Biomedical Engineering Reference
In-Depth Information
wave moving through the liquid. In order to have manageable equations to estimate the
viscosity of the liquid from acoustic measurements using these systems it is important to
generate planar standing waves from which acoustic parameters can be readily obtained.
Mert et al. (2004) described a model and the experimental conditions under which the
assumption of standing planar waves stands.
If a tube with rigid walls is considered the propagation of unidirectional plane sound waves
though the liquid contained in the rigid tube can be described by the following equation
(Kinsler et al., 2000):
2
2
p
p
2
1
C
(1)
2
2
t
x
where
p
is the acoustic pressure,
t
is time,
x
distance and
C
1
the speed of the sound in the
testing liquid. The solution of Equation (1) can be expressed in terms of two harmonic waves
travelling in opposite directions and whose composition gives place to the formation of
standing waves (Kinsler et al., 2000):
ˆ
ˆ
itkLx
(
)
itkLx
(
)
pxt
(,)
p e
p e
(2)
p
+
is the amplitude of the wave traveling in the direction
+x
whereas
p
-
is the amplitude of
the wave traveling in the direction
-x
and
i
is the imaginary number equal to
. The
ˆ
complex wave number
includes the attenuation due to the viscosity of the
liquid, is the frequency of the wave. The corresponding wave velocity can be obtained
from integration of the pressure derivative respect to the distance
x
, an equation that is
k
/
Ci
1
p
1
vxt
dt
known as the Euler equation (Temkin, 1981) and calculated as
(,)
), which
x
0
yields:
ˆ
ˆ
itkLx
(
)
itkLx
(
)
vxt
(,)
P e
P e
(3)
By applying the boundary conditions such that at
x
= 0 the fluid velocity is equal to the
velocity of the piston that creates the wave, which is
vt ve
. The other boundary
condition is derived from the practical situation of using an air space on the other end of the
tube, which mathematical provides a pressure release condition at
x = L
, written as
p(L, t) = 0
.
With these boundary conditions the coefficients
P
+
and
P
-
in Equation 3 can be calculated as:
it
(0, )
0
C
v
C
v
010
010
P
and
P
(4)
ˆ
ˆ
2cos
kL
2cos
kL
where is the density of the liquid, in repose, and
v
0
the amplitude of the imposed wave.
An analysis made by Temkin (1981) showed that two absorption mechanisms produce the
attenuation of the sound energy. One is due to the attenuation effects produced by the
dilatational motion of the liquid during the acoustic wave passage. The other term arises
from the grazing/friction motion of the liquid on the wall of the tube. It can be shown that
the attenuation due to wall effects is significantly larger than the attenuation due to the
