We use the same development as above to obtain the dimensionless profile celerity for a
fixed value of φ on the diffusion wave profile, which in simplified form is
-Gt /Gx
x
cdif =
=
(23)
~
.
t)2
1/ 2
πt
x + ckt
(x + ck
ck
ckt2 - ck
erfc
exp
D
(4Dt)1/ 2
4Dt
The forms of the linear diffusion wave and dynamic wave solutions have little apparent
resemblance to each other. The kinematic wave celerity appears in each term of the diffu-
sion wave solution (eq 20), but not in the dynamic wave solution (eq 14). The dynamic
wave celerities in the downstream c+ and upstream c directions both appear in z of the
dynamic wave solution, and c+ provides an upper bound on the speed of disturbances
moving downstream. There is no similar restriction on disturbances traveling downstream
in the diffusion wave solution. The diffusion coefficient D is an important parameter of the
diffusion wave solution, while ηc0 and η are corresponding parameters of the dynamic
wave solution. The Froude number F0 appears often in the dynamic wave solution, but is
absent from the diffusion wave solution unless the inertial diffusion coefficient is used.
NONLINEAR MONOCLINAL AND MONOCLINALDIFFUSION WAVES
Nonlinear monoclinal wave solutions that are analogous to the linear dynamic wave
and diffusion wave solutions will now be developed and compared. The term "monoclinal
wave" refers to the classical solution, and "monoclinaldiffusion wave" is the solution de-
veloped after neglecting the inertia terms of momentum equation (eq 2). A monoclinal
wave profile does not exist for the kinematic wave equation because diffusion is not
present to balance nonlinear steepening. For completeness we summarize the monoclinal
wave development of Whitham (1974), emphasizing the contribution of inertia to the solu-
tion.
We seek a solution that depends on a single variable X = x Ut, where U is the constant
profile celerity, and rewrite eq 1 and 2 as
-Uy′ + vy′ + yv′ = [y(U - v)]′ = 0
(24)
v2
(v - U )v′ + gy′ = gS0 - 2
(25)
Cy
*
respectively, where Chezy conveyance coefficients, dimensionless C and dimensional C,
*
are related as
C
C =
.
*
g
The first term on the left side of eq 25 follows from the inertia terms of eq 2. Integration of
eq 24 yields
y(U v) = B
(26)
where the constant of integration B represents a wave overrun unit discharge.
The profile celerity U is obtained from eq 26 using the flow states on either side of the
wave as
1/ 2
yf /2 - y0/2
(
)
3
3
v y - v0 y0
2
v0
U= f f
= g C* S0
= vf + v + v
2
(27)
y -y
yf - y0
0
0
f
f
7
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