Biomedical Engineering Reference
In-Depth Information
p
a
p
a
p
b
p
b
Fig. 4.6.
The labia can gain energy from the airflow when the two modes of motion
described in the text are active. The reason is that when they have a convergent
profile, the average pressure between the labia is closer to the bronchial pressure,
whereas the interlabial pressure is closer to atmospheric pressure for a divergent
profile. Consequently, if the the labia are displaced away from each other while
forming a convergent profile and displaced towards each other while forming a
divergent profile, the labia gain energy from the airflow in each cycle. In the first
part of the cycle, there is a strong force in the same direction as the velocity. In
the second part of the cycle (when the labia move towards each other), the force
(which opposes the velocity) is weaker
if the labial movements are coordinated in such a way that they form a
convergent profile when they are moving away from each other, there will be
a force acting on the labia in the direction of their velocities. Depending on the
value of the bronchial pressure
p
b
, this force may be capable of generating an
amount of work that overcomes the energy losses, giving rise to self-sustained
oscillations.
Computing the average pressure per unit
M
(see (4.1)) between the labia
gives rise to the following expression:
p
f
=
p
b
1
,
a
a
1
−
(4.12)
where
p
b
is the sublabial pressure per unit
M
. The technical details are
somewhat involved, but the basic idea is that the coordinated modes are able
to sustain oscillations of the membranes, transferring energy from the airflow
to the oscillating system [Titze 1988, Laje et al. 2002a]. In fact, Newton's
equations now would read
x
=
y,
y
=
−kx
+
−by − cx
2
y
+
p
b
1
−
,
a
a
1
(4.13)
which, after making the approximation 1
y
, gives rise to the equa-
tions of the simple model (4.5). Notice that these equations are similar to
those describing a mass attached to a spring, which actually was the first
quantitative model of sound production for birdsong [Fletcher 1988].
−
a
2
/a
1
∼
