500 BELL SYSTEM TECHNICAL JOURNAL 



easy to show that the number of independent arbitrary constants of 

 integration is 2n. These are determined by the 2n boundary condi- 

 tions to be satisfied at the physical terminals of the system. In general 

 these boundary conditions specify 2n relations among the impressed 

 voltages, the terminal impedances, and the line currents and voltages. 

 While the evaluation of the constants of integration from these 2n 

 relations is formally straightforward, it is actually a matter of very 

 considerable complexity if the system is composed of a large number 

 of wires; furthermore, the evaluation of the propagation constants 

 7i, • • • In presents great diflficulties in such cases. 



The results of the foregoing formal analysis may be summarized 

 as follows: In a system of n parallel wires there are in general w modes 

 of propagation, corresponding to the n roots 71, • • • 7n of the general- 

 ized telegraph equation; these may be termed the normal modes of 

 propagation. Except when special boundary conditions obtain, the 

 current in each and every wire is made up of component waves of all 

 n modes of propagation, and the distribution of energy among the n 

 modes is determined by the boundary conditions at the terminals of 

 the wires. A characteristic and fundamental property of the normal 

 modes of propagation is that a normal mode of propagation is the 

 type which can exist alone. That is to say, if the boundary condi- 

 tions have a particular set of values, the currents in all the wires may 

 be made up of one mode only; unless, however, the particular condi- 

 tions obtain, the currents involve components of all modes. 



The existence of n modes of propagation in a system of n parallel 

 wires follows from the fact that the determinant is of the wth order in 

 7^ and therefore has n roots. In certain cases of practical importance, 

 however, we may have multiple roots, so that the number of distinct 

 modes of propagation is reduced. For example, in the ideal case of 

 perfect conductors and perfect ground conductivity, Ljjjpjj = Ljklpjk 

 = i/c" and therefore 7 = iwjc, where c is the velocity of propagation in 

 the medium. In this case, only one mode of propagation exists, 

 namely, unattenuated transmission with the velocity of propagation 

 of light in the dielectric medium; thus, for the direct wave, 



/y = AjB-^l' (14) 



and the n constants Ai, • • • An are independent. 



Another case of some interest is that in which the wires are all 

 alike, so that Wu = W22 = • • • ninn = m and furthermore mjk = m' 

 (a condition which is partially realizable by a properly designed system 

 of transpositions). In this case equation (12) becomes 



1 



