Finite Difference Schemes on Hexagonal Grids for Thin Linear Plates with Finite Volume Boundaries

The thin plate is a key structure in various musical instruments, including many percussion instruments and the soundboard of the piano, and also is the mechanism underlying electromechanical plate reverberation. As such, it is a suitable candidate for physical modelling approaches to audio effects and sound synthesis, such as finite difference methods—though great attention must be paid to the problem of numerical dispersion, in the interest of reducing perceptual artefacts. In this paper, we present two finite difference schemes on hexagonal grids for such a thin plate system. Numerical dispersion and computational costs are analysed and compared to the standard 13-point Cartesian scheme. An equivalent finite volume scheme can be related to the 13-point Cartesian scheme and a 19-point hexagonal scheme, allowing for fitted boundary conditions of the clamped type. Theoretical modes for a clamped circular plate are compared to simulations. It is shown that better agreement is obtained for the hexagonal scheme than the Cartesian scheme.