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The wave-based approaches naturally capture all wave phenomena in enclosures, including interference and diffraction, and they allow for a detailed modeling of the interior boundary. Solution of the acoustic-wave equation represents the most strict modeling, a second- order partial differential equation that characterizes the wave propagation or its frequency-domain version, the Helmholtz equation. Solving these equations is challenging, particularly for complex geometries.
Practical wave-based solvers, therefore, apply some discrete grids of the space and/or time. These methods include the finite-difference/finite-volume time-domain (FDTD/FVTD), the time-domain spectral element (TD-SEM), the finite-element (FEM), and the boundary-element (BEM) methods. Botteldooren (1994) first applied the FDTD in room- acoustic simulations, and there has been significant progress in the ensuing years, such as on how to model complex geometries (Bilbao, 2013). Pind et al. (2019) investigated an attractive TD-SEM approach with geometrical flexibility because it is also able to incorporate a complex-valued frequency-dependent boundary. The recent special issue of The Journal of the Acoustical Society of America (Savioja and Xiang, 2019) reported the latest progress in incorporating source directivities in FDTD simulations, which involves the perceptual study of inherent dispersion errors specific in FDTD modeling. The special issue also includes an approach using the discontinuous Galerkin method, further development of the FEM incorporating source, and receiver directivity for auralization.
These techniques typically discretize the space into a grid, and the density of this grid determines the bandwidth that can be simulated. After that, the various techniques compute the solution iteratively for the whole grid. For auralization purposes, it is beneficial to store all these results, and while moving, use some interpolation between responses from different grid points such that the most correct response for any receiver location is obtained. Note that the FDTD and FEM use a volumetric grid, whereas the BEM utilizes a surface grid such that responses are stored at surfaces and can then be gathered and integrated to a given location. This is a straightforward operation for monophonic responses, but for spatial reproduction, the situation is more complex because it requires storing some spatial information instead of monophonic responses.
Geometrical-Acoustic Room Simulations
Since the 1960s, room-acoustic modeling, in principle, has employed geometrical acoustics. The geometrical acoustics basically assumes that sound waves propagate along straight lines like light rays, and all the wave phenomena, such as diffraction and interference, are neglected (Savioja and Svensson, 2015). Computational simulations based on the geometrical acoustics are highly efficient but less accurate when compared with the wave- based models. These geometrical-acoustic methods have been in room acoustic practice and research for over 50 years since Krokstad et al. (1968) published their seminal work on acoustic ray tracing. Another key method in the geometrical-acoustic regimen is the image-source method. Although it was applied in room acoustics as early as Eyring (1930), the widespread adaptation of the image- source method is attributable to Allen and Berkley (1979).
Geometrical room-acoustic simulations did not include an auralization capability until Pösselt (1987) incorporated the head-related impulse responses into image source-based room simulations, the end results of which were binaural audible samples that could be rendered using a set of headphones (see also Blauert and Pösselt, 1988). Pösselt’s (1987) pioneer modeling effort, although in a rectangular room, opened up opportunities for computer simulations (first based on a geometrical- acoustic principle) to create a computer model of the sonic environment for a listener as if (s)he were sitting in the simulated space, listening the sound field using her/ his own ears. Blauert et al. (1990) provided a brief review on binaural room simulation. Binaural room simulation itself includes binaural rendition for auditory perception through acoustic room simulations. A stream of research activities on both fundamental and application levels of auralization followed, leading to a boom in room acoustic modeling and auralization research as illustrated in a special issue of Applied Acoustics (Naylor, 1993).
In Image-Source Methods, we present the fundamental concepts underlying key geometrical acoustics methods. The techniques presented heavily rely on an overview by
Savioja and Svensson (2015).
Image-Source Methods
The image-source method recursively constructs the image sources to the sound source in the room. A sound source is image reflected against all interior surfaces,
Winter 2020 • Acoustics Today 51
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