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

42 CHAPTER 3. MEASURING ACOUSTIC IMPEDANCE
angular frequency and the air density. This treatment assumes that all higher modes excited at
the reference plane are evanescent and do not couple to any higher modes in other parts of the
object.
Incidentally, we may rewrite (3.11) in terms of the pressure and flow. The plane-wave flow
is continuous at a duct discontinuity (U true = U meas = U) and so the true plane wave pressure is
given by
N∑
p true ≈ p meas −U (F 0,n ) 2 Z n,n (c) . (3.13)
For the case of an axially symmetric impedance head comprised of an acoustic resistance
of annular cross section and a microphone, both located at the reference plane, Fletcher et al.
(2005) provide an equation similar to (3.11):
Z true ≈ Z meas −
n=1
n=1
N∑ ωρ
J 0 ( α na
), (3.14)
2πk n M n R
where a and R are, respectively, the radius of the annulus and the entry radius of the object
under study, and α n is the nth zero of the first-order Bessel function of the first kind. The factor
M n is given by
M n = R2
2 J 0 2 (α n). (3.15)
In both of these cases, the number of higher modes included in the sum, N, should be
chosen so that the sum converges. Van Walstijn et al. used 40 and Fletcher et al. 100 higher
modes to be certain of achieving a good approximation, although this will of course depend on
the size of the diameter mismatch.
Brass & Locke (1997) discuss the effect of the evanescent wave on impedance measurements
of the ear canal with a loudspeaker and microphone in close proximity.
An impedance determined by applying the above-mentioned corrections will usually be
less accurate than one measured with a matching impedance head, as turbulent losses are not
taken into account in the multi-modal model.
3.5.4 Random noise (acoustical or electrical)
Each transducer signal is contaminated by random noise. Whether this is of acoustical or electrical
origin is usually unimportant. This noise often has an overall 1/f n -dependance, where
0 < n < 1 and the quality of measurements may be improved by increasing the power in the
lower frequencies.
3.5.5 The ‘singularity factor’
The sensitivity of any impedance head to errors in the input quantities varies over frequency.
In the two-microphone method, for example, the head becomes ‘singular’ when the microphone
spacing is an integral multiple of λ/2 and in this vicinity large errors in impedance result
from small measurement errors. Conversely, for a microphone spacing of λ/4, the head is
least sensitive to errors in the measured quantities. This effect is conveniently represented by
the function SF (for ‘singularity factor’), defined by Jang & Ih (1998, their equation (16)), and