272 BELL SYSTEM TECHNICAL JOURNAL 



that the solutions of elementary theory, when valid, are only particular 

 solutions, and therefore do not, in general, represent the complete wave. 

 ^ In taking up this problem it is necessary to discard the simple 

 concepts underlying classical transmission theory and attack the 

 problem, ah initio, by aid of Maxwell's equation. Otherwise stated, 

 our problem is to find solutions of the wave equation which satisfy 

 the boundary conditions at the surfaces of the conductors, that is, the 

 continuity of the tangential component of E and H, and therefore 

 represent physically possible waves. 



To put the matter otherwise, we shall place ourselves in the position 

 of a mathematician, unacquainted with circuit theory or classical 

 transmission theory, for whom the laws governing propagation of 

 electromagnetic waves are formulated only by Maxwell's equations. 

 His procedure in developing the theory of transmission along wires 

 would be totally different from the way the theory has actually been 

 developed. Starting with Maxwell's equations he would find that 

 the electric and magnetic vectors satisfy a partial differential equation 

 called the wave equation. He would then search for particular solu- 

 tions of the wave equation which satisfy the geometry and electrical 

 constants of the system, and therefore represent physically possible 

 waves. The results of such a mode of approach to the problem are 

 sketched below. 



To formulate the problem concretely, consider a system of n parallel 

 conductors, parallel to the (plane) surface of the earth, and extending 

 along the positive X axis. The conductors may have any cross- 

 sectional shape desired, but it is expressly assumed that they do not 

 vary electrically or geometrically along the axis of transmission X 

 (except at points of discontinuity or the terminals) ; that is to say, the 

 transmission system is uniform along the axis of transmission. 



Now in any medium of conductivity a, permeability /i and dielectric 

 constant e, the electric and magnetic vectors satisfy the wave equation * 



where 



v^ = Airaniu} — (jp'lv^, 



o) = 27r times the frequency, 



= v=n:. 



and F may be any component electric or magnetic vector. 

 * See reference (11). 



