Biomedical Engineering Reference
In-Depth Information
With the help of Newton's laws for the air in our original imaginary cube,
and ignoring the effects of viscosity, the restitutive effect of this imbalance
may be written as follows:
ρ
0
∆
x
∂
2
D
∂t
2
=
−
[
P
(
x
+∆
x, t
)
−
P
(
x, t
)]
∂p
∂x
∆
x,
=
−
(1.3)
where
P
=
P
0
+
p
is the pressure and
p
is the (nonuniform) pressure pertur-
bation, or
acoustic pressure
. In addition, assuming that the pressure pertur-
bations are linear functions of the density perturbations (which holds if the
density perturbations are small enough), we can write the equation of state
κ
ρ
0
p
=
ρ
e
,
(1.4)
where
κ
is the adiabatic bulk modulus.
So far, we have a conservation law (1.2), a force law (1.3) and an equation
of state (1.4). With these ingredients, we can write an equation for
p
only. If
we differentiate (1.2) twice with respect to
t
, we obtain
∂
2
ρ
e
∂t
2
∂
2
∂t
2
∂D
∂x
.
=
−
ρ
0
(1.5)
On the other hand, the differentiation of (1.3) with respect to
x
gives us
∂
2
D
∂t
2
∂
2
p
∂x
2
∂
∂x
ρ
0
=
−
.
(1.6)
Writing
ρ
e
in terms of
p
and equating both expressions, we obtain the acoustic
wave equation
∂
2
p
∂t
2
∂
2
p
∂x
2
=
c
2
,
(1.7)
where
c
=
κ/ρ
0
is the speed of sound, which is 343 m/s in air at a tempera-
ture of 20
◦
C and atmospheric pressure. This is the simplest equation describ-
ing sound propagation in fluids. Some assumptions have been made (namely,
sound propagation is lossless and the acoustic disturbances are small), and
the reader may feel suspicious about them. However, excellent agreement
with experiments on most acoustic processes supports this lossless, linearized
theory of sound propagation. It is interesting to notice that the same equa-
tion governs the behavior of the variable
D
(displacement) and the particle
velocity
v
=
−
∂D/∂t
.
1.1.4 Sound Waves
Sound waves are constantly hitting our eardrums. They arrive in the form of
a constant perturbation (such as the buzz of an old light tube) or a sudden
