Non-linear behaviour of tone holes in musical instruments : an experimental study

Typical pressure amplitudes observed in woodwinds correspond to acoustic oscillation in open-tone-hole with large displacements involving flow separation at sharp edges. For high Strouhal numbers, fluid particle displacements small compared to the tone-hole diameter, one observes local vortex shedding near the edges. At low Strouhal numbers the formation of free jets alternatively leaving and entering the pipe is observed. The resulting nonlinear contribution to the resistance and reactance of the tone-hole impedance depends on the ratio of tone-hole diameter and the Stokes layer thickness (Shear number Sh). High Shear numbers Sh = O(10²) are typical for musical instruments. At high Shear numbers (Sh > 14) and intermediate Strouhal numbers (Sr = O(1)) one observes positive contribution to the end-correction (reactance) due to local vortex shedding near the edges. A negative contribution to the reactance end-correction is observed at low Strouhal numbers (Sr < 1). Undercutting of tone- holes is common practice when tuning musical instruments. This involves chamfering of the edges on the inner pipe side. It does significantly reduce the nonlinear resistance end correction at high Shear numbers. This effect is not sensitive to the size of the chamfer.