WIRE TRANSMISSION THEORY 275 



which will be termed complementary modes of propagation. In general, 

 the corresponding complementary waves differ from the principal 

 waves in that they are not quasi-plane and are very rapidly attenuated. 

 Consequently it appears that as regards the currents and charges, and 

 the fields near the conductors, the effect of the complementary waves 

 is usually appreciable only in the neighborhood of the physical 

 terminals of the system so that at a distance from the terminals, 

 usually small, they are represented with sufficient accuracy by the 

 principal waves alone. At a great distance from the conductors, 

 however, it appears that the errors resulting from ignoring the fields 

 of the complementary waves may be large ; in fact the complementary 

 waves must be expressly included to take into account the phenomena 

 of radiation. 



The practical as distinguished from the theoretical importance of 

 the foregoing resides in the fact that the principal waves corresponding 

 to those of elementary theory represent the transmission phenomena 

 accurately only at some distance from the physical terminals of the 

 line and then only in the neighborhood of the wires. This defect 

 may be of small practical consequence when the conductors all consist 

 of wires of small cross section. When, however, conductors of large 

 cross sections, or the ground, form part of the transmission system, 

 the theory may be quite inadequate for some purposes. In particular, 

 in calculating inductive disturbances in neighboring transmission 

 systems at a considerable distance it may lead to large errors. 



The discussion given above is based in part on a mathematical 

 analysis of simple representative systems, in part on inferences from 

 physical considerations. Unfortunately a direct frontal attack and 

 rigorous solution of the general problem appears impossible. For 

 example, in addition to finding the infinitely many modes of propaga- 

 tion the corresponding infinitely many complementary waves must 

 be so chosen as to satisfy the boundary conditions at the physical 

 terminals. In the classical theory these boundary conditions are 

 simply the continuity of currents and potentials; in the rigorous 

 formulation of the problem they are the continuity of Ey, Ez, Hy, Ht 

 throughout the entire boundary plane (x = 0). Even to formulate 

 these conditions involves specifying the impressed field throughout 

 the plane and this is never given explicitly in technical transmission 

 problems. While, therefore, the theory sketched above leads to 

 inferences and conclusions of importance, the writer is convinced 

 that some more powerful and indirect mode of attack on the problem 

 must be devised; a rather hopeful possibility along this line will be 

 briefly described. 



