LIGHTNING PROTECTION OF BURIED TOLL CABLE 271 



where 



^ o o -v/2 (-^4) 



Comparison with (21) and (23) shows that in this case: 



^(•^^'^) = o/"\^ ^g(^-^^ (45) 



2(o! - ^) 



In the above solution, /j, was assumed constant. Actually it changes 

 slightly with frequency and, for reasons mentioned before, it is accurate 

 enough for practical purposes to replace iw with 1// when calculating /x. 



The maximum voltage is obtained at x = 0, i.e., at a point opposite one 

 end or the other of the lightning channel, and comparison with (22) shows 

 that this voltage differs from that obtained in the case of a direct stroke 

 by the factor /x, since /3' <C a^ so that the second term may be neglected in 

 (23). The above factor has the following approximate value: 



M = .14.-:r^., (46) 



/r + y- 



Since each cloud has an equal and oppositely charged image at the dis- 

 tance // below the surface of the ground, the electric field between cloud and 

 ground is substantially equal to that between the clouds when 5 = 2h. 

 For a discharge to take place between clouds, rather than to the earth, the 

 length 5 would, therefore, have to be less than 2h; so that with y = the 

 factor would not be expected to exceed n = .28. Thus, for the cable previ- 

 ously considered, failures due to discharges between clouds would not be 

 expected except for currents in excess of 100,000 amperes in a lightning 

 channel approximately above and parallel to the cable. 



Maximum voltages of opposite signs are obtained at the two ends of the 

 lightning channel, the voltage at the mid-point being zero. As the distance 

 X from one end of the lightning channel increases, the voltage diminishes 

 rather slowly in the same manner as shown in Fig. 5 for Vo{x, /). 



1.9 Stratified Earth Structures 



In the foregoing, the earth was assumed to have a uniform resistivity p. 

 In many cases the average resistivity near the surface along a route may be 

 substantially greater or smaller than the resistivity at greater depths, and 

 this condition may affect the nature of lightning troubles, as will be shown 

 below. 



The function Qo{r) appearing in equation (33) represents the earth 



