Simulation of low frequency acoustic waves in small rooms : An SBP-SAT approach to solving the time dependent acoustic wave equation in three dimensions

Abstract: Low frequency acoustic room behaviour can be approximated using numerical methods. Traditionally, music studio control rooms are built with complex geometries, making their eigenmodes difficult to predict mathematically. Hence, a summation-by-parts method with simultaneous-approximation-terms is derived to approximate the time dependent acoustic wave equation in three dimensions. The derived model is limited to rectangular prismatic rooms but planned to be expanded to handle complex geometries in the future. Semi-reflecting boundary conditions are used, corresponding to tabulated reflection and absorption properties of real. walls. Two speakers are modeled as omnidirectional point sources placed on a boundary, to mimic common studio setups. Through tests and examination of eigenvalues of the matrices in the method, conditions for stability and reflection coefficients are derived. Simulations of sound pressure distribution produced by the model correlate well to room mode theory, suggesting the model to be accurate in the application of predicting low frequency acoustic room behaviour. However, the convergence rate of the model turns out to be lower than expected when point sources are introduced. Future steps towards applying the model to real music studio control rooms include modeling the walls as changes in density and wave speed rather than boundaries of the domain. This would potentially allow more complex geometries to be modeled within a larger, rectangular domain.