### 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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