Woodwind Tone Hole Acoustics and the Spectrum Transformation Function.

This report describes an investigation of woodwind musical instrument tone holes and their effect on the radiated spectrum, the total dissipation, the stability of oscillation, the psychoacoustical cues important in perception, and the tuning and response of the instrument. Varying tone hole proportions significantly affect the radiative and frictional damping near a single hole, the mutual interactions between holes, the onset of streaming and turbulence near the holes, and the perceived woodwind timbre. The interconnections between related fields are explored through a brief review of sound production in woodwinds plus more extensive reviews of room and psychological acoustics. A theoretical and experimental discussion of the spectrum transformation function from the mouthpiece into the room relates all these fields. Also, considered are differences between cylindrical and conical bore woodwinds, the systematic shifts in saxophone spectra produced by the beating of the reed, the coupling of many closely spaced tone holes to the room excitation, the role of the player, and the results pertaining to computer music synthesis. The complicated acoustical flow inside the main air column near a single tone hole has been examined using a Green function, integral equation approach. A variational formulation allows explicit calculation of the open and closed hole impedance parameters needed in the transmission line description of a woodwind, and experiments have verified the theory in detail. Major acoustical topics considered are listed below. The effective length t(,e) of an open hole, relevant for instrument design and modification, is calculated and measured in terms of the main bore diameter 2a, hole diameter 2b, and the height t of the hole chimney; the effect of a hanging pad is a semi-empirical correction on t(,e). When the fundamental plane-wave mode of the main air column oscillation is at a pressure node, both the open and closed hole series impedances are negative inertances whose values depend on the tone hole proportions. An open hole at a pressure node can radiate as a dipole when (b/a) is large and (t/2b) is small. Dissipative losses vary significantly with the frequency of oscillation and the tone hole geometry. Lowering the pad height above a tone hole increases the dissipation. Acoustical streaming through holes is very important for t << 2b, and the associated nonlinear dissipation can destroy the oscillation on poorly designed woodwinds. This unexpected phenomenon is critical in the playing behavior of some flutes, clarinets, and other woodwinds. The onset of streaming occurs at all dynamical levels and more easily for instruments whose spectra are in a 1:3:5 frequency ratio, rather than a 1:2:3 ratio. The streaming is most important for low register tones for which the usual dissipation is also the largest relative to the radiative dissipation, due to losses at the sharp edges inside the bore near the tone holes. Mutual interactions between holes separated by a distance 2s are most pronounced for large diameter holes (2b/2s not small). Holes interact externally via radiation, and internally via higher-order evanescent modes excited at the intersection of the main bore with each tone hole. The non-radiative dissipation increases, and the air column resonances are slightly shifted due to the presence of these interactions. Applications are discussed and numerous additional experiments are proposed which are relevant to woodwinds and their design, and the perception of listeners in rooms.