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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Implicit finite difference schemes for the 3-D wave equation using a 27-point stencil on the cubic grid are presented, for use in room acoustics modelling and artificial reverberation. The system of equations that arises from the implicit formulation is solved using the Jacobi iterative method. Numerical dispersion is analysed and computational efficiency is compared to second-order accurate 27-point explicit schemes. Timing results from GPU implementations demonstrate that the proposed algorithms scale over their explicit counterparts as expected: by a factor of M + 2, where M is a fixed number of Jacobi iterations (eight can be sufficient in single precision). Thus, the accuracy of the approximation can be improved over explicit counterparts with only a linear increase in computational costs, rather than the quartic (in operations) and cubic (in memory) increases incurred when oversampling the grid. These implicit schemes are advantageous in situations where less than 1% dispersion error is desired.

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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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Time-domain finite difference (FD) and digital waveguide mesh (DWM) methods have seen extensive exploration as techniques for physical modelling sound synthesis and artificial reverberation. Various formulations of these methods have been unified under the FD framework, but many discrete boundary models important in room acoustics applications have not been. In this paper, the finite volume (FV) framework is used to unify various FD and DWM topologies, as well as associated boundary models. Additional geometric insights on existing stability conditions provide guidance into the FV meshing pre-processing step necessary for the acoustic modelling of irregular and realistic room geometries. DWM “1-D” boundary terminations are shown, through an equivalent FV formulation, to have a consistent multidimensional interpretation that is approximated to second-order accuracy, however the geometry and wall admittances being approximated may vary from what is desired. It is also shown that certain re-entrant corner configurations can lead to instabilities and an alternative stable update is provided for one problematic configuration.

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