NETWORKS FOR IMPEDANCE FUNCTIONS 401 



These substitutions result in the following asymptotic formula for the 

 total impedance Z at radius a 



27? , ./ J?2\ 



— cos fcx + 1 1 1 + ~2 ) sm fcrc 



Z = KM "^ ^. "^ (3-12) 



cos kx -\ sin kx 



where A' = - — 1 and x = —iya. 

 a 



To find an equivalent network of the first kind to represent Z, we deal 



with the admittance, Y = 1/Z. It is instructive and saves much work 



to put Y in the form of exponential functions, with the substitution 



V — Vo 

 P = — r — 



which is the reflection coefficient at both inside and outside cylindrical 

 surfaces of the cavity. By this means we obtain 



y - HM ^' ;(f^;^p (3-13) 



This is now identical in form to the formula (2-13) of example 2, where 

 the £'-vector was radially, instead of axially, directed. In fact, since 



ikx = y(b — a) 

 and 



comparison ^vith the similar formulas of example 2 shows that all the 

 results of that example can be made to apply to the present one merely 

 by changing the dimensional parameters as follows : 



(3-14) 



b — a 



