Vibrations of the Reed

• Some of the earliest research on musical instrument reeds was conducted by Helmholtz, who concluded that pipes excited by inwardly striking reeds “of light material which offers but little resistance” produce tones at frequencies corresponding to the resonant frequencies of the pipe, which are much lower than the natural frequency of the reed itself (Helmholtz, 1954, p. 390).

• The lowest resonance frequency of a typical clarinet reed falls approximately in the range 2-3 kHz, while normal playing frequencies for clarinets are below 1 kHz.

• A mass-spring system driven at a frequency well below resonance is said to be stiffness dominated and its displacement amplitude will approach $f/k$, where $k$ is the spring constant and $f$ is the applied force.

• Thus, a common simplification for woodwind instruments has been to neglect the effect of the mass altogether, which is equivalent to assuming an infinite reed resonance frequency, and to model the reed system as a memory-less system as depicted in Fig. 8 (McIntyre et al., 1983; Backus, 1963; Nederveen, 1969).

  Figure 8: The single-reed as a mechanical spring blown closed.

  

• Observations by Backus (1961) of reed motion and mouthpiece pressure in an artificially blown clarinet appear to confirm that the reed is primarily stiffness controlled.

• Assuming the force on the reed is equal to $A_{r} \cdot p_{\Delta}$, where $A_{r}$ is the effective surface area of the reed exposed to $p_{\Delta}$, the reed tip opening with respect to its equilibrium position $H$ is given by Hooke's law as
    $$x = H - \frac{A_{r} \cdot p_{\Delta}}{k}.$$ (19)

• In this case, the motion of the reed is exactly “in-phase” with the pressure ($p_{\Delta}$) acting on it.