SOME RESULTS 0\ CYLINDRICAL CAVITY RESONATORS 419 



mum. W'c can, however, first differentiate (12) by setting 



dW _ dW dW dip 

 dr dr d(p dr 



dW 



and then substitute from (13). However, when (13) is satisfied, -— = 0. 



o<p 



This process yields 



dW ^ r (2 - 3 cos^ <p) 

 dr IT- 9 sin- (p cos^ (p 



This shows -r- to be positive, when cosV < I • Hence -— = corresponds 

 dr dr 



to a maximum, rather than a minimum.* If cos-(p < f, that is,^ > 35°16', 



then r should be as small as possible. The smallest r is 3.83, for the TE 



01;/ modes. For r = 3.83, and (p > 35°, from (13) there is obtained P > 



0.75. 



s 



The analysis thus indicates that, for values of P = ()- greater than 0.75, 



A 



the TE 01;/ mode yields the smallest ratio W/P or V/Q. 



An interesting and simple relation between /a and R for minimum W/P 

 can easily be derived from the foregoing equations. Substitute (14) back 

 into (6), thereby obtaining 



■ *^^ (15) 



3 a cos^ p 



Now use (7) with (15) to eliminate cos p, replace k by 27r/X, and r by 3.83, 

 its numerical value for the TE 01;; modes. This gives 



^] R = 2.23 



or by substituting X = - , c = 3 X 10 , 



(fa)- R - 20.1 X 10-0. 



This useful relation was first discovered by W. A. Edson. 



Some further discussion is of interest. It is realized that a number of 

 points have not been taken care of in a manner entirely satisfactory mathe- 

 matically, but nevertheless important practical results have been obtained. 

 As an example, since p and r can assume only discrete values, there are 



* It is for this reason that the determination of the stationary values of ]V{r, [>, f), 

 subject to the constraint P(r, p, ^) = constant, by La Grange multipliers fails to yield 

 the desired least value of W/P. 



