Acoustic wave scattering by spatio-temporal interfaces

Space-time materials are obtained by modulating a physical medium with a traveling-wave perturbation of one or several of its constitutive parameters, such as the density or the bulk modulus in the case of acoustic materials. When this modulation has the form of a moving and abrupt (subwavelength) transition between two parameter values, we refer to a spatio-temporal interface, which may be considered as a building block for more complex space-time materials. This work considers the problem interaction and scattering of acoustic waves with a single spatio-temporal interface, and a sequence of two interfaces forming a slab. Several regimes defined by the relation between the sound propagation velocities and the interface velocity (namely subsonic, intersonic, and supersonic regimes) are discussed. Analytical expressions for the frequency conversions and scattering coefficients are obtained, and compared with numerical simulations based on an equivalent FTFD squeme.

💡 Research Summary

The paper investigates acoustic wave interaction with a moving, abrupt discontinuity—termed a spatio‑temporal interface—and with a pair of such interfaces forming a slab. Starting from the Galilean‑transformed mass‑continuity and momentum equations, the authors derive the dispersion relation (ω − v·k)² = (c k)² for a uniformly moving medium. The interface is modeled as a step change in sound speed c(x,t)=c₁+(c₂‑c₁)θ(x‑vt), where v is the interface velocity. Three regimes are identified based on the relative magnitude of v and the sound speeds c₁, c₂:

1. Subsonic (|v| < min(c₁,c₂)) – an incident wave generates a reflected wave in the first medium and a forward‑propagating transmitted wave in the second. Enforcing pressure and particle‑velocity continuity yields frequency‑conversion ratios Ω_r = (1‑v/c₁)/(1+v/c₁) and Ω_f = (1‑v/c₁)/(1‑v/c₂). The pressure‑based scattering coefficients are R = (z₂‑z₁)/(z₁+z₂) and F = 2z₂/(z₁+z₂), identical to the static case.

2. Supersonic (|v| > max(c₁,c₂)) – the interface outruns the wave in both media; only two waves exist, both in the second medium, propagating forward and backward relative to the interface. The continuity conditions become sums of pressures and velocities on the right side, leading to new expressions for the frequencies and scattering amplitudes; no reflected wave appears in the original medium.

3. Intersonic (min < |v| < max) – one medium supports propagation while the other does not, producing a mixed situation that the authors treat by matching phase at the moving boundary.

The analytical formulas are validated with a Centered‑in‑Time Finite‑Difference Time‑Domain (CIT‑FDTD) scheme. By applying a Galilean shift, the moving interface is represented as a stationary interface with a uniform background flow, simplifying the numerical implementation. Simulations confirm the predicted frequency shifts and scattering coefficients across all regimes, especially reproducing the transmission‑only behavior in the supersonic case and the partial blockage in the intersonic case.

Extending the analysis to a slab (two parallel interfaces moving with the same speed) reveals Fabry‑Pérot‑like resonances. Multiple reflections inside the slab give rise to transmission peaks whose positions depend on slab thickness, interface velocity, and the impedance contrast. The derived expressions show how the slab can be engineered to achieve desired frequency conversion or broadband transmission, highlighting its potential as a building block for space‑time metamaterials.

Overall, the work provides a comprehensive theoretical framework for acoustic scattering at moving discontinuities, delivers closed‑form expressions for frequency conversion and scattering coefficients in all velocity regimes, and corroborates them with robust time‑domain simulations, thereby laying a solid foundation for designing dynamic acoustic devices and space‑time crystals.