### Wave Equation Applications

The ideal-string wave equation applies to any perfectly elastic medium
which is displaced along one dimension. For example, the air column
of a clarinet or organ pipe can be modeled using the one-dimensional
wave equation by substituting air-pressure deviation for string
displacement, and longitudinal volume velocity for transverse string
velocity. We refer to the general class of such media as
*one-dimensional waveguides.* Extensions to two and three
dimensions (and more, for the mathematically curious), are also
possible (see §C.14)
[518,520,55].

For a physical string model, at least three coupled waveguide models
should be considered. Two correspond to transverse-wave vibrations
in the horizontal and vertical planes (two
*polarizations* of
planar vibration); the third corresponds to *longitudinal
waves*. For bowed strings, *torsional waves* should also be
considered, since they affect bow-string dynamics
[308,421]. In the piano, for key ranges in
which the hammer strikes three strings simultaneously, *nine*
coupled waveguides are required per key for a complete simulation
(not including torsional waves); however, in a practical,
high-quality, virtual piano, one waveguide per coupled string
(modeling only the vertical, transverse plane) suffices quite well
[42,43]. It is difficult to get by with fewer
than the correct number of strings, however, because their detuning
determines the entire amplitude envelope as well as beating and
aftersound effects
[543].

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