Biomedical Engineering Reference
In-Depth Information
tract (such as lengths, widths and cross sections) and parameters related to
sound (such as the frequency and speed of the sound, and the air density).
A rigid-walled waveguide can propagate only plane waves if its cross-
sectional dimensions are much smaller than the sound wavelength
[Kinsler et al. 1982]. This is the case for the avian trachea. In the species
we are considering, and as a first approximation, the trachea is a very narrow
tube around 2 cm long and 1 mm wide, with a
cutoff frequency
(that is, the
frequency below which every propagating wave is a plane wave) of around
100 kHz. We can then assume that the sound waves in the avian trachea
are plane waves, and therefore make use of the very simple boundary condi-
tions (6.5) and (6.6) to account for the superposition of forward-emitted and
backward-reflected waves.
However, we do not expect the sound wave to be a plane wave very near
the source. As it is emitted by the glottis, the sound wave is of a rather diverg-
ing nature; for the sake of simplicity, we assume that the glottis behaves as
a local emitter of
spherical
diverging waves. Recall now our discussion of the
concept of flow in Chap. 2. Time variations of the flow
U
(
t
) (which is related
to the air particle velocity
v
through
U
=
vA
) induce pressure perturbations
s
(
t
) at the vocal-tract input. For a spherical sound wave, assuming that the
pressure perturbations
s
(
t
) and air particle velocity
v
(
t
) are both harmonic
functions of time, the relationship between them can be written as
s
(
t
)=
zv
(
t
)
,
(6.9)
where
z
=
R
+
iωI
is the
complex specific acoustic impedance
.
R
and
I
are
called the
specific acoustic resistance
and the
inertance
, respectively:
(
kd
)
2
1+(
kd
)
2
R
=
ρc
,
(6.10)
d
1+(
kd
)
2
I
=
ρ
,
(6.11)
where
c
and
ρ
are the sound speed and the air density, and
d
is the anatom-
ical parameter shown in Fig. 6.2. Note that both
R
and
I
depend on the
frequency
f
of the sound wave through the wavenumber
k
=2
πf/c
. Note
also that
R
and
I
are not independent quantities; they are related through
R
=4
π
2
f
2
(
d/c
)
I
. A plot of
R
and
I
as a function of the frequency
f
is
shown in Fig. 6.3.
The parameters
R
and
I
are the coupling coe
cients we are looking for
(
v
can be written in terms of the variables
x
and
x
describing labial motion).
The key issue is that the pressure perturbations
s
(
t
) have two different contri-
butions: the first one in phase with the flow, scaled by
R
, and the second one
in phase with flow derivative and scaled by
I
(recall the discussion leading to
(2.12)). The nature of the coupling depends on the balance between these two
contributions (“resistive” coupling when the term containing
R
dominates,
and “inertive” coupling otherwise), which in turn depends on the frequency
