No abstract

The sounding mechanism of a cylindrical pipe fitted with a clarinet mouthpiece (CPCM) is studied. The main aim of this paper is to introduce a reliable and minimal model which satisfactorily reproduces nonlinear vibrations excited in the air column and associated hysteretic transitions between them when the blowing pressure is varied as a control parameter. Such phenomena are commonly observed for woodwind reed instruments blown artificially, as reported by Idogawa et al. The noteworthy point of our model is that a reflection function consists of two main inverted peaks, one with a long delay, which represents the reflection from the open pipeend, and the other with a short delay, which originates from the irregular bore geometry of the mouthpiece. Our numerical calculation is carried out using the Schumacher model [Acustica 48 (1981), 71], including such a reflection function. Many kinds of nonlinear vibrations, periodic and quasi-periodic vibrations, as well as the hysteretic transitions between them are obtained numerically. Our results reproduce those observed experimentally quite well. They strongly suggest that the sound wave reflection due to the irregularity of the mouthpiece is one of the essential properties of woodwind sounding.

The sounding mechanism of a simple clarinetlike system has been investigated experimentally and numerically. The system consists of a cylindrical pipe fitted with the clarinet mouthpiece. By artificial blowing, it produced various periodic and quasiperiodic sounds, among which the lowest pitch was C♯3. In particular, hysteresis of the vibratory state transitions was observed when the flowing pressure was increased and decreased under the same lip adjustment. This was also observed for the artificially blown clarinet. The behavior of the simple system is analyzed numerically by using the Schumacher model [Acustica 48, 71–85 (1981)]. The numerical model is characterized as a double-delayed feedback system represented by a simple reflection function consisting of two separated narrow Gaussian peaks. One of the peaks is due to the main reflection from the open end of the cylindrical air column, and the other, which introduces a short delay, originates from the inside discontinuity of the mouthpiece. This numerical model reproduces well nonlinear vibrations of the clarinetlike system, as well as the hysteretic transition between vibratory states.