276 BELL SYSTEM TECHNICAL JOURNAL 



where Ki = {Z/Vy is the characteristic impedance of the insulated sheath 

 without breakdown. 



From (61) and (62) 



V, . .,.. Fo 



Ax =4 A,= no) - ^ (63) 



Ai Ai 



So that; 



-Vx , Vo , ~T^i -Vx\ 



7(.v) =I{0)c-'^ ^'-^{e-'^' -e-^') (64) 



Ai 



For a rubber insulated cable the breakdown voltage T^o would be in the 

 order of 30,000 volts and the characteristic impedance K\ would be in the 

 order of 100 ohms. The maximum current which could flow on the sheath 

 without breakdown, Fo/i^i , is then about 300 amperes, while in the case of 

 an average lightning stroke the current 7(0) would be about 15,000 amperes 

 (i.e. 30,000 amperes total). The first term in (6-i) gives the attenuation 

 along a sheath in direct contact with the ground. The current given by 

 this term would diminish from 15,000 amperes to about 2,000 amperes 

 within a distance of \ mile or so, for a typical lightning stroke wave shape 

 and an earth resistivity of 1000 meter-ohms. At distances of several miles 

 from the stroke point the first term will vanish and the current will be de- 

 termined by the second term, since exp ( — Fi.v) will vanish much more 

 slowly than exp (— F.v). 



In the case of a stroke to ground the impressed electric force in the ground 

 along the sheath is 



£o(-v) = -dVlx)ldx (65) 



where V ^ is the earth potential due to the lightning stroke current and is 

 given by 



Fe(x) = . , /^ .,. , (66) 



lr{x- -f y-y 



Instead of equation (58), the following equation is obtained for the cur- 

 rents in the sheath 



Writing £oCv) — cEo{x) + (1 — f)£oCv), the solution of the latter equation 

 may be written as the sum of two solutions of the form given by (6) and (7). 

 After the constants Aq , Bo applying to the current /o and the constants Ai 

 and Bi applying to the current /i have been determined from the boundary 



