Linear dynamic wave and diffusion wave solution comparisons cannot fully resolve the
relationship between these models because large-amplitude flow increases of practical
interest must be described by the nonlinear equations. This deficiency can be remedied in
part by considering the monoclinal rising wave, a nonlinear dynamic wave analytical solu-
tion (Chow 1959, Henderson 1966, Whitham 1974, Hunt 1987, and Agsorn and Dooge
1991). The monoclinal wave profile is an arbitrarily large transition between low steady,
uniform flow downstream and high steady flow upstream. This profile represents the bal-
ance between nonlinear wave steepening and diffusion, and has a known constant celerity
that increases with wave amplitude and the kinematic wave celerity. A comparison of a
corresponding nonlinear diffusion wave equation solution with the monoclinal wave
would identify temporally persistent inertial effects. The monoclinal wave solution does
not describe profile development nor provide the time and travel distance required to
attain a steady form. However, the combination of linear wave and monoclinal wave analy-
ses would quantify most aspects of relative dynamic wavediffusion wave solution
behavior.
The purpose of this report is to utilize analytical solutions to better understand the struc-
ture and relative behavior of the dynamic wave and diffusion wave unsteady river flow
models. An abrupt flow increase between initial and final steady flows is used as an up-
stream boundary condition to maximize the contribution of inertia. We compare linear
dynamic wave and diffusion wave solutions in a series of subcritical flow case studies.
Equations for the celerity of points along each profile are derived for comparison and to
explore the relationships between these profile celerities and the dynamic wave and kine-
matic wave celerities. This development provides the capability to trace selected profile
points on the xt plane, another means to compare the solutions. We also compare linear
wave profiles to depict relative behavior through time, and give small-amplitude mono-
clinal profiles to assess progress toward equilibrium. Our nonlinear monoclinal wave analysis
uses the same case studies as for linear waves, but considers a range of wave amplitudes. The
nonlinear diffusion wave equations are solved to obtain monoclinal-diffusion profiles
for comparison with the monoclinal wave. Relative steepening near the leading edge of the
monoclinal profile is caused by flow inertia that persists through time. Nonlinear effects
increase with wave amplitude, progressively separating monoclinal profile shapes, celeri-
ties, rating curves, and the Froude numbers and flow energy gradients along the profile
from those of linear waves. Dimensionless dependent variables of the linear and monocli-
nal solutions provide ease of comparison, while dimensional independent variables dis-
tance and time complement physical intuition. For generality we also develop and com-
pare fully dimensionless monoclinal and monoclinaldiffusion profiles.
LINEAR RIVER WAVE EQUATIONS
The continuity and momentum equations of unsteady flow in a wide rectangular open
channel with no lateral inflow or outflow are well known (Stoker 1957, Mahmood and
Yevjevich 1975):
y
y
v
+v +y =0
(1)
x
x
t
y
v
v
+v
+g
+ g(Sf - So ) = 0
(2)
t
x
x
where depth y and cross-sectional average velocity v are dependent variables, Sf is the flow
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