Back and Through the Looking Glass - Space-time scattering of elastic waves

<div data-page-number="1" aria-label="Page 1" data-listening-for-double-click="true" data-loaded="true"> <div><span>Due to causality wave scattering in time is simpler than scattering in space: In</span><br><span>contrast to multiple spatial boundaries, there are no infinite reflections between</span><br><span>temporal boundaries.</span> <span>Salem and Caloz, 2015 [1]</span> <span>showed that wave scatter-</span><br><span>ing can be simplified by constructing a time-space cross-mapping. We identify</span><br><span>the cross-mapped wavefields as the Focusing functions developed in data-driven</span><br><span>geophysical imaging.</span> <span>Experimentally,</span> <span>Bacot et al, 2016 [2]</span> <span>have shown that</span><br><span>time modulation of the medium properties of a capillary-gravity wave results in</span><br><span>time-refraction and time-reflection of the original wave. This experimental re-</span><br><span>sult should hold true for any system obeying Alembert&#8217;s equation. This should</span><br><span>in principle allow us to physically compute wavefields for the single-sided inverse</span><br><span>scattering problem through forward scattering experiments. We set up a sim-</span><br><span>ple comb-like discrete system for time-modulated 1D elastic wave propagation.</span><br><span>Elastic beams act as the masses and an electrostatic force as the springs of our</span><br><span>system. The effective coupling stiffness between the beams is modulated in time</span><br><span>through a variation of the electrostatic force. A Galerkin based wave propaga-</span><br><span>tion model shows that an experimental realization of hundreds of beams can be</span><br><span>achieved through micro-machining. Through time-modulations of the system&#8217;s</span><br><span>wavespeed a broadband excitation is refracted and reflected everywhere in space.</span><br><span>Time-scattering preserves the wave vector</span> <span>k</span><span>, which implies that the frequency</span><br><span>&#969;</span> <span>is not conserved.</span> <span>To elucidate the dispersion relation at time boundaries,</span><br><span>we employ a correction method for spatial dispersion.</span> <span>Herefore, a correction</span><br><span>method for time-dispersion in finite difference simulations developed by</span> <span>Koene</span><br><span>et al 2018 [3]</span> <span>is mapped to the spatial dimension of our meta-material.</span><br><span>[1] Salem, Mohamed A., and Christophe Caloz. &#8220;Space-Time Cross-Mapping</span><br><span>and Application to Wave Scattering.&#8221; ArXiv:1504.02012 [Physics], April 7, 2015.</span><br><span>http://arxiv.org/abs/1504.02012. [2] Bacot, Vincent, Matthieu Labousse, An-</span><br><span>tonin Eddi, Mathias Fink, and Emmanuel Fort. &#8220;Time Reversal and Holography</span><br><span>with Spacetime Transformations.&#8221; Nature Physics 12, no.</span> <span>10 (October 2016):</span><br><span>972&#8211;77. https://doi.org/10.1038/nphys3810. [3] Koene, Erik F M, Johan O A</span><br><span>1</span> <div>&#160;</div> </div> </div><div data-page-number="2" aria-label="Page 2" data-listening-for-double-click="true" data-loaded="true"> <div>&#160;</div> <div><span>Robertsson, Filippo Broggini, and Fredrik Andersson. &#8220;Eliminating Time Dis-</span><br><span>persion from Seismic Wave Modeling.&#8221; Geophysical Journal International 213,</span><br><span>no. 1 (April 1, 2018): 169&#8211;80. https://doi.org/10/gcz9wb.</span></div> </div>