Biomedical Engineering Reference
In-Depth Information
difference between our metaphor and the description of sound. The proper
variables to describe the problem will be the density (or pressure) and velocity
of the small portion of air, and not the positions and velocities of individual
molecules [Feynman et al. 1970].
1.1.3 Sound as a Physical Phenomenon
The physics of sound involves the motion of some quantity of gas in such a
way that local changes of density occur, and that these changes of density
lead to changes in pressure. These nonuniform pressures are responsible for
generating, in turn, local motions of portions of the gas.
In order to describe what happens when a density perturbation is gener-
ated, let us concentrate on a small portion of air (small, but large enough to
contain many molecules). We can imagine a small cube of size ∆
x
, and our
portion of air enclosed in this imaginary volume. Before the sound phenom-
enon is established, the air is at a given pressure
P
0
, and the density
ρ
0
is
constant (in fact, the value of the pressure is a function of the value of the
density). Before a perturbation of the density is introduced, the forces acting
on each face of the cube are equal, since the pressure is uniform. Therefore,
our portion of air will be in equilibrium. We insist on the following: when
we speak about a small cube, we are dealing with distances larger than the
mean free path. Therefore, the equilibrium that we are referring to is of a
macroscopic nature; on a small scale with respect to the size of our imaginary
cube, the particles move, collide, etc.
Now it is time to introduce a kinematic perturbation of the air in our small
cube, which will be responsible for the creation of a density perturbation
ρ
e
.
We do this in the following way: we displace the air close to one of the faces
at a position
x
by a certain amount
D
(
x, t
) (in the direction perpendicular to
the face), and the rest of the air is also displaced in the same direction, but
by a decreasing amount, as in Fig. 1.1. That is, the air at a position
x
+∆
x
is
displaced by an amount
D
(
x
+∆
x, t
), which is slightly less than
D
(
x, t
). As
the result of this procedure, the air in our imaginary cube will be found in
a volume that is compressed, and displaced in some direction. We now have
a density perturbation
ρ
e
. Conservation of mass in our imaginary cube (that
is, mass before displacement = mass after displacement) leads us to
ρ
0
∆
x
=(
ρ
0
+
ρ
e
)[(
x
+∆
x
+
D
(
x
+∆
x, t
))
−
(
x
+
D
(
x, t
))]
=(
ρ
0
+
ρ
e
)
∆
x
+
∂D
∂x
∆
x
∂D
∂x
∆
x
+
ρ
e
∆
x
+
ρ
e
∂D
=
ρ
0
∆
x
+
ρ
0
∂x
∆
x.
(1.1)
Let us keep only the linear terms by throwing away the term containing
ρ
e
∂D/∂x
as a second-order correction, since we can make the displacement
