Biomedical Engineering Reference
In-Depth Information
and for the two shear waves by
a
¼f
0
;
2
b
2
;
0
g;
b
¼f
0
;
b
2
;
0
g;
and
a
¼f
0
;
0
;
2
b
3
g;
b
¼f
0
;
0
;
b
3
g;
respectively.
Problems
9.6.1. Which wave modes propagate in a porous medium saturated with a very
low-density gas (i.e. He)? How many wave modes of each type propagate?
9.6.2. Which wave modes propagate in a porous medium when the porosity of the
porous medium is zero? How many wave modes of each type propagate?
9.6.3. Using Fig.
9.6
please explain what is meant by dispersion of fast and slow
waves plotted vs. frequency.
9.6.4. Explain the rise and fall of dispersion described in problem 9.6.3 in terms the
properties of the fluid (specify frequency range).
9.6.5. Using symbolic algebraic software show that the vanishing of the first of the
determinants (
9.42
) provides the two roots given by (
9.45
) and (
9.46
).
9.6.6. Using symbolic algebraic software show that the vanishing of the second and
third of the determinants (
9.42
) provides the two roots given by (
9.47
) and
two zero roots as well.
9.6.7. Explain how the amplitudes
a
and
b
of the two waves are determined once
the wave velocity is calculated in a typical problem of the type of problems
9.4.1 or 9.4.2.
9.6.8. Determine the wave velocities and the polarization vectors or eigenvectors
associated with a harmonic wave propagating along an axis of material
symmetry
e
1
in cancellous bone with a porosity of 0.2. Use the properties
specified for cancellous bone in Example 9.6.1.
9.6.9. Determine the wave velocities and the polarization vectors or eigenvectors
associated with a harmonic wave propagating along an axis of material
symmetry
e
1
in cancellous bone with a porosity of 0.5. Use the properties
specified for cancellous bone in Example 9.6.1.
9.7 Propagation of Waves in a Direction That Is Not
a Principal Direction of Material Symmetry; Quasi-Waves
In this section the theoretical framework for poroelastic waves is extended to the
propagation of waves along a general direction in orthotropic porous media, not
waves propagating the in the specific direction of a material symmetry axis consid-
ered in the previous section, which are called pure waves to distinguish them from
the wave types considered in the present section. The kind of waves under consid-
eration here are called quasi-longitudinal waves or quasi-shear waves as their
amplitudes or polarization vectors are neither parallel nor perpendicular to the
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