Nonlinear regeneration in reed instruments with variable up- and down-stream air column impedances
Peter L. Hoekje, A. H. Benade · The Journal of the Acoustical Society of America · 1984
A feedback oscillator, consisting of an active device with transconductance A, and a damped resonator of impedance Z, is operated at constant amplitude by an external driver. The oscillation runs at an energy “profit” or “loss” if ZA is above or below breakeven (ZA = 1). The rate of energy gain or loss, as a function of ZA, is largest in the feedback regions just above or below the energy breakeven value. In the nonlinear regenerative oscillations of reed wind instruments, energy is transferred via heterodyne effects between several frequencies. In this case, Z is dominated by the sum of the input impedances of the air columns upstream and downstream of the reed. At each component frequency, the energy balance is describable in terms of the single frequency oscillator. If (ZA)n is below breakeven at the frequency of the nth spectral component, and the magnitude Of Zn is decreased, the energy balance at this frequency causes less of a drain on the overall oscillation; if (ZA)n above breakeven, and the magnitude of Zn is increased, there is less energy production at this frequency. Finally, the effect on the spectral component of the pressure signal on one side of the reed at the frequency of the nth partial, when the input impedance is varied on the opposite side of the reed, can only be large when (ZA)n is near breakeven. Experimental evidence will be presented. Musical implications will be summarized. [Work supported by NSF.]