Anisotropy of effective velocity for elastic wave propagation in two-dimensional phononic crystals at low frequencies

The propagation of elastic waves in two-dimensional phononic crystals is investigated theoretically in the long-wavelength limit. Exact analytical formulas for effective velocities of elastic waves are derived according to the method presented by Krokhin et al. [Phys. Rev. Lett. 91, 264302 (2003)]. We find that the effective velocities for the propagating modes polarized in-plane (the SV and L waves) are distinct anisotropic, while the anisotropy for the mode polarized out-plane (the SH wave) is very little. The influences of material parameters, filling fraction and lattice shape on the anisotropy are analyzed in details by numerical simulations. Phononic crystals with square lattice have much higher anisotropy than those with hexagonal lattice, and with large contrast between material parameters reveal high anisotropy. At low filling fraction, anisotropy increases as the filling fraction increases, while as the filling fraction closes to the limitation, anisotropy decreases.