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Time-domain wave scattering in an unbounded two-dimensional acoustic mediumby sound-hard scatterers is considered. Two canonical geometries, namely asplit-ring resonator (SRR) and an array of cylinders, are used to highlight thetheory, which generalises to arbitrary scatterer geometries. The problem issolved using the generalised eigenfunction expansion method (GEM), whichexpresses the time-domain solution in terms of the frequency-domain solutions.A discrete GEM is proposed to numerically approximate the time-domain solution.It relies on quadrature approximations of continuous integrals and can bethought of as a generalisation of the discrete Fourier transform. The solutionthen takes a simple form in terms of direct matrix multiplications. In parallelto the GEM, the singularity expansion method (SEM) is also presented andapplied to the two aforementioned geometries. It expands the time-domainsolution over a discrete set of unforced, complex resonant modes of thescatterer. Although the coefficients of this expansion are divergent integrals,we introduce a method of regularising them using analytic continuation. Theresults show that while the SEM is usually inaccurate at $t=0$, it convergesrapidly to the GEM solution at all spatial points in the computational domain,with the most rapid convergence occurring inside the resonant cavity.

Generalised eigenfunction expansion and singularity expansion methods for two-dimensional acoustic time-domain wave scattering problems