Acoustic streaming, dimensional analysis of nonlinearities, and tone hole mutual interactions in woodwinds

The transmission line description of a woodwind assumes that the linearized wave equation of acoustics is accurate, and that the woodwind tone holes are independently represented in the transmission line by shunt and series impedances. Additional interactions exist between the tone holes when the interhole separation becomes small relative to the main bore diameter. Internal hole interactions, expressed via the main bore evanescent modes, are most important in woodwinds with large diameter holes such as the saxophone and Boehm flute. Tone holes interact externally to a lesser degree when the radiated sound field from one hole modifies the near field of neighboring holes. Acoustic streaming and convective nonlinearities are components of the sound field in woodwinds under playing conditions. The presence of the nonlinear sound field is correlated with changes in timbre and the stability of oscillation, and is more pronounced for tone holes whose height is much smaller than the hole diameter. Dimensional analysis, using the Reynolds number associated with the steady streaming velocity and the Strouhal number associated with the strength of the convective nonlinearity along the streamlines near the tone hole edges, is used to estimate the importance of sound field nonlinearities in various woodwinds. Observation and dimensional analysis are in agreement that nonlinearities are most important for the low register tones of a woodwind. The tone hole geometries of woodwind families are tabulated and discussed.