1352 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 



dinary waveguide, i.e., "fast" modes as contrasted with the well-known 

 "slow" modes used in traveling- wave tubes. ^' ^ To solve the problem we 

 follow the procedure set up by Stratton^ for the ordinary cylindrical 

 waveguide boundary problem. The fields E and H are derived from an 

 electric Hertz vector IT and a magnetic Hertz vector 11* by 



^ = vxvxn- ^■w/xV X n* 

 H= {a + iwe)'^ X ii + V X V xn* 



(2) 



where 



(3) 



(4) 



n = ezU, 



n* = Ln* 



and, assuming a time dependence exp (iwt), 



00 



ni V~^ i 7" / V N —ihz—inB 



z = 2^ anJn{tir)e 



n=— 00 



00 



ne V"* ejr (2)/ 5, \ —ihz—inB 



n=— 00 



00 



n*i V~^ 7 i T / <, \ —ihz—inB 



z = 2^ OnJn{hr)e 



n=— 00 



Jl= — 00 



In these expressions 



5-2 2 7,2 



f 1 = CO )Ltoeo — h 



5-2 2 / / ■ ,/\ 7 2 



e — ie" = e/eo — ia/weo 



where the interior region is assumed to have permittivity eo and perme- 

 ability ^0 , while the exterior region has permittivity e, permeability no , 

 and conductivity a. The superscripts i and e refer to the interior and ex- 

 terior regions respectively, and the a's and 6's are amplitude coefficients. 



6 J. R. Pierce, Proc. I.R.E., 35, pp. 111-123, 1947. 



* S. Sensiper, Electromagnetic Wave Propagation on Helical Conductors, Sc.D. 

 thesis, M.I.T., 1951. In Appendix B of this reference, Sensiper shows that when 

 the interior and exterior media are the same, only slow waves will exist except in 

 special cases. Fast guided waves become possible if the conductivity of the exterior 

 medium is sufficiently high. 



'J. A. Stratton, Electromagnetic Theory, McGraw-Hill, New York, 1941, pp. 

 524-527. Note that Stratton uses the time dependence exp (—icot). 



