High-Precision Calculation for the Cut-Off Wave Numbers for Waveguides of Arbitrary Cross Sections with Inner Conductors

A method for the accurate calculation of the cut-off wavenumbers of a waveguide with an arbitrary cross section and a number of inner conductors is demonstrated. A secure basis for formulating the spectral problem relies upon concepts of integral and infinite-matrix (summation) operator-valued functions depending nonlinearly on the frequency spectral parameter; whilst a variety of methods might be employed in determining the cut-off wavenumbers so specified, the Method of Analytical Regularization provides a route to the construction of an algorithm that is well-conditioned (and hence reliable). The algorithm is based on a mathematically rigorous solution of the homogeneous Dirichlet problem for the Helmholtz equation in the interior of the waveguide, excluding the regions occupied by the inner conductor boundaries; it results in a highly efficient method of calculating the cut-off wavenumbers and the corresponding non-trivial solutions representing the modal distribution. The ability to calculate the cut-off wavenumbers with any prescribed and proven accuracy provides a secure basis for treating the wavenumbers so found as “benchmark solutions”.