Flute acoustics: measurement, modelling and design - School of ...

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Table 3.2: Several techniques for calibration of impedance heads, with selected references and notes. Calibration Loads Refs. Notes (a) TMTC technique L L (b) resonance analysis of long tube 2L L = 1 - 2 m (c) semi-infinite pipe (d) resonance-free loads ∞ ∞ Gibiat & Laloë (1990), van Walstijn et al. (2005) Bruneau (1987), Dalmont (2001b) Wolfe et al. (2001a) • ‘complete’ calibration with three known loads • several sets of calibration loads needed to cover wide f -range • complex wavenumber need not be known if a fourth ‘negative length’ load is used • after initial measurement of Z for long tube, calibration parameters determined from oscillations in α and k • complex wavenumber need not be known if an extra short tube is used • data obtained only at resonances and antiresonances of long tube—low frequency limit determined by length of tube • almost purely resistive load—impedance insensitive to complex wavenumber • used for calibration of volume flow sources • complete calibration as in (a) • valid for all frequencies, due to lack of resonances • the flange calibration may be omitted if a model of the impedance head is available CHAPTER 3. MEASURING ACOUSTIC IMPEDANCE 36
CHAPTER 3. MEASURING ACOUSTIC IMPEDANCE 37 resonant pipes explicitly depends on a theory for wall losses. Further, the temperature and humidity must be accurately determined, and the test pipes must be very accurately machined. If an extra calibration is available, then the complex wavenumber need not be known; this is the approach of van Walstijn et al. (2005), who employ the three calibrations of Gibiat & Laloë plus a ‘negative length’ tube, realised by defining the reference plane some distance from the first microphone. Dalmont (2001b) presents a calibration technique for impedance heads with a volume flow source, which may also be extended to the two-microphone case (Dalmont 2001a) (calibration loads (b) in Table 3.2). The method is based on resonance analysis of a long closed tube. The impedance of the tube is measured using the uncalibrated head and the attenuation coefficient and wavenumber are derived from the measurement. When plotted against frequency these will show periodic oscillations with amplitude proportional to any errors in calibration. The three calibration constants are estimated from these oscillations, and the procedure is repeated until the oscillations are tolerably small. The main advantage of this procedure is that it does not depend on exact knowledge of the complex wavenumber. The main disadvantage is that it only yields calibration data at each resonance frequency of the calibration tube. These may be interpolated, but the lowest frequency that can be measured is limited by the length of the calibration tube (and is around 80 Hz for a 2 m tube—and higher for shorter tubes). 3.4.2 General calibration technique using up to three resonant-free loads Here, we calibrate a three-microphone impedance head in a method similar to that used by Gibiat & Laloë. However, to obviate the need to know the complex wavenumber precisely, we use three loads without any resonances: a quasi-infinite impedance; an almost purely resistive impedance; and a flange (calibration laods (d) in Table 3.2). The resistive impedance is in our case a pipe so long that the reflected wave returns reduced in amplitude by 80 dB or more. For lower frequencies, with loss less than 80 dB, it suffices to deliver the signal in pulses of duration T with T < 2L/c, where L is the length of the pipe. The three calibration loads (∞, Z ′ and Z ′′ ) are measured and yield the measurement vectors b, b ′ and b ′′ . The measurements are made and the output spectrum optimised (see §3.6) using either a theoretical matrix A derived from (3.2) or one derived from a previous calibration on the same measurement head. We then have the three calibration equations [ ] 1 pA = b (3.4) 0 p ′ A p ′′ A [ [ ] 1 1/ ¯Z ′ ] 1 1/ Z ¯′′ = b ′ (3.5) = b ′′ (3.6) where the bar represents a reduced impedance ( ¯Z ≡ Z Z 0 ) and p, p ′ and p ′′ are the pressures at the reference plane during measurement of each of the three calibration loads. Dividing each
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Page 3 and 4: To Renée iii
Page 5 and 6: v I am grateful to my parents, Ross
Page 7 and 8: vii Contents Dedication . . . . . .
Page 9 and 10: 1 Chapter I Introduction The world
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CHAPTER 6. MATERIAL AND SURFACE EFF
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CHAPTER 7. THE EMBOUCHURE HOLE AND
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CHAPTER 7. THE EMBOUCHURE HOLE AND
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D4 E4 F4 F#4 G4 G#4 A4 A#4 B4 C5 C#
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107 Chapter VIII Software implement
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CHAPTER 8. SOFTWARE IMPLEMENTATION
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CHAPTER 9. APPLICATIONS AND FURTHER
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Tuning (cents) CHAPTER 9. APPLICATI
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Mean tuning (cents re A 400) Tuning
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APPENDIX A. IMPEDANCE SPECTRA 137 C
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APPENDIX A. IMPEDANCE SPECTRA 141 A
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APPENDIX A. IMPEDANCE SPECTRA 145 D
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189 Appendix B Program listings B.1
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331 Appendix C Quantifying music Th
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333 References Åbom, M. & Bodén,
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REFERENCES 335 Dubos, V., Kergomard
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REFERENCES 337 Strong, W. J., Fletc
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