With a few assumptions it is easy to calculate the resonances
of a cylindrical cavity . The solutions here are from Jackson [2].
The first assumption is that the ends are plane and perpendicular to the
axis of the cylinder. Also the walls of the cavity are taken to have
infinite conductivity and the cavity is filled with a lossless dielectric.
The resonate frequencies of the TM modes are given by
where Xmn is the nth root of a Bessel function (J), R is the radius, d is the length, m and p = 0, 1, 2, .... and n = 1, 2, 3, ... The TE modes are given by the same equation with J' instead of J. The resonance positions were calculated for the first few TE and TM modes of the bellows. These positions can be compared to a measurement of the transmitted power as a function of frequency. The approximations are reasonably close to the measured values despite the approximations and the difference between the bellows and an actual cylindrical cavities. These approximations would be the starting point for finding which resonance to use and how long the cavity should be to pass the desired frequency.
| Modes | Calc Frequency (GHz) | Meas Frequency (GHz) |
| TE211 | 7.3 | 7.5 |
| TM111 | 9.0 | 9.2 |
| TE212 | 10.1 | 10.0 |
| TM112 | 11.6 | 11.1 |
| TE311 | 12.0 | 12.8 |
| TE312 | 13.6 | 13.6 |