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 §G.12) [488,491,50].

For a physical string model, at least three coupled waveguide models should be considered. Two corresponding to transverse-wave vibrations in the horizontal and vertical planes (two polarizations of vibration in a plane); the third corresponds to longitudinal waves. For bowed strings, torsional waves should also be considered, since they affect bow-string dynamics [289,397]. 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 [39,40]. 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 [520].