270 BELL SYSTEM TECHNICAL JOURNAL 



eliminated from (5), we get a set of n homogeneous equations in I or 

 F, the determinant of which must vanish for a non-trivial solution. 

 This determinant is a function of 7^ and it has in general « roots in 7-, 

 indicating 71 possible modes of propagation, with corresponding char- 

 acteristic impedances kjk- The general solution is then of the form 



Here Ajk, Bjk are integration constants, the number of independent 

 constants being 2w. These are determined by the 2w boundary con- 

 ditions at the physical terminals of the system ; that is, the continuity 

 of the 11 currents and n potentials. The solution represents n direct 

 and n reflected current waves, which in general are propagated with 

 different attenuations and different phase velocities. 



The conclusions derivable from the classical theory sketched above 

 may be summarized as follows: In a system of n parallel conductors 

 there are in general n modes of propagation, that is, n direct and n 

 reflected waves, which may be termed the normal modes of propaga- 

 tion. The distribution of the wave energy among the n modes of 

 propagation or n component waves, is determined by the boundary 

 conditions, which are essentially the continuity of the currents and 

 potentials of the n wires. The system is supposed to be completely 

 specified by the self and mutual series impedances and the self and 

 mutual shunt admittances of the n conductors, while in the solution 

 for the waves the physical and electrical characteristics of the line 

 enter only through the propagation constants and corresponding 

 characteristic impedances. 



Before analyzing the theoretical basis of the preceding elementary 

 theory, and showing its limitations, an interesting and practically 

 important extension will be briefly touched on. The equations of the 

 theory given above presuppose that the impressed electromotive forces 

 are concentrated at the terminals of the system, and that in the line 

 itself the electric and magnetic fields are due entirely to the currents 

 and charges of the conductors, and consequently that the distribution 

 of current and charge is determined entirely by their own fields. 

 Suppose, however, that the system is in addition exposed throughout 

 its length to an impressed field, from some disturbing source; then the 

 preceding theory must be modified to take into account the effect of 

 this additional field. To take the simplest case, consider a single wire 

 parallel to the surface of the earth (ground return circuit). Let us 

 suppose that this wire is exposed to an arbitrary field specified by an 



