Effects of bending portions of the air column on the acoustical resonances of a wind instrument

The need to keep long wind musical instruments compact imposes the bending of portions of the air column. Although manufacturers and players mention its effects as being significant, the curvature is generally not included in physical models and only a few studies, in only simplified cases, attempted to evaluate its influence. The aim of the study is to quantify the influence of the curvature both theoretically and experimentally. A multimodal formulation of the wave propagation in bent ducts is used to calculate the resonances frequencies and input impedance of a duct segment with a bent portion. From these quantities an effective length is defined. Its dependence on frequency is such that, compared to an equivalent straight tube, the shift in resonance frequencies in a tube with bent sections is not always positive, as generally stated. The curvature does not always increase the resonances frequencies, but may decrease them, resulting in a complex inharmonicity. An experimental measurement of the effect of the curvature is also shown, with good agreement with theoretical predictions.