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Due to causality wave scattering in time is simpler than scattering in space: Incontrast to multiple spatial boundaries, there are no infinite reflections betweentemporal boundaries. Salem and Caloz, 2015 [1] showed that wave scatter-ing can be simplified by constructing a time-space cross-mapping. We identifythe cross-mapped wavefields as the Focusing functions developed in data-drivengeophysical imaging. Experimentally, Bacot et al, 2016 [2] have shown thattime modulation of the medium properties of a capillary-gravity wave results intime-refraction and time-reflection of the original wave. This experimental re-sult should hold true for any system obeying Alembert’s equation. This shouldin principle allow us to physically compute wavefields for the single-sided inversescattering problem through forward scattering experiments. We set up a sim-ple comb-like discrete system for time-modulated 1D elastic wave propagation.Elastic beams act as the masses and an electrostatic force as the springs of oursystem. The effective coupling stiffness between the beams is modulated in timethrough a variation of the electrostatic force. A Galerkin based wave propaga-tion model shows that an experimental realization of hundreds of beams can beachieved through micro-machining. Through time-modulations of the system’swavespeed a broadband excitation is refracted and reflected everywhere in space.Time-scattering preserves the wave vector k, which implies that the frequencyω is not conserved. To elucidate the dispersion relation at time boundaries,we employ a correction method for spatial dispersion. Herefore, a correctionmethod for time-dispersion in finite difference simulations developed by Koeneet al 2018 [3] is mapped to the spatial dimension of our meta-material.[1] Salem, Mohamed A., and Christophe Caloz. “Space-Time Cross-Mappingand Application to Wave Scattering.” ArXiv:1504.02012 [Physics], April 7, 2015.http://arxiv.org/abs/1504.02012. [2] Bacot, Vincent, Matthieu Labousse, An-tonin Eddi, Mathias Fink, and Emmanuel Fort. “Time Reversal and Holographywith Spacetime Transformations.” Nature Physics 12, no. 10 (October 2016):972–77. https://doi.org/10.1038/nphys3810. [3] Koene, Erik F M, Johan O A1 Robertsson, Filippo Broggini, and Fredrik Andersson. “Eliminating Time Dis-persion from Seismic Wave Modeling.” Geophysical Journal International 213,no. 1 (April 1, 2018): 169–80. https://doi.org/10/gcz9wb. |
