In an ideal concert hall, the reverberant response should be smooth, dense, and free of overly prominent resonances and reflections. While this is theoretically impossible at all frequencies in a typical concert-hall geometry, the reverberant response can be improved in a variety of ways. In particular, it is desirable that reflected sound waves in the hall scatter as uniformly as possible throughout the audience. That is, rather than having specular reflections, which are analogous to light reflecting from a mirror, we prefer diffuse reflections--more analogous to the scattered light which illuminates the daytime sky.
In the 1970s, Schroeder proposed methods of designing highly diffusing surfaces based on maximum-length sequences and quadratic residue sequences [SchroederSchroeder1975,SchroederSchroeder1979]. These so-called quadratic residue diffusers (QRD) have been widely applied to the design of recording studios and concert halls. In the 1960s, Schroeder also initiated the topic of artificial reverberation [Schroeder and LoganSchroeder and Logan1961,SchroederSchroeder1970], in which digital filter structures (particularly allpass filters) were used to simulate ``colorless'' reverberation. The use of allpass filters guaranteed an equal reverberant response at all frequencies, while a QRD guarantees an equal reflection strength at some number of frequencies and reflection angles.
In 1993, Van Duyne and Smith introduced an efficient way of modeling wave propagation in a membrane using a 2-D digital waveguide mesh, and showed that it coincided with a standard finite difference approximation scheme for the 2-D wave equation [Van Duyne and SmithVan Duyne and Smith1993b]. The mesh has also been applied to the problem of artificial reverberation [SmithSmith1985,Savioja, Backman, Järvinen, and TakalaSavioja et al.1995,Huang, Serafin, and SmithHuang et al.2000,Laird, Masri, and CanagarajahLaird et al.1999,Murphy and HowardMurphy and Howard2000,Murphy, Newton, and HowardMurphy et al.2001].
In this paper, the 2-D digital waveguide mesh is extended to include a diffusing boundary based on a Schroeder quadratic residue diffuser. First we review quadratic residue diffusers and the 2-D digital waveguide mesh, followed by implementation details and simulation results.