WAVEGUIDE MODES IN GYROMAGNETIC MEDIA 425 



of n, in marked contrast to the case of ^ > 0. The solutions remain identi- 

 cal, as before, if both k and /; are simultaneously reversed in sign, but 

 differ if only one of the two quantities is nnersed. 



Since n = is a branch point, the limiting condition as m approaches 

 zero for \'alues ju < differs from that for the re^■erse case. Efjuation (32) 

 is now satisfied in the limit of small m by the real zeros of ./„(pi). Since 

 these roots are finite, A% , ecjual to —{ — nYpi/R, tends towards zero for 

 all modes. Since the formulae developed in this paper always presume 

 large wa\'e numbers, we may infer a ^•anishing \'alue of A'j to simply 

 represent a value which is small relative to the reciprocal of the wave- 

 guide radius. In any event. A,, is no longer singular at m = 0, and there 

 is no resonance in the approach from negati^'e values of m- 



In sum, the features of the circular guide strongly resemble those of 

 the rectangular guide in the region of pi > 0. This was to be anticipated 

 by the "wrapped wall" construction where the wa^'e is tightly bound 

 to the wall. The wrapped equivalences do not hold in the region ^ < 

 since, with harmonic transverse dependence, the waA-e is no longer 

 bound to the wall. This lack of equivalence is manifested in the matter 

 of ordering modes. For a rectangular waveguide of finite aspect ratio, we 

 find from (25) that there are but a finite number of modes corresponding 

 to each \-alue of m for ju < 0. The circular guide differs in pro\'iding an 

 infinite number of modes corresponding to each A'alue of n. Further, 

 whereas the circular guide covers the entire range of | m | < | « |, (25) in- 

 dicates that the various modes of the rectangular guide covers a more 

 restricted range determined by the guide aspect ratio. 



IV. CONCLUSIONS 



The waveguide beha\ior analyzed in this paper has been experi- 

 mentally observed^ and good correlation has been obtained. From the 

 viewpoint expressed of forming a guide cross-section by wrapping a wall 

 to which a surface wave is bound, we may anticipate that the unusual 

 behavior observed in the two types of guides examined is probably 

 characteristic of manj^ other structures. 



It is not clear, at this time, if the complete set of modes of either the 

 rectangular or circular guides have been exhausted. We already observe 

 that an infinite number of modes propagate simultaneously so that 

 scattering problems become considerably more complex than in the usual 

 cases. It is felt by the author that the field of waveguide analysis calls 

 for new methods and techniques of modal synthesis when ferrite loaded 

 structures are considered. 



