258 BELL SYSTEM TECHNICAL JOURNAL 



to cables buried at depths up to one meter or so, the propagation constant 

 for a cable buried at infinite depth being larger than that given above by 

 a factor of v2. 



It is assumed that the current is propagated from the cable up the light- 

 ning channel with infinite velocity. The voltage between the cable con- 

 ductors and the sheath obtained in this manner reaches a crest value after 

 some 50 to 100 microseconds, or after the current in an actual lightning 

 channel has traveled from the ground to the cloud. The error due to this 

 assumption is thus probably quite small as regards the crest voltage, al- 

 though the wave front will be somewhat slower when the actual velocity 

 of propagation is considered. 



1.3 Direct Strokes — Voltage for Siimsoidal Current 



The current along the sheath gives rise to an electric force along the 

 latter. The electric force along the inner surface of the sheath is given by: 



E{x) = zl(x) = J ze-'' (4) 



where z is the mutual-impedance of the sheath-earth and core-sheath cir- 

 cuits. 



The latter mutual impedance is equal to the ratio of electric force along 

 the inner surface of the sheath at any point, to the total current along the 

 sheath at the same point, and for low frequencies equals the direct-current 

 resistance of the sheath. It is given by the following slightly approximate 

 formula : 



Z = R{i wyy/s'mh {ioiy)' (5) 



where: co = 27r/and 



R = Unit length d-c resistance of sheath, ohms/meter 



7 = v8/2iraR 



6 = Thickness of sheath, meter 



a = Radius of sheath, meter 



V = Intrinsic Inductivity of sheath 

 = 1.256 X 10~ henrys/meter 

 The current in the core-sheath circuit and the voltage between core and 

 sheath due to an impressed field E{x) along the core (inner surface of 

 sheath) are obtained from the following equations, which are the general 

 solutions of the transmission line equation for the core-sheath circuit. 



/(.t) = [A -f P(x)]e-'-'^ - [B + ()(.v)]/°^ (6) 



^ U{x) = Ko[A + P{x)]e-'°'- + Ko[B -f (2(.v)]e'°^ (7) 



