

1380 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 



For this polarization the reflection coefficient increases steadily from its 

 value at normal incidence to unity at grazing incidence. Thus one has an 

 optical interpretation of the anomalous attenuation-frequency behavior 

 of circular electric waves. 



If instead of varying the frequency one imagines the wall resistance 

 varied at a fixed frequency, he can easily convince himself that there 

 usually exists a finite value of resistance which maximizes the attenua- 

 tion constant of a given mode. An idealized illustrative example has been 

 worked out by Schelkunoff. He considers the propagation of transverse 

 magnetic waves between parallel resistance sheets, and shows that if the 

 sheets are far enough apart the attenuation constant increases from zero 

 to a maximum and then falls again to zero, as the wall resistance is made 

 to increase from zero to infinity. It may be instructive to consider that 

 maximum power is dissipated in the lossy walls when their impedance is 

 matched as well as possible to the wave impedance, looking normal to 

 the walls, of the fields inside the guide. 



In conclusion we mention a couple of theoretical questions which are 

 suggested by the numerical results of Section IV. 



(1) Limit modes. It has been seen that the limit which a given lossy 

 mode approaches as the jacket conductivity becomes infinite may not 

 be unique. Can rules be given for determining limit modes when the 

 manner in which | e' — ?'e | approaches infinity is specified? 



(2) Behavior of modes as a — ^ 0. It is known^^ that the number of true 

 guided waves (i.e., exponentially propagating waves whose fields vanish 

 at large radial distances from the guide axis) possible in a cylindrical 

 waveguide is finite if the conductivity of the exterior medium is finite. 

 The number is enormously large if the exterior medium is a metal; but 

 the modes presumably disappear one by one as the conductivity is de- 

 creased. If the conductivity of the exterior medium is low enough and if 

 its permittivity is not less than the permittivity of the interior medium, 

 no true guided waves can exist. At what values of conductivity do the 

 first few modes appear in a guide of given size, and how do their propa- 

 gation constants behave at very low conductivities? 



The complete theory of lossy -wall waveguide would appear to present 

 quite a challenge to the applied mathematician. Fortunately the en- 

 gineering usefulness of helix waveguide does not depend upon getting 

 immediate answers to such difficult analytical questions. 



13 Reference 9, pp. 484-489. 



" G. M. Roe, The Theory of Acoustic and Electromagnetic Wave Guides and 

 Cavity Resonators, Ph.D. thesis, U. of Minn., 1947, Section 2. 



