A special case of digital waveguide networks known as the
*digital waveguide mesh*
has also been proposed for use in artificial reverberation systems
[399,522].

As discussed in §2.4, a digital waveguide (bidirectional delay
line) can be considered a computational acoustic model for traveling
waves in opposite directions. A *mesh* of such waveguides in 2D
or 3D can simulate waves traveling in *any* direction in the
space. As an analogy, consider a tennis racket in which a rectilinear
mesh of strings forms a pseudo-membrane.

A major advantage of the waveguide mesh for reverberation applications is that wavefronts are explicitly simulated in all directions, as in real reverberant spaces. Therefore, a true diffuse field can be developed in the late reverberation. Also, the echo density grows with time and the mode density grows with frequency in a natural manner for the 2D and 3D mesh. Finally, the low-frequency modes of the reverberant space can be simulated very precisely (for better or worse).

The computational cost of a waveguide mesh is made tractable relative
to more conventional finite-difference simulations by (1) the use of
multiply-free scattering junctions and (2) very coarse meshes. Use of
a coarse mesh means that the ``physical modeling'' aspects of the mesh
are only valid at low frequencies. As practical matter, this works
out well because the ear cannot hear mode tuning errors at high
frequencies. There is no error in the mode dampings in a lossless
reverberator prototype, because the waveguide mesh is lossless by
construction. Therefore, the only errors relative to an ideal
simulation of a lossless membrane or space are (1) mode tuning error,
and (2) finite band width (cut off at half the sampling rate). The
tuning error can be understood as due to *dispersion* of the
traveling waves in certain directions
[522,402]. Much progress has been
made on the problem of correcting this dispersion error in various
mesh geometries (rectilinear, triangular, tetrahedral, etc.)
[525,401,402].

See §C.14 for an introduction to the digital waveguide mesh and a few of its properties.

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University