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

CHAPTER 2. THEORY AND LITERATURE REVIEW 18
(2.17c). The condition r v = 10 corresponds to a tube diameter of approx. 1 mm. Since this is
much smaller than typical bores or tone holes used in flutes, the simplified expressions given
above have been used in this thesis.
2.2.8 Radiation impedance
The impedance at the end of an open pipe is small but not negligible, and depends on the
frequency, the extent and geometry of flanging around the open end and the presence of any
occluding objects (such as an overhanging keypad or a player’s finger). In this section I will
compare various theoretical and experimental determinations of the radiation impedance for
the following cases:
• an unflanged pipe
• an infinite flange
• a circular flange
• a cylindrical flange
• a disk poised over a circular flange.
The real and imaginary parts of the radiation impedance for both flanged and unflanged
pipes are shown in Figure 2.4, in terms of the dimensionless quantity ka, where a is the radius
of the pipe. In the low frequency limit ka ≪ 1, the impedance is purely reactive and equal to the
impedance of a short, ideally open section of pipe. This leads to the concept of a length correction.
To calculate the impedance of an open pipe, it is sufficient at low frequencies to calculate
the impedance of a slightly longer pipe with a zero load impedance. For higher frequencies the
length correction must be allowed to vary with frequency to give a true representation of the
radiation impedance. The small imaginary part of the end correction (due to radiation losses
from the end of the pipe) must also be included. Following Dalmont et al. (2001), we shall
denote the frequency-dependent end correction by ˜δ and the complex, frequency dependent
end-correction by ˜δ ∗ .
The complex, frequency dependent end correction is related to the real, frequency dependent
end correction and to the modulus of the reflection coefficient according to
˜δ ∗ = ˜δ + i ln(|R|)
2k
(2.18)
and the impedance is just the impedance of an ideally open pipe of length ˜δ ∗ :
Z = iZ 0 tan(k ˜δ ∗ ). (2.19)
2.2.8.1 Unflanged pipe
The low-frequency, lossless end correction for an open unflanged pipe is (Dalmont et al.
2001)
δ open = 0.6133a. (2.20)