Corrections for Woodwind Tone-Hole Calculations

The resonance frequencies of a woodwind instrument can be calculated by modelling it as a lattice of series and parallel tube pieces. For greater accuracy, corrections due to local changes in compliance and inertance by the presence of holes must be taken into account. The compliance change due to a closed hole is determined by its volume. The inertance change is dependent on both hole volume and shape. A quick estimate of the inertance is obtained when approximating the flow as stratified and summing the results of the parallel layers using values from accurately obtainable two-dimensional results. This solution has a systematic error. A correct solution is obtained by a finite difference method, but its accuracy is limited by the number of elements which can be handled by a computer. Results of both approaches are compared. For a closed cylindrical hole, the volume contributing to the (negative) inertance change is found to be approximately 28% of the hole radius times its area. The inner end correction for an open hole is found to vary between 82% and 16% of the hole radius when the ratio of hole and tube radii varies between zero and unity.