Simulation and Calculation of Undercutting Effect of the Toneholes of Woodwind Instruments on the Shift of Eigenfrequencies of the Air Duct

The influence of the presence and value of the radius of curvature at the junctions of the toneholes of woodwind instruments with the main air duct (undercutting) on the shift of its eigenfrequencies is considered. A methodology and formulas for numerical calculation are presented, which allow one to determine the value of the effective radius for open and closed toneholes with a variable cross-section. Based on the dependences obtained, using the transmission-matrix method, a calculation of the eigenfrequencies of an air bore with a single tonehole was carried out and compared with the results of computer simulation in the COMSOL Multiphysics 5.6 program. It is shown that an increase in the degree of undercutting of the tonehole leads to an increase in its effective radius, which increases the resonance frequencies in the case of an open hole and decrease it in the case of a closed tonehole. Averaging the acoustic mass (for open hole) and volume (for closed hole) over the longitudinal sections of the hole, which doesn’t have rotational circular symmetry in the area of connection with the main bore, gives better results in finding eigenfrequencies in comparison with the computer simulation.