Biomedical Engineering Reference
In-Depth Information
(i.e., the upper traces are not multiples of the lower ones). To the right of
the dotted line, the situation is different: even though there is no harmonic
relationship between them, there are common temporal dynamics. The bird
has modified its vocal organ and/or its tract in order to make the sources
interact. The result is the appearance of subharmonics, as can be observed
in this complex sonogram. In this case, both complex neural instructions and
complex physics are in play to generate a unique (and, in some cases, proven
to be biologically relevant [Aubier et al. 2000]) sound.
6.4.2 Modeling Two Acoustically Interacting Sources
A quantitative description of this phenomenon requires us to take into ac-
count the dynamics of the two pairs of labia. The simplest model to describe
the two pairs can be expressed in terms of the variables
x
i
and
y
i
(
i
=1
,
2),
describing the deviations from the rest position and the velocities of the labia
in each of the sources:
x
1
=
y
1
,
y
1
=
cx
1
−
k
1
x
1
−
by
1
−
y
1
−
f
01
+
p
g
1
,
(6.22)
x
2
=
y
2
,
y
2
=
cx
2
−
k
2
x
2
−
by
2
−
y
2
−
f
02
+
p
g
2
,
(6.23)
where
k
i
are the restitution constants for each side,
b
is the dissipation con-
stant,
c
is the coe
cient of the nonlinear terms bounding the labial motion,
and
f
0
i
are force terms controlled by the activity of the dorsal muscles, all
per unit mass of the labia. Finally,
p
gi
are the averaged interlabial pressures
on each side.
The terms describing the pressure between the labia on each side,
p
gi
,
depend on both the sublabial pressure
p
s
and the pressure at the input of
the trachea
p
i
. Considering that the dynamics of each pair of labia can be
described in terms of a flapping model (as discussed in Chap. 4), we can write
a
a
1
p
gi
=
p
s
−
(
p
s
−
p
i
)
,
(6.24)
i
where
p
i
stands for the air pressure at the input of the trachea, which is
common to both sources. The ratios
r
i
≡
(
a
2
/a
1
)
i
between the glottis exit
and entrance areas can be approximated by a linear function of the velocity
of the
i
th labium, i.e.,
r
i
∼−
y
i
; this term is therefore directly responsible for
the onset of the oscillations, since it is capable of compensating dissipative
losses.
We further assume that the main constribution to the pressure fluctua-
tions at the input of the trachea is given by the time derivatives of the air-
flow (inertive coupling). With this and some additional assumptions (among
