A model of a single reed instrument is studied in which the reed is described as an Euler-Bernoulli beam, and the air flow through the instrument is calculated using the Navier-Stokes equations. The hypothetical instrument resembles a clarinet, but is smaller than a real clarinet to keep the numerical modeling feasible on available supercomputers. This article explores the conditions under which the frequency of the reed oscillations and the emitted sound are determined by the resonant frequency of the bore of the instrument. The effect of the contact between the reed and the player's lips is also studied, and quantitative results for the air density and pressure in the mouthpiece and throughout the instrument bore are also presented.
Direct numerical solutions of the Navier-Stokes (NS) equations have been used to study the air flow in a recorder. When the recorder is driven with a DC flow into the windway, we observe the familiar oscillating flow out of the window adjacent to the labium. The associated flow at the open end is also studied. For a pipe with a square cross-section, we find the flow at the open end to be small but nonzero, consistent with recent studies using speckle pattern interferometry by Coyle and Moore. In addition, we have analyzed the NS results for the air density so as to compare directly with the interferometry experiments. We also describe simulations for circular pipes. [Work supported by NSF Grant PHY1513273.]
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