#### Yusuke Naka^{1}, Assad A. Oberai^{2}, and Barbara G. Shinn-Cunningham^{3}

*1 Department of Aerospace and Mechanical Engineering, Boston University
2 Department of Mechanical, Aerospace, and Nuclear Engineering, Rensselaer Polytechnic Institute
3 Boston University Hearing Research Center*

#### Introduction

Many acoustic events in our everyday lives occur in reverberant environments, from small offices to large concert halls. In such an enclosure, echoes and reverberation impact auditory perception in many ways, distorting auditory spatial cues, rendering speech less intelligible, and providing cues for source distance and room characteristics. In order to investigate the influence of room acoustics on perception, computer simulations of sound that a listener would hear in the room is invaluable. By using the binaural room impulse responses (BRIRs), the impulse responses from a source position to the listener’s left and right ears, the effects of reverberation can be added to any anechoic sound source.

A BRIR can be obtained by solving the wave equation in a room using a numerical method. For the accuracy of the numerical method, the dispersion (phase) and dissipation (magnitude) errors resulting from spatial and temporal discretization are more important than the formal order of accuracy in a Taylor series expansion of the differential operators in space and time. Efforts have been made in various application fields, such as in computational aero-acoustics (CAA), to reduce the dispersion and dissipation errors by optimizing the discretization parameters in space and time [1, 2]. However, since the dispersion and dissipation errors accumulate over time, more accurate numerical methods are required in room acoustics, where reverberation makes it necessary to calculate BRIRs over long durations.

#### Optimal Space-Time (OST) Finite Difference Scheme

In this study, a new approach for constructing low dispersion- and dissipation-error finite difference schemes for the scalar wave equation in the time domain is developed. The numerical parameters (spatial and temporal finite difference coefficients) are determined by minimizing the total error (space and time), unlike the previous approaches, in which the discretization schemes are optimized only in either space or time, and spatial and temporal discretizations are not coupled. The resulting scheme is referred to as the optimal space-time (OST) finite difference scheme. Using the OST approach, finite difference stencils and time integration schemes, with remarkable accuracy, are developed in both two and three dimensions. For example, in three dimensions, a 25-point stencil with a 10-stage time-integration scheme is developed that incurs an error of less than 2 percent in propagating a high wavenumber plane wave (6 points per wavelength) at moderate CFL (unity) through a distance of 1,000 wavelengths.

#### Results

The OST scheme is applied to the wave equation in a 3m x 3m x 3m cubic room with a simple model of a listener. The acoustic pressure at all finite difference grid points are calculated at each time step, and the wave front (isosurface) is visualized from these results. The visualized images show the interesting wave phenomena such as diffraction around the listener and reflection from the walls. The BRIRs (the acoustic pressure at listener’s left and right ears as a function of time) are also obtained. These BRIRs are converted to audio files to be used for psychoacoustic experiments or for virtual audio rendering, for example. Simulations were executed on 512 nodes of Boston University’s IBM Blue Gene. The visualization was accomplished using RSI IDL, Alias Maya, SGI Performer and custom code.

#### Visualization Movie

This movie was originally shown in stereo on our traveling high-resolution stereo display at the Supercomputing 2006 conference at a resolution of 2048 x 1536. For the web version here, it has been downsampled to 720 x 540 mono and is available in the Microsoft AVI and Apple Quicktime video formats.

The visualization process is described in the Results section above and plays for 20 seconds. (23MB AVI, 177MB QuickTime)

#### References

[1] C.K.W. Tam and J.C. Webb, “Dispersion-relation-preserving finite difference schemes for computational acoustics,” Journal of Computational Physics, 107 (2), 262-281, 1993.

[2] F.Q. Fu, M.Y. Hussaini, and J.L. Manthey, “Low-dissipation and low-dispersion Runge-Kutta schemes for computational acoustics,” Journal of Computational Physics, 124 (1), 177-191, 1996.