Numerically stable method for wave-field calculation and band-gap estimation in layered phononic crystals

High-frequency wave propagation in layered waveguides composed of periodic unit-cells is studied. Each unit-cell consists of a certain number of various elastic layers. Wave motion in the considered elastic structures (phononic crystals) is characterized by frequency bands, known as forbidden zones or band-gaps, where the full reflection of time-harmonic plane waves incident from an external half-space is observed. Using the transfer matrix method, the mathematical model of elastic wave propagation in anisotropic layered phononic crystals between two half-spaces is developed. A stable numerical algorithm for calculation of wave fields in the cells of phononic crystals is proposed. The amplitude coefficients of transmitted longitudinal and shear body waves are expressed in terms of transfer-matrix eigenvalues. A classification of band-gaps and pass-bands in layered anisotropic phononic crystals is proposed. It is based on the analysis of Bloch wavenumbers (expressed via the eigenvalues of the unit-cell transfer matrix) and the derived semi-analytical asymptotic expression of the transmission coefficients. Each Bloch wave has attenuation within the band-gaps of first kind. Waves propagating without attenuation are not excited within the band-gaps of second kind due to their specific polarization and boundary conditions at the interface between the multi-layered structure and the external half-space. By changing the parameters of the incident field, for instance, angles of incidence, band-gaps are converted into low transmission pass-bands where transmission coefficients are rather small. Therefore, a low transmission pass-band is similar to the traditional band-gap from an engineering point of view. The numerical results for energy transmission coefficient and Bloch waves are given at different angles of incidence; they demonstrate the formation of band-gaps in anisotropic phononic crystals.