SOME RESULTS ON CYLINDRICAL CAVITY RESONATORS 413 



from which A — 0.262,, B = 0.057. These values give better agreement 



with the 180 tabulated values of r. 



There is a two-fold degeneracy in a circular cylinder for modes with 



■^ > 0, which is removed, for example, when the cylinder is made elliptical. 



The total number of modes, then, counting degeneracies twice, is about 2N, 



which brings (2) in line with the general result that, in any cavity resonator, 



Stt V 

 the total number of modes is of the order -— r^ . 



3 Ao 



Minimum Volume of Circular Cylinder for Assigned Q 



In practical applications of resonant cavities, the conditions of operation 

 may require high values of Q which can be attained only by the use of high 

 order modes. The total number of modes, most of which are undesired, 

 can then be reduced only by making the cavity volume as small as possible, 

 consistent with meeting the requirement on Q. 



It will be shown that, for a cylinder, operation in the TE 01m mode very 

 probably gives the smallest volume for an assigned Q. 



Statement of Problem 



When the relative proportions (the shape) of a cavity and the mode of 

 oscillation are fixed, both the Q and the volume, V, of the cavity are func- 

 tions of the operating wavelength, X. Since we are primarily interested 

 in the relationship between Q and V, with X fixed, some simplification can 

 be made by eliminating X as a parameter. This may be done by a change of 



8 V 



variables to () - and — , respectively; to simplify the typography, these 



A A 



quantities will be denoted by single symbols: 



We are, consequently, interested in the following specific problem: 

 In a circular cylindrical resonator, which is the optimum mode 



family and what is the corresponding shape to obtain the smallest 



value of W for a preassigned value of P? 



A rigorous solution cannot be obtained by the methods of elementary 

 calculus, since P is not a continuous function of the mode of oscillation. 

 However, a possible procedure is to assume continuity, and examine the 

 relation between P and W under this assumption. If sufficiently positive 

 results are obtained, the conclusions may then be carried over to the dis- 

 continuous (i.e., the physical) case with reasonable assurance that, except 



