980 



THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 



Fig. 18 — (a) &- versus u when 1 — (p/2) < o- < ao . (b) /3_ versus ro under the 

 same circumstances. 



}/>{- — 1). When this occurs, o-„ > 1 — (p/2), so that there will be a 



V 

 value 0- between cr„_i and o-„ beyond which the denominator no longer 



decreases through zero as li -^ qo , but approaches a finite positive value, 

 Fig. 18(a). Accordingly /S" approaches a finite positive value, and cut-off 

 of the extreme right hand branch. Fig. 16(c), no longer occurs. The cor- 

 responding j8 versus fo branch is as in Fig. 18(b). 



As n — >• 00 (o-„ -^ o-Q = \/l + (pV-l) ~ p/2) the number of branches 

 increases to infinity. This situation resembles that in the completely 

 filled waveguide (Part I), where we found an infinity of modes ("Shape- 

 resonances") in the range o-q < tr < 1. In the present case, however, 

 they are to be found in the range 1 — p < a < o-q • 



When cr = (To + 0, X is infinite and negative. The function Z^{^ is 

 then constant and equal to unity. A is less than unity, and the denomi- 

 nator of equation (27) has no zeros. The ^ versus fo curve is now "nor- 

 mal" again, see Fig. 12(a). As a increases further, the curve falls (since 

 A decreases steadily to — 1 as o- -^ =o), and no more quaUtative changes 

 occur. 



3.3 Cylindrical Waveguides 



As pointed out before, the fact that the propagation problem in the 

 cylindrical case can always be integrated in terms of Whittaker functions 

 when the fields show no angular variation is an accident, and in view of 

 the lack of numerical tables, not a particularly fortunate one. Only in 

 special cases (like that of the slow-wave helix) is the text-book informa- 

 tion on these functions of any great utility. In general, it will be more 

 convenient to solve the differential equations numerically. However, 

 for completeness, we shall state some of the formal results for a cylin- 

 drical waveguide containing a cylinder of circumferentially magnetized 

 ferrite, and propagating a TEo mode. 



