Biomedical Engineering Reference
In-Depth Information
This vector
a
makes an angle of 104
with the direction of propagation
n
.This
vector
b
makes an angle of 107
with the direction of propagation
n
. Thus this is a
quasi-shear wave. For the 893 m/s wave speed the polarization vectors are
a
¼f
;
:
;
:
g
a
1
;
b
¼f
:
;
:
;
:
g
a
1
:
1
2
30
11
9
11
08
12
73
17
55
This vector
a
makes an angle of 129
with the direction of propagation
n
.This
vector
b
makes an angle of 110
with the direction of propagation
n
. Thus this is a
quasi-longitudinal slow wave.
Problems
9.7.1. What is the word used to describe waves composed of a mixture of longitu-
dinal and wave modes?
9.7.2. For the mixed waves of the previous question, what is the particle's polari-
zation vector orientation relative to the vector representing the direction of
wave propagation?
9.7.3. Determine the wave velocities and the polarization vectors or eigenvectors
associated with a harmonic wave propagating in a direction that is not an axis
of material symmetry in cancellous bone with a porosity of 0.2. Let the
direction that is not an axis of material symmetry be
n
¼
{1, 1, 1}(1/
3).
√
Use the properties specified for cancellous bone in Example 9.4.1.
References
Biot MA (1941) General theory of three-dimensional consolidation. J Appl Phys 12:155-164
Biot MA (1955) Theory of elasticity and consolidation for a porous anisotropic solid. J Appl Phys
26:182-185
Biot MA (1956a) Theory of propagation of elastic waves in a fluid saturated porous solid I low
frequency range. J Acoust Soc Am 28:168-178
Biot MA (1956b) Theory of propagation of elastic waves in a fluid saturated porous solid II higher
frequency range. J Acoust Soc Am 28:179-191
Biot MA (1962a) Mechanics of deformation and acoustic propagation in porous media. J Appl
Phys 33:1482-1498
Biot MA (1962b) Generalized theory of acoustic propagation in porous dissipative media. J Acoust
Soc Am 28:1254-1264
Cardoso L, Cowin SC (2011) Fabric dependence of quasi-waves in anisotropic porous media.
J Acoust Soc Am 129(5):3302-3316
Cardoso L, Cowin SC (2012) Role of the structural anisotropy of biological tissues on poroelastic
wave propagation. Mech Mater 44:174-188
Cardoso L, Teboul F, Sedel L, Meunier A, Oddou C (2003) In vitro acoustic waves propagation in
human and bovine cancellous bone. J Bone Miner Res 18(10):1803-1812
Cardoso L, Meunier A, Oddou C (2008) In vitro acoustic wave propagation in human and bovine
cancellous bone as predicted by the Biot's theory. J Mech Med Biol 8(2):1-19
Cowin SC (1985) The relationship between the elasticity tensor and the fabric tensor. Mechanics
of Materials 4: 137-47.
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