Wave Propagation over Parallel Wires. 609 



in the following section of tin's paper 1ms one substantial 

 advantage which gives it an interest extending beyond the 

 specific problem : it is quite generally applicable to problems 

 in wave propagation along conducting systems in which the 

 surfaces of the conductors are generated by lines parallel t*> 

 the axis of propagation. For example, it has been successfully 

 applied by the writer to the problem of wave propagation 

 along a wire parallel to the plane surface of a semi-infinite 

 solid of finite conductivity ; the corresponding practical 

 problem is, of course, transmission over a ground return 

 circuit. Again, it has been applied to quantitatively 

 investigate the effect of a concentric ring of iron armour 

 wires on submarine cable transmission. 



From an engineering standpoint the most important 

 effect in parallel wire transmission is the dissipation of 

 energy in non-magnetic wires. Consequently formulas 

 for the alternating current resistance of the wire have 

 been worked out in detail, and the functions involved have 

 been computed and graphed, the data being collected in 

 section lit of this paper. The a.c. resistance of the wire 

 is expressed in the form 



R = CRo, 



where R is the a.c. resistance of the wire when the return 

 conductor is concentric (and is therefore calculable from 

 well-known formulae and tablesj, and C is a correction factor 

 which formulates the modifying effect of the current in the 

 adjacent wire. This is termed the proximity effect correction 

 factor, following a usage suggested by Kennelly *. The 

 correction factor approaches an upper limit m , which is 

 a function of the parameter k only (ratio of radius to inter- 

 axial separation between wires), which it approaches in 

 accordance with an asymptotic formula derived from the 

 asymptotic expansion of the Bessel functions involved. By 

 aid of the data of section III the calculation of C is reduced 

 to a very simple matter. 



II. Mathematical Analysis and Derivation of Formula'. 



The conducting system under consideration, as stated, 

 consists of two long similar and equal parallel wires of 

 circular cross-section, in which equal and opposite currents 

 are flowing. The radius of the wire is denoted by a, its 

 conductivity and permeability by A and /u, respectively, and 



* Kennelry, Laws, and Pierce, Proceedings A.I.E.E. 1915, iu>. 1749 



1813. ° '" 



