Much of the discussion thus far has been based on rather ideal behaviors and/or system properties. Despite that, these analyses provide good ``first-order'' descriptions for the various components of the reed mechanism and together, capture the fundamental response of the overall system. Below, we discuss a number of incremental refinements that have been proposed to these equations.
- The motion of the reed is expected to induce a flow contribution of its own proportional to
- The flow that enters the reed channel from the upstream side is therefore considered to divide into a component ud entering the downstream air column and the volume flow swept out by the reed,
- This extra flow component has been shown to have some significant influence on playing frequencies as predicted from analytical and numerical simulations (Coyle et al., 2015).
- Stewart and Strong (1980) and Sommerfeldt and Strong (1988) modeled the reed as a damped, driven, nonuniform bar using a fourth-order differential equation. In this way, changes in effective mass and stiffness with bending along the curvature of the lay are automatically incorporated.
- Avanzini and van Walstijn (2004) also modeled the reed as a clamped-free bar using finite difference techniques and derived appropriate parameters from measurements and tuning. These results were then used to develop a non-linear lumped model (van Walstijn and Avanzini, 2007).
- More recent models of reed stiffness include an extra force term which only becomes active when the reed displacement goes beyond a certain distance, thus effectively producing a non-linear spring effect (Chatziioannou and van Walstijn, 2012).
- An additional force on the reed caused by higher localized volume flow within the reed channel was proposed by Worman (1971) and had been included in a few later reed analyses (Benade, 1976; Keefe, 1990b; Schumacher, 1981). For such a force to exist, the reed channel height must be non-uniform or the flow must separate at the channel entrance and then subsequently reattach at a further point in the channel. Hirschberg et al. (1990) point out that there are incompatibilities in the derivation of the ``Bernoulli'' force that make its application to the clarinet reed mechanism questionable.
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