MUTUAL IMPEDANCE OF GROUNDED WIRES 409 



(3) n. = C(A'^ - ^0-^) ^ COS J^ -Ijrp. e'^^Vdp, 



where the time factor e'^"' is omitted throughout. Jo is the Bessel 

 function of order zero, and the constants k and ko are the propagation 

 constants in the earth and in the air for plane waves varying with the 

 time as e"*"". Their values in Heaviside units are given by Sommerfeld 

 as 



(4) A'- = - (ew- + -/o-oj), ko- = - eoco-, 



where e and eo are the dielectric constants of the earth and of the air, 

 respectively, a is the conductivity of the earth, assumed uniform, and 

 c is the velocity of light. In both media the permeability is taken as 

 unity. Also 



(5) N = k-<p' - ko- + ko-^p- ~ k\ 



(6) N' = Vp- - ko' + Vp^ - k\ 



and C is a constant measuring the electric moment of the doublet. 



We now replace the doublet by a short element of wire dS carrying a 

 current 7e*"', and at the same time we assume that e and eo are both 

 negligible, so that all displacement currents are neglected. This is a 

 simplification which is ordinarily made as a first approximation at 

 power frequencies for the shorter transmission lines. Then, introduc- 

 ing c.g.s. electromagnetic units, in which the conductivity of the 

 earth is X, and noting that we have changed the sign of w, formulae 

 (4)-(6) become 



(7) k- = — i4Tr\co = — 7^, 



(8) ^0^ = 0, 



(9) A^ = - TP, 



(10) iV' = P + Vp^ + 7-, 

 and the constant C is such that 



r^h 2 1 



(11) —T^ = - X current X effective length of doublet 



^ Ids 

 2w\' 



