890 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 



constant of the principal mode (lowest even mode) when the walls are 

 very good conductors. 



If the walls were perfectly conducting we should have Z{y) = 0, 

 and the lowest root 70 of (11) would be given by 



(<^o — llyb = 0, or To = o-o . (14) 



The principal mode between perfectly conducting sheets is just an un- 

 disturbed slice of the plane TEM wave which could propagate in an 

 unbounded medium. If Z{yn) is not rigorously zero, but still so small that 



1^^^^ «1, (15) 



and if Ziy) does not vary rapidly with 7 in the neighborhood of 70 , 

 then the lowest root of (11) is given approximately by 



7" = (To + 2aoZ(yn)/r}nb. (1(3) 



If Z(7ii) is so small that we have the further inequality 



Z{yo) 



« 1, (17) 



CoOTJo 



then (IG) yields the approximation 



7 = <r„ + Z(yo)/-noh, (18) 



where the second term is the first-order change in y due to the finite 

 impedance of the walls. If we formally set ^o = (this does not actually 

 restrict us to perfect dielectrics since we could still assume eo or /xo to be 

 complex), we have 



a = ?co\/Moe(i , V = Vmo/^o • (19) 



If the mediimi between the sheets is lossless, the attenuation and phase 

 constants of the principal mode become 



a = Re 7 = Re Z(yo)/voh, (20) 



iS := Ira 7 = coV/ioeo + Im Z(y„)/r}ob. (21) 



Although the fields of the principal mode between perfectly conducting 

 walls are entirely transverse to the direction of propagation, if the walls 

 are not perfectly conducting there will also be a small longitudinal com- 

 ponent Es of electric field associated with this mode. The leading terms 

 in the expressions for the field components, as obtained from equations 

 (8), (9), and (16), are 



