Biomedical Engineering Reference
In-Depth Information
In general, numerical tools are needed as soon as nonlinear effects are
taken into account. Acoustics (in the regimes that we have been exploring) is
a linear theory and, therefore, complex problems can be expressed in terms
of simple problems. For example, the pressure waves established in a tube
which is forced by a complicated excitation at one end can be computed
by adding the pressure waves established when the tube is forced by a sum
of simple (harmonic) excitations at the end. Whenever the physical system
under study is linear, this procedure can be followed. In the problem of sound
generation by an oscillating device coupled to a tube, this strategy can be
followed as long as the whole system behaves linearly. For these cases, a most
useful concept can be defined:
impedance
.
7.4.1 Definition of Impedance
At the core of the concept of impedance lies the following feature of forced
linear oscillators: a linear oscillator forced by a harmonic function of time
will ultimately end up following the driver at the forcing frequency. However,
there will be phase lags betwen the forcing and the forced system. Also, the
amplitude of the response will depend on how similar the forcing frequency
and the natural frequency of the oscillating system are. But after a transient,
the forced oscillator will follow the forcing.
As an example, let us consider a simple harmonic oscillator, whose dis-
placement
x
from equilibrium is governed by the equation
m
d
2
x
dt
2
+
b
dx
dt
+
kx
=
f
(
t
)
,
(7.26)
whenever it is forced by a driving force
f
(
t
). An arbitrary forcing
f
(
t
)can
be decomposed in terms of harmonic functions as
f
(
t
)=
F
n
e
iω
n
t
,
(7.27)
as discussed in Chap. 1. Since the system under study is linear, a sum of
solutions is also a solution. Therefore, the building block for understanding
the general problem (7.26) is the solution of
m
d
2
x
dt
2
+
b
d
x
dt
+
k
x
=
Fe
iωt
,
(7.28)
where the real part of the complex forcing is Re(
Fe
iωt
)=
F
cos(
ωt
), and
therefore
x
is a complex variable whose real part represents the physical
displacement of the oscillator (in this section, boldface fonts are used to
indicate complex numbers). Since we know that, ultimately, the forced system
will oscillate at the same frequency as the forcing, we can propose as a solution
x
=
A
e
iωt
. The advantage of using these complex numbers is that taking a
temporal derivative is now trivial (it is simply a multiplication by
iω
), and
therefore we can write
