420 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1957 



This is the circular guide which has been the subject of considerable 

 analysis by Suhl and Walker. It is instructive, however, to repeat the 

 analysis of this case, in the small guide limit, showing more pointedly its 

 behavior from the viewpoint of combinations of the two types of waves 

 in the medium. 



The character of transmission in undersized circular waveguide is 

 very similar to that of the undersized rectangular case. We may demon- 

 strate the physical significance of this statement by the following argu- 

 ment. The excitation in a rectangular waveguide, for | /x | < \ k\ and 

 M > 0, is essentially that of a surface wave bound very tightly to a 

 single wall. Considering this wall alone, which may now be extended to 

 arbitrary dimensions but Avith k^ kept large, it may be wrapped upon 

 itself either about the magnetic field as an axis or containing the mag- 

 netic field peripherally. In either event, the wrapped guide must start 

 and terminate at the same phase, requiring a multiplicity of 2ir around 

 the circumference, and the wave must thus continue to have a large 

 kg value. Considering the large value of k^ and the state of excitation of 

 the ferrite, the small circular guide may propagate. 



Analysis will demonstrate that propagation also takes place in the 

 region ^Lt < 0. The quantity k^^ is real and k^^ imaginary, see (26), leading 

 to a case essentially similar to that of the rectangular waveguide. The 

 analogy is appropriate to the case of h/a of finite value for which the 

 rectangular guide requires the appearance of both refractions. We now 

 proceed to obtain the field solutions for the circular guide. 



Referring to (9) for the plane wave solution of the electric field, let 

 us define to within a constant multiplier. 



Ey' 



£ 



—i{kyy+kzz) 



(27) 



Eg 

 where, for the case of large kg (9, 12, 13) 



^(x) ^ l^2± E''' = 



'X 



(1) _ p(2) 

 U — -C'2 



(1) _ • -i 7?(2) 



E^'^ = E'^' = -1 



Er = iiT' EY' = i I 



fcj^ _ . _J ky^ _ ^ 



Kz Kg 



We shall consider here, of the two possible wrapped-wall structures, 

 that case in w^hich the magnetic field is applied axially as shown in Fig. 3. 

 Referring to Fig. 4, the cylindrical drical electric wave satisfying Max- 



