Biomedical Engineering Reference
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where
m
r
f
o
i
m
r
f
o
i
L
ij
¼ r
J
ik
þ
R
ik
r
f
d
ij
;
N
¼
M
r þ r
f
Q
11
J
11
þ
R
11
2
C
11
;
(9.44)
or
v
2
v
o
c
o
;
¼
(9.45)
where the notation
s
N
2
C
11
N
þ
r
f
ð
MQ
11
Þ
r
f
L
11
v
o
¼
c
o
¼
r
f
L
11
;
(9.46)
2
2
r
f
L
11
has been employed. The vanishing of the second and third of the determinants of the
2 by 2 matrices in (
9.42
) provide a zero root and a nonzero root from each
determinant. The two nonzero roots are the squared shear wave speeds
Q
22
L
22
m
r
f
o
i
Q
33
L
33
m
r
f
o
i
v
2
v
2
¼
J
22
þ
R
22
and
¼
J
33
þ
R
33
:
(9.47)
The vectors
a
and
b
for the fast and slow waves are given by
a
¼f
a
1
;
0
;
0
g;
b
¼f
b
1
;
0
;
0
g;
(9.48)
where
a
1
and
b
1
are related by the following two equivalent expressions for the fast wave
v
o
þ
c
o
Þ
v
o
þ
c
o
Þ
C
11
r
f
ð
M
ðr
f
J
11
þð
i
m=oÞ
R
11
Þð
a
1
¼
Q
11
;
b
1
¼
b
1
(9.49)
rð
v
o
þ
c
o
Þ
r
f
ð
v
o
þ
c
o
Þ
C
11
and the next two equivalent expressions for the slow wave,
v
o
c
o
Þ
v
o
c
o
Þ
C
11
r
f
ð
M
ðr
f
J
11
þð
i
m=oÞ
R
11
Þð
a
1
¼
Q
11
;
b
1
¼
b
1
(9.50)
rð
v
o
c
o
Þ
r
f
ð
v
o
c
o
Þ
C
11
and for the two shear waves by
b
2
i
m
o
a
¼f
0
;
a
2
;
0
g;
b
¼f
0
;
b
2
;
0
g;
a
2
¼
J
22
þ
R
22
and
b
3
;
i
m
o
a ¼f
0
;
0
;
a
3
g;
b ¼f
0
;
0
;
b
3
g;
a
3
¼
J
33
þ
R
33
(9.51)
respectively.
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