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Wind instruments containing a resonator (i.e., pipe) with an open end are expected to exhibit an acoustic standing wave characterized by a density oscillation whose amplitude falls to zero a short distance beyond the end of the resonator. An extrapolation of this amplitude based on the behavior inside the resonator yields an “effective” node of the standing wave (i.e., a point at which the extrapolated amplitude vanishes), and the distance from the end of the resonator to the location of this effective node (which is commonly referred to as simply a “node”) is known as the “end correction.” Recent work using a novel optical technique involving optical speckle patterns surprisingly suggested instead that a node is located inside the resonator with unexpected structure in the standing wave amplitude just beyond the end of the resonator. We have studied this problem by numerically solving the Navier-Stokes equations and find that the effective node of the density oscillation is located at the expected position outside the resonator with no unexpected structure in the functional form of the standing wave. We also show how pressure gradients and the flow pattern found near the end of the resonator can account for the unexpected behavior observed in the experiments. This sensitivity of optical interference effects to flow structure may give a new experimental way to investigate vorticity and other complex flows found in the mouthpiece of a musical instrument and in other situations. 

Results are presented from a modeling study of the clarinet in which the air flow through the instrument is calculated using the Navier-Stokes equations. The reed is modeled as an Euler-Bernoulli beam with damping whose motion is driven by the pressure in the mouthpiece. Damping of the reed due to its contact with the lip is studied and shown to be crucial to achieve oscillations in which the reed vibrates at the lowest resonant frequency of the instrument, producing sound at that frequency. This finding is consistent with previous studies in which a clarinet is excited with an artificial blowing machine.