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 :
Search WWH ::




Custom Search