432 BELL SYSTEM TECHNICAL JOURNAL 



TE Omn modes, and r{\ — rj) -^ {m — l)x for all other TE modes. For 

 the former modes, r becomes very large as r; — > 1, that is, as the inner con- 

 ductor fills the cavity more and more, the frequency gets higher and higher. 

 For the TE (In modes, however, as the inner conductor grows, the f re- 

 queue}' falls to a limiting value. This is discussed in more detail by 

 Borgnis.^^ 



Figure 7 shows r(l — 77) vs 77, for a few of the lower modes; the scale for 77 

 between 0.5 and 1.0 is collapsed since this region does not appear to be of 

 great engineering interest. A different procedure is used for the roots of 

 the TE (hi modes. Figure 8 is a direct plot of r vs 77 for a few of the lower 

 modes. In this case, r -^ f as 77 -h^ 1. 



Distribuiion of Normal Modes 



The calculation of the distribution of the resonant modes for the coaxial 

 case follows along the lines of that for the cyhnder, as given previously. 

 The difference lies in the distribution of the roots r, which now depend upon 

 the parameter r,. The determination of this latter distribution offers 

 difficulties. There is some evidence, however, that the normal modes will 

 follow, at least to a first approximation, the same law as the cylinder, viz.: 



V 

 N = 4.4 ^ 



Ao 



with some doubt regarding the value of the coefficient. 



- in Coaxial Resonator 

 X 



The integrations needed to obtain this factor are relatively straightfor- 

 ward, but a little complicated. The final results are given in Fig. 1. 



The defining equation is (16); the components of H are given in Fig. 1. 

 The integrations can be done with the aid of integrals given by McLachlan^^ 

 and the following indefinite integral : 



which can be verified by differentiation, remembering that y = Zi{x) is a 



solution of y" + - y' -f ( 1 - - ) y = 0. 

 X \ x-J 



