Biomedical Engineering Reference
In-Depth Information
ω
2
m
A
+
i
A
ωb
+
A
k
=
F.
−
(7.29)
This equation allows us to compute
x
and
v
=
d
x
/dt
, since it determines
the complex amplitude
A
of the displacement. Notice that this gets us close
to computing the only unknown quantities of the problem: how strong the
reaction to the forcing will be (given by the modulus of this complex number),
and the relative phase between the forcing and the displacement (given by
the phase of this number). It is customary to represent this information by
computing the ratio between the complex forcing and the complex velocity
of the system,
f
v
,
Z
=
(7.30)
where
Z
is called the
impedance
[Kinsler et al. 1982]. In general, it is a ratio
between a complex forcing and the resultant complex speed of the system,
at the point where the force is applied. Therefore, this number allows us to
compute the velocity with which the forced system will react to a forcing.
This provides a general framework for the definition of the specific acoustic
impedance presented in Chap. 6.
7.4.2 Impedance of a Pipe
In acoustics, it is natural to use this concept to take account of situations in
which the geometry of the boundaries confines the pressure wave to a limited
region of space. If a pressure wave is established in a pipe of cross section
S
,
for example, we can write
p
=
A
e
i
(
ωt
+
k
(
L−x
))
+
B
e
iωt−k
(
L−x
)
,
(7.31)
where we have assumed that the diameter of the pipe is much smaller than
the sound wavelength. This allows us to approximate the wave as a planar
wave. Using the relations described in Chap. 1 to relate the pressure to the
velocity of the air,
1
ρ
0
∂p
∂x
dt ,
v
=
−
(7.32)
we can compute the impedance of the system. At the end
x
=
L
, we find that
Z
L
=
ρ
0
cS
A
+
B
A
B
,
(7.33)
−
while at
x
= 0, we find that
Z
0
=
S
p
/
u
is given by
Z
0
=
ρ
0
cS
A
e
ikL
+
B
e
−ikL
A
e
ikL
B
e
−ikL
.
(7.34)
−
Following [Kinsler et al. 1982], we combine these expressions to eliminate the
amplitudes
A
and
B
, writing
