Introduction to Acoustics

To a first approximation, opening a side hole creates
a pressure node at that position, shortening the effective
length of the instrument and raising the modal frequencies.
However, as described in Fletcher and Rossing
([15.5], Sect. 15.2) and in detail by Benade ([15.133],
Chaps. 21 and 22), the influence of the side holes is in
practice strongly dependent on the hole size, position
and frequency, as summarised below.
At low frequencies, when the acoustic wavelength
is considerably longer than the size and spacing of the
tone holes, one can account for the effect of the tone
holes by considering their equivalent capacitance when
closed and their inductance when open, as illustrated
schematically in Fig. 15.80.
Because the walls of wind instruments and particularly
woodwind instruments have a significant thickness,
the tone holes when shut introduce additional small volumes
distributed along the length of the vibrating air
column. Each closed hole will introduce an additional
volume and equivalent capacitance Cc-hole = πb 2 /γ P0,
which will perturb the frequencies of the individual partials
upwards or downwards by a small amount that
will depend on its position relative to the pressure
and displacement nodes and the closed volume of the
hole. In severe cases, the perturbations can be as large
as a few per cent (one semitone is 6%), which requires
compensating changes in bore diameter along
the length of the instrument, to retain the harmonicity
of the partials. However, this is essentially a problem
l'/l
1
0.8
0.6
0.4
0.2
0
0 0.2 0.4 0.6 0.8 1
b/a
Fig. 15.81 Low-frequency (kl ≪ 1) fractional reduction of
effective length of a cylindrical end-pipe as a function of
hole to cylinder radius, for additional lengths of 10 (lower
curve) and 20 (upper curve) times the tube radius in length.
The side wall thickness is 0.4 times the tube radius
Musical Acoustics 15.3 Wind Instruments 615
that depends on geometrical factors involving the air
column alone. Once solved, like all acoustic problems
involving the shape and detailed design, instruments can
be mass-produced with almost identical acoustic properties,
quite unlike the problems that arise for stringed
instruments.
The more interesting situation is when the holes are
opened, introducing a pressure node at the exit of the
tone hole and shortening the effective acoustical length
of the instrument. An open hole can be considered as
an inductance, L ∼ ρ(t + 0.6b)/πb 2 , where the effective
length of the hole is increased by the unflanged
hole end-correction. Neglecting radiation losses from
the hole [Fletcher and Rossing [15.5], (15.21, 22)], the
effective impedance Z ∗ of an open-ended cylindrical
pipe of length l and radius a shunted by the inductive
impedance of a circular hole of radius b set into the wall
of thickness t is given by
1
∼
Z∗ =
πb 2
iωρ(t + 0.6b)
πa 2
+ πa2
iρc0 tan kl
. (15.121)
iρc0 tan kl ′
Closed holes Open holes
Fig. 15.82 Schematic representation of the influence of
open holes on the first four partials of a woodwind instrument,
with the effective length indicated by the intercept
neff on the axis of the extrapolated incident wave (after
Benade [15.133])
neff
neff
neff
neff
Part E 15.3