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

58 CHAPTER 4. FINGER HOLE IMPEDANCE SPECTRA AND LENGTH CORRECTIONS
and U, are then each some linear combination of the two loudspeaker signals:
p = αVr+ βV r
(4.5)
U = δVr+ γV r . (4.6)
(The proportionality constants may be calculated using models of each source, impedance
head and any connecting conduit, but this is unnecessary for the present.) For a measurement
of Z s on the ‘no-hole’ system, we require U = 0, or (using (4.6)) r 2 s =− γ δ
. Likewise, for a measurement
of Z a , we require p = 0, or (from (4.5)) r 2 a =− β α
. The four proportionality constants
may be determined from the four measurements p s and U s (measured with r = r s , initially 1)
and p a and U a (measured with r = r a , initially i). The adjusted splitting ratios r ′ s and r ′ a are
then given by
√
r ′ r s V s U a − r a V a U s
s = r s r a , (4.7)
r a V s U a − r s V a U s
and
√
r ′ r s V s p a − r a V a p s
a = r s r a . (4.8)
r a V s p a − r s V a p s
Note that the loudspeakers are assumed to be linear in the above calculation of the splitting
ratios. To reduce the effect of loudspeaker non-linearity very strong resonances in the system
should be avoided so that the magnitudes of V s and V a are similar. For this reason, acoustic
wool is placed between each microphone array and loudspeaker.
4.2.4 Measurement optimisation
After an initial measurement of Z s or Z a , we can change the output spectrum V to optimise the
measurement in a manner similar to that described in §3.6.
Some simplifying assumptions make this easier. For a measurement of Z s , we assume that
Z a = 0. Then Z s = p U where p = p 1+p 2
2
and U = U 1 +U 2 . For a measurement of Z a , we assume
that Z s =∞(when there is a pressure node at the hole there is zero flow through the hole and
Z s is not ‘seen’). Then Z a = p U where p = p 1 − p 2 and U = U 1−U 2
2
. The errors in p and U, ∆p and
∆U, are obtained in both cases by straight-forward propagation of errors, as is the error ∆Z in
Z (for simplicity the subscript has been dropped here).
Having determined the error in Z , the output spectrum is modified by multiplying with
either of the correction functions C 1 (3.22) or C 2 (3.24).
4.2.5 Frequency range and dynamic range
Impedance spectra were measured between 1 and 3 kHz. This frequency range was chosen as
it encompasses the upper range of woodwind instruments, where the finger hole series and
shunt impedances have most effect. Below 1 kHz the series and shunt impedances are near or
beyond the limits of the dynamic range of the system, at least for small holes. Also, the microphone
separation of the impedance heads measures impedance optimally at the centre of this
range. During measurements on finger holes, the control pipe with no hole was measured repeatedly
to check for drift and to gauge the dynamic range of the system. The system dynamic