Biomedical Engineering Reference
In-Depth Information
2
3
Q
11
00
0
Q
22
0
00
Q
33
4
5
;
Q
¼
(9.38)
In the coordinate system of the principal axes of material symmetry and at the
vector
n
{1, 0, 0}, the four 3 by 3 submatrices that form the 6 by 6 matrix in
equation (
9.30
) are given by
¼
2
3
v
2
Q
11
r
0
0
4
5
;
v
2
1
Q
r
¼
Q
22
r
v
2
(9.39)
0
0
Q
33
r
v
2
0
0
2
4
3
5
;
C
11
r
f
v
2
0
0
C r
f
v
2
1 ¼ C
T
r
f
v
2
1 ¼
r
f
v
2
(9.40)
0
0
r
f
v
2
0
0
v
2
i
m
o
Mn
n
r
f
J
þ
R
2
4
3
5
:
imv
2
o
r
f
v
2
J
11
M
R
11
0
0
imv
2
o
¼
r
f
v
2
J
22
0
R
22
0
i
m
v
2
o
r
f
v
2
J
33
0
0
R
33
(9.41)
Substitution of the four 3 by 3 matrices above into the 6 by 6 determinant (
9.31
)
reveals that 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
};
a
1
b
1
¼
v
2
C
11
r
f
v
2
Q
11
r
v
2
0
;
i
m
o
C
11
r
f
v
2
M
r
f
J
11
þ
R
11
a
2
b
2
¼
v
2
r
f
v
2
Q
22
r
v
2
0
;
i
m
o
r
f
v
2
r
f
J
22
þ
R
22
a
3
b
3
¼
Q
33
r
v
2
r
f
v
2
v
2
:
0
(9.42)
r
f
v
2
i
m
o
r
f
J
33
þ
R
33
Requiring that the determinants of these 2 by 2 matrices 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,
s
N
2
C
11
N
þ
r
f
ð
MQ
11
Þ
r
f
L
11
v
2
¼
r
f
L
11
;
(9.43)
2
2
r
f
L
11
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