Biomedical Engineering Reference
In-Depth Information
Theoretical velocity of longitudinal waves - isotropy
Phase velocity of shear waves - isotropy
4000
2000
P1 Fast Wave S1 Shear Wave
S2 Shear Wave
P2 Slow Wave 1800
1600
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0
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0 0 0 0 0 0 0
Porosity %
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0
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Porosity %
Fig. 9.2 A plot of the ultrasound wave velocities as a function of the porosity from the theoretical
model. P1 is the fast wave, P2 the slow wave, S1 and S2 are the two propagating shear waves,
while S3 and S4 are non-propagating shear waves with null velocity. These results are plotted for a
medium with isotropic fabric. The velocity of propagation for all directions is the same
Phase Velocity along the axes of symmetry - anisotropy
Phase Velocity along the axes of symmetry - anisotropy
4000
2000
P1 F1
P2 F1
P1 F2
P2 F2
P1 F3
P2 F3
S1 F1
S2 F1
S1 F2
S2 F2
S1 F3
S2 F3
1800
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0
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0 0 0 0 0 0 0
Porosity %
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0
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40 50
Porosity %
60
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100
Fig. 9.3 A similar plot to Fig. 9.2 , but for a medium with anisotropic fabric. Propagation along the
three different principal directions is shown
The six-ultrasound wave velocities as a function of the porosity from the model
are shown in Fig. 9.2 for an isotropic material. These six possible wave velocities
are the six roots of ( 9.42 ). In the figure the fast wave is denoted by P1 (P for
pressure), the slow wave by P2, S1, and S2 denote the two propagating shear waves
(S for shear), while S3 and S4 are non-propagating shear waves with zero velocity.
These results are plotted for a medium for the isotropic case, thus the velocity of
propagation is the same for all directions. In Fig. 9.3 , the six-ultrasound wave
velocities are plotted again as a function of porosity, but for an anisotropic medium.
Propagation along all three principal directions are shown for each wave as a solid,
dashed, and dotted line. In these two figures one can see that the response of the
compression or P waves is the most varied. The fast wave has a high velocity at low
porosity and that velocity drops to the velocity of sound in water, about 1,500 m/s,
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