The elastic wave propagation in two-dimensional phononic crystal at low frequencies and the anisotropy of effective velocity

The propagation of elastic waves in two-dimensional phononic crystal is investigated theoretically in the long-wavelength limit by the MST method. Exact analytical formulas for effective velocities of elastic waves are derived according to the behavior of the Mie scattering solution and the lattice sum in the long-wavelength limit. It has been found that the coupling effect between the two second-order components of the cylindrical scattered wave changes with the propagation orientation. Therefore whether the effective velocities of the propagating modes are anisotropic depends on whether the second-order components of the Mie scattering are indispensable.