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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