In wind instrument bores, the primary mode of wave propagation is along the central axis of the tube. Equations describing this wave motion are possible if a coordinate system can be found in which one coordinate surface coincides with the walls of the given pipe and in which the wave equation is separable (Fletcher and Rossing, 1991, p. 187). There are 11 coordinate systems in which the Helmholtz equation is separable. One-parameter waves, however, are possible only in rectangular, circular cylindrical, and spherical coordinates, which correspond to pipes of uniform cross-section and conical horns, respectively (Putland, 1993). Wave propagation in the other separable coordinate systems must be comprised of an admixture of orthogonal modes and be a function of more than one coordinate. In order to perfectly meet the requirements of a musically useful wind instrument bore as discussed above, one-parameter wave propagation is necessary. Thus, cylindrical pipes and conical horns are the two obvious choices for wind instrument bores. All wind instrument air columns are based on shapes roughly corresponding to cylinders or cones. Further, accurate representations of wave propagation in actual, imperfectly shaped bores can be well approximated in terms of cylindrical and conic sections.
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