WIRE TRANSMISSION THEORY 273 



We now suppose that solutions of the type 



/7 = /(3;, s)g(i"'-yx) (11) 



exist, where f{y, z) is a two-dimensional wave function satisfying the 

 two-dimensional wave equation 



(|-' + £)^=(*^--')-^- ('^> 



In other words we search for exponentially propagated waves of this 

 type; that is, waves which involve the spatial coordinate x only 

 exponentially. It is well known that solutions of this type exist when 

 the transmission system is uniform along the X axis. 



The mathematical analysis of the problem outlined above is dealt 

 with in detail in my paper 'The Rigorous and Approximate Theories 

 of Electrical Transmission along Wires' (ref. 11) and the outstanding 

 conclusions of that analysis are as follows : 



The form of the differential equations of classical transmission 

 theory is rigorously valid, that is, the system is specified rigorously 

 by its self and mutual series impedances and shunt admittances, 

 only for the ideal case of a system consisting of perfect conductors 

 embedded in a perfect dielectric. In this case j^^ — 7^ = in the 

 dielectric; v"^ = 00 in the conductors, and the propagation constant 

 7 is icA^/v, indicating unattenuated transmission with the velocity of 

 light, V = l/VeAt. The wave is a pure plane guided wave, and the 

 electric and magnetic fields are derivable from two wave functions, 

 one a linear function of the conductor charges and the other a linear 

 function of the conductor currents, the determination of which, in 

 terms of the geometry of the system, is reduced to the solution of a 

 well-known potential problem. 



Such a system, the ideal for guided wave transmission, is of course 

 unrealizable, since there are always losses in both conductors and 

 dielectric. For efficient transmission, however, the losses must be 

 small and the guided wave must approximate the plane wave of the 

 ideal case. Let us suppose, therefore, that the losses in the system are 

 so small that in the dielectric, in the neighborhood of the conductors, 

 we can set j/2 — y = 0, and that in the conductors the conductivity is 

 so high that v^ — 7^ may be replaced by v^ without appreciable error. 

 Under the circumstances where these approximations are valid it is 

 found that the electric and magnetic fields in the dielectric and the 

 current distribution over the cross-sections of the conductors are 

 likewise derivable from two wave functions which are linear functions 

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