Ab initio calculations of the oscillations of a clarinet

Abstract

The internal oscillating functions volume velocity, pressure, and reed displacement have been calculated for a clarinet. The data used, all in the literature, are the mechanical properties of the reed, the geometrical measurements of the instrument, the properties of air, properties of tone holes [A. H. Benade, J. Acoust. Soc. Am. 41, 1609 (1967)], and data and calculations on the flow of air through a slit. From these, the input impedance Z(ω) is calculated, and is used to find an effective reflection function r(t). The latter is used in an algorithm [first used in calculating bowed string oscillations: M. E. McIntyre and J. Woodhouse, Acoustica 41 (in press)] that permits rapid numerical integration of the integral equations of motion of the system. The normal behavior of clarinets is observed in both chalumeau and clarion registers. Multiphonics are also calculated. Steady state behavior agrees qualitatively with the previous calculations [R. T. Schumacher, Acustica 40, 298 (1978)] to the extent that the latter are comparable with the present more elaborate model.

Principles of sound production in wind instruments

“…Schumacher developed a physical model of the clarinet where the reed is regarded as a spring-mass system with one degree of freedom, and a reflection function calculated from the shape of an actual instrument is used [3]. This model can generate synthesized sound very similar to the actual one throughout the entire pitch range.…”

Parameters Influencing the Noise Emission of Planing Machines

“…Particle velocity can be measured directly with a hot-wire anemometer by Particle Image Velocimetry (PIV) or Laser Doppler Velocimetry (LDV) or indirectly with a microphone pair [2][3][4][5][6][7][8][9][10][11][12]. In this study, the indirect method of sound intensity measurement is applied.…”

Recent progress in the acoustics of wind instruments.

“…Once the complexities of this generator have been understood, it is possible to model the whole system using a time-domain formalism developed by McIntyre, Schumacher and Woodhouse [6] and applied explicitly to reeddriven woodwind instruments by Schumacher [7]. This approach allows the calculation of both steady-state and transient behavior and thus, in principle, the complete behavior of the instrument.…”

“…The mouthpiece cup provides an auxiliary resonance that emphasizes the impedance peaks in the normal playing range of the instrument. The formalism of Schumacher [7], discussed above, can be used to model the interaction of the instrument air column with the pressure-driven lip generator and, as before, most current interest centers on the operation of this generator.…”

“…The reeds of woodwinds, on the other hand, are strongly damped by the players lips and normally operate at frequencies well below their resonance, giving a flat frequency response and a sounding frequency that is controlled predominantly by the pipe resonator. The numerical approach of Schumacher [7,10] then gives a relation between the flow U and the pressure p in the instrument mouthpiece of the form…”