416 BELL SYSTEM TECHNICAL JOURNAL 



We next substitute this expression for <p in W and calculate — assuming 



dr 



now that r is continuous, and find that W has no minimum. Practically, 



this means that the smallest value of r should be used, i.e., the TEOln mode. 



Finally, since from Fig. 2 it is seen that the envelope is reasonably smooth 



8 



for values of ^ - > 1, the expression for <p derived on the assumption of 



continuous p is used to obtain a simple relation of great utility in practical 

 cavity design. 



Details of solution 



In (3), since R must be finite for a physical cylinder, < tan (p < oo , 

 < sin v? < 1, and < cos v? < 1. Hence we may always divide by 

 sin (p or cos <p. Note that (p ranges between 0° and 90°, 



From Fig. 1, 



2d2\1/2 



whence 



^ ^ 2r(l + p'R') 



, . 2prR 



k sin (p = — — 



a 



(6) 



^ cos ^ = — . (7) 



We define W by: 



a 



3 ,3 



X3 4R 87r3 ^^^ 



Substituting (6) and (7) in (8), 



pr^ 1 



W = ^-2 —2 r— . (5') 



47r cos cp sin <p ^ 



Substitution of (3) into the expression for Q- (= P) for the TE modes as 



A 



given in Fig. 1 yields, after some manipulation 



2x 3 

 COS 



(p -\- - sin^ ^ + ( COS ^ — - sin ^ ) (^/r)^sin^ <p 

 P \ P / 



