274 BELL SYSTEM TECHNICAL JOURNAL 



of the conductor charges and currents respectively. The first of these 

 is determined in terms of the geometry of the system by the solution 

 of the same two-dimensional potential problem as in the ideal case, 

 while the second is determined in terms of the geometry and electrical 

 constants of the system, by a generalized two-dimensional potential 

 problem.* Otherwise stated, to the approximations explained above 

 the system may be regarded as specified by self and mutual series 

 impedances and self and mutual shunt admittances, and these are 

 calculable by the solution of the two-dimensional potential problems. 

 The solution of the differential equations leads, precisely as in the 

 classical theory, to an wth order equation in 7^, indicating n modes 

 of propagation. Moreover, the n corresponding waves, which will, 

 for reasons explained below, be termed the principal waves, are quasi- 

 plane. This means that, in the dielectric the axial electric intensity 

 is in general small compared with the electric intensity in the plane 

 normal to the axis of transmission; or, more broadly stated, the 

 departure of the waves from true planarity is due entirely to dissipa- 

 tion in conductors and dielectric. A plane wave is here understood 

 to mean a wave in which Ex = Hx = 0. 



Now it is important to observe that in arriving at the foregoing 

 result we have introduced at the outset approximations and assump- 

 tions regarding the order of magnitude of the propagation constant 7 

 which depend on the assumption that the transmission losses are small. 

 Fortunately these assumptions are justified, and the resulting approxi- 

 mate solutions are valid to a high degree of accuracy, in those systems 

 which can be employed for the efficient guided transmission of electro- 

 magnetic energy; otherwise stated, the mathematical restrictions 

 correspond to the actual requirements for efficient transmission. If, 

 however, either the conductors or the dielectric become sufficiently 

 imperfect, the approximations introduced and the resulting wave 

 solutions become increasingly inaccurate and unreliable. 



Suppose now that we attack the problem in a still more fundamental 

 way: discard the assumptions regarding the order of magnitude of 7, 

 introduced above, and attempt to deal with the problem and the 

 solution of the wave equation in its general form. The case then is 

 entirely different and vastly more complicated. In general, the solu- 

 tion can not be carried out, but a few simple systems have been studied 

 and the results of this analysis may be generalized as follows: * in a 

 system of n parallel conductors there exist, in addition to the n principal 

 modes ot propagation, an w-fold infinity of other modes of propagation, 



*See reference (8). 

 ' See reference (5). 



