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.