Dynamic Poroelasticity - Continuum Mechanics of Anisotropic Materials - page 247

at very high porosity. The slow wave is at or near zero velocity at very high and
very low porosities and climes to a peak near 60% porosity. The nonzero shear
wave velocities are highest at the lowest porosity and decrease linearly to zero at the
highest porosity.
The experimental identification of the fast and slow waves is illustrated in
Figs. 9.4 and 9.5 . Figure 9.4 contains four panel figures with plots of amplitude
vs. time for a wave passing thought different media. Figure 9.5 contains four panel
figures with plots the frequency spectrum vs. time for a wave passing thought the
same four different media. For both figures, panels 1 and 4 the media in the
container is water, but the placement of the emitting and receiving transducers is
different. In panel 2 there is a fluid saturated porous specimen and in panel 3 there is
the porous specimen but no water. In each figure these various combinations of test
media and transducer setups are illustrated in the small cartoons to the left or right
of each plot of amplitude vs. time. In each cartoon the emitting and receiving
transducers are indicated as tubes (bottom and top of the cartoon) applied to the
specimen in the water or air filled container. In panel 1 the shape of the wave or
frequency spectrum is determined by its passage through the small space between
transducers. In panel 2 the shape of the wave or frequency spectrum is determined
by its passage through a fluid saturated porous specimen between transducers. In
panel 3 there is no water, just the porous specimen, so the shape of the wave or
frequency spectrum is determined by its passage through the unsaturated porous
specimen between transducers. Panel 4 is the same as panel 1 except the space
between the transducers is the same as if the specimen were there. Notice that, in
this case, the shape of the wave or frequency spectrum is almost the same as in
panel 1 but it is displaced to a greater time. It is only in the second panel that the
waveform shape and frequency spectrum are determined by passage through the
saturated porous specimen. In this panel one can see that the first part of the
waveform is similar to that of the fast wave and the last part is similar to that of
the slow wave. Panel 4 shows the passage of only the wave in the fluid because there
is no porous specimen in the container; this situation is similar to that of the slow
wave in the porous media shown in the second panel. Panel 3 shows the passage of a
wave propagating within an unsaturated porous specimen; this wave is similar to
the fast wave shown in the second panel. These data suggest that under the tested
conditions of porosity, the propagation of the fast wave is mainly related to the solid
phase of the medium, while the slow wave is characteristic of the fluid phase
(Cardoso et al. 2003 ).
A plot of the fast (top three curves) and slow (lower three curves) wave speeds as
a function of frequency for different degrees of anisotropy at a porosity of 50 % in
cancellous bone tissue is shown in Fig. 9.6 . In ultrasonic measuring systems the
viscous effects of the pore fluid damp out the genesis of the slow wave and its
potential observation below the critical frequency. The amplitude damping of both
waves also occurs at frequencies above the viscous frequency, which is 10 4 times
the critical frequency, making the observation of both waves above the viscous
frequency challenging (Cardoso et al. 2008 ).