Impedance scaling for small angle transitions

Based on the parabolic equation approach to Maxwell's equations, we have derived scaling properties of impedance that apply either to (1) structures of general shape at high frequencies, or (2) small angle transitions at all frequencies. Applying these scaling properties to impedance/wakefield calculation of long, small angle, beam pipe transitions, like one often finds in insertion regions of storage rings, one can greatly reduce the computer resource required. We have tested the scaling with wakefield simulations of 2D and 3D models of such transitions, and have found that it appears to work well.