V2
HT =
+y+z
(2.166)
2g
or
Q2
HT =
+y+z
(2.167)
2gA 2
Then assuming a wide channel for simplicity
q2 dy dy dz
dHT
+ 3
=
+
(2.168)
dx
gy dx dx dx
The term dHT/dx is the slope of the energy gradeline Sf, by assumption. For short distances and
small changes in y the energy gradient can be evaluated using the Manning or Chezy velocity
equations.
When Chezy's equation (Equation 2.84) is used the expression for dHT/dx is
q2
dHT

= Sf = 2 3
(2.169)
dx
Cy
The term dy/dx is the slope of the water surface Sw, and dz/dx is the bed slope So. For steady
flow, the bed slope is (from Equation 2.84)
 q2
So = 2 3
(2.170)
Co y o
where the subscript "o" indicates the steady uniform flow values.
When Equations 2.169 and 2.170 are substituted into Equation 2.168, the familiar form of the
gradually varied flow equation is obtained:
C 2 y
3
1  o o
C y
dy
= So
(2.171)
3
dx
yc
1 y
If Manning's equation is used to evaluate Sf and So, Equation 2.171 becomes
2
10 / 3
1  n y o
no y
dy
= So 3
(2.172)
dx
y
1 c
y
2.51