Low frequency impedance of tapered transitions with arbitrary cross sections

We study the impedance of a tapered transition at small frequencies for an arbitrary shape of the transition cross section. Our approach does not require a symmetry axis in the system (unlike round geometry). We show that the calculation of the impedance reduces to finding a few auxiliary potential functions that satisfy two-dimensional Poisson equations with Dirichlet boundary conditions. In simple cases such solutions can be obtained analytically; for more complicated geometries they can easily be found numerically. We apply our method to axisymmetric geometry and reproduce results known from the literature. We then calculate the impedance of a taper with rectangular cross section in which the vertical dimension of the cross section is a slowly changing function of the longitudinal coordinate. Finally, we find a transverse kick experienced by a beam passing near a conducting wall with a variable distance from the beam to the wall.