Small‐Vibration Theory of the Clarinet

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 foun...