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

24 CHAPTER 2. THEORY AND LITERATURE REVIEW
In most practical situations, the geometry of a keypad placed over a tone hole differs from
the simplified geometries considered here, and some empirical corrections may be needed before
applying these results to real tone holes. Coltman (1979) measured the reactances of real
flute tone holes but did not separate the length corrections into inside and outside corrections.
2.2.9 Transmission line theory
Many of the methods and results of electrical transmission line theory can be applied to an
analysis of woodwind instruments, with the acoustic pressure p taking the place of electric
potential and the volume flow U replacing electric current. A small section of air within an
instrument then has an inertance (analogous to electrical inductance) and a compliance (analogous
to electrical capacitance). Since the wavelength in woodwind instruments is not large
compared to the dimensions of the instrument, we must use transmission lines (or transfer
matrices) to model the bore, rather than simple lumped elements. Lumped elements are useful,
however, in analysing components that are small compared to the wavelength, such as tone
holes.
In applying transmission line theory to a woodwind instrument, one must first represent
the instrument bore as a succession of cylindrical and conical elements. This is exact in the
limit as the length of the elements approaches zero.
Tone holes are represented in the model as T- or Π-circuits with impedance elements given
by e.g. §2.2.10. The transfer matrix for an instrument relates the pressure and volume flow at
the output to the same quantities at the input, and may be derived by multiplying together
the transfer matrices of each element. The transfer matrices for cylindrical and conical pipe
sections are given in §2.2.5 and §2.2.6. The transfer matrix for a tone hole with series and shunt
impedances Z a and Z s is (Keefe 1990)
[ ]
1 Za
T =
, (2.33)
1
Z s
1
provided Z a ≪ Z s . I discuss Z a and Z s in the next section.
2.2.10 Tone hole equivalent circuits
At the junction between a tone hole and the bore, impedance corrections must be added to the
one-dimensional model. In most cases, these can be added as length corrections (frequencydependent
in accurate calculations) to the hole segment and bore segments on each side. Figure
2.7 shows the geometry and dimensions of a simple cylindrical side hole. Dimensions a and
b are the bore and hole radius respectively, and for convenience we shall use γ = b/a. Shown
in Figure 2.8 is the substitution circuit for a symmetrical side hole between two tube sections.
The impedance of the side hole itself is a summation of three impedances:
1. Z h , the impedance of a tube with dimensions equal to the hole dimensions (as defined
in Figure 2.7)
2. Z r , the radiation impedance (see §2.2.8)
3. Z m , the ‘matching volume’ correction.