1

β′ =

y

2

(2.105)

y' v dy

o

V (y o - y′)

2

If we substitute Equations 2.78 and 2.104 into Equation 2.107 and integrate, the result is the

y 2

2y′

yo

yo

1

β′ =

o

In - 2 In + 2 -

(2.106)

y′

2

y o - y′

y′

yo

y

In 11.11 o

ks

Similarly, the energy coefficient for a vertical section unit width is

1

α′ =

yo

v 3 dy

(2.107)

y′

V 3 (y o - y′)

or

y 3

6y′

2

yo

yo

yo

1

α′ =

In - 3

In

+ 6 In - 6 +

o

(2.108)

y′

y′

y o - y′

2

y′

yo

y

In 11.11 o

ks

These equations (Equations 2.106 and 2.108 are rather complex, so a graph of α′ and β′ vs yo/ks

has been prepared. The relations are shown in Figure 2.12.

For the entire river cross-section (shown in Figure 2.13) Equation 2.45 can be written

[

0 v dy dw ]

1

y

W

3

α=

(2.109)

o

0

3

Q

A

A

where W is the top width of the section, w is the lateral location of any vertical section, yo is the

depth of flow at location w, and v is the local velocity at the position y, w. The total discharge is Q

and the total cross-sectional area is A.

2.29

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