amplitude 0.1y0, representing fully diffused linear wave profiles at large times, are also
given for each case in Figure 5. The most prominent feature of these solutions is the low
profile slope, and the extended time indicated for the linear wave to attain this profile. Only
the higher Froude number linear profiles at 36,000 s even approach the small-amplitude
monoclinal profiles.
ANALYSIS OF THE MONOCLINAL SOLUTIONS
The numerator of the depth gradient in eq 34 does not change sign along the profile, but
a sign change in the denominator can result from the presence of inertia. This sign change
indicates a monoclinal wave profile that turns back upstream, becomes unstable, and forms
a shock. In contrast, the monoclinaldiffusion profile cannot become unstable because ycr is
not present. Initial monoclinal wave instability occurs at the toe of the profile when the
denominator goes to zero, and
1/ 3
B2
y = y0 = ycr
(37)
.
g
Following Hunt (1987) we evaluate the stability limit eq 37 using eq 28, and after some
algebra, the depth ratio across the wave yr is obtained as a function of F0
-2
1 1
1/ 2
yr = - +
+ F0
(38)
.
2 4
The depth ratio range of stable monoclinal wave profiles decreases as Froude number in-
creases toward 2. Conversely F0 can be obtained as a function of yr with eq 35 as
(yr1/2 + 1) .
=
F0
(39)
yr
The stability limit in eq 37, evaluated using eq 26, yields U = v0+c0 as the maximum
profile celerity prior to instability. The range of stable profile celerities, bounded below by
the celerity of lower-order kinematic waves and bounded above by the celerity of higher-
order dynamic waves, can be written in dimensionless form as
U 2
1
1<
< 1 + .
(40)
ck 3 F0
Dimensionless profile celerity and overrun discharge given in Figure 6 increase continu-
ously with wave amplitude from lower limits of 1 and 0.5, respectively, at yr = yf/y0 = 1. The
upper limits are indicated for selected values of F0 by dots that follow from eq 40.
The difference between the monoclinal and monoclinaldiffusion solutions can be para-
meterized by considering the bracketed term in eq 33 rewritten using dimensionless vari-
ables
2
F0 yr
1/2
3
yr + 1
y
B2
(41)
1-
= 1 - cr = 1 -
[(yr - 1)y + 1]3
y
g y3
~
where dimensionless depth y , defined analogously to φ in eq 13, varies between 0 and 1.
~
~
14
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