dynamic forerunner indicate a progressively diminishing shock amplitude. Afterward, f sep-
arates from the profile and no longer contributes to the solution. Overall, the x-t traces indi-
cate general agreement between the dynamic wave and diffusion wave solutions following
attenuation of the initial shock.
The case I traces of all corresponding dynamic wave and diffusion wave profile points are
essentially identical. The dynamic forerunner f in case I begins to separate from the profile in
the first 5 km, and leads by increasing distances farther downstream. At early time in case II
the diffusion profile celerity of the leading edge exceeds that of f, causing minor differences
between the traces. These differences disappear about 5 km from the origin, and thereafter the
traces of all points are identical. The forerunner and profile in case II progressively separate
beyond 10 km from the origin. The high F0 in case III reduces the rate of spread of the dynamic
profile, causing the diffusion traces of the front half to lead and of the back half to lag the
dynamic traces, and these trends persist. With the inertial diffusion coefficient the traces in
case III agree closely beyond 8 km, and the profile progressively separates from f beyond 15
km. In case IV a shock amplitude of 0.1 is carried by the forerunner for 16 km, with f separat-
ing from the profile beyond 25 km. Excess profile spreading in the diffusion solution, caused
by initial celerity differences and high diffusion coefficient, persists throughout and was not
greatly improved by the inertial diffusion coefficient. The case V dynamic wave and diffusion
wave traces compare similarly to those of case IV, except that differences higher on the profile
do not persist.
We can compare dynamic wave and diffusion wave profiles through time on a single figure
by using a moving x-coordinate system with origin at φ = 0.5. Comparisons of these origin x-
values through time (xdyn, xdif) are given in Figure 4. The origin traces of the dynamic wave
model were similar for cases with the same kinematic wave celerity. The absolute value of the
difference in origin position between the dynamic wave and diffusion wave models was al-
Figure 4. Comparison between origin positions of the moving coordinate system in the linear diffusion
wave (xdif) and dynamic wave (xdyn) models for all cases.
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