Biomedical Engineering Reference
In-Depth Information
the “area”). In fact, the flow is defined as the product of the area and the
velocity of the objects:
U
=
v
·
A
.
(2.1)
Here we let the velocity
and
the area be vectors, because the important thing
here is the cross section; that is, the effective area faced by the velocity. This
can be seen by expressing the dot product as
U
=
vA
cos
α
,where
α
is the
angle between the velocity and the direction perpendicular to the area.
Flow is an appropriate concept for stating conservation laws. For a general
closed surface
S
, we can write the flow
U
across it as
U
=
v
·
d
a
.
(2.2)
S
According to our previous discussion, this should be proportional to the vari-
ation of the mass within the volume enclosed by the surface
S
:
1
ρ
0
∂m
∂t
U
=
−
,
(2.3)
where
ρ
0
stands for the constant equilibrium density of the fluid. Changing
from mass to density by means of a volume integral (
V
ρdV
=
m
), we can
write
1
ρ
0
∂
∂t
U
=
−
ρdV
(2.4)
V
=
v
·d
a
,
(2.5)
S
and since
S
v
d
a
=
V
∇·
·
v
dV
(Gauss's theorem),
1
ρ
0
∂ρ
∂t
,
∇·
v
=
−
(2.6)
where
v
stands for the
particle velocity
. As is known in acoustics,
v
is related
to the displacement
D
introduced in Chap. 1 through
v
=
∂
D
/∂t
.Thesymbol
∇
is a compact notation for the vector spatial derivative. This equation does
not provide us with more physics than its one-dimensional version (1.2).
However, it is more general and will allow us to advance beyond a particular
geometry. Similarly, we can write Newton's second law in a more general way
than in (1.3):
∂
v
∂t
.
∇
p
=
−
ρ
0
(2.7)
As in the one-dimensional case, we can relate the density fluctuations to the
pressure perturbations by means of the linearized equation of state (1.4).
Simple algebra then allows us to derive the acoustic-pressure wave equation
