Biomedical Engineering Reference
In-Depth Information
direction of propagation
n
, in contrast to the case with pure longitudinal waves or
pure shear waves. The waves under consideration here are said to be
quasi-
longitudinal waves
if their amplitudes or polarization vectors make an angle of
45
or less with the direction of propagation
n
,ortobe
quasi-shear waves
their
amplitudes or polarization vectors make an angle of more than 45
with the
direction of propagation
n
. The theory developed in the previous sections is
applicable to quasi-waves as well as to pure waves, but the algebra is much simpler
for pure waves.
Example 9.5.1: Wave Propagation in a Direction That Is Not a Principal Direction
of Symmetry
Determine the wave velocities and the polarization vectors or eigenvectors for
cancellous bone associated with a harmonic wave propagating in a direction that
is not an axis of material symmetry. Let the axis of material symmetry be
e
1
and the
direction that is not an axis of material symmetry be
n
¼
{1, 1, 1}(1/
3). The
√
properties of the specified cancellous bone those of Example 9.6.1.
Using the data from Example 9.6.1 and
n
3)[1, 1, 1]
T
,the6by6
determinant of the wave speeds (
9.31
) in this case is given by
¼
(1/
√
10
6
10
6
10
6
10
6
10
6
10
6
2
:
46
1
:
5
v
2
9
:
6
7
:
08
4
:
407
v
2
0
:
44
0
:
44
10
6
10
6
5
v
2
10
6
10
6
10
6
v
2
10
6
9
:
6
2
:
053
1
:
0
:
59
0
:
44
0
:
44
0
:
44
0
:
708
10
6
0
:
59
10
6
1
:
50
10
6
1
:
5
v
2
0
:
44
10
6
0
:
44
10
6
0
:
44
10
6
v
2
1
;
000
¼
0
0
:
44
10
6
v
2
0
:
44
10
6
0
:
44
10
6
0
:
59
10
6
2v
2
0
:
59
10
6
0
:
59
10
6
10
6
10
6
10
6
10
6
10
6
10
6
0
:
44
0
:
44
v
2
0
:
44
0
:
59
0
:
59
2v
2
0
:
59
10
6
10
6
10
6
v
2
10
6
10
6
10
6
2v
2
0
:
44
0
:
44
0
:
44
0
:
59
0
:
59
0
:
59
and the wave speeds are 1,692, 1,127, 1,039, and 893 m/s. For the 1,692 m/s wave
speed the polarization vectors are
a
¼f
1
;
0
:
74
;
0
:
40
g
a
1
;
b
¼f
0
:
42
;
0
:
29
;
0
:
122
g
a
1
:
This vector
a
makes an angle of 19
with the direction of propagation
n
.This
vector
b
makes an angle of 156
with the direction of propagation
n
. Thus this is the
quasi-longitudinal fast wave. For the 1,127 m/s wave speed the polarization vectors
are
a ¼f
1
;
1
:
30
;
0
:
007
g
a
1
;
b ¼f
0
:
56
;
0
:
59
;
0
:
06
g
a
1
:
This vector
a
makes an angle of 96
with the direction of propagation
n
.This
vector
b
makes an angle of 91
with the direction of propagation
n
. Thus this is a
quasi-shear wave. For the 1,039 m/s wave speed the polarization vectors are
a
¼f
1
;
0
:
94
;
3
:
52
g
a
1
;
b
¼f
1
:
09
;
0
:
106
;
1
:
17
g
a
1
:
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