260 BELL SYSTEM TECHNICAL JOURNAL 



For sufficiently low frequencies, so that {io^y)' < 1, /co < i/pK, iw < Ro/Lo , 

 and sinh (icjoy)' = (/aj7)"(l + iu)y/6), expression (12) becomes 



/ R 1 



^^^^ - 2 a^ + ^M^a;)^(/ + a;7 '6)^ ^^"^^ 



where 



a = v/2p, jS = RoCo 



For small values of time, corresponding to large values of /co, the function 

 S'(t) defined before, as obtained by operational solution of (13) is 



(i)' 



5'(/) = 7?7- ^ ( -J e-^'- (15) 



For large values of time, corresponding to small values of iu\ the function 

 is obtained by operational solution of (14) and equals: (Reference 10, pair 

 542) 



where, with {Gt/yY = u: 



h{ii) = —ie " erf Ciu) = —j^ e ' e cIt (17) 



erf being the error function. 



Values of the function Ii{u) are given in Table I. 



In Fig. 2, curve 1 shows the function S'(f) calculated from (15), and 

 curve 2 that calculated from (16), for a cable of 1.4" diameter, using con- 

 stants as indicated in figure. The constants apply to a cable on which 

 measurements have been made of the voltage between sheath and core 

 conductors, at a location where the measured earth resistivity was 400 

 meter-ohms. The function S'(t) corresponding to equation (12) is obtained 

 with sufficient accuracy by drawing a transition curve, 3, between curves 

 1 and 2. 



If the impedance s is taken equal to the direct-current resistance R of the 

 sheath and if the velocity of propagation along the sheath and along the 

 core are assumed to be infinite, so that T = {iuaY and To = {io^lHy, the 

 following expression is obtained 



^'« - 2-(?T7) fe)' (>«) 



In the following it will be shown that (18) is accurate enough for practical 

 purposes. 



The wave shape of the current in lightning strokes may be approximated 

 by an expression of the form: 



/(/) = /(e,-"' - e-''). (19) 



