16 BELL SYSTEM TECHNICAL JOURNAL 



must vanish. In the dielectric reference to equations (10) shows that 

 if Ez = Mz = 0, a finite solution requires that 



l'" — 7" = (J 



or, since X = in the dielectric, 



7 = iwjv. 



That is to say, the plane wave is propagated with the velocity of light 

 V, without attenuation. 



Reference to equations (8) and (9) shows that the boundary condi- 

 tions can be satisfied by setting 0=0, writing 



$ = 0-exp {iwt — iwzjv), 



and determining the function which satisfies Laplace's equation in 

 two dimensions. 



. + — J cA = 0, 



and is constant over the cross section of the conductors. 



From the relation F = (iu/v)^ it is also easily shown that the 

 electric and magnetic forces are both in planes normal to Z and that 

 these vectors are normal to each other and in time phase. The flow 

 of energy is therefore parallel to the Z-axis everywhere. We therefore 

 have a pure plane guided wave of unit power factor; the ideal for the 

 electrical transmission of energy. 



Ill 



We now take up the much more complicated problem arising 

 when the conductivity X of the conductors is finite and when the 

 dielectric media themselves may be dissipative. In attacking this 

 general problem we shall be guided throughout by the fact that the 

 wave solution we are seeking must approximate, more or less closely, 

 to the ideal plane wave * if the system is to efficiently transmit elec- 

 trical energy. We shall therefore introduce ab initio approximations 

 which must be valid in all efficient transmission systems. These 

 approximations cannot be all justified a priori; their justification must 

 come a posteriori from the fact that the final solution satisfies the 

 original assumptions and approximations. 



* It is to be noted that the solution sought is the principal wave. (See "The 

 Radiated and Guided Energy in Wire Transmission," Trans. A. I. E. E., 1924.) 

 This wave does not, in general, completely represent the phenomena, except at a 

 considerable distance from the physical terminals of the transmission system, and 

 then only in the neighborhood of the conductors. 



