Biomedical Engineering Reference
In-Depth Information
Z 0
ρ 0 cS
Z L 0 cS + i tan( kL )
1+ i ( Z L 0 cS )tan( kL ) .
=
(7.35)
In order to compute the input impedance of the system (i.e., the im-
pedance at x = 0), we have to compute (or estimate) the impedance at
x = L . If we are interested in a tube closed at x = L , this is really easy.
At the closed end there is no air displacement, regardless of the pressure.
Therefore, the output impedance is infinite, and we obtain
Z 0 =
0 cS cot( kL ) .
(7.36)
Notice that the zeros of the impedance mean that, for a forcing with a
bounded amplitude, the velocity of the system can diverge. Therefore, we can
interpret these zeros as the resonances of the system.
This is a beautiful example of the power of linear methods ...when they
are valid. We can obtain analytic expressions for the conditions for resonance
in terms of the system's parameters, allowing us to understand general re-
lationships between geometry and acoustics. The problem of the open tube
can be approached similarly. Unfortunately, the condition Z L = 0 is an over-
simplification. The output impedance will be some radiation impedance (we
need some force to move the air outside the tube), which is not so easy to
compute [Kinsler et al. 1982].
In acoustics, we can define the acoustic impedance z of a fluid that is acting
on a surface area A as the acoustic pressure divided by the volume velocity
of the fluid (which is the product of the area and the particle velocity). This
quantity is useful for discussing the properties of the transmission of pressure
fluctuations through pipes of various geometries. The reason is that it allows
us to find the impedance of a complex arrangement of tubes in terms of the
impedances of the individual tubes.
Let us consider the example illustrated in Fig. 7.6, where we show a pipe
that branches into two pipes, denoted as pipes 1 and 2. Let us assume that the
branching pipes have acoustic impedances z 1 and z 2 , and that the pressures
at the junction are P 1 and P 2 . Finally, let us denote by P 0 the pressure at
the bifurcating branch. Continuity of the pressure allows us to write
P 0 = P 1 = P 2 ,
(7.37)
while the continuity of the volume velocity allows us to write
U 0 = U 1 + U 2 ,
(7.38)
which, combined, allows us to write that
1
z 0
1
z 1
1
z 2
=
+
.
(7.39)
A very nice discussion of the impedances of avian vocal organs is presented
in [Fletcher and Tarnopolsky 1999]. The vocal-tract input impedances are
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