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

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.