270 BELL SYSTEM TECHXICAL JOURNAL 



JM, where M is the unit length mutual impedance of the lightning channel 

 and the sheath. The resulting sheath current I(x) is obtained from (6) 

 and (7), when E is replaced by £ , To by T and A'o by K, the characteristic 

 impedance of the sheath-earth circuit. The constants A and B are found 

 by observing the voltage between sheath and ground is zero at .v = s/2. 

 The electric force along the core is given by E(x) = RI(x), and the voltage 

 between the core conductors and the sheath is obtained by a second ap- 

 plication of (6) and (7), the constants .1 and B being determined from the 

 condition that the latter voltage must equal zero at x = s/2. The voltage 

 between the core conductors and the sheath at the distance x along the cable 

 beyond one end or the other of the lightning channel projection on the 

 cable is then: 



^,, , JRMT's 



L{x) = 



r -Tflj- -vx 



2Z(r- - To 



l^s (' - ^""'^ -T^'^'- ^""'] (^') 



the sign of the voltage beyond one end of the channel being opposite to 

 that beyond the other end. 



Since F » To , the last bracket term may be neglected. It was shown 

 previously, that attenuation along the core-sheath circuit within a distance 

 of one mile, which is representative of the length s, is quite small, so that 

 1 — e~ °'' ~ T(tS. With these modifications: 



JRMT-s ^ ^ 



The earth-return impedances M and Z are given by the following 

 approximate expressions: " 



M = '^ . ^? . (41) 



Itt (If + y-)(iu}a)' 



where a is defined as before, log = logt and: 



It = height of lightning channel above ground 



y — horizontal separation of lightning channel from cable 



The expression for M holds when a{Jf -(- y")" > 5, a condition which is 

 satisfied in the important part of the frequency range. 

 Inserting (41) and (42) in (40): 



