Biomedical Engineering Reference
In-Depth Information
Q
ik
rd
ik
v
2
C
ik
r
f
d
ik
v
2
ð
Þ
a
k
þð
Þ
b
k
¼
0
;
(9.25)
b
k
¼
i
m
o
C
ki
r
f
d
ik
v
2
Mn
k
n
i
r
f
J
ik
v
2
R
ik
v
2
ð
Þ
a
k
þ
0
;
(9.26)
where the following notation has been introduced:
Z
ijkm
n
m
n
j
;
Q
ik
¼
C
ik
¼
M
ij
n
j
n
k
:
(9.27)
Q
is the acoustical tensor from elastic wave propagation and
C
represents the
interaction of the displacement fields
u
and
w
. Rewritten in matrix notation
equations (
9.25
) and (
9.26
) take the form
v
2
1
r
f
v
2
1
ð
Q
r
Þ
a
þð
C
Þ
b
¼
0
;
(9.28)
v
2
i
m
o
C
T
r
f
v
2
1
ð
Þ
a
þ
Mn
n
r
f
J
þ
R
b
¼
:
0
(9.29)
These equations represent an eigenvalue problem, the squares of the wave
speeds
v
2
representing the eigenvalues and the vectors
a
and
b
representing the
eigenvectors. Rewriting (
9.28
) and (
9.29
) as a scalar 6 by 6 matrix formed from the
four 3 by 3 matrices that appear in (
9.28
) and (
9.29
) and also representing the two
3D vectors
a
and
b
as one 6D vector, the following representation is obtained:
a
b
¼
v
2
1
r
f
v
2
1
Q
r
C
v
2
0
:
(9.30)
C
T
i
m
o
r
f
v
2
1 Mn
n
r
f
J
þ
R
Please note that the scalar 6 by 6 matrix operating on the vectors
a
and
b
is
symmetric; the 3 by 3 matrices along the diagonal are symmetric and, even though
C
is not symmetric, having the transpose of
C
is the lower left 3 by 3 matrix and the
3 by 3 matrix
C
itself in the upper right makes the 6 by 6 matrix symmetric. Since
the right hand side of this linear system of equations is a zero 6D vector, it follows
from Cramer's rule that, in order to avoid the trivial solution, it is necessary to set
the determinant of the 6 by 6 matrix equal to zero, thus
¼
0
:
v
2
1
C r
f
v
2
1
Q r
v
2
(9.31)
i
m
o
C
T
r
f
v
2
1 Mn
n
r
f
J
þ
R
This condition will provide six (four nonzero) values of the possible squares of
the wave speeds
v
2
in the direction
n
. In each direction there will be four nontrivial
wave speeds, two representing shear waves and one each representing the Biot fast
and slow waves. For each value of a squared wave speed
v
2
substituted back into
(
9.30
), two 3D vectors
a
and
b
will be determined subject to the condition that they
are both unit vectors. The ease with which these calculations are described does not
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