Generalised eigenfunction expansion and singularity expansion methods for canonical time-domain wave scattering problems

TL;DR

This paper compares the generalized eigenfunction expansion (GEM) and singularity expansion methods (SEM) for solving time-domain wave scattering problems, demonstrating their applications to string and mass-spring systems with resonances.

Contribution

It introduces the application of GEM and SEM to canonical wave scattering problems, including derivations and analysis of their equivalence and effectiveness.

Findings

GEM reduces to d'Alembert's formula without scatterers

Both GEM and SEM effectively model scattering by mass-spring systems

The methods handle complex resonant behaviors in wave scattering

Abstract

The generalised eigenfunction expansion method (GEM) and the singularity expansion method (SEM) are applied to solve the canonical problem of wave scattering on an infinite stretched string in the time domain. The GEM, which is shown to be equivalent to d'Alembert's formula when no scatterer is present, is also derived in the case of a point-mass scatterer coupled to a spring. The discrete GEM, which generalises the discrete Fourier transform, is shown to reduce to matrix multiplication. The SEM, which is derived from the Fourier transform and the residue theorem, is also applied to solve the problem of scattering by the mass-spring system. The GEM and SEM are also applied to the problem of scattering by a mass positioned a fixed distance from an anchor point, which supports more complicated resonant behavior.