Acoustics in 2D Spaces of Constant Curvature

  • Michael M. Tung
  • José M. Gambi
  • María L. García del Pino
Conference paper
Part of the Mathematics in Industry book series (MATHINDUSTRY, volume 26)

Abstract

Maximally symmetric spaces play a vital role in modelling various physical phenomena. The simplest representative is the 2-sphere \({\mathbb S}^2\), having constant positive curvature. By embedding it into (2 + 1)D spacetime with Lorentzian signature it becomes the prototype of homogeneous and isotropic spacetime of constant curvature with constant scale factor: the Einstein cylinder \({\mathbb R}\times {\mathbb S}^2\). This work outlines a variational approach on how to model acoustic wave propagation on this particular curved spacetime. On the Einstein cylinder, the analytical solutions of the wave equation for the acoustic potential are shown to reduce to solutions of a differential equation of Sturm-Liouville type and simple harmonic time and angular dependence. Moreover, we discuss the implementation of such an underlying curved spacetime within an acoustic metamaterial—an artificially engineered material with remarkable properties exceeding the possibilities found in nature.

Notes

Acknowledgements

M. M. T. wishes to thank the Spanish Ministerio de Economía y Competitividad and the European Regional Development Fund (ERDF) for financial support under grant TIN2014-59294-P.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  • Michael M. Tung
    • 1
  • José M. Gambi
    • 2
  • María L. García del Pino
    • 2
  1. 1.Instituto de Matemática MultidisciplinarUniversitat Politècnica de València, Camino de Vera, s/nValenciaSpain
  2. 2.Gregorio Millán InstituteUniversidad Carlos III de MadridLeganés (Madrid)Spain

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