Modelling acoustics on the Poincaré half-plane

[EN] Novel advances in the field of metamaterial research have permitted the engineering of devices with extraordinary characteristics. Here, we explore the possibilities in transformation acoustics to implement a model for the simulation of acoustic wave propagation on the Poincaré half-plane¿the simplest model possessing hyperbolic geometry and also of considerable historical interest. We start off from a variational principle on the given spacetime manifold to find the design description of the model in the laboratory. After examining some significant geometrical and physical properties of the Poincaré half-plane model, we derive a general formal solution for its acoustic wave propagation. A numerical example for the evolution of the acoustic potential on a rectangular region of the Poincaré half-plane concludes this discussion.M.M.T. wishes to thank the Spanish Ministerio de Economia y Competitividad and the European Regional Development Fund (ERDF) for financial support under grant TIN2014-59294-P.Tung, MM. (2018). Modelling acoustics on the Poincaré half-plane. Journal of Computational and Applied Mathematics. 337:366-372. https://doi.org/10.1016/j.cam.2017.10.037S36637233

Metamaterial acoustics on the (2+ 1)D Einstein cylinder

[EN] The Einstein cylinder is the first cosmological model for our universe in modern history. Its geometry not only describes a static universe-a universe being invariant under time reversal-but it is also the prototype for a maximally symmetric spacetime with constant positive curvature. As such, it is still of crucial importance in numerous areas of physics and engineering, offering a fruitful playground for simulations and new theories. Here, we focus on the implementation and simulation of acoustic wave propagation on the Einstein cylinder. Engineering such an extraordinary device is the territory of metamaterial science, and we will propose an appropriate tuning of the relevant acoustic parameters in such a way as to mimic the geometric properties of this spacetime in acoustic space. Moreover, for probing such a space, we derive the corresponding wave equation from a variational principle for the underlying curved spacetime manifold and examine some of its solutions. In particular, fully analytical results are obtained for concentric wave propagation. We present predictions for this case and thereby investigate the most significant features of this spacetime. Finally, we produce simulation results for a more sophisticated test model which can only be tackled numerically.This work has been supported by the Spanish Ministerio de Economia y Competitividad, the European Regional Development Fund (ERDF) under grant TIN2017-89314-P, and the Programa de Apoyo a la Investigacion y Desarrollo 2018 (PAID-06-18) of the Universitat Politecnica de Valencia under grant SP20180016.Tung, MM. (2021). Metamaterial acoustics on the (2+ 1)D Einstein cylinder. Mathematics. 9(17):1-11. https://doi.org/10.3390/math9172079S11191