Beam coupling impedances of obstacles protruding into beam pipe

The beam coupling impedances of small obstacles protruding inside the vacuum chamber of an accelerator are calculated analytically at frequencies for which the wavelength is large compared to a typical size of the obstacle. Simple formulas for a few important particular cases, including both essentially three-dimensional objects like a post or a mask and axisymmetric irises, are presented. The analytical results are compared and agree with three-dimensional computer simulations. These results allow simple practical estimates of the broad-band impedance contributions from such discontinuities