The maximum E is then obtained by substituting eq 46 into 45 and simplifying
(
)3
yr - 1
3/2
4
=
Emax
.
(
)
(47)
27 yr (yr - 1)2 yr /2 - 1
1
The location and value of Emax are given in Figure 18. Emax increases nearly linearly over
the yr range, and its location rapidly approaches the leading edge as yr increases from 1.
Both energy gradient and Froude number vary continuously along the monoclinal wave
profile, with amplitudes proportional to yr. Froude numbers that exceed 1 can occur on
stable profiles when F0 is large, as in case III.
CONCLUSIONS
The presence or absence of the inertia terms distinguishes dynamic wave and diffusion
wave models of unsteady river flow. Analytical solutions of the linear dynamic wave and
diffusion wave equations were compared for a small instantaneous increase from an initial
steady, uniform flow condition throughout the channel to a higher steady flow velocity,
and depth at the upstream boundary. The comparison used case studies that represented a
wide range of flow depth, velocity, channel slope, and wave diffusion coefficient, and
spanned the range of subcritical Froude numbers. Analytical solutions were also obtained
for nonlinear monoclinal wave and monoclinaldiffusion wave equations, and compari-
sons were again made using the same case studies with a wide range of wave amplitudes.
The linear solution comparisons focused on the evolution of the dynamic wave and diffu-
sion wave profiles with time and distance, while the nonlinear solution comparisons inves-
tigated the effects of wave amplitude and persistent inertia.
The linear solution comparisons included the celerity and the trace on the x-t plane of
selected points from the wave profile, and the complete dynamic and diffusion profiles at
selected times. The initial shock traveled downstream with the dynamic forerunner at c+,
the maximum celerity in subcritical flow. A limitation of the diffusion wave solution is
premature replacement of this shock by a profile having a range of point celerities that
exceed c+ near the leading edge. The diffusion wave and dynamic wave profiles remain
distinct until after the shock attenuates and their profile celerities converge. In cases where
the Froude number approaches 1, this convergence requires a diffusion coefficient cor-
rected for inertia. Points near the leading edge of each profile travel faster than those higher
on the profile, causing diffusion. These differences diminish over time and distance, and all
dynamic and diffusion profile celerities asymptotically approach that of a kinematic wave.
General agreement of the linear diffusion wave and dynamic wave solutions after attenua-
tion of the shock is indicated in all cases by common profile celerities and x-t traces, and by
profile covergence with time. The role of the characteristics in the linear solution becomes
negligible following shock attenuation at time and distance scales that increase with both η
and the Froude number.
The analysis of the nonlinear monoclinal wave solutions linked important inertial effects
at large time with increasing Froude number. As F0 increases corresponding monoclinal
and monoclinaldiffusion profiles separate near the leading edge, and these differences
increase with wave amplitude. Monoclinal profile instability occurs at higher F0, but mono-
clinaldiffusion profiles are always stable. General dimensionless monoclinaldiffusion
profiles exist for each depth ratio with distance scaled by y0/S0, but monoclinal profiles
deviate from these general profiles at higher F0. Several effects of wave amplitude on mon-
oclinal waves were also identified. The celerity of a small-amplitude monoclinal wave
equals that of a kinematic wave, and it increases continuously from this lower limit with
23
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