892 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 



and 



dE^/dp - dEp/ds = io^fxH^ , (28) 



from which we can ehmiuate Ep and Ez to obtain 



d'H(f) 1 dH^ H^ d H^ 2 , . 



-T-T, — r — 7— — —r -\ — TT- = <^ ■"<*• UW 



op- p op p- oz- 



If we assume a wave tra\eliug in the positive ^-direction with propaga- 

 tion constant 7 and write 



H,{p, z) = R{p)e-'\ (30) 



we find that (29) becomes 



P dp \ dp/ \ p-/ 



where k is given by (6) as before. But (31) is just the equation satisfied 

 "by modified Bessel functions of order one and argument Kp, so 



R(p) = AhiKp) + BKr(Kp), (32) 



w^here A and B are arbitrary constants. The other field components can 

 be obtained from H^ using (27) ; the results are 



H^ = [Ah{Kp) + BK^{kpW, 



Ep = -^r- [AhiKp) + BK,{Kp)]e''% ,_. 



g -\- lo:e (33) 



E. ^ -—"^ [AhiKp) - BK,{Kp)]e-'\ 

 g + ?we 



For cylinth-ieal fields of the type that we are considering, the wave 

 impedances looking in the positive and negative p- and ^-directions at a 

 typical point are defined to be, respectively, 



Zt = -E, //, , Z+ = E/H^, 



(34) 

 Z; = E, H,, Z7 = -E,/H^. 



We shall now discuss the propagation of circular transverse magnetic 

 waves in a homogeneous region of space whose electrical constants are 

 <o , Mo , ^0 (or (To , T/o), and which is bounded by coaxial cylinders of radii 

 pi and p-2 , where p2 > pi , as shown in Fig. 2. We suppose that the radial 

 impedances looking from the main dielectric into the inner and outer 

 cylinders are, respectively, 



Z7|p=p, = Zi{y), ^J|p=po = Zi{y). (35) 



