764 BELL SYSTEM TECHNICAL JOURNAL 



1 is determined by 



The functions A, B, and C are plotted in Figs. 1, 2, and 3 for some 

 values of the parameters often to be found in practice. 



In these formulas X is the ground conductivity in electromagnetic 

 c.g.s. units, X the separation between wires in centimeters, / the time in 

 seconds, and j = V— 1. The functions ker' and kei' are related to 

 the Bessel function of the second kind for imaginary arguments de- 

 fined by G. N. Watson, " Bessel Functions " as follows 



ker'(2) ±ikei'(s) = - j=^'''Ki(zj^'i^) 



Values of these functions are tabulated in Table I of " Bessel Functions 

 for A-C Problems " by H. B. Dwight A. I. E. E. Trans. 1929 pp. 81 2-820. 

 The induced voltage is in units of abvolts per cm. which is trans- 

 formed to volts per mile by the factor 1.61 X 10~*. 



Part II 



The second part of this paper will be devoted to a discussion of the 

 theory leading to the above results. 



Consider a system of two wires, 1 and 2, wire 1 being of infinite 

 length, parallel with each other, with the heights hi and Jh above 

 earth and separated by a distance x. The general problem is to 

 calculate the voltage on wire 2 as a function of time due to the sudden 

 flow of a current in wire 1, this current being zero before / = 

 and /(/) thereafter. Let the voltage on wire 2 due to a unit current 

 step, that is, a current equal to zero before t = and unity after 

 / = 0, be denoted by Znit), then the voltage due to a current I{t) is 

 given by 



Vnit) = j^J' Z:,{r)I(t - T)dr.' (1) 



The fundamental quantity thus necessary in the solution of the 

 problem is Zuit). This quantity completely determines the voltage 

 Vi2{i) for all types of disturbing currents. Zio{t) may be written as a 

 Fourier integral : 



1 r+'° pj<^t 



Zi2(t) =i- V- Zi2(co)(/a,, (2) 



where Zuioo) is the mutual impedance for periodic earth currents and 

 ^ Carson, Electric Circuit Theory' and the Operational Calculus, page 16. 



