Numerical methods for solving the wave equation in large enclosures with application to room acoustics

Many acoustical events in our everyday lives occur in enclosures, in which echoes and reverberation impact auditory perception in many ways. Numerical models play an important role in analyzing the acoustic behavior in reverberant environments, as they allow systematic control of physical parameters that affect human perception. Most numerical models used in architectural and room acoustics are based on the ray acoustics approximation, which leads to reasonable computational costs, at the expense of excluding certain important wave phenomena such as diffraction. In order to obtain more accurate results, an approach based on solving the wave equation is appropriate. However, standard numerical methods for solving the wave equation are not practical for room acoustics applications, since these problems are acoustically large and incur prohibitively large computational costs. Even in a small room, a sound wave with audible high-frequency content must propagate for about 10,000 wavelengths before it decays to an inaudible level. To address this, three new, efficient ways of simulating acoustic responses in a room are developed in this study. First, a method for calculating the resonant frequencies and normal modes in a rectangular room with arbitrary wall impedance is developed that uses the interval Newton/generalized bisection (IN/GB) method for solving the acoustic eigenvalue equation. The second approach applies the finite element method in the frequency domain, but uses a Dirichlet-to-Neumann (DtN) map to model empty, rectangular portions of the room, thereby truncating the effective computational domain. Finally, a finite difference method with minimal dispersion and dissipation errors is developed in the time domain. The parameters in the discretization in both space and time are optimized to minimize these errors. This method has been implemented on the IBM Blue Gene platform at Boston University, and allows for the calculation of the impulse response in a practical size of room (3 m x 3 m x 3 m) at relatively high frequencies (5 kHz) for the entire duration of the reverberation (1--3 seconds). Initial results indicate that this methodology has the ability to serve as a tool for conducting psychoacoustic experiments in reverberant spaces.