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

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48 CHAPTER 3. MEASURING ACOUSTIC IMPEDANCE (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.
3.8. RESULTS AND DISCUSSION 49 the measurement of the resistive impedance and has minima corresponding to nodes of the standing wave for measurements of the 200 mm closed pipe. The choice of the exponent w in C 1 (f ) (3.22) determines which part of the spectrum will be measured with the greatest precision. For w = 1, the fractional error will be constant for all impedances. For w > 1, the impedance minima will be determined with greater precision, and for w < 1 the impedance maxima will be given preference. The case where w > 1 is particularly useful for measuring the impedance of instruments in the flute family, which play near impedance minima, whereas w < 1 is useful for reed and lip valve instruments such as the clarinet which play near impedance maxima. In some cases (such as when measuring calibration loads) it is not desirable to use the function C 1 (f ) to modify the output spectrum, but we may still like to compensate for the system responses and the ‘singularity factor’ of the head. In these cases, we modify the output spectrum to ensure that the acoustic energy density at the reference plane is the same as it would be during a (hypothetical) measurement of Z 0 with ∆Z /Z = K where K is independent of frequency. The acoustic energy density at the reference plane during a measurement of an impedance Z is proportional to ɛ = 1 2 (|p|2 +|Z 0 U| 2 ). For a measurement of Z 0 , ɛ 0 =|Z 0 U| 2 = |∆p|2 +|Z 0 ∆U| 2 K 2 , (3.23) where (3.21) was used for K with the substitution p = Z 0 U. The correction factor C 2 (f ) used to modify the output spectrum will be proportional to the square root of the energy ratio ɛ 0 ɛ .We use √ ɛ0 C 2 (f ) = K ɛ = √ |∆p| 2 +|Z 0 ∆U| 2 2 |p| 2 +|Z 0 U| 2 , (3.24) where the factor K has no effect on the output waveform (being independent of frequency) but is used to ensure that C 2 (f ) = ∆Z /Z when Z = Z 0 . 3.8.2 Effect of calibration Figure 3.5 illustrates the effects of calibrating with varying numbers of known calibration loads. Measured impedance spectra are shown for a closed 15 mm pipe, 200 mm long. To simulate the effect of using unmatched microphones, in Figure 3.5 (a) and (b) the signals from the second and third microphones were scaled by 1.2 and 0.8, respectively. The measured impedance spectrum before calibration (i.e. assuming unity gain for each microphone) is shown in Figure 3.5 (a), while in Figure 3.5 (b) the impedance was calculated after calibrating on the quasi-infinite impedance load (using (3.7) and (3.10)). The pre-calibration measurement deviates significantly from the theoretical impedance, while calibrating with only one load (b) improves the accuracy significantly. In Figure 3.5 (c) the measured impedance spectrum using the matched microphone signals is shown before and after calibration with two known loads. While the two spectra are similar in many respects, the size and frequency of extrema (particularly maxima) are significantly different. For this measurement, adding the flange calibration made very little difference. The measurement made without using the two calibration loads shows similar features to those of the precise measurement over most of the range, but has noticeable errors in the magnitude of extrema (particularly maxima) and smaller errors in the frequencies at which
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Flute acoustics: measurement, model
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To Renée iii
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vi I am grateful to my parents, Ros
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340 REFERENCES Botros, A., Smith, J
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342 REFERENCES Kob, M. & Neuschaefe
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