Wave propagation in strongly curved musical instrument tubes

Nederveen (1969) calculated the wave impedance Z and phase velocity v in a curved pipe. These are altered relative to a straight of the same cross section by the amounts φ = +Δv/v and ψ = −ΔZ/Z. We have shown that Rostafinski's exact result (1972) reduces to Nederveen's in the long wavelength limit. We have performed experiments to separately measure φ and ψ along with the junction discontinuity impedance Zδ = jω(pc/πɑ²)δ between curved and straight pipes. Two curved pipes were used, each composed of five butted semitorus sections of tube diameter 2a = 17.5 mm, and length 2S = 39.9 mm with midline radius of curvature R = 12.7 mm (baritone crooks). One pipe was formed as a continuous $212$ turn helix, the other as a sinuous duct of alternating curvatures. Resonance frequencies were measured with these tubes connected in various configurations to segments of straight tubing of the same ID. Preliminary data shows φ ≃ 0.058, ψ ≃ 0.059, δ ≃ .029a. Our values are smaller than Nederveen's prediction, ψ = −φ = 0.089, whereas similar experiments by Brindley (1973) showed larger values than theory. Work continues to resolve the discrepancies.