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

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CHAPTER 3. MEASURING ACOUSTIC IMPEDANCE 46 of acoustically-absorbing wool. If one assumes a fully reflective termination, i.e. neglecting the low reflection termination, at 120 Hz the reflected wave returns with a loss of at least 76 dB (Fletcher & Rossing 1998). The actual loss at 120 Hz will be greater than this lower-bound due to absorption by the acoustic wool. The loss will also increase at higher frequencies due to viscothermal effects. Thus if there were equal power at each of the approx. 1400 frequencies, each reflected component would lie below the effective resolution of the ADC (∼ 105 dB). The quasi-infinite flange is a square perspex plate of side 600 mm in the centre of which is a hole for mounting on the measurement head (for the end effect of a square flange see Dalmont et al. 2001). Over the frequency range of interest the impedance of a flange is lower than that of the resistive impedance load and so these three loads give complementary information. Preliminary experiments have shown that the inclusion of the third load changes the calibration very little, thereby justifying the assumptions made about propagation in the waveguide, and so it was omitted for most measurements. 3.8 RESULTS AND DISCUSSION 3.8.1 Effect of optimising the output signal As discussed earlier, equal distribution of energy among frequencies does not result in equal distribution of errors in the measurements, and the size of the largest error can be substantially reduced by adjusting the output spectrum for particular circumstances and loads. Figure 3.4 shows the magnitude of and (absolute) fractional error in the measured impedance for three sequential impedance measurements. The error was calculated using (3.19)– (3.21). In Figure 3.4 (a), the impedance Z of the resistive impedance load is measured with a non-ideal source. The measured impedance spectrum is not completely flat since this measurement was performed before calibration. The error function ∆Z /Z in Figure 3.4 (a) has broad features corresponding primarily to loudspeaker and conduit resonances. The output spectrum is multiplied by the error function in Figure 3.4 (a) and used to measure an open pipe (Figure 3.4 (b)). The output spectrum is again multiplied by the error function and the pipe is measured a second time. The error in the resulting measurement (Figure 3.4 (c)) is uniformly distributed over frequency. The errors shown in Figure 3.4 (b) and (c) have been divided by a factor of 10 (the signal amplitude was deliberately kept smaller than optimum to increase the apparent errors). It is worth noting that in the uncorrected measurement (b) very high or very low impedances are measured with a greater error than are impedances close to Z 0 . This is unfortunate in the case of musical instruments where we are particularly interested in impedance maxima and/or minima. On subsequent iterations, more power is put into these frequencies and the error is thereby reduced. As can be seen in Figure 3.4 (c), the error increases for impedances close to Z 0 (the error is also more apparent here since the slope of the curve is not as steep as at impedance extrema). However, the maximum error over the entire spectrum is reduced by a factor of approx. 10 through this optimisation procedure. Also shown in Figure 3.4 is the sound pressure spectrum measured by the microphone closest to the reference plane for each measurement. This is relatively uniform over frequency for
CHAPTER 3. MEASURING ACOUSTIC IMPEDANCE 47 (a) 15 mm resistive load (b) open pipe — 1 st meas. (c) open pipe — 2 nd meas. 10 8 |Z| (Pa s m -3 ) 10 7 10 6 10 5 10 -1 error / 10 error / 10 ∆Z / Z 10 -2 10 -3 100 P 1 (dB) 50 0 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 Frequency (kHz) Figure 3.4: Optimising the output signal for measurements on 15 mm pipes. Impedance magnitude (upper panel) and fractional error (middle panel) are shown for (a) the 15 mm resistive impedance load measured with a non-ideal source and for an open pipe of length 200 mm measured with a deliberately very low signal (b) before and (c) after optimisation. The error spectra in (b) and (c) have been scaled for comparison with (a). Also shown for each measurement is the sound pressure in dB re 20 µPa (lower panel) as measured by the microphone nearest to the reference plane.
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Flute acoustics: measurement, model
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 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 145 D
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APPENDIX A. IMPEDANCE SPECTRA 161 E
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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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