The Reed as a Mass-Spring-Damper System

• While the normal playing range of clarinets and saxophones is typically well below the first resonance frequency of the reed, extended range techniques are becoming an increasingly common part of contemporary performance practice.

• In reality, the reed has some non-zero mass (the effective mass may vary with displacement and lip position) so that some phase delay will occur as the vibrating frequency of the reed increases. This behavior could have an important affect on the response of the instrument.

• The most common approach is to model the reed as a simple damped mechanical oscillator, as depicted in Fig. 15, with an equation of motion of the form
    $$m \frac{d^{2}x}{dt^{2}} + b \frac{dx}{dt} + k x(t) = A_{r} \, p_{\Delta}(t),$$ (29)
  where $m$ is the equivalent reed mass, $k$ is the reed spring constant, and $b$ is the damping factor.

  Figure 15: The single-reed as a mechanical oscillator blown closed.

  

• A substantial portion of the damping comes from the player's lower lip.

• Again, the reed motion is “displacement limited” by the mouthpiece facing.

• The natural frequency of the system in the absence of damping and for constant reed parameters is $\omega_{r} = \sqrt{k/m}$. Equation (29) is commonly expressed as
    $$\ddot{x}(t) + g_r \dot{x}(t) + \omega^{2}_{r} x(t) = \frac{p_{\Delta}(t)}{\mu_r},$$ (30)
  where $g_r$ is the reed damping coefficient and $\mu_r$ is the reed's dynamic mass per unit area.