SOME RESULTS ON CYLINDRICAL CAVITY RESONATORS 411 



3. Limitation of frequency range of a tunable cavity in the TE Oln mode 

 as set by ambiguity. 



4. Resonant frequencies of an elliptic cylinder. 



5. Resonant frequencies and Q of higher order modes of a coaxial reso- 

 nator. 



6. Fins in a circular cylinder. 



Approximation Formula for Number of Resonances in a 

 Circular Cylinder 



From Fig. 1, the resonant frequencies of the cylindrical cavity are ob- 

 tained from the equation: 



In which r is written in place of f/m , to simplify the equations. The dis- 

 tribution of the resonant frequencies, starting with the lowest, can be 

 approximated by a continuous function 



where N represents the total nunter of resonances up to a frequency /o 

 or a wavelength Xo . This is bcur.d lo be en approxirraticn, since the true 

 function F is discontinuous (or stepped) by virtue of the resonances being a 

 series of discrete values. For practical purposes, if /*' fits the stepped curve 

 so that the steps fluctuate above and below F, it will be a useful approxi- 

 mation. 



Derivation of such a formula as applied to the acoustic resonances of a 

 rectangular box has recently been a subject of investigation by Bolt^ and 

 Maa.'" Only slight modifications of their method need be made to apply 

 to the {^resent situation. 



MuUiply (1) thru by (- 



TTflA" 2 , /wan 

 .7) -•■ +[-2L 



Hence, if a point ( r, — — J is plotted on the A'l' plane the distance from the 



origin to this point will be — - and hence a measure of the resonant fre- 



c 



quency. If all such points are plotted, they will form a lattice represent- 

 ing all the possible modes of resonance. The problem, then, is to find the 



number of lattice ]X)ints in a quadrant of a circle with radius, R = — — . 



