PROPAGATION OF PERIODIC CURRENTS 541 



Thus: 



When 7' = 7: j{s) = 0, J{s) = 0. 



For some cases, particularly those where the attenuation is neglected, 

 it is advantageous to express the square-bracketed factors in equations 

 (134), (138), (141), (145) partially in terms of hyperbolic sines. 



Appendix I 



Derivations of Equations (25) and (90) 



Equation (25) 



Let the primary or impressed field of force be specified by an electric 

 intensity /„ parallel to the axis of the wire (and to the surface of the 

 earth), and an electric intensity /„ normal to the surface of the earth 

 and measured downward. We denote by /„. the value of /„ at the axis 

 of the wire,-^ and by /„ its value at the surface of the ground in the 

 plane which is normal to the ground and which includes the axis of 

 the wire. The impressed or primary potential Foi the wire, due to the 

 impressed field, is then 



F= Cjndy, 



'Jo 



where // is the height of the wire above ground and the integral is 

 taken along the vertical from the wire (v = 0) to ground (y = h). 



Due to the impressed field, specified above, a current / flows in the 

 wire and a corresponding superposed current distribution is induced in 

 the ground. The resultant axial electric intensity at the surface of 

 the wire is then s„/ (where s,,. is the internal impedance of the wire, 

 per unit length); correspondingly the resultant electric intensity along 

 the surface of the ground is fy — Zgl. Application of the second 

 curl law to a contour composed of two verticals from the wire to 

 ground and the line segments dx in the surfaces of the wire and ground 

 gives 



(. +s)/-f +^= -^ 

 which is preferably written as 



■-^ ^"^ dx dt ' ^^ 



where z — z,c + Sy is the internal impedance of the circuit, per unit 

 length; V is the resultant potential of the wire; and is the resultant 

 magnetic flux threading the contour, per unit length. But we have 



-''/„: is assumed to be sensibly constant over the cross-section of the wire. 



