900 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 



(60) into 



Finally we obtain from (59) and (64) an expression for the impedance 

 Zo looking into a plane stack of ?i double layers when the nth layer is 

 backed by a surface whose impedance is Zn , namely 



Eo ^ ^ZnJKie"^ + gge"""^) + K1K2 sh n V 

 Ho ~ Zn sh nV + K^ie""'' + /^2e"'^) 



^0 = TT = — ~ — ; — ir^ — ■ y^^) 



For the cylindrical geometry, matters are a good deal more compli- 

 cated. If we consider waves having field components H^ , Ep , Ez in a 

 homogeneous, isotropic shell bounded bj'' coaxial cylindrical surfaces, 

 and assume a propagation factor 6"*^^ Maxwell's equations (27) and (28) 

 may be written 



Ep = [y/{g + i^e)]H, , (66) 



and 



d{-pH^)/dp = -(fir + iwi)pEz , 



(67) 

 dE,/dp = -[K/{g + ^■coe)p](-pi7^). 



If desired, we might identify E^ with "voltage" and —pH^ with "current" 

 and regard equations (67) as describing a nonuniform radial 'transmis- 

 sion line, having series impedance K/{g + icoe)p per unit length and shunt 

 admittance {g + io>t)p per unit length. Since, how^ever, in equations (34) 

 we have already defined the radial wave impedance to be a field ratio 

 without the extra factor of p, we shall carry out the analysis of the 

 present paper directly in terms of the field components Ez and — H^ . 



From the general expressions (33) for the fields in cylindrical co- 

 ordinates, we can show that the matrix relation between the tangential 

 field components E^ , —H^ at two radii pi and p2 is given by 



EipiY 



-H{p,) 

 UpiiKoiTu + Knloi) Vpi^Pii^oiIoi — Ko2loi)\ I Eipi) 



— (A^i/i2 - /Vi2/n) Kp-2iKnh2 + KoJn) }\-H{p2) 



Vp 



, (68) 



