We investigate theoretically the band structure of a phononic crystal of finite thickness constituted of a periodical array of cylindrical dots deposited on a thin plate of a homogeneous material. We show that this structure can display a low-frequency gap, as compared to the acoustic wavelengths in the constituent materials, similarly to the case of locally resonant structures. The opening of this gap requires an appropriate choice of the geometrical parameters, and in particular the thickness of the homogeneous plate and the height of the dots. However, the gap persists for various combinations of the materials constituting the plate and the dots. Besides, the band structure can exhibit one or more higher gaps whose number increases with the height of the cylinders. We discuss the condition to realize waveguiding through a linear defect inside the phononic crystal dots. The numerical simulations are performed by using the finite difference time domain and the finite element methods.

We introduce a supercell plane wave expansion ͑SC-PWE͒ method for the calculation of elastic band structures of two-dimensional phononic crystal plates. We compute the band structure of solid-solid and air-solid two-dimensional phononic crystal plates. The air is modeled as a low impedance medium ͑LIM͒ with very low density and very high velocities of sound. We investigate the influence of the constituent materials, of the plate thickness, and of the geometry of the array on the band structure. We establish the range of validity of the SC-PWE method in terms of the rate of convergence with respect to the number of plane waves and contrast in physical properties of the matrix and inclusion materials. We show that for high contrast solid-solid phononic crystal plates, our SC-PWE method, as other PWE-based methods introduced to date, suffers from convergence difficulties. In the case of air ͑modeled as the LIM͒ holes-solid plates, we demonstrate that the SC-PWE method leads to fast convergence for a wide range of values of solid physical properties. With these constituent materials, we find that the largest absolute forbidden bands occur in the band structure of the phononic crystal plate provided the thickness of the plate is of the order of magnitude of the periodicity of the array of inclusions. We demonstrate the existence of guided modes in an air-silicon phononic crystal plate containing a linear defect.

The feasibility of tuning the band structure of phononic crystals is demonstrated by employing magnetostrictive materials and applying an external magnetic field. Band structures are calculated with a plane wave expansion method that accounts for coupling between the elastic behavior and the magnetic field through the development of elastic, piezomagnetic, and magnetic permeability effective tensors. We show the contactless tunability of the absolute band gaps of a two-dimensional phononic crystal composed of an epoxy matrix and Terfenol-D inclusions. The tunable phononic crystal behaves like a transmission switch for elastic waves when the magnitude of an applied magnetic field crosses a threshold.