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

CHAPTER 2. THEORY AND LITERATURE REVIEW 25
(a)
(b)
Figure 2.9: Flow and potential lines for a closed side hole at a pressure node (a) and for an open
side hole and closed right hand tube piece (b) (Nederveen et al. 1998).
‘matching volume’ correction) plus the impedance correction Z i . This ‘inner’ correction can
be thought of as ‘internal radiation’ and in the limiting case of a very small hole is equal to the
end correction for an infinite flange. If the side hole is long enough that the evanescent higher
modes evoked at the inside and outside ends do not interact, then Z a and Z i are the same for
open and closed holes. For very short holes (such as are found on e.g. the modern flute) Z a will
differ depending on whether the hole is open or closed, since the extent of flow widening may
be reduced by the presence of the keypad. The substitution circuit shown in Figure 2.8 may
also not strictly hold, due to interaction between the inside and outside fields.
Keefe (1982b) applied modal decomposition to the problem of a cylindrical side-branch
in a cylindrical tube and gives formulae for the imaginary parts of the impedances in the T-
circuit describing the hole. Keefe did not separate the components in the shunt impedance as
shown in Figure 2.8, giving instead formulae for Z s , the symmetrical or shunt impedance for the
hole. Dubos et al. (1999) extended this work, correcting some errors and also treating the hole
as a three-port. The results of Dubos et al. are more directory comparable to results of other
authors. Nederveen et al. (1998) used the finite difference method to treat the same problem
and derived fit-formulae similar to those of Dubos et al. Dalmont et al. (2002) compare the
results of both theoretical papers and give some experimental results for the equivalent circuit
of a side hole at low and high levels.
Recently, a hybrid method using the method of moments and finite difference methods has
been used to analyse undercut tone holes in woodwinds (Poulton 2005).