Numerical study on role of foot of a flue organ pipe: relative phases in oscillations among pipe, foot and jet

Recent numerical studies showed that the foot of flue organ pipes closed at one end acts as a Helmholtz resonator and influences the acoustic oscillation in the pipe. Namely, the relative phase of oscillations between the pipe and foot changes depending on the resonance frequency of the foot. This fact can be explained by the theory of forced harmonic oscillators, in which the pipe works as a force driving the foot. However, in the damping-controlled regime, in which the resonance frequency of the foot, fH, is nearly equal to the frequency of acoustic oscillation, fa, the amplitude of the acoustic oscillation is smaller than those in the mass-controlled regime fH < fa and the stiffness-controlled regime fH > fa, even though the pressure oscillation in the foot lags behind that in the pipe by π/2. In this paper, to investigate this problem and to explore the role of the foot in detail, using a three-dimensional model of a closed flue organ pipe, we numerically investigate the relative phases among the pressure oscillation in the pipe, that in the foot and the jet velocity. We conclude that total balance among them is crucial for sustaining acoustic and fluid oscillations in the sound generation process. The oscillating jet drives the pipe with appropriate timing to sustain the acoustic oscillation for the stiffness-controlled and mass-controlled regimes. In contrast, the jet inefficiently drives the pipe with inappropriate timing for the damping-controlled regime.