Trapped Modes in the Vacuum Chamber of an Arbitrary Cross Section

It has been shown that a small discontinuity such as an enlargement or a hole on circular waveguides can produce trapped electromagnetic modes with frequencies slightly below the waveguide cutoff. The trapped modes due to multiple discontinuities can lead to high narrow-band contributions to the beam-chamber coupling impedance, especially when the wall conductivity is high enough. To make more reliable estimates of these contributions for real machines, an analytical theory of the trapped modes is developed in this paper for a general case of the vacuum chamber with an arbitrary single-connected cross section. The resonant frequencies and coupling impedances due to trapped modes are calculated, and simple explicit expressions are given for circular and rectangular cross sections. The estimates for the LHC are presented.