Influence of a toroidal bend on wind instrument tuning

Abstract

In a toroidal bend in a cylindrical wind instrument, with the same cross section as the cylindrical part, the inertance for longitudinal waves is reduced. This induces a change in the resonance frequency. The magnitude of this change depends on the length and the sharpness of the bend and on its position in the sound field. A transition to a cylinder gives an additional inertance correction. The present study compares known results from literature and adds new facts, employing analytical and numerical methods. Possible causes of the discrepancies between measurements and theory as reported in the literature are considered. The theory was verified by measurements on some toroidal bends in between cylindrical tube pieces. Corrections were applied for diameter differences between the torus and the rest of the pipe system. Within experimental uncertainty, a satisfactory correspondence with theory was found. The results obtained in the present study determine which diameter reduction in the bend can compensate the effects of the bend on the tuning. a ratio of inner and outer radii of a bendϭ(1ϩb)/ (1Ϫb) b parameter indicating ͑local͒ bend sharpnessϭy/R 0 B bend sharpness for a toroidal bendϭr 0 /R 0 c sound speed, m/s d diameter, m f frequency, s Ϫ1 h mesh size of finite difference grid, m k wave number, m Ϫ1 K bulk modulus of air, Pa L tube length, m M inertance, kg/m 4 s p acoustic pressure, Pa Q resonator quality r radial coordinate in bore, m r 0 radius of bore, m R 0 radius of curvature of center line of a bend, m S cross-sectional area of bore, m 2 T temperature,°C u particle velocity, m/s V volume, m 3 x position coordinate, m y half-width of curved rectangular duct, m Z acoustic impedance, kg/m 4 s ␣ wall damping term ⑀ geometrically determined factor tangential coordinate in circular section angular wave number density of air, kg/m 3 Bapparent air density in bend, kg/m 3 ͑i͒ angle coordinate in bend, ͑ii͒ geometrically determined factor Φ total angle of bend length of torus measured along the center line, m phase angle angular frequencyϭ2 f , s Ϫ1 z/r 0

Helmholtz resonator with a spiral neck

“…c o m / l oc a t e / a p a c o u s t where p is the pressure, x is the angular frequency, q is the medium density, / is the curvature angle, and R is the distance from the point to the curvature center. Nederveen [7] assumed the radial dependence of the pressure is small at low frequency which meant the pressure is the same over the cross section; this is valid for the frequency range discussed in this paper. Therefore, the particle velocity is:…”

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

“…Several authors attempted to quantify the effect of the curvature on the tuning. Nederveen 2,3 and later Keefe and Benade 4 showed that in a bent duct, in the long wavelength limit, the inertance is lowered relative to its value in a straight duct having the same cross-section. The curvature induces an increase in resonance frequencies.…”

Effects of a curved vocal tract with grid-generated tongue profile on low-order formants

“…Morse and Ingard 8 along with Sundberg et al 9 assumed that acoustic propagation at low frequencies follows a wave mode characterized by the conformal grid characteristic of streamline flow: 10 the wavefronts are defined by curved lines of constant-pressure in two dimensions ͑2Ds͒, curved surfaces in three dimensions ͑3Ds͒, that intersect the vocal-tract walls at right angles. 8,9,11,12 The flow in an acoustic wave is a local back-and-forth disturbance, not a bulk streaming flow, but it is assumed for the stated conditions that the acoustic particle velocities are aligned with streamlines under steady potential flow. Pressure measurements of cast models of the vocal tract 13 along with a finite-element model 14 ͑FEM͒ confirm those assumptions for frequencies up to 4 kHz for the shapes studied, although another FEM study showed a departure from such a plane-wave propagation mode at 1.5 kHz in the case of the large front cavity for /r/.…”