Biomedical Engineering Reference
In-Depth Information
and degree of asymmetry. And how do these pressure waves sound when
the labial motion is complex? Well... not very nice. It is interesting that the
presence of subharmonics can be found in juvenile birds and also in the crying
of human babies. It is tempting to speculate on the evolutionary advantages
of irritating mom and dad to get their attention [Robb 1988]. Can physics be
an ally in this process?
6.4 Choosing Between Two Models
In Fig. 6.1b, we were presented with a most interesting and puzzling effect: a
complex sound whose sonogram displayed evidence of subharmonicity. How-
ever, we have discussed two different mechanisms which are capable of ac-
counting for this effect: coupling between the sources and the tract (Sect. 6.2),
and a complex spatial excitation of the oscillating tissues (Sect. 6.3). Is there
a way to find the mechanisms at play for a particular bird? The unveiling of
the physical processes involved in each of these different mechanisms allows
us to predict the outcome of experiments designed to allow us to choose be-
tween these competing models. In particular, we shall investigate what could
be expected to occur if a bird producing complex vocalizations sings in a
heliox atmosphere (i.e., one in which nitrogen, comprising 80% of ordinary
air, is replaced by the less dense helium). Experiments of this sort has been
used to show that some birds actively coordinate the passive-filter charac-
teristics of their vocal tract with the output of the syrinx [Nowicki 1987]
and to discern between competing proposed mechanisms for song production
[Ballintijn and ten Cate 1998].
According to (6.9), the pressure fluctuations induced by the source at the
input of the tract depend on the air particle velocity
v
and its derivative
dv/dt
through the coe
cients
R
and
I
. The air velocity at the input of the tract
can be written as
v
(
t
)=
v
0
a
2
(
t
)
/A
0
(see Fig. 6.2) owing to mass conservation,
where
v
0
=
2
P
s
/ρ
. This defines a dependence of the air particle velocity
on the density of the medium. On the other hand, the coe
cients
R
and
I
depend on the density, as shown in (6.10) and (6.11). Therefore, the pressure
perturbation
s
(
t
) can finally be written as
s
(
t
)=
αf
(
x, dx/dt
)+
βdf
(
x, dx/dt
)
/dt ,
(6.17)
where
f
(
x, dx/dt
) is a function that depends on the kinematics of the labia,
and
α
and
β
scale with the density as
ρ
1
/
2
c
−
1
∼
α
∼
ρ,
(6.18)
ρ
1
/
2
.
β
∼
(6.19)
Since a heliox atmosphere has a density 33% that of ordinary air, we obtain
α
Heliox
=0
.
3
α
air
,
(6.20)
β
Heliox
=0
.
5
β
air
,
(6.21)
