Fourier Methods for Harmonic Scalar Waves in General Waveguides

TL;DR

This paper introduces Fourier-based semi-analytical techniques for solving wave scattering in complex waveguides, offering a physically insightful and computationally efficient alternative to finite element methods especially at low frequencies.

Contribution

It develops new conformal mapping and Fourier analysis methods for waveguide problems, enhancing analytical understanding and inverse engineering capabilities.

Findings

Good agreement with finite element solutions at low and medium frequencies

Fourier methods provide added physical insight at low frequencies

Complement finite element analysis as a waveguide simulation tool

Abstract

A set of semi-analytical techniques based on Fourier analysis is used to solve wave scattering problems in variously shaped waveguides with varying normal admittance boundary conditions. Key components are newly developed conformal mapping methods, wave splitting, Fourier series expansions in eigen-functions to non-normal operators, the building block method or the cascade technique, Dirichlet-to-Neumann operators, and reformulation in terms of stable differential equations for reflection and transmission matrices. For an example the results show good correspondence with a finite element method solution to the same problem in the low and medium frequency domain. The Fourier method complements finite element analysis as a waveguide simulation tool. For inverse engineering involving tuning of straight waveguide parts joining complicated waveguide elements, the Fourier method is an attractive alternative including time aspects. The prime motivation for the Fourier method is its added physical understanding primarily at low frequencies.