394 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 



From this, 



P = '-^ (2-11) 



Having in mind microwave applications, where the moduH of the argu- 

 ments of the Bessel functions are >3000, we take 



Uz) ^ KM 

 Ii(z) K,(z) 

 so that 



= 1 



^ + 7>^ = .f (- + r) + ^log-^ (2-12) 



Iir \a 0/ 27r a 



Also, we have in mind onty air dielectric and assume any loss therein to 

 be negligible ; that is, we assume G = 0. 



All further approximations that are made are either 



1/2 



1 - A 



= 1 -f A or (1 + 2A)"- = 1 + A 



where, for an air-space enclosed by copper walls, and for frequencies on 

 the order of 30,000 megacycles, A is on the order of 10~ . For cavities 

 made of other materials, the results obtained may not be sufficiently ac- 

 curate and the problem would have to be reviewed from the start. In 

 particular, the results do not hold for a cavity having walls of magnetic 

 material, because we assume here that the permeabiht}^ of the metal walls 

 is the same as that of air; i.e., ^t = )Uo . 



To obtain an equivalent network of the first kind, we deal with the ad- 

 mittance, which is, from (2-2), 



V - ^ - U (1 - P)(l - pe '' ) fr, .o\ 



where Ho = 1/Ko . 



The poles of Y are then the zeros of 1 — pe~~''^ ^ which are obtained by 

 successive approximations. We first make a close estimate of the zeros by 

 assuming that the impedance of the short-circuiting plugs is zero; that 

 is, we assume, Za = 0, whence p = — 1. To obtain this estimate, we have 

 to solve 



<y/i = ^ ( 1 + A ) = T,in (w = ±1, ±2, ±3 • • • ) (2-14) 

 V \ da/ 



where 



2ah log (b/g) 

 a -1- 6 



