dy S_{o}  S_{c} F ^{2}
___ _{=} ___________
dx 1  F ^{2}
 (6) 
which is strictly valid for the following condition: P /T
= P_{c} /T_{c} . This latter
condition is generally satisfied in a hydraulically wide channel,
for which T is asymptotically equal to P.
For ease of expression, the flowdepth gradient is renamed S_{y} = dy/dx. Solving for Froude number from Equation 6:
S_{o}  S_{y}
F ^{2} = _________
S_{c}  S_{y}
 (7) 

Since F ^{2} > 0, the flowdepth gradient must satisfy the following inequalities:
S_{o} ≥ S_{y} ≤ S_{c}
 (8) 

S_{o} ≤ S_{y} ≥ S_{c}
 (9) 

which effectively limits the flowdepth gradient to values outside the range encompassed by S_{o} and S_{c}. Furthermore, Equation 6 can be alternately expressed as follows:
S_{y} ( S_{o} / S_{c} )  F ^{2}
___ _{=} ______________
S_{c} 1  F ^{2}
 (10) 

Equation 10 is the GVF equation in terms of bed slope S_{o} , critical slope
S_{c} , and Froude number F. The bed slope could be positive
(steep, critical, or mild), zero (horizontal), or negative (adverse). The critical
slope (Equation 5) and Froude number squared (Equation 3) are always positive.
3. CLASSIFICATION OF WATERSURFACE PROFILES
Equation 10 is used to develop a classification of watersurface profiles based solely
on the three dimensionless parameters: S_{y} /S_{c} ,
S_{o} /S_{c} , and F. For the sake of completeness,
subcritical flow is defined as that for which the flow depth is greater than the critical
depth (F ^{2} < 1) (Chow 1959; Henderson 1966). Paralleling this widely accepted
definition, subnormal flow is defined as that for which the flow depth is greater than
the normal depth [F ^{2} < S_{o} /S_{c} ].
Supernormal flow is defined as that for which the flow depth is smaller than
the normal depth [F ^{2} > S_{o} /S_{c} ]
(USDA SCS 1971). Table 1 shows the four (4) classes of watersurface profiles and the twelve (12)
possible profiles.
Table 1. Possible classes and types of watersurface profiles.

CLASS 1: SUBCRITICAL/SUBNORMAL

 Steep: S_{1}
 Critical: C_{1}
 Mild: M_{1}

CLASS 2A: SUPERCRITICAL/SUBNORMAL


CLASS 2B: SUBCRITICAL/SUPERNORMAL

 Mild: M_{2}
 Horizontal: H_{2}
 Adverse: A_{2}

CLASS 3: SUPERCRITICAL/SUPERNORMAL

 Steep: S_{3}
 Critical: C_{3}
 Mild: M_{3}
 Horizontal: H_{3}
 Adverse: A_{3}

A summary of the twelve possible watersurfaces profiles is shown in Table 2.
The classification follows directly from the governing equation (Equation 10).
It is seen that the general class of profile (Class 1, 2A, 2B, or 3)
determines the sign of S_{y} /S_{c} (Column 2) and,
thus, the classification of either backwater or drawdown (Column 3). Also, the general class
of profile determines the feasible range of S_{o} /S_{c}
(Column 4) and, thus, the existence of specific profiles types (Steep, Critical, Mild, Horizontal,
or Adverse) within each general type. Note that not all combinations
of S_{y} /S_{c} and
S_{o} /S_{c} are feasible.
Unlike the description available in standard references (Chow 1959; Henderson 1966),
the flowdepth gradient ranges (Table 2, Columns 7 and 8) are now complete for all
twelve watersurfaces profiles. Significantly, the flowdepth gradient S_{y} is
shown to be outside the range encompassed by S_{c} and S_{o} .
Figure 1 shows a graphical representation of flowdepth gradient ranges in the watersurface
profiles. The arrow shows the direction of computation. For instance, the depth gradient for
the S_{3} profile (supercritical/supernormal) decreases from S_{c}
(a finite positive value) to 0 (asymptotic to normal flow). Likewise, the depth gradient
for the C_{1} (subcritical/subnormal) and C_{3}
(supercritical/supernormal) profiles is constant and equal to S_{o} = S_{c} .
Online watersurface profile calculators are enabled in Table 2.
Table 2. Classification of watersurface profiles.
[Click on any profile type on Col. 9 to link to online watersurface profile calculator]

No. (1)
 S_{y} /S_{c} (2)
 Profile (3)
 S_{o} /S_{c} (4)
 Slope (5)
 Depth relations (6)
 S_{y} varies
 Profile type (9)

From (7)
 To (8)

1. SUBCRITICAL/SUBNORMAL FLOW^{ 1}: 1 > F ^{2} < S_{o} / S_{c}

1
 Positive
 Backwater
 > 1
 Steep
 y > y_{c} >
y_{n}
 S_{o}
 ∞
 S_{1}

2
 Positive
 Backwater
 = 1
 Critical
 y > y_{c} = y_{n}
 S_{o} = S_{c}
 S_{o} = S_{c}
 C_{1}

3
 Positive
 Backwater
 < 1; > 0
 Mild
 y > y_{n}
= y_{c}
 S_{o}
 0
 M_{1}

2A. SUPERCRITICAL/SUBNORMAL FLOW ^{2}: 1 < F ^{2} < S_{o} / S_{c}

4
 Negative
 Drawdown
 > 1
 Steep
 y_{c} > y >
y_{n}
  ∞
 0
 S_{2}

2B. SUBCRITICAL/SUPERNORMAL FLOW ^{3}: 1 > F ^{2} > S_{o} / S_{c}

5
 Negative
 Drawdown
 < 1; > 0
 Mild
 y_{n} > y >
y_{c}
  ∞
 0
 M_{2}

6
 Negative
 Drawdown
 = 0
 Horizontal
 y > y_{c} ;
y_{n} → ∞
  ∞
 S_{o} = 0
 H_{2}

7
 Negative
 Drawdown
 < 0
 Adverse
 y > y_{c} ;
y_{n} → ∞
  ∞
 S_{o} < 0
 A_{2}

3. SUPERCRITICAL/SUPERNORMAL FLOW ^{4}: 1 < F ^{2} > S_{o} / S_{c}

8
 Positive
 Backwater
 > 1
 Steep
 y_{c} > y_{n}
> y
 S_{c}
 0
 S_{3}

9
 Positive
 Backwater
 = 1
 Critical
 y_{c} = y_{n} >
y
 S_{o} = S_{c}
 S_{o} = S_{c}
 C_{3}

10
 Positive
 Backwater
 < 1; > 0
 Mild
 y_{n} > y_{c}
> y
 S_{c}
 ∞
 M_{3}

11
 Positive
 Backwater
 = 0
 Horizontal
 y_{c} > y ;
y_{n} → ∞
 S_{c}
 ∞
 H_{3}

12
 Positive
 Backwater
 < 0
 Adverse
 y_{c} > y ;
y_{n} → ∞
 S_{c}
 ∞
 A_{3}

^{1} Given that S_{o} /S_{c} > F ^{2} > 0, no horizontal or adverse profiles are possible in subcritical/subnormal flow.
^{2} Given that S_{o} /S_{c} > 1, no critical, mild, horizontal or adverse profiles are possible in supercritical/subnormal flow.
^{3} Given that S_{o} /S_{c} < 1, no steep or critical profiles are possible in subcritical/supernormal flow.
^{4} Given that S_{o} /S_{c} is not limited, all five profiles are possible in supercritical/supernormal flow.


Figure 1. Graphical representation of flowdepth gradient ranges in watersurface profiles.
4. SUMMARY
The gradually varied flow equation is expressed in terms of the critical
slope S_{c} . In this way, the flow depth gradient dy/dx
is shown to be strictly limited to values outside of the range encompassed
by S_{o} and S_{c}. This completes the definition of
depthgradient ranges for all watersurface profiles. For instance,
the flowdepth gradient for the S_{3} profile decreases
from S_{c} (a finite positive value) to 0 (asymptotic to
normal depth). Likewise, the flow depth gradient for the C_{1}
and C_{3} profiles is constant and equal to S_{o} = S_{c}.
Table 3 shows a summary of watersurface profiles. Online calculators are provided to round up the experience.
Table 3. Summary of watersurface profiles.
[Click on any image to enlarge]

Family
 Character
 Rule
 S_{o} > S_{c}
 S_{o} = S_{c}
 S_{o} < S_{c}
 S_{o} = 0
 S_{o} < 0

1
 Retarded (Backwater)
 1 > F ^{2} < S_{o} / S_{c}
 S_{1} 
C_{1} 
M_{1} 
 
 
2A
 Accelerated (Drawdown)
 1 < F ^{2} < S_{o} / S_{c}
 S_{2}
 
 
 
 

2B
 Accelerated (Drawdown)
 1 > F ^{2} > S_{o} / S_{c}
 
 
 M_{2}
 H_{2}
 A_{2}

3
 Retarded (Backwater)
 1 < F ^{2} > S_{o} / S_{c}
 S_{3}
 C_{3}
 M_{3}
 H_{3}
 A_{3}

REFERENCES
Chow, V. T. (1959). Openchannel hydraulics. McGrawHill, New York.
Henderson, F. M. (1966). Open channel flow. MacMillan, New York.
USDA Soil Conservation Service. (1971). Classification system for varied flow in prismatic channels. Technical Release No. 47 (TR47), Washington, D.C.
