Mutual Impedance of Grounded Wires for Horizontally 

 Stratified Two-Layer Earth * 



By JOHN RIORDAN and ERLING D. SUNDE 



A general formula is derived for the mutual impedance of wires embedded 

 in a conducting medium and lying in one of two parallel planes of discon- 

 tinuity in the conductivity. The general formula is quite complicated, but 

 simplifies in a number of important cases, and summarizes the published 

 mutual impedance formulas relating to two-layer earth and its special cases. 

 The two most important cases are obtained by making the conductivity of 

 one or the other of the outer regions zero. The special case in which the 

 conductivity of the outer region adjoining the wires is zero gives the mutual 

 impedance of any thin grounded wires lying on the surface of a horizontally 

 stratified earth having conductivities Xi and X2 at depths less than or greater 

 than b, respectively. When the conductivity of the other outer region is 

 zero, the formula gives the mutual impedance of wires lying in the plane of 

 separation, at the depth b. The formulas in both cases involve integrals 

 which apparently cannot be evaluated in closed form: for practical applica- 

 tion the use of curves of the kind given is suggested for approximate 

 numerical results. The formulas for the special cases, of which there are 11, 

 together with some of their limiting forms, are tabulated together for ready 

 reference. 



I 



THE general formula for the mutual impedance of wires embedded 

 in a conducting medium and lying in one of two parallel planes 

 of discontinuity in the conductivity with displacement currents 

 neglected is of the following form: 



Z Sa — 



The integrations are extended in the double integral over the two 

 wires 5 and 5 extending between points A, B, and a, b, respectively, 

 whose elements dS and ds are separated by distance r and include the 

 angle e between their directions. Q(r) and P(r) are functions of the 

 frequency, the conductivities, and of the separation b between the 

 planes of discontinuity in the conductivities as well as of r. 



For wires on the surface of a horizontally stratified earth having 

 conductivities Xi and X2 at depths less than or greater than b, 

 respectively, Q{r) and P(r) are given by: 



* A preliminary report of some of the results of this paper has been given in a 

 letter to the Editor of the Physical Review, Vol. 37, No. 10, pp. 1369-1370 (May 15, 

 1931). 



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