#### The Digital Waveguide Mesh for Reverberation

A special case of digital waveguide networks known as the*digital waveguide mesh*has also been proposed for use in artificial reverberation systems [396,518].

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 [518,399]. Much progress has been made on the problem of correcting this dispersion error in various mesh geometries (rectilinear, triangular, tetrahedral, etc.) [521,398,399]. See §C.14 for an introduction to the digital waveguide mesh and a few of its properties.

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Digital Waveguide Reverberators