  We introduce 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.
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 in 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.
