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Math. Modelling</a> <a href="/?q=in%3A525142" title="Articles in this Issue">138, Article ID 115749, 22 p. (2025)</a>. </div> <div class="abstract"></div> <div class="clear"></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A74Kxx" title="MSC2020">74Kxx</a> </td> <td class="space"> Thin bodies, structures </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A74Jxx" title="MSC2020">74Jxx</a> </td> <td class="space"> Waves in solid mechanics </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35Qxx" title="MSC2020">35Qxx</a> </td> <td class="space"> Partial differential equations of mathematical physics and other areas of application </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Acircular-edge+waves">circular-edge waves</a>; <a href="/?q=ut%3Arainbow+reflection">rainbow reflection</a>; <a href="/?q=ut%3Atopological+interface+state">topological interface state</a>; <a href="/?q=ut%3Acustomizable+local+mode">customizable local mode</a>; <a href="/?q=ut%3Aacoustic+device+development">acoustic device development</a></div> <!-- Modal used to show zbmath metadata in different output formats--> <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;"> <!-- alert for connection errors etc --> </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function"> <!-- Button trigger metadata modal --> <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 08005570" data-ciurl="/ci/08005570" data-biburl="/bibtex/08005570.bib" data-amsurl="/amsrefs/08005570.bib" data-xmlurl="/xml/08005570.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/08005570.pdf" title="Zbl 08005570 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.apm.2024.115749" aria-label="DOI for “Flexural edge waves and customizable local modes of circular plates with metasurface”" title="10.1016/j.apm.2024.115749">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Cummer, S. A.; Christensen, J.; Alu, A., Controlling sound with acoustic metamaterials, Nat. Rev. Mater., 1, 16001, 2016; Cummer, S. A.; Christensen, J.; Alu, A., Controlling sound with acoustic metamaterials, Nat. Rev. Mater., 1, 16001, 2016</td> </tr><tr> <td>[2]</td> <td class="space">Ma, G.; Sheng, P., Acoustic metamaterials: from local resonances to broad horizons, Sci. Adv., 2, Article 1501595 pp., 2016; Ma, G.; Sheng, P., Acoustic metamaterials: from local resonances to broad horizons, Sci. Adv., 2, Article 1501595 pp., 2016</td> </tr><tr> <td>[3]</td> <td class="space">Surjadi, J. U.; Gao, L.; Du, H.; Li, X.; Xiong, X.; Fang, N. X.; Lu, Y., Mechanical metamaterials and their engineering applications, Adv. Eng. Mater., 21, Article 1800864 pp., 2019; Surjadi, J. U.; Gao, L.; Du, H.; Li, X.; Xiong, X.; Fang, N. X.; Lu, Y., Mechanical metamaterials and their engineering applications, Adv. Eng. Mater., 21, Article 1800864 pp., 2019</td> </tr><tr> <td>[4]</td> <td class="space">Hussein, M. I.; Leamy, M. J.; Ruzzene, M., Dynamics of phononic materials and structures: historical origins, recent progress, and future outlook, Appl. Mech. Rev., 66, Article 040802 pp., 2014; Hussein, M. I.; Leamy, M. J.; Ruzzene, M., Dynamics of phononic materials and structures: historical origins, recent progress, and future outlook, Appl. Mech. Rev., 66, Article 040802 pp., 2014</td> </tr><tr> <td>[5]</td> <td class="space">Chen, N.; Jiang, P.; Shi, P.; Xu, Y.; Yang, Z., A fish-skeleton-like metastructure isolator for low-frequency vibration isolation, Appl. Acoust., 203, Article 109224 pp., 2023; Chen, N.; Jiang, P.; Shi, P.; Xu, Y.; Yang, Z., A fish-skeleton-like metastructure isolator for low-frequency vibration isolation, Appl. Acoust., 203, Article 109224 pp., 2023</td> </tr><tr> <td>[6]</td> <td class="space">Chen, N.; Yang, Z.; Zuo, A.; Jiang, P.; Jin, S.; Xu, Y., Bandgap regulations of longitudinal wave for a nonlinear metastructure isolator with high-static-low-dynamic stiffness, Compos. Struct., 327, Article 117706 pp., 2024; Chen, N.; Yang, Z.; Zuo, A.; Jiang, P.; Jin, S.; Xu, Y., Bandgap regulations of longitudinal wave for a nonlinear metastructure isolator with high-static-low-dynamic stiffness, Compos. Struct., 327, Article 117706 pp., 2024</td> </tr><tr> <td>[7]</td> <td class="space">Zhou, W.; Wu, Y.; Zuo, L., Vibration and wave propagation attenuation for metamaterials by periodic piezoelectric arrays with high-order resonant circuit shunts, Smart Mater. Struct., 24, Article 065021 pp., 2015; Zhou, W.; Wu, Y.; Zuo, L., Vibration and wave propagation attenuation for metamaterials by periodic piezoelectric arrays with high-order resonant circuit shunts, Smart Mater. Struct., 24, Article 065021 pp., 2015</td> </tr><tr> <td>[8]</td> <td class="space">Liu, F.; Shi, P.; Xu, Y.; Cao, L.; Shen, Y.; Yang, Z., Total reflection of flexural waves by circular meta-slab and its application in vibration isolation, Int. J. Mech. Sci., 212, Article 106806 pp., 2021; Liu, F.; Shi, P.; Xu, Y.; Cao, L.; Shen, Y.; Yang, Z., Total reflection of flexural waves by circular meta-slab and its application in vibration isolation, Int. J. Mech. Sci., 212, Article 106806 pp., 2021</td> </tr><tr> <td>[9]</td> <td class="space">Liu, F.; Yang, Z.; Shi, P.; Shen, Y.; Xu, Y., Enhanced vibration absorption of plates with circular metasurface composed of lossy acoustic black hole subunits, J. Sound Vib., 536, Article 117047 pp., 2022; Liu, F.; Yang, Z.; Shi, P.; Shen, Y.; Xu, Y., Enhanced vibration absorption of plates with circular metasurface composed of lossy acoustic black hole subunits, J. Sound Vib., 536, Article 117047 pp., 2022</td> </tr><tr> <td>[10]</td> <td class="space">Oh, J. H.; Seung, H. M.; Kim, Y. Y., Doubly negative isotropic elastic metamaterial for sub-wavelength focusing: design and realization, J. Sound Vib., 410, 169-186, 2017; Oh, J. H.; Seung, H. M.; Kim, Y. Y., Doubly negative isotropic elastic metamaterial for sub-wavelength focusing: design and realization, J. Sound Vib., 410, 169-186, 2017</td> </tr><tr> <td>[11]</td> <td class="space">Li, K.; Rizzo, P.; Bagheri, A., A parametric study on the optimization of a metamaterial-based energy harvester, Smart Mater. Struct., 24, Article 115019 pp., 2015; Li, K.; Rizzo, P.; Bagheri, A., A parametric study on the optimization of a metamaterial-based energy harvester, Smart Mater. Struct., 24, Article 115019 pp., 2015</td> </tr><tr> <td>[12]</td> <td class="space">Wen, Z.; Jin, Y.; Gao, P.; Zhuang, X.; Rabczuk, T.; Djafari-Rouhani, B., Topological cavities in phononic plates for robust energy harvesting, Mech. Syst. Signal Process., 162, Article 108047 pp., 2022; Wen, Z.; Jin, Y.; Gao, P.; Zhuang, X.; Rabczuk, T.; Djafari-Rouhani, B., Topological cavities in phononic plates for robust energy harvesting, Mech. Syst. Signal Process., 162, Article 108047 pp., 2022</td> </tr><tr> <td>[13]</td> <td class="space">Gliozzi, A. S.; Miniaci, M.; Bosia, F.; Pugno, N. M.; Scalerandi, M., Metamaterials-based sensor to detect and locate nonlinear elastic sources, Appl. Phys. Lett., 107, Article 161902 pp., 2015; Gliozzi, A. S.; Miniaci, M.; Bosia, F.; Pugno, N. M.; Scalerandi, M., Metamaterials-based sensor to detect and locate nonlinear elastic sources, Appl. Phys. Lett., 107, Article 161902 pp., 2015</td> </tr><tr> <td>[14]</td> <td class="space">Tian, Y.; Shen, Y.; Rao, D.; Xu, W., Metamaterial improved nonlinear ultrasonics for fatigue damage detection, Smart Mater. Struct., 28, Article 075038 pp., 2019; Tian, Y.; Shen, Y.; Rao, D.; Xu, W., Metamaterial improved nonlinear ultrasonics for fatigue damage detection, Smart Mater. Struct., 28, Article 075038 pp., 2019</td> </tr><tr> <td>[15]</td> <td class="space">Liu, Z.; Shan, S.; Cheng, L., Nonlinear-Lamb-wave-based plastic damage detection assisted by topologically designed metamaterial filters, Struct. Health Monit., 22, 1828-1843, 2023; Liu, Z.; Shan, S.; Cheng, L., Nonlinear-Lamb-wave-based plastic damage detection assisted by topologically designed metamaterial filters, Struct. Health Monit., 22, 1828-1843, 2023</td> </tr><tr> <td>[16]</td> <td class="space">Yu, N.; Genevet, P.; Kats, M. A.; Aieta, F.; Tetienne, J.-P.; Capasso, F.; Gaburro, Z., Light propagation with phase discontinuities: generalized laws of reflection and refraction, Science, 334, 333-337, 2011; Yu, N.; Genevet, P.; Kats, M. A.; Aieta, F.; Tetienne, J.-P.; Capasso, F.; Gaburro, Z., Light propagation with phase discontinuities: generalized laws of reflection and refraction, Science, 334, 333-337, 2011</td> </tr><tr> <td>[17]</td> <td class="space">Yu, N.; Capasso, F., Flat optics with designer metasurfaces, Nat. Mater., 13, 139-150, 2014; Yu, N.; Capasso, F., Flat optics with designer metasurfaces, Nat. Mater., 13, 139-150, 2014</td> </tr><tr> <td>[18]</td> <td class="space">Sun, S.; He, Q.; Xiao, S.; Xu, Q.; Li, X.; Zhou, L., Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves, Nat. Mater., 11, 426-431, 2012; Sun, S.; He, Q.; Xiao, S.; Xu, Q.; Li, X.; Zhou, L., Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves, Nat. Mater., 11, 426-431, 2012</td> </tr><tr> <td>[19]</td> <td class="space">Aieta, F.; Kats, M. A.; Genevet, P.; Capasso, F., Multiwavelength achromatic metasurfaces by dispersive phase compensation, Science, 347, 1342-1345, 2015; Aieta, F.; Kats, M. A.; Genevet, P.; Capasso, F., Multiwavelength achromatic metasurfaces by dispersive phase compensation, Science, 347, 1342-1345, 2015</td> </tr><tr> <td>[20]</td> <td class="space">Chen, A. L.; Wang, Y.; Wang, Y.; Zhou, H.; Yuan, S., Design of acoustic/elastic phase gradient metasurfaces: principles, functional elements, tunability, and coding, Appl. Mech. Rev., 74, Article 020801 pp., 2022; Chen, A. L.; Wang, Y.; Wang, Y.; Zhou, H.; Yuan, S., Design of acoustic/elastic phase gradient metasurfaces: principles, functional elements, tunability, and coding, Appl. Mech. Rev., 74, Article 020801 pp., 2022</td> </tr><tr> <td>[21]</td> <td class="space">Zhu, H.; Semperlotti, F., Anomalous refraction of acoustic guided waves in solids with geometrically tapered metasurfaces, Phys. Rev. Lett., 117, Article 034302 pp., 2016; Zhu, H.; Semperlotti, F., Anomalous refraction of acoustic guided waves in solids with geometrically tapered metasurfaces, Phys. Rev. Lett., 117, Article 034302 pp., 2016</td> </tr><tr> <td>[22]</td> <td class="space">Colombi, A.; Ageeva, V.; Smith, R. J.; Clare, A.; Patel, R.; Clark, M.; Colquitt, D.; Roux, P.; Guenneau, S.; Craster, R. V., Enhanced sensing and conversion of ultrasonic Rayleigh waves by elastic metasurfaces, Sci. Rep., 7, 6750, 2017; Colombi, A.; Ageeva, V.; Smith, R. J.; Clare, A.; Patel, R.; Clark, M.; Colquitt, D.; Roux, P.; Guenneau, S.; Craster, R. V., Enhanced sensing and conversion of ultrasonic Rayleigh waves by elastic metasurfaces, Sci. Rep., 7, 6750, 2017</td> </tr><tr> <td>[23]</td> <td class="space">Kim, M. S.; Lee, W.; Park, C. I.; Oh, J. H., Elastic wave energy entrapment for reflectionless metasurface, Phys. Rev. Appl., 13, Article 054036 pp., 2020; Kim, M. S.; Lee, W.; Park, C. I.; Oh, J. H., Elastic wave energy entrapment for reflectionless metasurface, Phys. Rev. Appl., 13, Article 054036 pp., 2020</td> </tr><tr> <td>[24]</td> <td class="space">Kim, S. Y.; Bin Oh, Y.; Lee, J. S.; Kim, Y. Y., Anomalous mode-converting reflection of elastic waves using strip-type metagratings, Mech. Syst. Signal Process., 186, Article 109867 pp., 2023; Kim, S. Y.; Bin Oh, Y.; Lee, J. S.; Kim, Y. Y., Anomalous mode-converting reflection of elastic waves using strip-type metagratings, Mech. Syst. Signal Process., 186, Article 109867 pp., 2023</td> </tr><tr> <td>[25]</td> <td class="space">Shi, P.; Liu, F.; Xu, Y.; Yang, Z., Tunable elastic metasurface based on adjustable impedances for Gaussian beam manipulation, Int. J. Mech. Sci., 249, Article 108268 pp., 2023; Shi, P.; Liu, F.; Xu, Y.; Yang, Z., Tunable elastic metasurface based on adjustable impedances for Gaussian beam manipulation, Int. J. Mech. Sci., 249, Article 108268 pp., 2023</td> </tr><tr> <td>[26]</td> <td class="space">Lin, Z.; Xu, W.; Xuan, C.; Qi, W.; Wang, W., Modular elastic metasurfaces with mass oscillators for transmitted flexural wave manipulation, J. Phys. D Appl. Phys., 54, Article 255303 pp., 2021; Lin, Z.; Xu, W.; Xuan, C.; Qi, W.; Wang, W., Modular elastic metasurfaces with mass oscillators for transmitted flexural wave manipulation, J. Phys. D Appl. Phys., 54, Article 255303 pp., 2021</td> </tr><tr> <td>[27]</td> <td class="space">Lee, S. W.; Oh, J. H., Single-layer elastic metasurface with double negativity for anomalous refraction, J. Phys. D Appl. Phys., 53, Article 265301 pp., 2020; Lee, S. W.; Oh, J. H., Single-layer elastic metasurface with double negativity for anomalous refraction, J. Phys. D Appl. Phys., 53, Article 265301 pp., 2020</td> </tr><tr> <td>[28]</td> <td class="space">Liu, F.; Yang, Z.; Shi, P.; Shen, Y.; Cao, L.; Xu, Y., Refraction of flexural waves by ultra-broadband achromatic meta-slab with wavelength-dependent phase shifts, J. Appl. Mech., 89, Article 041003 pp., 2022; Liu, F.; Yang, Z.; Shi, P.; Shen, Y.; Cao, L.; Xu, Y., Refraction of flexural waves by ultra-broadband achromatic meta-slab with wavelength-dependent phase shifts, J. Appl. Mech., 89, Article 041003 pp., 2022</td> </tr><tr> <td>[29]</td> <td class="space">Liu, Y.; Liang, Z.; Liu, F.; Diba, O.; Lamb, A.; Li, J., Source illusion devices for flexural lamb waves using elastic metasurfaces, Phys. Rev. Lett., 119, Article 034301 pp., 2017; Liu, Y.; Liang, Z.; Liu, F.; Diba, O.; Lamb, A.; Li, J., Source illusion devices for flexural lamb waves using elastic metasurfaces, Phys. Rev. Lett., 119, Article 034301 pp., 2017</td> </tr><tr> <td>[30]</td> <td class="space">Tang, Y.-F.; Lin, S.-Y., Reconfigurable source illusion device for airborne sound using an enclosed adjustable piezoelectric metasurface, Chin. Phys. B, 32, Article 034306 pp., 2023; Tang, Y.-F.; Lin, S.-Y., Reconfigurable source illusion device for airborne sound using an enclosed adjustable piezoelectric metasurface, Chin. Phys. B, 32, Article 034306 pp., 2023</td> </tr><tr> <td>[31]</td> <td class="space">Chen, Z.; Shao, S.; Negahban, M.; Li, Z., Tunable metasurface for acoustic wave redirection, focusing and source illusion, J. Phys. D Appl. Phys., 52, Article 395503 pp., 2019; Chen, Z.; Shao, S.; Negahban, M.; Li, Z., Tunable metasurface for acoustic wave redirection, focusing and source illusion, J. Phys. D Appl. Phys., 52, Article 395503 pp., 2019</td> </tr><tr> <td>[32]</td> <td class="space">Shen, Y.; Xu, Y.; Liu, F.; Wang, F.; Yang, Z., 3D-printed meta-slab for focusing flexural waves in broadband, Extreme Mech. Lett., 48, Article 101410 pp., 2021; Shen, Y.; Xu, Y.; Liu, F.; Wang, F.; Yang, Z., 3D-printed meta-slab for focusing flexural waves in broadband, Extreme Mech. Lett., 48, Article 101410 pp., 2021</td> </tr><tr> <td>[33]</td> <td class="space">Shen, Y.; Xu, Y.; Liu, F.; Yang, Z., Focusing and self-bending of flexural waves by meta-slabs with subunits of varying thicknesses, (15th Symposium on Piezoelectricity, Acoustic Waves and Device Applications, SPAWDA 2020, April 16, 2021 - April 19, 2021, 2021, Institute of Electrical and Electronics Engineers Inc.: Institute of Electrical and Electronics Engineers Inc. Zhengzhou, Henan Province, China), 314-318; Shen, Y.; Xu, Y.; Liu, F.; Yang, Z., Focusing and self-bending of flexural waves by meta-slabs with subunits of varying thicknesses, (15th Symposium on Piezoelectricity, Acoustic Waves and Device Applications, SPAWDA 2020, April 16, 2021 - April 19, 2021, 2021, Institute of Electrical and Electronics Engineers Inc.: Institute of Electrical and Electronics Engineers Inc. Zhengzhou, Henan Province, China), 314-318</td> </tr><tr> <td>[34]</td> <td class="space">Xu, Z.; Wang, D.; Shi, Y.; Qian, Z.; Assouar, B.; Chuang, K.-C., Arbitrary wavefront modulation utilizing an aperiodic elastic metasurface, Int. J. Mech. Sci., 255, 2023; Xu, Z.; Wang, D.; Shi, Y.; Qian, Z.; Assouar, B.; Chuang, K.-C., Arbitrary wavefront modulation utilizing an aperiodic elastic metasurface, Int. J. Mech. Sci., 255, 2023</td> </tr><tr> <td>[35]</td> <td class="space">Laude, V.; Robert, L.; Daniau, W.; Khelif, A.; Ballandras, S., Surface acoustic wave trapping in a periodic array of mechanical resonators, Appl. Phys. Lett., 89, Article 083515 pp., 2006; Laude, V.; Robert, L.; Daniau, W.; Khelif, A.; Ballandras, S., Surface acoustic wave trapping in a periodic array of mechanical resonators, Appl. Phys. Lett., 89, Article 083515 pp., 2006</td> </tr><tr> <td>[36]</td> <td class="space">Assouar, M. B.; Oudich, M., Dispersion curves of surface acoustic waves in a two-dimensional phononic crystal, Appl. Phys. Lett., 99, Article 123505 pp., 2011; Assouar, M. B.; Oudich, M., Dispersion curves of surface acoustic waves in a two-dimensional phononic crystal, Appl. Phys. Lett., 99, Article 123505 pp., 2011</td> </tr><tr> <td>[37]</td> <td class="space">Oudich, M.; Assouar, M. B., Surface acoustic wave band gaps in a diamond-based two-dimensional locally resonant phononic crystal for high frequency applications, J. Appl. Phys., 111, Article 014504 pp., 2012; Oudich, M.; Assouar, M. B., Surface acoustic wave band gaps in a diamond-based two-dimensional locally resonant phononic crystal for high frequency applications, J. Appl. Phys., 111, Article 014504 pp., 2012</td> </tr><tr> <td>[38]</td> <td class="space">Xu, Y.; Peng, P., High quality broadband spatial reflections of slow Rayleigh surface acoustic waves modulated by a graded grooved surface, J. Appl. Phys., 117, Article 035103 pp., 2015; Xu, Y.; Peng, P., High quality broadband spatial reflections of slow Rayleigh surface acoustic waves modulated by a graded grooved surface, J. Appl. Phys., 117, Article 035103 pp., 2015</td> </tr><tr> <td>[39]</td> <td class="space">De Ponti, J. M.; Colombi, A.; Ardito, R.; Braghin, F.; Corigliano, A.; Craster, R. V., Graded elastic metasurface for enhanced energy harvesting, New J. Phys., 22, Article 013013 pp., 2020; De Ponti, J. M.; Colombi, A.; Ardito, R.; Braghin, F.; Corigliano, A.; Craster, R. V., Graded elastic metasurface for enhanced energy harvesting, New J. Phys., 22, Article 013013 pp., 2020</td> </tr><tr> <td>[40]</td> <td class="space">Colquitt, D. J.; Colombi, A.; Craster, R. V.; Roux, P.; Guenneau, S. R.L., Seismic metasurfaces: sub-wavelength resonators and Rayleigh wave interaction, J. Mech. Phys. Solids, 99, 379-393, 2017; Colquitt, D. J.; Colombi, A.; Craster, R. V.; Roux, P.; Guenneau, S. R.L., Seismic metasurfaces: sub-wavelength resonators and Rayleigh wave interaction, J. Mech. Phys. Solids, 99, 379-393, 2017</td> </tr><tr> <td>[41]</td> <td class="space">Colombi, A.; Colquitt, D.; Roux, P.; Guenneau, S.; Craster, R. V., A seismic metamaterial: the resonant metawedge, Sci. Rep., 6, 27717, 2016; Colombi, A.; Colquitt, D.; Roux, P.; Guenneau, S.; Craster, R. V., A seismic metamaterial: the resonant metawedge, Sci. Rep., 6, 27717, 2016</td> </tr><tr> <td>[42]</td> <td class="space">Chaplain, G. J.; De Ponti, J. M.; Colombi, A.; Fuentes-Dominguez, R.; Dryburg, P.; Pieris, D.; Smith, R. J.; Clare, A.; Clark, M.; Craster, R. V., Tailored elastic surface to body wave Umklapp conversion, Nat. Comm., 11, 2020; Chaplain, G. J.; De Ponti, J. M.; Colombi, A.; Fuentes-Dominguez, R.; Dryburg, P.; Pieris, D.; Smith, R. J.; Clare, A.; Clark, M.; Craster, R. V., Tailored elastic surface to body wave Umklapp conversion, Nat. Comm., 11, 2020</td> </tr><tr> <td>[43]</td> <td class="space">Pu, X.; Palermo, A.; Marzani, A., Topological edge states of quasiperiodic elastic metasurfaces, Mech. Syst. Signal Process., 181, Article 109478 pp., 2022; Pu, X.; Palermo, A.; Marzani, A., Topological edge states of quasiperiodic elastic metasurfaces, Mech. Syst. Signal Process., 181, Article 109478 pp., 2022</td> </tr><tr> <td>[44]</td> <td class="space">Liu, F.; Yang, Z.; Shi, P.; Shen, Y.; Xu, Y., Sound radiation attenuation by circular total-reflection elastic metasurface composed of subunits with cubic profiles, Chinese J. Aeronaut., 36, 515-528, 2023; Liu, F.; Yang, Z.; Shi, P.; Shen, Y.; Xu, Y., Sound radiation attenuation by circular total-reflection elastic metasurface composed of subunits with cubic profiles, Chinese J. Aeronaut., 36, 515-528, 2023</td> </tr><tr> <td>[45]</td> <td class="space">Konenkov, Y. K., A Rayleigh-type flexural wave, Sov. Phys. Acoust., 6, 122-123, 1960; Konenkov, Y. K., A Rayleigh-type flexural wave, Sov. Phys. Acoust., 6, 122-123, 1960</td> </tr><tr> <td>[46]</td> <td class="space">Thurston, R. N.; McKenna, J., Flexural acoustic-waves along edge of a plate, IEEE Trans. Son. Ultrason., SU21, 296-297, 1974; Thurston, R. N.; McKenna, J., Flexural acoustic-waves along edge of a plate, IEEE Trans. Son. Ultrason., SU21, 296-297, 1974</td> </tr><tr> <td>[47]</td> <td class="space">Norris, A. N., Flexural edge waves, J. Sound Vib., 171, 571-573, 1994; Norris, A. N., Flexural edge waves, J. Sound Vib., 171, 571-573, 1994 · <a href="/1064.74580" class="nowrap">Zbl 1064.74580</a></td> </tr><tr> <td>[48]</td> <td class="space">Siddiqui, M. A.; Hawwa, M. A., Flexural edge waves in a Kirchhoff plate carrying periodic edge resonators and resting on a Winkler foundation, Wave Motion, 103, Article 102720 pp., 2021; Siddiqui, M. A.; Hawwa, M. A., Flexural edge waves in a Kirchhoff plate carrying periodic edge resonators and resting on a Winkler foundation, Wave Motion, 103, Article 102720 pp., 2021 · <a href="/1524.74332" class="nowrap">Zbl 1524.74332</a></td> </tr><tr> <td>[49]</td> <td class="space">Chaplain, G. J.; Makwana, M. P.; Craster, R. V., Rayleigh-Bloch, topological edge and interface waves for structured elastic plates, Wave Motion, 86, 162-174, 2019; Chaplain, G. J.; Makwana, M. P.; Craster, R. V., Rayleigh-Bloch, topological edge and interface waves for structured elastic plates, Wave Motion, 86, 162-174, 2019 · <a href="/1524.74226" class="nowrap">Zbl 1524.74226</a></td> </tr><tr> <td>[50]</td> <td class="space">Xu, Y.; Shen, Y.; Liu, F.; Cao, L.; Yang, Z., Modulated flexural edge waves in a plate with its free edge structured by an array of grooves, J. Sound Vib., 536, Article 117173 pp., 2022; Xu, Y.; Shen, Y.; Liu, F.; Cao, L.; Yang, Z., Modulated flexural edge waves in a plate with its free edge structured by an array of grooves, J. Sound Vib., 536, Article 117173 pp., 2022</td> </tr><tr> <td>[51]</td> <td class="space">Shen, Y.; Xu, Y.; Liu, F.; Yang, Z., Metasurface-guided flexural waves and their manipulations, Int. J. Mech. Sci., 257, Article 108538 pp., 2023; Shen, Y.; Xu, Y.; Liu, F.; Yang, Z., Metasurface-guided flexural waves and their manipulations, Int. J. Mech. Sci., 257, Article 108538 pp., 2023</td> </tr><tr> <td>[52]</td> <td class="space">Liu, T.; An, S.; Gu, Z.; Liang, S.; Gao, H.; Ma, G.; Zhu, J., Chirality-switchable acoustic vortex emission via non-Hermitian selective excitation at an exceptional point, Sci. Bull., 67, 1131-1136, 2022; Liu, T.; An, S.; Gu, Z.; Liang, S.; Gao, H.; Ma, G.; Zhu, J., Chirality-switchable acoustic vortex emission via non-Hermitian selective excitation at an exceptional point, Sci. Bull., 67, 1131-1136, 2022</td> </tr><tr> <td>[53]</td> <td class="space">Armani, D. K.; Kippenberg, T. J.; Spillane, S. M.; Vahala, K. J., Ultra-high-Q toroid microcavity on a chip, Nature, 421, 925-928, 2003; Armani, D. K.; Kippenberg, T. J.; Spillane, S. M.; Vahala, K. J., Ultra-high-Q toroid microcavity on a chip, Nature, 421, 925-928, 2003</td> </tr><tr> <td>[54]</td> <td class="space">Yu, D.; Humar, M.; Meserve, K.; Bailey, R. C.; Chormaic, S. N.; Vollmer, F., Whispering-gallery-mode sensors for biological and physical sensing, Nat. Rev. Method. Prime., 1, 83, 2021; Yu, D.; Humar, M.; Meserve, K.; Bailey, R. C.; Chormaic, S. N.; Vollmer, F., Whispering-gallery-mode sensors for biological and physical sensing, Nat. Rev. Method. Prime., 1, 83, 2021</td> </tr><tr> <td>[55]</td> <td class="space">Chen, W.; Ozdemir, S. K.; Zhao, G.; Wiersig, J.; Yang, L., Exceptional points enhance sensing in an optical microcavity, Nature, 548, 192, 2017, -+; Chen, W.; Ozdemir, S. K.; Zhao, G.; Wiersig, J.; Yang, L., Exceptional points enhance sensing in an optical microcavity, Nature, 548, 192, 2017, -+</td> </tr><tr> <td>[56]</td> <td class="space">Wiersig, J., Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: application to microcavity sensors for single-particle detection, Phys. Rev. Lett., 112, Article 203901 pp., 2014; Wiersig, J., Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: application to microcavity sensors for single-particle detection, Phys. Rev. Lett., 112, Article 203901 pp., 2014</td> </tr><tr> <td>[57]</td> <td class="space">Mansfield, E. H., The Bending and Stretching of Plates, 1989, Cambridge University Press: Cambridge University Press UK; Mansfield, E. H., The Bending and Stretching of Plates, 1989, Cambridge University Press: Cambridge University Press UK · <a href="/0755.73003" class="nowrap">Zbl 0755.73003</a></td> </tr><tr> <td>[58]</td> <td class="space">Su, W. P.; Schrieffer, J. R.; Heeger, A. J., Solitons in polyacetylene, Phys. Rev. Lett., 42, 1698-1701, 1979; Su, W. P.; Schrieffer, J. R.; Heeger, A. J., Solitons in polyacetylene, Phys. Rev. Lett., 42, 1698-1701, 1979</td> </tr><tr> <td>[59]</td> <td class="space">Xiao, M.; Zhang, Z. Q.; Chan, C. T., Surface impedance and bulk band geometric phases in one-dimensional systems, Phys. Rev. X, 4, Article 021017 pp., 2014; Xiao, M.; Zhang, Z. Q.; Chan, C. T., Surface impedance and bulk band geometric phases in one-dimensional systems, Phys. Rev. X, 4, Article 021017 pp., 2014</td> </tr><tr> <td>[60]</td> <td class="space">Yang, Z.; Zhang, B., Acoustic Type-II Weyl Nodes from Stacking Dimerized Chains, Phys. Rev. Lett., 117, Article 224301 pp., 2016; Yang, Z.; Zhang, B., Acoustic Type-II Weyl Nodes from Stacking Dimerized Chains, Phys. Rev. Lett., 117, Article 224301 pp., 2016</td> </tr><tr> <td>[61]</td> <td class="space">Huo, S.-Y.; Yang, Q.-S.; Chen, J.-J.; Li, H.-K.; Yao, L.-C.; He, F.-C.; Fu, C.-M., Elastic three-dimensional phononic topological insulators with Dirac hierarchy, Sci. China-Phys. Mech. Astron., 67, 2024; Huo, S.-Y.; Yang, Q.-S.; Chen, J.-J.; Li, H.-K.; Yao, L.-C.; He, F.-C.; Fu, C.-M., Elastic three-dimensional phononic topological insulators with Dirac hierarchy, Sci. China-Phys. Mech. Astron., 67, 2024</td> </tr><tr> <td>[62]</td> <td class="space">Chien, C.-C.; Velizhanin, K. A.; Dubi, Y.; Ilic, B. R.; Zwolak, M., Topological quantization of energy transport in micromechanical and nanomechanical lattices, Phys. Rev. B, 97, Article 125425 pp., 2018; Chien, C.-C.; Velizhanin, K. A.; Dubi, Y.; Ilic, B. R.; Zwolak, M., Topological quantization of energy transport in micromechanical and nanomechanical lattices, Phys. Rev. B, 97, Article 125425 pp., 2018</td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. 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