Biomedical Engineering Reference
In-Depth Information
Fig. 1.1.
Propagation of air density perturbations. (
top
) The air in a small imag-
inary cube is initially in equilibrium. We now “push” from the left, displacing the
left face of the imaginary cube and compressing the air in the cube. (
bottom
)A
density perturbation is created by the push, leading to an imbalance of forces in
the cube. The forces now try to restore the air in the cube to its original position.
At the same time, however, the portion of air in the “next” cube will be pushed in
the same direction as the first portion was, propagating the perturbation
and hence the density perturbation as small as we want. Solving for
ρ
e
, (1.1)
now reads
∂D
∂x
.
ρ
e
=
−
ρ
0
(1.2)
By virtue of the way we have chosen to displace the air (a decreasing dis-
placement), air has accumulated within the cube, which means that we have
created a positive density fluctuation.
What can we say about the dynamics of the problem now? Since we have
created a nonuniform (and increasing) density in the direction of the dis-
placements, we have established an increasing pressure in the same direction.
By doing this, we have broken the equilibrium of forces acting on our por-
tion of air. We have moved the faces, but by doing so, we have created an
imbalance of density and pressure that tries to take our portion of air back
to its original position, in a restitutive way. Another consequence is seen in
the fate of a second portion of air, close to the original one in the direction in
which we generated the compression. The imbalance of pressures around the
new portion of air will lead to new displacements in the direction in which
we generated our original perturbation, as shown in Fig. 1.1: a picture that
does not differ much from the propagation of “pushes” discussed before.
