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The generalised eigenfunction expansion method (GEM) and the singularityexpansion method (SEM) are applied to solve the canonical problem of wavescattering on an infinite stretched string in the time domain. The GEM, whichis 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. Thediscrete GEM, which generalises the discrete Fourier transform, is shown toreduce to matrix multiplication. The SEM, which is derived from the Fouriertransform and the residue theorem, is also applied to solve the problem ofscattering by the mass-spring system. The GEM and SEM are also applied to theproblem of scattering by a mass positioned a fixed distance from an anchorpoint, which supports more complicated resonant behavior.

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