Impedance of Beam Pipes with Smooth Shallow Corrugations

The beam impedance of monopole modes in a pipe with periodic, smooth and shallow corrugation has been calculated. The approach uses an orthogonal TM field expansion with Bessel functions and an impedance boundary condition, which takes into account resistive wall effects. The approach is valid for corrugations with period lengths larger, smaller and of the same order as the free space wavelength. The synergetic effect of the surface resistivity and sinusoidal or non-sinusoidal corrugations has been investigated. It is shown that these effects are nonlinear with respect to the beam impedance but they are well approximated by a superposition of the equivalent surface impedances. The concept of the equivalent surface impedance, which is directly related to the beam impedance, is introduced. Wake functions and wake potentials are calculated for several examples either by a pole expansion method (for perfect conducting surfaces) or by a Fourier transformation. The convergence of the approach is tested numerically and for one example the wake potential is compared with a direct numerical computation. The wake field effects due to surface roughness and conductivity in the TESLA-FEL beam pipe are estimated. This estimation takes into account statistical properties of a real measured surface.