Biomedical Engineering Reference
In-Depth Information
FIGURE 16.7
Oblique longitudinal waves at a liquid-liquid interface. The
x
-axis lies along the boundary, and
the
z
-axis is normal to the boundary.
which can be used to find the angle
y
T
. Equation (16.16a) enables the determination of
y
R
.
From the previous relations, the wavenumber components along
z
are
incident
k
Iz
¼
k
1
cos
y
I
ð
16
:
17a
Þ
reflected
k
Rz
¼
k
1
cos
y
R
ð
16
:
17b
Þ
transmitted
k
Tz
¼
k
2
cos
y
T
ð
16
:
17c
Þ
which indicate that the effective impedances at different angles are
r
1
c
Z
1
cos
1
Z
1y
¼
y
I
¼
ð
16
:
18a
Þ
cos
y
I
and
r
2
c
2
cos
Z
2
cos
Z
2y
¼
y
T
¼
ð
16
:
18b
Þ
y
T
Here, the impedance is a function of angle that reduces to familiar values at normal inci-
dence and otherwise becomes larger with angle. When the incident wave changes direction
as it passes into medium 2, the bending of the wave is called refraction. For semi-infinite
fluid media joined at a boundary, each medium is represented by its characteristic imped-
ance given by Eqs. (16.18a) and (16.18b). Then, just before the boundary, the impedance
looking toward medium 2 is given by Eq. (16.18b). The reflection coefficient there is given
by Eq. (16.15a),
RF
¼
Z
2
y
Z
1
y
Z
y
¼
Z
2
cos
y
I
Z
1
cos
y
T
ð
16
:
19a
Þ
y
þ
Z
Z
2
cos
y
I
þ
Z
1
cos
y
T
2
1
where the direction of the reflected wave is along
y
R
, and the transmission factor along
y
T
is
2
Z
2y
2
Z
2
cos
y
I
TF
¼
y
¼
ð
16
:
19b
Þ
Z
y
þ
Z
Z
2
cos
y
I
þ
Z
1
cos
y
T
1
2
To solve these equations,
y
T
is found first from Eq. (16.16b). This liquid-liquid interface is
often used to model waves approximately at a tissue-to-tissue boundary.
