Error estimates of asymptotic expansions in periodic waveguides

Abstract

ABSTRACT The filtering problem of a periodically corrugated rectangular metallic waveguide of length l is used as an example to set an upper bound on the validity of the various orders of asymptotic approximation of the electromagnetic field in the waveguide. We limit the analysis to the first two orders of asymptotic expansion in powers of a small parameter δ characterising the amplitude of the periodic corrugations to minimise the detail, but the procedure is applicable to higher orders of approximation without loss of generality. The asymptotic expansions are obtained via the perturbation method of multiple scales. Following conventional practice in antenna work, a phase error of π/8 is used to set an upper bound on the phase error for each order of approximation. This bound is found by setting the difference between the results of simulation using the HFSS software package and the phase of either the first-order or the second-order approximation of the electromagnetic field equal to π/8. This bound sets an upper limit on δ for a given length I which gives the same results for a fixed (δ × I) product for the first-order expansion.