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.