pith. sign in

arxiv: 2411.18556 · v4 · submitted 2024-11-27 · ⚛️ physics.app-ph

Symmetry-driven Phononic Metamaterials

Simon Yves , Michel Fruchart , Romain Fleury , Gal Shmuel , Vincenzo Vitelli , Michael R. Haberman , Andrea Al\`u This is my paper

Pith reviewed 2026-05-23 17:28 UTC · model grok-4.3

classification ⚛️ physics.app-ph
keywords phononic metamaterialssymmetry engineeringphononic crystalsnon-reciprocal acousticselastic waveswave controlmetamaterial design
0
0 comments X

The pith

Symmetry principles at micro and meso scales enable precise tailoring of phononic wave responses in synthetic media.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper reviews recent advances in phononic crystals and metamaterials, showing how control over symmetry classes shapes mechanical vibration behavior. It begins with broken spatial symmetries and explores their combination with time symmetries to create non-reciprocal effects. Further sections examine multiple symmetry classes together for unusual wave transport, concluding with directions for future symmetry-based designs in sound and vibration control.

Core claim

Controlling different symmetry classes at the microscopic and mesoscopic scales in synthetic media offers a powerful tool to precisely tailor phononic responses for advanced acoustic and elastodynamic wave control.

What carries the argument

Broken spatial symmetries and their interplay with time symmetries, which together induce non-reciprocal and exotic phononic phenomena.

If this is right

  • Precise tailoring of acoustic isolation and sensing devices.
  • Non-conservative phenomena in elastodynamic systems.
  • Exotic wave transport from combined symmetry classes.
  • Future designs based on symmetry engineering for energy harvesting and imaging.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The symmetry framework may apply to designing thermal phonon control in materials.
  • Fabrication techniques could be developed to enforce specific symmetry breaks at scale.
  • Connections to active metamaterials where time symmetry is dynamically altered.

Load-bearing premise

Symmetry principles serve as the primary organizing framework for progress in phononic media design.

What would settle it

Demonstration of a phononic metamaterial achieving advanced wave control without reliance on symmetry classes at micro or meso scales.

Figures

Figures reproduced from arXiv: 2411.18556 by Andrea Al\`u, Gal Shmuel, Michael R. Haberman, Michel Fruchart, Romain Fleury, Simon Yves, Vincenzo Vitelli.

Figure 1
Figure 1. Figure 1: Symmetry-driven phononics. Identification and selective breaking of the various symmetries characterizing phononic materials and meta-structures, both at the microscopic and macroscopic scales. These broken symmetries enable enhanced control over acoustic and elastodynamic wave propagation. In this review we focus on different symmetry classes, namely spatial symmetries, reciprocity, energy conservation an… view at source ↗
Figure 2
Figure 2. Figure 2: Phononic phenomena controlled by spatial symmetries. a, Willis coupling between momentum (Pz) and strain ∂xuz stemming from a resonant meta-atom with broken spatial inversion symmetry, in the context of flexural waves in a structured beam. b, Experimental verification of the existence of Willis coupling for air-borne acoustic waves resulting from a scatterer with broken spatial inversion symmetry. c, Exper… view at source ↗
Figure 3
Figure 3. Figure 3: Phononic phenomena controlled by time-symmetry and reciprocity. a, Non-reciprocal phonon transmission induced by the combination of spatial asymmetry and nonlinearity stemming from the acoustic radiation pressure at the interface between air and water. b, Acoustic circulator based on a three-port cavity (left panel) with embedded flow leads zero transmission from port 1 to 2, and full transmission from por… view at source ↗
Figure 4
Figure 4. Figure 4: Phononic phenomena controlled by time symmetry and energy conservation. a, A two-level non-Hermitian system made of two tightly-coupled acoustic cavities with controllable asymmetric loss (top panel) which allows to demonstrate the coalescence of the two modes at the exceptional point, evidenced by the merging of the two transmission peaks depending on the loss amount (bottom panel). b, An invisible acoust… view at source ↗
Figure 5
Figure 5. Figure 5: Phononic phenomena controlled by global symmetries. a, Correspondence between a conventional resonant cavity made of two mirrors with reflection phases φL,R, described by a scattering matrix S(ω) (left panel) and the virtual cavity at the edge of a gapped phononic crystal, whose reflection phase φR(ω) depends on the operating frequency, hosting a boundary mode. b, Flow-induced time-reversal symmetry breaki… view at source ↗
read the original abstract

Phonons are quasiparticles associated with mechanical vibrations in materials. They are at the root of the propagation of sound and elastic waves, as well as of thermal phenomena, which are pervasive in our everyday life and in many technologies. The fundamental understanding and control of phonon responses in natural and artificial media are key in the context of communications, isolation, energy harvesting and control, sensing and imaging. It has recently been realized that controlling different symmetry classes at the microscopic and mesoscopic scales in synthetic media offers a powerful tool to precisely tailor phononic responses for advanced acoustic and elastodynamic wave control. In this Review, we survey the recent progress in the design and synthesis of artificial phononic media, namely phononic crystals and metamaterials, guided by symmetry principles. Starting from tailored broken spatial symmetries, we discuss their interplay with time symmetries for non-reciprocal and non-conservative phenomena. We also address broader concepts that combine multiple symmetry classes to induce exotic phononic wave transport. We conclude with an outlook on future research directions based on symmetry engineering for the advanced control of phononic waves.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. This manuscript is a review surveying recent progress in the design of phononic crystals and metamaterials, organized around symmetry principles. It starts from tailored broken spatial symmetries, discusses their interplay with time symmetries for non-reciprocal and non-conservative wave phenomena, addresses combinations of multiple symmetry classes for exotic transport, and concludes with an outlook on future symmetry-engineering directions for phononic wave control.

Significance. If the cited literature is represented accurately and comprehensively, the review supplies a coherent organizing framework that highlights symmetry control at micro- and mesoscopic scales as a design tool for acoustic and elastodynamic applications. The paper contains no new derivations, data, or falsifiable predictions; its value is therefore synthetic rather than generative.

major comments (1)
  1. [Abstract] Abstract: the central organizing premise—that symmetry classes at microscopic and mesoscopic scales constitute a 'powerful tool' for tailoring phononic responses—is stated as an established perspective but is not accompanied by any explicit discussion of regimes or metrics where symmetry-based design is demonstrably superior (or inferior) to alternative approaches such as topology optimization or graded-index methods.
minor comments (2)
  1. The abstract and introduction would benefit from a brief statement of the temporal scope (e.g., references published after 2015) and the approximate number of works surveyed to allow readers to gauge coverage.
  2. Section headings and subheadings should be numbered consistently to facilitate cross-referencing in a review of this length.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment and recommendation of minor revision. We address the single major comment below with a proposed revision to the abstract.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central organizing premise—that symmetry classes at microscopic and mesoscopic scales constitute a 'powerful tool' for tailoring phononic responses—is stated as an established perspective but is not accompanied by any explicit discussion of regimes or metrics where symmetry-based design is demonstrably superior (or inferior) to alternative approaches such as topology optimization or graded-index methods.

    Authors: We agree that the abstract would benefit from a brief qualification of scope. The review is organized around symmetry principles as a unifying design lens, supported by the surveyed literature, rather than a comparative methodology study. In revision we will add one sentence to the abstract noting that symmetry-based approaches supply physical insight and analytical tractability that complement numerical techniques such as topology optimization, particularly for protected modes or non-reciprocal transport, while acknowledging that quantitative superiority depends on the specific application constraints. revision: yes

Circularity Check

0 steps flagged

Literature survey with no derivations or predictions

full rationale

This paper is a descriptive review that surveys existing progress in phononic crystals and metamaterials organized around symmetry principles. It advances no original equations, derivations, fitted parameters, or falsifiable predictions. The abstract and structure explicitly frame the work as a survey of prior results, with the central perspective on symmetry control presented as an established viewpoint rather than a new proposition derived within the manuscript. No load-bearing steps reduce to self-definition, fitted inputs, or self-citation chains.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

As a review paper, the central claim rests on the compilation and organization of prior literature rather than new derivations. No free parameters or invented entities are introduced by the authors.

axioms (1)
  • domain assumption Symmetry classes at microscopic and mesoscopic scales can be independently controlled in synthetic media to tailor wave responses
    Invoked throughout the abstract as the guiding principle for the survey structure.

pith-pipeline@v0.9.0 · 5735 in / 1180 out tokens · 37765 ms · 2026-05-23T17:28:28.218417+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

297 extracted references · 297 canonical work pages

  1. [1]

    Deymier, P. A. Acoustic metamaterials and phononic crystals, vol. 173 (Springer Science & Business Media, 2013)

    work page 2013

  2. [2]

    Haberman, M. R. & Guild, M. D. Acoustic metamaterials. Phys. Today 69, 42–48 (2016)

    work page 2016

  3. [3]

    & Sheng, P

    Ma, G. & Sheng, P. Acoustic metamaterials: From local resonances to broad horizons. Sci. Adv. 2, e1501595 (2016)

    work page 2016

  4. [4]

    A., Christensen, J

    Cummer, S. A., Christensen, J. & Alù, A. Controlling sound with acoustic metamaterials. Nat. Rev. Mater. 1, 1–13 (2016)

    work page 2016

  5. [5]

    Assouar, B. et al. Acoustic metasurfaces. Nat. Rev. Mater. 3, 460–472, DOI: 10.1038/s41578-018-0061-4 (2018)

    work page doi:10.1038/s41578-018-0061-4 2018

  6. [6]

    V ., Guenneau, S

    Craster, R. V ., Guenneau, S. R., Muamer, K. & Wegener, M. Mechanical metamaterials.Reports on Prog. Phys. (2023)

    work page 2023

  7. [7]

    Pierce, A. D. Acoustics, 3rd Edition (Springer, Cham, Switzerland, 2019)

    work page 2019

  8. [8]

    Achenbach, J. D. Wave propagation in elastic solids(Elsevier, Amsterdam, Netherlands, 1973)

    work page 1973

  9. [9]

    Willis, J. R. Variational and related methods for the overall properties of composites. Adv. Appl. Mech. 21, 1–78 (1981)

    work page 1981

  10. [10]

    Willis, J. R. The construction of effective relations for waves in a composite. Comptes Rendus Mécanique 340, 181 – 192, DOI: https://doi.org/10.1016/j.crme.2012.02.001 (2012). Recent Advances in Micromechanics of Materials

    work page doi:10.1016/j.crme.2012.02.001 2012

  11. [11]

    Quan, L., Ra’di, Y ., Sounas, D. L. & Alù, A. Maximum Willis Coupling in Acoustic Scatterers.Phys. Rev. Lett. 120, 254301, DOI: 10.1103/PhysRevLett.120.254301 (2018)

    work page doi:10.1103/physrevlett.120.254301 2018

  12. [12]

    Willis, J. R. Effective constitutive relations for waves in composites and metamaterials. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 467, 1865–1879, DOI: 10.1098/rspa.2010.0620 (2011)

    work page doi:10.1098/rspa.2010.0620 2010

  13. [13]

    W., Briane, M

    Milton, G. W., Briane, M. & Willis, J. R. On cloaking for elasticity and physical equations with a transformation invariant form. New J. Phys. 8, 248 (2006)

    work page 2006

  14. [14]

    Milton, G. W. & Willis, J. R. On modifications of Newton’s second law and linear continuum elastodynamics.Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 463, 855–880, DOI: 10.1098/rspa.2006.1795 (2007)

    work page doi:10.1098/rspa.2006.1795 2006

  15. [15]

    F., Alù, A

    Sieck, C. F., Alù, A. & Haberman, M. R. Origins of Willis coupling and acoustic bianisotropy in acoustic metamaterials through source-driven homogenization. Phys. Rev. B 96, 104303, DOI: 10.1103/PhysRevB.96.104303 (2017)

    work page doi:10.1103/physrevb.96.104303 2017

  16. [16]

    R., Norris, A

    Pernas-Salomón, R., Haberman, M. R., Norris, A. N. & Shmuel, G. The electromomentum effect in piezoelectric Willis scatterers. Wave Motion 102797, DOI: https://doi.org/10.1016/j.wavemoti.2021.102797 (2021)

    work page doi:10.1016/j.wavemoti.2021.102797 2021

  17. [17]

    Willis metamaterial on a structured beam

    Liu, Y .et al. Willis metamaterial on a structured beam. Phys. Rev. X 9, 011040 (2019)

    work page 2019

  18. [18]

    B., Sieck, C

    Muhlestein, M. B., Sieck, C. F., Alù, A. & Haberman, M. R. Reciprocity, passivity and causality in Willis materials. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 472, DOI: 10.1098/rspa.2016.0604 (2016)

    work page doi:10.1098/rspa.2016.0604 2016

  19. [19]

    & Shmuel, G

    Pernas-Salomón, R. & Shmuel, G. Fundamental Principles for Generalized Willis Metamaterials. Phys. Rev. Appl. 14, 064005, DOI: 10.1103/PhysRevApplied.14.064005 (2020)

    work page doi:10.1103/physrevapplied.14.064005 2020

  20. [20]

    & Park, N

    Koo, S., Cho, C., Jeong, J.-h. & Park, N. Acoustic omni meta-atom for decoupled access to all octants of a wave parameter space. Nat. Commun. 7, 13012 (2016)

    work page 2016

  21. [21]

    & Shvets, G

    Fietz, C. & Shvets, G. Current-driven metamaterial homogenization. Phys. B: Condens. Matter 405, 2930 – 2934, DOI: https://doi.org/10.1016/j.physb.2010.01.006 (2010). Proceedings of the Eighth International Conference on Electrical Transport and Optical Properties of Inhomogeneous Media

    work page doi:10.1016/j.physb.2010.01.006 2010

  22. [22]

    Willis, J. R. Exact effective relations for dynamics of a laminated body. Mech. Mater. 41, 385–393, DOI: http: //dx.doi.org/10.1016/j.mechmat.2009.01.010 (2009). 14/31

    work page doi:10.1016/j.mechmat.2009.01.010 2009

  23. [23]

    Melnikov, A. et al. Acoustic meta-atom with experimentally verified maximum Willis coupling. Nat. Commun. 10, 3148, DOI: 10.1038/s41467-019-10915-5 (2019)

    work page doi:10.1038/s41467-019-10915-5 2019

  24. [24]

    & Christensen, J

    Merkel, A., Romero-García, V ., Groby, J.-P., Li, J. & Christensen, J. Unidirectional zero sonic reflection in passive pt-symmetric willis media. Phys. Rev. B 98, 201102 (2018)

    work page 2018

  25. [25]

    & Alù, A

    Peng, Y .-G., Mazor, Y . & Alù, A. Fundamentals of acoustic willis media.Wave Motion 112, 102930 (2022)

    work page 2022

  26. [26]

    & Alù, A

    Esfahlani, H., Mazor, Y . & Alù, A. Homogenization and design of acoustic Willis metasurfaces. Phys. Rev. B 103, 054306 (2021)

    work page 2021

  27. [27]

    B., Sieck, C

    Muhlestein, M. B., Sieck, C. F., Wilson, P. S. & Haberman, M. R. Experimental evidence of Willis coupling in a one-dimensional effective material element. Nat. Commun. 8, 15625 EP – (2017)

    work page 2017

  28. [28]

    T., Yang, M

    Lau, J., Tang, S. T., Yang, M. & Yang, Z. Coupled Decorated Membrane Resonators with Large Willis Coupling.Phys. Rev. Appl. 12, 014032, DOI: 10.1103/PhysRevApplied.12.014032 (2019)

    work page doi:10.1103/physrevapplied.12.014032 2019

  29. [29]

    Sounas, D. L. & Alù, A. Extinction symmetry for reciprocal objects and its implications on cloaking and scattering manipulation. Opt. Lett. 39, 4053–4056 (2014)

    work page 2014

  30. [30]

    J., Goldsberry, B

    Lawrence, A. J., Goldsberry, B. M., Wallen, S. P. & Haberman, M. R. Numerical study of acoustic focusing using a bianisotropic acoustic lens. The J. Acoust. Soc. Am. 148, EL365–EL369, DOI: 10.1121/10.0002137 (2020)

    work page doi:10.1121/10.0002137 2020

  31. [31]

    Li, J. et al. Highly efficient generation of angular momentum with cylindrical bianisotropic metasurfaces. Phys. Rev. Appl. 11, 024016 (2019)

    work page 2019

  32. [32]

    Li, J., Shen, C., D’iaz-Rubio, A., Tretyakov, S. A. & Cummer, S. A. Systematic design and experimental demonstration of bianisotropic metasurfaces for scattering-free manipulation of acoustic wavefronts. Nat. Commun. 9, 1342, DOI: 10.1038/s41467-018-03778-9 (2018)

    work page doi:10.1038/s41467-018-03778-9 2018

  33. [33]

    Hao, Y ., Shen, Y ., Groby, J.-P. & Li, J. Experimental demonstration of Willis coupling for elastic torsional waves.Wave Motion 112, 102931 (2022)

    work page 2022

  34. [34]

    & Haberman, M

    Chen, Y . & Haberman, M. R. Controlling displacement fields in polar Willis solids via gauge transformations.Phys. Rev. Lett. 130, 147201 (2023)

    work page 2023

  35. [35]

    Sepehrirahnama, S., Oberst, S., Chiang, Y . K. & Powell, D. A. Willis coupling-induced acoustic radiation force and torque reversal. Phys. Rev. Lett. 129, 174501 (2022)

    work page 2022

  36. [36]

    Mason, W. P. Piezoelectric crystals and their application to ultrasonics. (Van Nostrand, New York, 1950)

    work page 1950

  37. [37]

    & Shmuel, G

    Pernas-Salomón, R. & Shmuel, G. Symmetry breaking creates electro-momentum coupling in piezoelectric metamaterials. J. Mech. Phys. Solids 134, 103770, DOI: https://doi.org/10.1016/j.jmps.2019.103770 (2020)

    work page doi:10.1016/j.jmps.2019.103770 2019

  38. [38]

    & Amir, O

    Kosta, M., Muhafra, A., Pernas-Salómon, R., Shmuel, G. & Amir, O. Maximizing the electromomentum coupling in piezoelectric laminates. Int. J. Solids Struct. 254-255, 111909, DOI: https://doi.org/10.1016/j.ijsolstr.2022.111909 (2022)

    work page doi:10.1016/j.ijsolstr.2022.111909 2022

  39. [39]

    & Shmuel, G

    Muhafra, A., Kosta, M., Torrent, D., Pernas-Salomón, R. & Shmuel, G. Homogenization of piezoelectric planar Willis materials undergoing antiplane shear. Wave Motion 108, 102833, DOI: https://doi.org/10.1016/j.wavemoti.2021.102833 (2022)

    work page doi:10.1016/j.wavemoti.2021.102833 2021

  40. [40]

    Muhafra, K., Haberman, M. R. & Shmuel, G. Discrete one-dimensional models for the electromomentum coupling. Phys. Rev. Appl. 20, 014042, DOI: 10.1103/PhysRevApplied.20.014042 (2023)

    work page doi:10.1103/physrevapplied.20.014042 2023

  41. [41]

    & Tol, S

    Danawe, H. & Tol, S. Electro-momentum coupling tailored in piezoelectric metamaterials with resonant shunts. APL Mater. 11, 091118, DOI: 10.1063/5.0165267 (2023). https://pubs.aip.org/aip/apm/article-pdf/doi/10.1063/5.0165267/ 18141427/091118_1_5.0165267.pdf

    work page doi:10.1063/5.0165267 2023

  42. [42]

    Huynh, H. D. et al. Maximizing electro-momentum coupling in generalized 2d Willis metamaterials. Extrem. Mech. Lett. 61, 101981, DOI: https://doi.org/10.1016/j.eml.2023.101981 (2023)

    work page doi:10.1016/j.eml.2023.101981 2023

  43. [43]

    Lee, J.-H., Zhang, Z. & Gu, G. X. Maximum electro-momentum coupling in piezoelectric metamaterial scatterers. J. Appl. Phys. 132, 125108 (2022)

    work page 2022

  44. [44]

    Lee, J.-H., Zhang, Z. & Gu, G. X. Reaching new levels of wave scattering via piezoelectric metamaterials and electro- momentum coupling. The J. Acoust. Soc. Am. 153, A163–A163, DOI: 10.1121/10.0018518 (2023)

    work page doi:10.1121/10.0018518 2023

  45. [45]

    P., Casali, M

    Wallen, S. P., Casali, M. A., Goldsberry, B. M. & Haberman, M. R. Polarizability of electromomentum coupled scatterers. In Proceedings of Meetings on Acoustics, vol. 46, DOI: 10.1121/2.0001597 (AIP Publishing, 2022). 15/31

    work page doi:10.1121/2.0001597 2022

  46. [46]

    & de Abajo, F

    Christensen, J. & de Abajo, F. J. G. Anisotropic metamaterials for full control of acoustic waves. Phys. Rev. Lett. 108, 124301, DOI: 10.1103/PhysRevLett.108.124301 (2012)

    work page doi:10.1103/physrevlett.108.124301 2012

  47. [47]

    Milton, G. W. & Willis, J. R. On modifications of newton’s second law and linear continuum elastodynamics.Proc. Royal Soc. A: Math. Phys. Eng. Sci. 463, 855–880 (2007)

    work page 2007

  48. [48]

    N., Jacob, Z., Narimanov, E., Kretzschmar, I

    Krishnamoorthy, H. N., Jacob, Z., Narimanov, E., Kretzschmar, I. & Menon, V . M. Topological transitions in metamaterials. Science 336, 205–209 (2012)

    work page 2012

  49. [49]

    & Zhang, X

    Li, J., Fok, L., Yin, X., Bartal, G. & Zhang, X. Experimental demonstration of an acoustic magnifying hyperlens. Nat. Mater. 8, 931–934 (2009)

    work page 2009

  50. [50]

    & Liu, Z

    Lu, D. & Liu, Z. Hyperlenses and metalenses for far-field super-resolution imaging. Nat. Commun. 3, 1205 (2012)

    work page 2012

  51. [51]

    H., Min Seung, H

    Oh, J. H., Min Seung, H. & Young Kim, Y . A truly hyperbolic elastic metamaterial lens.Appl. Phys. Lett. 104 (2014)

    work page 2014

  52. [52]

    & Huang, G

    Zhu, R., Chen, Y ., Wang, Y ., Hu, G. & Huang, G. A single-phase elastic hyperbolic metamaterial with anisotropic mass density. The J. Acoust. Soc. Am. 139, 3303–3310 (2016)

    work page 2016

  53. [53]

    H., Seung, H

    Lee, H., Oh, J. H., Seung, H. M., Cho, S. H. & Kim, Y . Y . Extreme stiffness hyperbolic elastic metamaterial for total transmission subwavelength imaging. Sci. Rep. 6, 24026 (2016)

    work page 2016

  54. [54]

    & Zhang, C

    Dong, H.-W., Zhao, S.-D., Wang, Y .-S. & Zhang, C. Broadband single-phase hyperbolic elastic metamaterials for super-resolution imaging. Sci. Rep. 8, 2247 (2018)

    work page 2018

  55. [55]

    Yves, S. et al. Moiré-driven topological transitions and extreme anisotropy in elastic metasurfaces. Adv. Sci. 9, 2200181 (2022)

    work page 2022

  56. [56]

    & Alu, A

    Gomez-Diaz, J. & Alu, A. Flatland optics with hyperbolic metasurfaces. ACS Photonics 3, 2211–2224 (2016)

    work page 2016

  57. [57]

    & Alù, A

    Quan, L. & Alù, A. Hyperbolic sound propagation over nonlocal acoustic metasurfaces. Phys. Rev. Lett. 123, 244303 (2019)

    work page 2019

  58. [58]

    & Lerosey, G

    Kaina, N., Lemoult, F., Fink, M. & Lerosey, G. Negative refractive index and acoustic superlens from multiple scattering in single negative metamaterials. Nature 525, 77–81 (2015)

    work page 2015

  59. [59]

    & Alù, A

    Quan, L. & Alù, A. Passive acoustic metasurface with unitary reflection based on nonlocality. Phys. Rev. Appl. 11, 054077 (2019)

    work page 2019

  60. [60]

    Hou, Z., Ding, H., Wang, N., Fang, X. & Li, Y . Acoustic vortices via nonlocal metagratings.Phys. Rev. Appl. 16, 014002 (2021)

    work page 2021

  61. [61]

    F., Jared, B

    Zhu, H., Patnaik, S., Walsh, T. F., Jared, B. H. & Semperlotti, F. Nonlocal elastic metasurfaces: Enabling broadband wave control via intentional nonlocality. Proc. Natl. Acad. Sci. 117, 26099–26108 (2020)

    work page 2020

  62. [62]

    Zhou, Z., Huang, S., Li, D., Zhu, J. & Li, Y . Broadband impedance modulation via non-local acoustic metamaterials. Natl. Sci. Rev. 9, nwab171 (2022)

    work page 2022

  63. [63]

    & Wang, Y .-S

    Zhou, H.-T., Fu, W.-X., Wang, Y .-F. & Wang, Y .-S. High-efficiency ultrathin nonlocal waterborne acoustic metasurface. Phys. Rev. Appl. 15, 044046 (2021)

    work page 2021

  64. [64]

    & Yang, T

    Yang, T., Lin, Z. & Yang, T. Experimental evidence of high-efficiency nonlocal waterborne acoustic metasurfaces.Adv. Eng. Mater. 25, 2200805 (2023)

    work page 2023

  65. [65]

    Nonlocal acoustic metasurface for ultrabroadband sound absorption

    Zhu, Y .et al. Nonlocal acoustic metasurface for ultrabroadband sound absorption. Phys. Rev. B 103, 064102 (2021)

    work page 2021

  66. [66]

    & Semperlotti, F

    Nair, S., Jokar, M. & Semperlotti, F. Nonlocal acoustic black hole metastructures: Achieving broadband and low frequency passive vibration attenuation. Mech. Syst. Signal Process. 169, 108716 (2022)

    work page 2022

  67. [67]

    Yuan, X., Li, Q., Wu, C., Huang, Y . & Wu, X. Broadband ventilated metamaterial absorber from non-local coupling. Extrem. Mech. Lett. 66, 102119 (2024)

    work page 2024

  68. [68]

    Kazemi, A. et al. Drawing dispersion curves: Band structure customization via nonlocal phononic crystals. Phys. Rev. Lett. 131, 176101 (2023)

    work page 2023

  69. [69]

    & Chaplain, G

    Moore, D., Sambles, J., Hibbins, A., Starkey, T. & Chaplain, G. Acoustic surface modes on metasurfaces with embedded next-nearest-neighbor coupling. Phys. Rev. B 107, 144110 (2023)

    work page 2023

  70. [70]

    & Starkey, T

    Chaplain, G., Hooper, I., Hibbins, A. & Starkey, T. Reconfigurable elastic metamaterials: Engineering dispersion with beyond nearest neighbors. Phys. Rev. Appl. 19, 044061 (2023)

    work page 2023

  71. [71]

    & Wegener, M

    Chen, Y ., Kadic, M. & Wegener, M. Roton-like acoustical dispersion relations in 3d metamaterials.Nat. Commun. 12, 3278 (2021). 16/31

    work page 2021

  72. [72]

    Iglesias Martínez, J. A. et al. Experimental observation of roton-like dispersion relations in metamaterials. Sci. Adv. 7, eabm2189 (2021)

    work page 2021

  73. [73]

    & Wegener, M

    Wang, K., Chen, Y ., Kadic, M., Wang, C. & Wegener, M. Nonlocal interaction engineering of 2d roton-like dispersion relations in acoustic and mechanical metamaterials. Commun. Mater. 3, 35 (2022)

    work page 2022

  74. [74]

    Zhu, Z. et al. Observation of multiple rotons and multidirectional roton-like dispersion relations in acoustic metamaterials. New J. Phys. 24, 123019 (2022)

    work page 2022

  75. [75]

    Phonon transmission through a nonlocal metamaterial slab

    Chen, Y .et al. Phonon transmission through a nonlocal metamaterial slab. Commun. Phys. 6, 75 (2023)

    work page 2023

  76. [76]

    Observation of chirality-induced roton-like dispersion in a 3d micropolar elastic metamaterial

    Chen, Y .et al. Observation of chirality-induced roton-like dispersion in a 3d micropolar elastic metamaterial. Adv. Funct. Mater. 2302699 (2023)

    work page 2023

  77. [77]

    & Fleury, R

    Bossart, A. & Fleury, R. Extreme spatial dispersion in nonlocally resonant elastic metamaterials. Phys. Rev. Lett. 130, 207201 (2023)

    work page 2023

  78. [78]

    Bliokh, K. Y . & Nori, F. Spin and orbital angular momenta of acoustic beams.Phys. Rev. B 99, 174310 (2019)

    work page 2019

  79. [79]

    & Chan, C

    Wang, S., Ma, G. & Chan, C. T. Topological transport of sound mediated by spin-redirection geometric phase. Sci. Adv. 4, eaaq1475 (2018)

    work page 2018

  80. [80]

    & Zhang, L

    Jiang, X., Li, Y ., Liang, B., Cheng, J.-c. & Zhang, L. Convert acoustic resonances to orbital angular momentum.Phys. Rev. Lett. 117, 034301 (2016)

    work page 2016

Showing first 80 references.