Nonetheless,
we conclude this section by giving predictions for
a granular solid having φL = φfp and φR = 0.99 to illustrate
how reflected waves resulting from spatially decreasing
porosity may also have important combustion consequences.
4.1 Case I
Figure 4 shows predictions in the piston-attached frame for
the evolution of reflected and transmitted waves due to the
nonlinear interaction of an incident compaction wave with a
region of steeply increasing porosity. For this case, the material
to the left of the interface is almost fully consolidated
(φL = 0.99) in its initial state, whereas the loose material
to the right is at the free-pour volume fraction for granular
HMX(φR = φfp ≡ 0.655). The incidentwave is propagating
from left to right in the figure, and is supported by a piston
having constant speed up = 100 m/s located at ξ = 0m.
Figure 4a gives the solid volume fraction history for approximately
0.9 ≤ t ≤ 19ms following impact; earlier profiles
for the incident wave are omitted as they are structurally no
different from those shown in the figure. The piston generates
a compaction wave having a steady structure that propagates
at speed D = 2, 590 m/swith respect to the laboratory frame.
Thiswave speed is less than the ambient acoustic speed of the
homogeneous solid (2,765 m/s) due to compaction induced
dissipation; as such, this wave structure does not possess a
shock, but is continuous having width δ = 2.72mm. Because
the left region initially has little porosity, only a small amount
of compaction occurs across the wave resulting in complete
material consolidation (i.e., φ = 1). There is predicted a
continuous increase in solid density, velocity, pressure, and
temperature across the wave, as seen in Fig. 4b–e.When the
incident wave encounters the loose material, part of its compaction