-1^ 



RADIO PROPAGATION 

 1 + t2 



[l-3-5---(2« - 1)>„, 



71 



ZTTHa/X I 



Z) = - _^-_e(2--<i/x)a-i/r) ^ z) / fl!^ , (40) 



1 — r'* n=0 \ '^''' / 



/'=:r-^g^2'^^'^/^)(i-^/^>L ( ^V/l )"[l-3-5---(2»-l)]/,„ (41) 

 1 — 7-^ „=i \ — livialKT / 



2ialf (42) 



where 



~2= ^ 



and r is in the first quadrant. The positive square root of i is to be 

 taken in equation (38). 



These expressions follow from those given by Wise '^ when the sign 

 of i is changed so that the impHed time factor is e'"' in accordance 

 with engineering practice instead of the e~^'^' employed by Sommerfeld 

 and Wise. Their expressions were derived for an antenna half in air 

 and half in the earth. To obtain the above expressions which apply 

 to antennas on the surface of the earth, Wise's expressions have been 

 multiplied by 2/(1 + t^), A corresponds to his expression (5), B to 

 his (12), C to his (8) and D to his (6). The quantities, a^ and Cn are 

 substantially unity except when r is not small. 



tanh-^V^ * yfe"-i 

 oi = 7= — = L. 



k n=i(2w - 1) ' 



{2n — 1)(2« — 3)(a„_i — an-2 



02 



3(ai - 1) 



Ci = 1, 



C2 



Cn — Cn—l 



(n - l)~k 

 3' 



(43) 



(2w - l){2n - 3) 



Cn-2, 



(44) 



ti{2n - 1)' 



(2« - 3)Dn-l - IDn-i 



where 



Do =1, Di = V/tanh-i V7 = ^ 



D2^ Di- I, D„ = 



k = 



^ " rr^ ~ e+ 1 - Half 



/« 



(45) 



(46) 



(47) 



