula is increased. The term 



1 



2irid/X 



I -|_ ^g-4xiAiA2/Xd 



has been omitted from the right side of equation (27) since it can be shown that 

 this term is always small compared with the remaining terms when lird/X "^ I . 

 In order to facilitate calculations by means of the reflection coefficient curves, — g> 

 is replaced by {R + \yd'^l2{h\-\-h2Y to which it is equal to the required order of 

 approximation. Another form of equation (27) that may be preferred in some cases 

 is given as equation (35) in the conclusions. This form results from substituting 

 for — gi its value (a — iby/2. 



