The second integral term in brackets in Equation 2.109 can be written
y′ v dy = α′ V (y o - y′)
y
3
3
(2.110)
o
Here α′ is the energy coefficient for the vertical section dw wide and yo deep, V is the
depth-averaged velocity in this vertical section and y′ = ks/30.2.
Now, Equation 2.109 can be written
A2 W
α = 3 0 α′ V 3 (y o - y′) dw
(2.111)
Q
Except for cases of low flow in gravel bed rivers, the term y′ is very small compared to yo so
A2 W
α = 3 0 α′ V 3 y o dw
(2.112)
Q
The discharge at a river cross-section is determined in the field by measuring the local depth and
two local velocities at each of approximately 20 vertical sections. In accordance with this general
stream gaging procedure, Equation 2.112 could be written as
A2
α = 3 i α′ v 3 y oi ∆w i
i
i
Q
or
A2
α = 3 i Σ α′ v i2 ∆Qi
(2.113)
i
Q
Here, the subscript i refers to the i-th vertical section, and ∆Qi is the river discharge associated
with the i-th vertical or
∆Qi = Vi y oi ∆w i
In a similar manner, the expression for β is
A
′
i βi Vi ∆Qi
β=
(2.114)
Q2
Now, with Equations 2.113 and 2.114, and Figure 2.12 we are in a position to compute α and β
for any river cross-section given the discharge measurement notes. A calculation example is
presented in Section 2.14 (SI) and 2.15 (English). It is important to recognize that for river flows
over floodplains, the correction factors α and β can be significantly larger than α′ and β′.
2.31
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