Small-Vibration Theory of the Clarinet
John Backus · The Journal of the Acoustical Society of America · 1963
A theory of the clarinet is developed based on the experimental observation that for weak tones the reed and air-column vibrations are nearly sinusoidal. The clarinet is assumed to be a cylindrical air column open at one end and closed at the other by a diaphragm containing a slit of variable width, corresponding to the aperture between reed and mouthpiece. A velocity potential appropriate to a tube with wall friction is assumed. The impedance of the slit as a function of opening and pressure across it is evaluated experimentally and checked against theory. The volume flow through the slit is calculated from the velocity potential and equated to the flow calculated from the slit impedance. The flow and impedance both depend on the pressure, which in turn is again calculated from the velocity potential. Expressions for the operating frequency and threshold blowing pressure are obtained by assuming the flow to consist of a small alternating component superimposed on a steady component. The frequency is found to be below the resonance of the system considered as a tube closed at the reed end, the shift varying nearly linearly with the slit opening and depending on the reed damping. The threshold blowing pressure is found to be proportional to the opening and to the reed stiffness. Operating frequencies and pressures for the artificially blown clarinet were measured experimentally and found to be in very good agreement with values calculated from the theory.