PROPAGATION OF PERIODIC CURRENTS 503 



or line is exposed throughout its length to an arbitrary impressed 

 disturbance. Furthermore, in the problem of crosstalk the departures 

 from the conditions of balanced symmetry are necessarily very small, 

 whereas the formulas are so general as to make it very difficult to 

 introduce the essential simplifications which follow from the condition 

 of small departures. For example, if the foregoing formulas are 

 applied, as they stand, to transposed lines, it is necessary to set up new 

 boundary conditions and evaluate a new set of integration constants 

 at every transposition point, since a transposition point is a discon- 

 tinuity. The difficulty of such a procedure is very great, aside from 

 the fact that it requires as a preliminary the calculation of the n 

 modes of propagation of the system in each transposition interval. 

 In view of these difficulties a more powerful method of attack is 

 required in the analytical investigation of the problems of crosstalk 

 and of interference in general. Fortunately this is furnished by the 

 solution of the problem dealt with in the next section : the propagation 

 of periodic currents over wires in an arbitrary impressed field of force. 



II 



Propagation of Periodic Currents Over Wires in an 

 Arbitrary Impressed Field of Force 

 We shall consider first the simplest case, namely, a single wire with 

 ground return. The impressed or disturbing field is assumed to be 

 periodic, of frequency a)/27r, so that the problem is a steady state one 

 and the time is involved only through the factor exp {iwt). The result- 

 ant field is made up of two parts: first that due to the primary im- 

 pressed field; and secondly that due to the current in and the charge 

 on the wire, and the corresponding induced currents and charges in the 

 ground. Let f{x) = / denote the component of the electric force of 

 the primary or impressed field parallel to the axis of the wire at its 

 surface,'' and F{x) = F the line integral of the impressed or primary 

 field from the surface of the wire to ground. It is then proved in 

 Appendix I that the differential equation of the problem is 



7 V ^ dx^ 



{y'-h)^=f' (25) 



the solution of which is ^ 

 / = e- 



A-h-^ rdyf(y)ey«] 



' . (26) 



''f(x) is assumed to be sensibly constant over the cross-section of the wire. 

 8 The lower limit, v, of integration is at our disposal. In case the line begins at 

 jc = 0, it may be convenient to take v = 0. 

 33 



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