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.

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In this paper, a physics-based model for a snare drum will be discussed, along with its finite difference simulation. The interactions between a mallet and the membrane and between the snares and the membrane will be described as perfectly elastic collisions. A novel numerical scheme for the implementation of collisions will be presented, which allows a complete energy analysis for the whole system. Viscothermal losses will be added to the equation for the 3D wave propagation. Results from simulations and sound examples will be presented.

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