Cylindrical Air Column Acoustics

The resonator of a musical instrument serves to emphasize certain desirable frequencies of sound, corresponding to its normal modes. A wind instrument air column is a subclass of general resonators and functions to encourage the production of sustained, quasi-periodic oscillations of sound within a feedback system. In order that the air column be musically useful, it is necessary that it support sound production over a wide range of frequencies. This is accomplished by allowing the tube length to be variable while also encouraging sound production based on higher normal modes. Acoustically speaking, two principle requirements need be met for a wind instrument resonator to be musically useful. It is first necessary that the ratio between the first and successively higher normal modes be independent of horn length. In this way, the perceived timbre or spectral content of sounds will remain similar over the full range of the bore. Further, this permits the production of sounds based on second and third resonances, or overblowing, over the full length of the tube. Secondly, a stable regenerative process associated with the nonlinear excitation mechanism in wind instruments is favored when the successive partial frequencies, particularly the first few, are related to the fundamental frequency by integer multiple ratios (Benade, 1959,1977).

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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