Biomedical Engineering Reference
In-Depth Information
In the first chapter, we discussed the concept of the spectral content of a
signal. In the case of the human voice, for example, the time variations of the
airflow induced by periodic obstructions caused by the vocal folds typically
give rise to signals that are spectrally very rich. By this we mean that they
can be written as a sum of many harmonics, as in the case of the triangular
wave illustrated in Chap. 1. Our vocal tract stresses some components and
attenuates others, modifying the timbre of the sound. How does this happen?
2.2.2 Traveling Waves
Once a source of pressure perturbations has been set oscillating, a propagative
phenomenon takes place, leading to a sound wave. As mentioned in Chap. 1,
sound waves are solutions to the wave equation (1.7). That is, a sound wave
can be mathematically described by a function of space and time
p
=
p
(
x, t
)
which fulfills (1.7). One way of achieving this is by choosing
p
to be a
traveling
wave
. Traveling waves are functions of the form
p
(
x, t
)=
p
(
kx
−
ωt
)
,
(2.14)
where
k
and
ω
are wave parameters such that
c
=
ω/k
is the velocity at
which the perturbation propagates, i.e., the sound velocity (see Chap. 1).
A well-known example of a traveling wave is the cosine function
p
(
x, t
)=
A
cos(
kx
−
ωt
). Notice that the particular combination of time and space
kx
ωt
in the argument of
p
is what makes this wave a traveling wave (see suc-
cessive snapshots in Fig. 2.3a). These waves describe all kind of propagative
−
(a)
(b)
0
0.25
0.5
0.75
1
0
0.25
0.5
0.75
1
x (arb. units)
x (arb. units)
Fig. 2.3.
Successive snapshots of waves. (
a
) Traveling wave. A traveling wave is
a function
p
T
(
x, t
) satisfying (1.7), with space and time appearing in a particular
combination:
p
T
(
x, t
)=
p
T
(
kx
−
ωt
). Notice that there are no points at rest, and,
further, that the wave is traveling to the right. A wave traveling to the left would
be
p
T
(
x, t
)=
p
T
(
kx
+
ωt
). (
b
) Standing wave. A standing wave is a function
p
S
(
x, t
) satisfying (1.7) with a factorized space and time dependence
p
S
(
x, t
)=
p
1
(
x
)
p
2
(
t
). Notice the existence of points at rest, or
nodes
: points that always have
zero amplitude, being at positions
x
such that
p
1
(
x
)=0
