96 BELL SYSTEM TECHNICAL JOURNAL 



practical applications amounts at most to 0.05 to 0.10. On the other 

 hand, with the wire close to the surface of the ground, the conducting 

 area of the ground return path is just one half the area available when 

 the ground extends indefinitely in all directions and the return im- 

 pedance is Zg^. In other words, the departure from circular symmetry 

 means only a very small increase in the ground return impedance. 

 In fact this increase is so small and the ground conductivity actually 

 so variable, the correction is hardly justified by the precision of the 

 data, so that, in most engineering applications, we may take Zg as 

 equal to Z„° with an error probably less than that involved in other 

 factors, and lack of precision in data. 



Derivation of Preceding Formulas 



The derivation of the preceding formulas is not without interest. 

 Since, however, this derivation is, in general, an adaptation of the 

 methods employed in my paper 'Wave Propagation in Overhead 

 Wires with Ground Return' {B. S. T. J., Oct., 1926) it will be out- 

 lined rather than given in detail. 



Take the axis of the wire as the origin and Y as the vertical axis; 

 then the surface of the ground is the plane y = h. Let a unit current 

 flow in the wire and take the axis of the wire as the Z axis. In the 

 ground (p = -^x^ -\- y^ = a) the axial electric intensity will be written 



^ = r. ^i F. K,{p'i-4i) + £' = E° + £', (4) 



Ao(a 4-Vi) 



where p' = VaV-x^ + y^ and K^ is the Bessel function of the second 

 kind, related to the Hankel function by the equation 



The first term on the right hand side of (4) represents the circularly 

 symmetrical distribution which would alone exist if the surface of 

 the ground were removed to an infinite distance, while E' is a secondary 

 distribution due to reflection at the surface of the earth {y = h) , 

 Inspection of equation (4) shows that when p = a, E is the required 

 return impedance Z. 



Strictly speaking £" should be written as 



E' = ^^ ^l ,- {Ko(p'i^i) + hK,{p'Ui)cos d 



+ Ji2Ki{p'i-^i) cos 2d -{- ■ • •], 



