are minor at all points after 1 km of travel distance. Initial differences between the profile
celerities in case II occur as a result of the shock in the dynamic wave solution for φ up to
0.5, and some differences persist for 5 km downstream. Similar profile celerity disagree-
ment is also evident in case III, but celerity convergence after shock attenuation is inexact,
with a slightly larger range remaining in the diffusion wave solution. With the modified
inertial diffusion coefficient (eq 12) all case III diffusion wave and dynamic wave profile
celerities converge by 6 km, the distance for shock attenuation below φ = 0.1. The large D in
case IV causes leading edge diffusion wave profile celerities to greatly exceed the dynamic
wave celerity. The shock persists for 17 km downstream, again delaying profile celerity
agreement. The profile celerity change in case IV with the inertial diffusion coefficient was
negligible. The celerity comparisons for case V are similar to those of case IV, with larger
initial dimensionless profile celerities in both solutions, and celerity agreement at all points
following shock attenuation at 17 km.
Traces on the x-t plane of selected wave profile points in Figure 3 help to visualize the
effects of profile celerity differences. The time scales used for each case are related by the
kinematic wave celerities given in Table 1. The dynamic wave f from the origin at t = 0,
termed the dynamic forerunner by Stoker (1957), carries the initial shock downstream at a
constant celerity c+. A positive value of F(x,t,φ) immediately behind the forerunner indi-
cates that a given φ is on the forerunner. The φ-traces that successively separate from the
Figure 3. Traces on the x-t plane of selected linear diffusion wave and dynamic wave profile points and
the dynamic forerunner f for all cases plus case III with the modified inertial diffusion coefficient. Note
the changes in the time scale between panels.
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