Longitudinal coupling impedances in a periodically loaded guide above cut off

A bunched circulating proton beam will cause electromagnetic fields to be generated in a metal wall vacuum structure. When there are regions of the vacuum pipe which have different geometrical properties, such as vacuum ports, probe regions, clearing electrode regions, etc., it is possible for the beam to excite modes which can travel, as in a periodically loaded waveguide, with the velocity of the beam bunch. In this case, a longitudinal beam instability can be set up, limiting the current which can be circulated. This limit is related to the ''coupling impedance'' of the electromagnetic structure. An approximate calculation of this coupling impedance is presented