Bound States to the Continuum: Time-varying Spoof Acoustic Surface Waves

TL;DR

The paper addresses how bound acoustic surface waves (ASWs) on a time-periodically modulated boundary can couple to bulk waves, effectively converting a bound state into radiation. It develops an operator-based theory for a 1D array of time-varying cavities, deriving a dispersion relation from the eigencondition $\det[ Z_{mp} + \delta_{mp} \langle \kappa^{-1} \rangle_m ] = 0$, with the impedance $Z$ built from a time-dependent reflection matrix $r_{mp}$ and a Floquet index operator $\Lambda$. For sinusoidal cavity-depth modulation, the reflection coefficients are given by $r_{mp} = \frac{\omega+p\Omega}{\omega+m\Omega} e^{i\chi_{mp}\langle d\rangle}\left[ J_{m-p}(q_{mp}) + \frac{e}{2}J_{m-p-1}(q_{mp}) + \frac{e}{2}J_{m-p+1}(q_{mp}) \right]$, with $q_{mp}=\chi_{mp}\alpha$, $\chi_{mp}=(2\omega+(m+p)\Omega)/c$, and $e=\alpha\Omega/c$, producing Floquet sidebands spaced by $\Omega$. The analysis shows that modulation opens $k_\parallel$-space gaps when sidebands intersect, with gap width scaling with the modulation depth, and that negative-frequency branches fold into the positive spectrum, enabling radiative leakage above the sound line; these predictions are corroborated by time-domain FEM simulations. Crucially, the framework applies to general time-dependent boundary conditions via a phase-only modulation and remains computationally efficient by truncating to a small number of Floquet channels. The results offer a versatile design tool for active acoustic metasurfaces and highlight the potential for non-mechanical implementations of time-modulated boundaries.

Abstract

We develop a theoretical framework for time-modulated acoustic metasurfaces comprising a line array of modulated cavities, and show that bound acoustic surface waves can undergo temporal diffraction from bound states localised at an interface into bulk waves. The dispersion relation is derived via an operator formalism that captures the spatio-temporal coupling between Floquet sidebands. We show that under periodic modulation of the cavity length sidebands spaced by the modulation frequency are produced (diffraction in time), enabling the coupling of bound surface acoustic waves with bulk radiation i.e. from a bound state \textit{to} the continuum. We observe the negative-frequency spectra as spatial reflections along the array via time-domain finite element simulations. Spectral $k$-gaps are observed at band crossings, with the width of the gap proportional to the modulation amplitude. The modulation enters solely through a time-dependent reflection phase, such that the framework applies generally to metasurfaces with programmable boundary conditions, beyond purely mechanical modulation.

Figures (3)

• Figure 1: Schematic of the time-modulated acoustic metasurface. An infinite one-dimensional array of identical, subwavelength cavities of width $a$ and time-dependent depth $d(t)$ is with cavity openings at $z=0$. The blue highlighted region depicts one unit cell of length $L$. The aperture spans $x\in[-a/2,a/2]$ within each period $L$. Arrows denote the modulation of the cavity depth, $d(t) = \langle d\rangle + \sigma(t)$, where $\sigma(t)$ is a small periodic displacement applied in-phase to all cavities. Acoustic waves propagate in the half-space $z>0$ above the metasurface.
• Figure 2: Numerical solution to analytical dispersion relation and reconstructed real pressure fields: (a,b) Unmodulated metasurface showing the acoustic surface wave (ASW) branch below the sound line (black dotted line) and its exponentially decaying pressure field, evaluated at $k_\parallel = 170~\mathrm{m^{-1}}$ and $f = 6.37$ kHz. (c,d) Weak modulation ($\alpha = 0.01d$, $f_\text{mod}=10$ kHz) generates Floquet sidebands spaced by the modulation frequency. Negative-index branches ($m<0$) are folded upward into the positive spectrum; the red circle marks the representative point ($k_\parallel = 170~\mathrm{m^{-1}}$, $f = 16.37$ kHz) used for the corresponding field plot. Sideband indices are labelled in (c). The pressure field shows a superposition of the evanescent ASW and a very faint radiative component. (e,f) Strong modulation ($\alpha = 0.1d$) enhances sideband coupling, producing hybridisation between reflected branches ($m'=-m-1$) and opening modulation-induced band gaps in $k_\parallel$ (inset), the size of which scales with the modulation amplitude. The red circle in (e) marks the same representative point ($k_\parallel = 170~\mathrm{m^{-1}}$, $f = 16.37$ kHz) used in (c), at which the corresponding field in (f) is reconstructed. The pressure field in (f) exhibits pronounced radiation into the bulk, the emission angle of which depends on $k_\parallel$, while the surface-bound envelope remains visible near the interface.
• Figure 3: Finite-element dispersion maps of the time–modulated cavity metasurface. Each panel shows the spatio-temporal Fourier transform of the pressure field recorded $10$ mm above a finite array of $80$ cavities, excited by a point source at one end of the array. Increasing the modulation amplitude from 1% (a) to 10% (b) shows strengthened higher-order sidebands. The same branches as in Fig. \ref{['fig:big']}(c) are labelled for reference. Overlaid points indicate the corresponding analytical sideband positions. Regions that appear as avoided crossings (momentum-space gaps) in the analytical dispersion manifest here with increased Fourier amplitude at the temporal band-edge (inset), consistent with parametric amplification in the driven, time-modulated system hooper2025quasi. The dashed line marks the sound line.