Longitudinal and Quadrupolar Coupling Impedance of an Elliptical Vacuum Chamber With Finite Conductivity in Terms of Mathieu Functions

The resistive wall impedance of an elliptical vacuum chamber in the classical regime with infinite thickness is known analytically for ultra-relativistic beams by means of the Yokoya form factors. Starting from the longitudinal electric field of a point charge moving at arbitrary speed in an elliptical vacuum chamber, which we express in terms of Mathieu functions, in this paper we take into account the finite conductivity of the beam pipe walls and evaluate the longitudinal and quadrupolar impedance for any beam velocity. We also obtain that the quadrupolar impedance of a circular pipe is different from zero, approaching zero only for ultra-relativistic particles. Even if some of the results, in particular in the ultra-relativistic limit, are already known and expressed in terms of form factors, this approach is the first step towards the calculation of the general problem of a multi-layer vacuum chamber with different conductivities and of elliptic cross section.