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

TL;DR

This work presents two complementary time-domain techniques for 1D wave scattering on stretched strings: the generalised eigenfunction expansion (GEM) and the singularity expansion method (SEM). GEM reinterprets the time-domain problem as a self-adjoint spectral problem, enabling a continuous superposition of frequency-domain generalized eigenfunctions and a practical discrete form that reduces to matrix multiplication. SEM, derived via the Fourier transform and residue calculus, expresses transient responses as a sum over complex resonances with absorbing modes, providing accurate large-time behavior under suitable scattering-theoretic conditions. The authors apply both methods to canonical problems—an infinite string, a mass-spring scatterer, and a mass near an anchor point—demonstrating agreement with standard solutions and highlighting the complementary strengths and convergence properties of GEM and SEM, including normalization of resonances and the role of Lax-Phillips theory in SEM accuracy.

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.

Figures (7)

• Figure 1: A schematic of the mass-spring scattering problem.
• Figure 2: String displacement (blue line) induced by an incident wave packet given by $f(x)=e^{-(x-5)^2}\cos(4x)$ and $g(x)=cf^\prime(x)$ at (a) $t=0$ s, (b) $t=5$ s, (c) $t=10$ s and (d) $t=15$ s. The physical parameters are $\mu=1$ kg m$^{-1}$, $T=1$ N, $k_s=80$ kg s$^{-2}$ and $M=5$ kg. The numerical parameters used to generate the figure are $[x_{\mathrm{min}},x_{\mathrm{max}}]=[-10,10]$ m, $\Delta x=0.025$ m,$[\omega_{\mathrm{min}},\omega_{\mathrm{max}}]=[-20,20]$ s$^{-1}$ and $\Delta\omega=40/799$ s$^{-1}$. The mass-spring system is shown symbolically. Note that animations corresponding to all figures in this paper are provided in the supplementary material.
• Figure 3: String displacement computed using the GEM (solid blue line) and SEM approximation (dashed red line) at (a) $t=0$ s, (b) $t=5$ s, (c) $t=10$ s and (d) $t=15$ s. The incident wave packet and physical parameters are identical to those in figure \ref{['fig1']}.
• Figure 4: String displacement computed using the GEM (solid blue line) and SEM approximation (dashed red line) at (a) $t=0$ s, (b) $t=5$ s, (c) $t=10$ s and (d) $t=15$ s. The wave motion is induced by an impulse of the form $f(x)=0$, $g(0)=1$ and $g(x)=0$ for $x\neq 0$. The physical parameters are identical to those in figure \ref{['fig1']}.
• Figure 5: String displacement calculated using the GEM (solid blue line) and SEM approximation (dashed red line) at (a) $t=0$ s, (b) $t=5$ s, (c) $t=10$ s and (d) $t=15$ s. Wave motion is induced by an incident wave packet given by $f(x)=e^{-(x-5)^2}$ and $g(x)=cf^\prime(x)$. The physical parameters are $\mu=1$ kg m$^{-1}$, $T=1$ N, $L=5$ m and $M=1$ kg. The numerical parameters used to compute the GEM solution are $[x_{\mathrm{min}},x_{\mathrm{max}}]=[-5,15]$ m, $\Delta x=0.025$ m $[\omega_{\mathrm{min}},\omega_{\mathrm{max}}]=[-20,20]$ s$^{-1}$ and $\Delta\omega=0.02$ s$^{-1}$. To compute the SEM approximation, the sum in \ref{['SEM_full_expansion']} was truncated to only contain the terms $-9\leq j \leq 8$. The mass is shown symbolically.