ELECTRICAL TRANSMISSION ALONG WIRES 15 



Another useful formulation of the field equations equivalent to and 

 directly deducible from (8) is 



(.'-7').1/.= -'^E^ + y^M,. 



In this formulation the problem is reduced to the determination of 

 the wave ftinctions Ez and Mz, and the propagation constant 7. 



It will be observed that, by virtue of the assumption that the wave 

 functions of (8), (9) and (10) involve / and z only through the common 

 factor exp {iwt — 7s), we can write 



P = fix, y)-ex-p (iut - yz), 



$ = 4>(x, 3')- exp (iut — yz), (11) 



E = e(x, 3') -exp (icot — yz), etc., 



where /, </>, e, etc., are two-dimensional functions of x and y alone, 

 and satisfy the two-dimensional wave equations 



In the following, therefore, we shall regard the wave functions 

 F, $, E, etc., as two-dimensional functions with the understanding 

 that the common factor exp (iwt — yz) is omitted for convenience. 



II 



Before taking up the discussion of the general problem in the light 

 of equations (8) and (10) we shall first consider a type of plane wave 

 propagation to which the transmission phenomena closely approximate 

 in an efficient transmission system. We consider the ideal trans- 

 mission system composed of any number of straight parallel perfectly 

 conducting conductors imbedded in a perfect dielectric. For such a 

 system we assume the possibility of plane wave propagation by 

 supposing that Ez and Mz are everywhere zero. By virtue of the 

 assumption of perfect conductivity, the electric force must vanish 

 inside the conductors, and at the surface the tangential component 



