### A View of Single-Reed Oscillation

To start the oscillation, the player applies a pressure at the mouthpiece which ``biases'' the reed in a ``negative-resistance'' region. (The pressure drop across the reed tends to close the air gap at the tip of the reed so that an increase in pressure will result in a net decrease in volume velocity--this is negative resistance.) The high-pressure front travels down the bore at the speed of sound until it encounters an open air hole or the bell. To a first approximation, the high-pressure wave reflects with a sign inversion and travels back up the bore. (In reality a lowpass filtering accompanies the reflection, and the complementary highpass filter shapes the spectrum that emanates away from the bore.)As the negated pressure wave travels back up the bore, it cancels the elevated pressure that was established by the passage of the first wave. When the negated pressure front gets back to the mouthpiece, it is reflected again, this time with no sign inversion (because the mouthpiece looks like a closed end to a first approximation). Therefore, as the wave travels back down to the bore, a

*negative*pressure zone is left behind. Reflecting from the open end again with a sign inversion brings a return-to-zero wave traveling back to the mouthpiece. Finally the positive traveling wave reaches the mouthpiece and starts the second ``period'' of oscillation, after four trips along the bore. So far, we have produced oscillation without making any use of the negative-resistance of the reed aperture. This is merely the start-up transient. Since in reality there are places of pressure loss in the bore, some mechanism is needed to feed energy back into the bore and prevent the oscillation just described from decaying exponentially to zero. This is the function of the reed: When a traveling pressure-drop reflects from the mouthpiece, making pressure at the mouthpiece switch from high to low, the reed changes from open to closed (to first order). The closing of the reed increases the reflection coefficient ``seen'' by the impinging traveling wave, and so as the pressure falls, it is amplified by an increasing gain (whose maximum is unity when the reed shuts completely). This process sharpens the falling edge of the pressure drop. But this is not all. The closing of the reed also cuts back on the steady incoming flow from the mouth. This causes the pressure to drop even more, potentially providing effective

*amplification*by more than unity. An analogous story can be followed through for a rising pressure appearing at the mouthpiece. However, in the rising pressure case, the reflection coefficient falls as the pressure rises, resulting in a progressive attenuation of the reflected wave; however, the increased pressure let in from the mouth amplifies the reflecting wave. It turns out that the reflection of a positive wave is boosted when the incoming wave is below a certain level and it is attenuated above that level. When the oscillation reaches a very high amplitude, it is limited on the negative side by the shutting of the reed, which sets a maximum reflective amplification for the negative excursions, and it is limited on the positive side by the attenuation described above.

*Unlike*classical negative-resistance oscillators, in which the negative-resistance device is terminated by a simple resistance instead of a lossy transmission line, a

*dynamic equilibrium*is established between the amplification of the negative excursion and the dissipation of the positive excursion. In the first-order case, where the reflection-coefficient varies linearly with pressure drop, it is easy to obtain an exact quantitative description of the entire process. In this case it can be shown, for example, that amplification occurs only on the positive half of the cycle, and the amplitude of oscillation is typically close to half the incoming mouth pressure (when losses in the bore are small). The threshold blowing pressure (which is relatively high in this simplified case) can also be computed in closed form.

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Single-Reed Instruments