Intercomponent Energy Exchange and Upstream/downstream Symmetry in Nonlinear SelfSustained Oscillations of Reed Instruments
Abstract
We discuss the nonlinear oscillations of reed woodwind and brass instruments and the roles played by the input impendances Z(,in) of the air columns upstream (Z(,u)) and downstream (Z(,d)) of the reed. Conventional linear theory shows that oscillation is favored when the feedback ZA is greater than one, where Z((omega)) is the parallel combination of the motional impedance of the reed and the sum (Z(,u) + Z(,d)), and A((omega)) is the small signal transconductance of the reed flow control valve. Thus, the system plays near the frequencies of the peaks of Z. By an impulse technique, Z(,in) = Z(,d) is measured for several brass and single reed wind instruments and shown to have peaks whose heights lie between 100 and 1000 acoustic ohms. Calculations and similar measurements for Z(,in) = Z(,u) of the player's windway, which reaches from the lips to the lungs, show a mode adjustable by the player to have a frequency anywhere in the range 500 to 1100 Hz, and a height anywhere in the approximate range of 75 to 300 ohms. Under the control of the player, Z(,u) thus has effects which may be inconsiderable, subtle, or dominant. At larger amplitudes, the nonlinearity of the flow control characteristic of the reed spreads energy among all of the frequency components of the oscillation. Each component is modeled as a linear feedback oscillator, driven by a current u('(e)) due to the nonlinear mixing of all the components of the oscillation. The flow of power from u('(e)) into each component is positive for small ZA and negative for large ZA, crossing zero near ZA = 1. An increment in Z(,u) or Z(,d) will cause a proportionate change in the corresponding pressure component on the same side so long as ZA < 1. The far side pressure will change significantly only near ZA = 1. The pressure and the power flow limit to constant values as ZA gets large. The nonlinearity also causes harmonic spectra, tends to remove details from the flow spectrum envelope, and promotes stability of the oscillation.
 Publication:

Ph.D. Thesis
 Pub Date:
 1986
 Bibcode:
 1986PhDT........87H
 Keywords:

 WOODWIND;
 BRASS;
 Physics: Acoustics