Introduction to Acoustics

616 Part E Music, Speech, Electroacoustics
Part E 15.3
Thus can be expressed in terms of an impedance of an
effectively reduced length l ′ .
l
For kl ≪ 1 ,
′
l =
�
t + 0.6b
1 +
l
�
a
� �
2 −1
.
b
(15.122)
The change in effective length introduced by the open
hole depends strongly on its area relative to that of the
cylinder, the thickness of the wall and its length from the
end. This gives the instrument designer a large amount
of flexibility in the positioning of individual holes on an
instrument. Figure 15.81 illustrates the dependence of
the effective pipe length on the ratio of hole to cylinder
radii for two lengths of pipe between the hole and end
of the instrument.
Not surprisingly, a very small hole with b/a ≪ 1has
a relatively small effect on the effective length of an
instrument. In contrast, a hole with the same diameter as
that of the cylinder shortens the effective added length
to about one hole diameter.
In practice, there will often be several holes open
beyond the first open tone hole, all of which can affect
the pitch of the higher partials.
Consider a regular array of open tone holes spaced
a distance d apart. The shunting kinetic inductance of
each open hole is in parallel with the capacitance associated
with the volume of pipe between the holes.
At low frequencies, such that ω ≪ 1/ √ LholeC0d, the
impedance is dominated by the hole inductance, so that
Input impedance
Pipe alone
Pipe plus
tone hole
lattice
Cutoff frequency
0 1000
2000
Frequency (Hz)
Fig. 15.83 Illustration of the cut-off-frequency effect, when
adding an addition length of tubing with an array of open
tone holes (after Benade [15.133])
each hole attenuates any incident wave by approximately
the ratio
�
d
�
a
� �
2 −1
∼ Lhole/ (Lhole + L0d) = 1 +
,
t + 1.5b b
(15.123)
where L0 is the inductance of the pipe per unit length.
Incident waves are therefore attenuated with an effective
node just beyond the actual hole as discussed above.
√
However, for frequencies such that ω ≫ 1/
LholeC0d, the impedance of the shunting hole inductance
is much larger than that of the capacitance of the
air column, so that the propagating properties of the incident
waves is little affected be the presence of the open
hole. There is therefore a crossover or cut-off frequency
ω ≈ 1/ � a
LholeC0d = c0
b
� 1
�1/2 teffd
, (15.124)
below which the incident waves are reflected to give
a pressure node just beyond the first hole of the array
and above which waves propagate increasingly freely
through the array to the open end of the instrument.
Figure 15.82 (Benade [15.133], Fig. 21.1) illustrates
the effect of an array of open holes on the first few partials
of a typical woodwind instrument, highlighting the
increase in acoustic length of the instrument (indicated
by the intercept of the extrapolated incident waveform)
Thumb
1
2
3
1
2
3
4
C D E F# G# A# C
Fig. 15.84 Soprano recorder fingering for the first seven
notes of a whole-tone scale (after Fletcher and Rossing
[15.5])