Acoustics in 2D Spaces of Constant Curvature

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.