Effective phononic crystals for non-Cartesian elastic wave propagation

Phononic crystals show novel characteristics when it comes to acoustic and elastic wave propagation control. Nevertheless, most studies on phononic crystals are based on a plane wave assumption because this allows for application of Bloch theorem and analysis of the infinite system based on a single unit cell. However, the plane wave assumption is not valid in the near field of a source, where the wave front takes cylindrical or spherical form. Here, we overcome this limitation by introducing the concept of effective phononic crystals, which combine periodicity with varying isotropic material properties to force periodic coefficients in the elastic equations of motion in a non-Cartesian basis. The periodic coefficients allow for band structure calculation using Bloch theorem. Using the band structure, we demonstrate band gaps and topologically protected interface modes can be obtained for cylindrically propagating waves. Through effective phononic crystals, we show how behaviors of Cartesian phononic crystals can be realized in regions close to sources, where near-field effects are non-negligible.