Biomedical Engineering Reference
In-Depth Information
These results may be assembled into a rather large 6 by 6 determinant,
10
6
5
v
2
10
6
v
2
3
:
48
1
:
0
0
1
:
322
0
0
2
:
32
10
6
1
:
5
v
2
v
2
0
0
0
0
1
:
58
10
6
1
:
5
v
2
v
2
0
0
0
0
1
;
000
¼
0
;
1
:
322
10
6
v
2
1
:
76
10
6
2v
2
0
0
0
0
v
2
2v
2
0
0
0
0
v
2
2v
2
0
0
0
0
but the division of this into three 2 by 2 determinants is more manageable.
Substitution of the four 3 by 3 matrices above into the 6 by 6 determinant (
9.31
)
reveals that this result may be expressed as three 2 by 2 matrices for the three sets of
components, {
a
1
,
b
1
}, {
a
2
,
b
2
}, and {
a
3
,
b
3
} (as accomplished in (
9.42
));
a
1
b
1
¼
10
6
5
v
2
10
6
v
2
3
:
48
1
:
1
:
322
1
;
000
0
;
10
6
v
2
10
6
2v
2
1
:
322
1
:
76
a
2
b
2
¼
10
6
v
2
2
:
32
1
;
000
0
;
v
2
2v
2
a
3
b
3
¼
10
6
:
v
2
1
58
1
;
000
0
:
v
2
2v
2
Requiring that the determinants of the three 2 by 2 matrices above vanish yields
four nontrivial solutions for the squared wave speed
v
2
. The vanishing of the first of
the determinants of these 2 by 2 matrices provides two roots of a quadratic equation
that represent the fast and the slow squared longitudinal wave speeds. The fast and
the slow squared longitudinal wave speeds are given by,
v
¼
1
;
629 m
=
s
;
v
¼
909 m
=
s
;
and the vanishing of the second and third of the determinants of the 2 by 2 matrices
above provides a zero root and a nonzero root from each determinant. The two
nonzero roots are the squared shear wave speeds
v
¼
1
;
523 m
=
s
and
v
¼
1
;
258 m
=
s
:
The vectors
a
and
b
for the fast and slow waves are given by
a
¼f
2
:
66
b
1
;
0
;
0
g;
b
¼f
b
1
;
0
;
0
g
and
a
¼f
0
:
211
b
1
;
0
;
0
g;
b
¼f
b
1
;
0
;
0
g
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