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.