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

TL;DR

This work advances time-domain 2D acoustic scattering by presenting a generalised eigenfunction expansion (GEM) and a singularity expansion method (SEM) and applying them to two canonical geometries: a split-ring resonator (SRR) and an array of circular cylinders. The discrete GEM reformulates the time-domain solution as a matrix-multiplication problem obtained through quadrature over space and frequency, enabling efficient computation once the frequency-domain solutions are known. The SEM provides a resonant-mode expansion in terms of complex frequencies, with a novel regularisation via analytic continuation to handle divergent normalization integrals in two dimensions. The two methods show strong qualitative agreement after transient excitation, with SEM converging rapidly to GEM inside resonant cavities, and the discrete GEM offering a flexible framework that generalises to arbitrary scatterer geometries. This combination yields robust, efficient tools for analyzing transient wave scattering and metamaterial-like resonances in 2D acoustics.

Abstract

Time-domain wave scattering in an unbounded two-dimensional acoustic medium by sound-hard scatterers is considered. Two canonical geometries, namely a split-ring resonator (SRR) and an array of cylinders, are used to highlight the theory, which generalises to arbitrary scatterer geometries. The problem is solved using the generalised eigenfunction expansion method (GEM), which expresses 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 be thought of as a generalisation of the discrete Fourier transform. The solution then takes a simple form in terms of direct matrix multiplications. In parallel to the GEM, the singularity expansion method (SEM) is also presented and applied to the two aforementioned geometries. It expands the time-domain solution over a discrete set of unforced, complex resonant modes of the scatterer. Although the coefficients of this expansion are divergent integrals, we introduce a method of regularising them using analytic continuation. The results show that while the SEM is usually inaccurate at $t=0$, it converges rapidly to the GEM solution at all spatial points in the computational domain, with the most rapid convergence occurring inside the resonant cavity.

Figures (9)

• Figure 1: Schematic of the SRR. The radius of the SRR is denoted $a$, its opening angle is given by $2\alpha$, and its orientation with respect to the positive $x$-axis is denoted $\beta$.
• Figure 2: Schematic of an array of $N=3$ circular cylinders of radius $a$. The centre-points of the cylinders are denoted by $(x_j,y_j)$ for $j=1,\dots,N$.
• Figure 3: (a,b) Resonant modes of a SRR with parameters $a=1$ m, $\alpha=\pi/4$ and $\beta=\pi$, at the complex resonant frequency (a) $(0.5923-0.1126\mathrm{i})$ s$^{-1}$ and (b) $(1.9437-0.0294\mathrm{i})$ s$^{-1}$. (c,d) Resonant modes of an array of four cylinders with radius $a=0.33$ m and with centre-points $(0.5,0.5)$ m, $(-0.5,0.5)$ m, $(0.5,-0.5)$ m and $(-0.5,-0.5)$ m, at the complex resonant frequencies (c) $(1.7648-0.3372\mathrm{i})$ s$^{-1}$ and (d) $8.2132-0.0410\mathrm{i})$ s$^{-1}$. In all panels, the colours indicate the values of $\mathrm{Re}(\phi_j(\mathbf{x}))$.
• Figure 4: Temporal evolution of the velocity potential $\phi$ in the presence of the SRR with the initial condition $f(x,y)=e^{-2(x^2+y^2)}$, computed using the GEM (a--e) and the SEM (f--j). The panels correspond to the times (a,f) $t=0$ s, (b,g) $t=2$ s, (c,h) $t=4$ s, (d,i) $t=6$ s and (e,j) $t=8$ s. The error (see \ref{['error_eq']}) in the GEM solution is $0.028$. Note that the colour scales are set using the GEM solution, thus panels showing the SEM solution at the corresponding times may contain regions where the colour has saturated due to the exponential growth of the modes.
• Figure 5: As for figure \ref{['fig:SRR_Gaussian']} with the initial disturbance $f(x,y)=\sin(y)e^{-2(x^2+y^2)}$. The error of the GEM solution is $0.029$.