Wave Propagation in Overhead Wires 

 with Ground Return 



By JOHN R. CARSON 



I 



THE problem of wave propagation along a transmission system 

 composed of an overhead wire parallel to the (plane) surface of 

 the earth, in spite of its great technical importance, does not appear 

 to have been satisfactorily solved.^ While a complete solution of 

 the actual problem is impossible, on account of the inequalities in the 

 earth's surface and its lack of conductive homogeneity, the solution 

 of the problem, where the actual earth is replaced by a plane homo- 

 geneous semi-infinite solid, is of considerable theoretical and practical 

 interest. The solution of this problem is given in the present paper, 

 together with formulas for calculating inductive disturbances in 

 neighboring transmission systems. 



The axis of the wire is taken parallel to the s-axis at height h 

 abov^e the .xs-plane and passes through the y-axis at point 0' as 

 shown in Fig. 1 herewith. The "image" of the wire is designated 

 by 0". 



For y>^ (in the dielectric) the medium is supposed to have zero 

 conductivity, while for y<0 (in the ground) the conductivity of the 

 medium is designated by X. The xz-plane represents the surface of 

 separation between dielectric and ground. 



We consider a wave propagated along the s-axis and the current, 

 charge and field are supposed to contain the common factor 

 exp { — Vz-\-iwt), which, however, will be omitted for convenience in 

 the formulas. The propagation constant, F, is to be determined. It is 

 assumed, ah mitio, as a very small quantity in c.g.s. units.- 



In the ground (t^O) the axial electric force is formulated as the 



1 See Rudenberg, Zt. f. Angewandt, Math. u. Mechanik, Band 5, 1925. In that 

 paper the current density in the ground is assumed to be distributed with radial 

 symmetry. The resulting formulas are not in agreement with those of the present 

 paper. Since this paper was set up in type I have learned that formulas equivalent 

 to equations (26), (28), (31) for the mutual impedance of two parellel wires were 

 obtained by my colleague, Dr. G. A. Campbell, in 1917. It is to be hoped that 

 his solution will be published shortly. 



* The simplifying assumptions introduced in this analysis are essentially the same 

 as those employed and discussed in "Wave Propagation Over Parallel Wires: The 

 Pro.ximity Effect," Phil. Mag., Vol. xli, April, 1921. 



539 



