MUTUAL IMPEDANCE OF GROUNDED WIRES 171 



Thus for low frequencies and short wires the main effect of two- 

 layer earth will consist of the effect on the d.-c. mutual resistance. 



Ill 



The mutual impedance of wires in a medium having two parallel 

 planes of discontinuity in the conductivity may be derived by extension 

 of certain results published by A. Sommerfeld,® who has obtained the 

 electric and magnetic fields of a horizontal electric doublet in a medium 

 having one plane of discontinuity; the doublet may be regarded as an 

 element dS of a wire of negligible diameter carrying a finite current and 

 the mutual impedance of wires obtained by double integration over 

 their lengths. The general formula may also be derived by extension 

 of the second method of derivation given by R. M. Foster (loc. cit.), 

 but for brevity this derivation is omitted here. 



In the following both rectangular coordinates (x, y, z) and cylindrical 

 coordinates (r, <^, z) are employed, with the origin in the upper hori- 

 zontal plane of discontinuity, z in the vertical direction and x in the 

 direction of the doublet. Electromagnetic c.g.s. units are used, and 

 the field variation with time taken as e*"', this factor being omitted 

 throughout. The fields are defined through "Hertzian Vectors,"^ 

 the rectangular components of which must individually satisfy the 

 wave equation: 



a^n d^u a^n 



and in terms of which 



E = c grad div n - ct^H, (2) 



where 



H = - - 72 curl n, (3) 



CO 



n = Hertzian vector = Ila;, IIj,, Hz, 



7^ = 47rX«co — ew^, 

 X, e = conductivity and dielectric constant, 

 w = IttJ = radian frequency, 

 c = velocity of light. 



* A. Sommerfeld, "Uber die Ausbreitung der Wellen in der drahtlosen Tele- 

 graphic," Annalen der Physik (4), 81, 1135-1153, December, 1926. 



^Abraham and Foppl, "Theorie der Elektrizitat," 7th ed., Leipzig and Berlin, 

 1922, Vol. I, § 79, page 322. 



