The hydraulic geometry relations of alluvial streams are useful in river engineering. The

forerunner of these relations are the regime theory equations of stable alluvial canals (see for

example, Kennedy 1895, Lacy 1930, and Leliavsky 1955). Hydraulic geometry relations were

developed by Leopold and Maddock (1953) for different regions in the United States and for

different types of rivers. In general the hydraulic geometry relations are stated as power

functions of the discharge:

W = a Qb

(5.4)

yo = c Qf

(5.5)

V = k Qm

(5.6)

QT = p Qj

(5.7)

Sf = t Q z

(5.8)

n = r Qy

(5.9)

where:

W

=

Channel width

yo

=

Channel depth

V

=

Average velocity of flow

QT

=

Total bed sediment load

Sf

=

Friction slope

n

=

Manning's roughness coefficient

Q

=

Discharge as defined in the following paragraphs

The coefficients a, c, k, p, t, r and exponents b, f, m, j, z, y in these equations are determined

from analysis of available data on one or more streams. From the continuity equation (Q =

WyoV), it is seen that

a x c xk = 1

(5.10)

and

b+ f +m=1

(5.11)

Leopold and Maddock (1953) have shown that in a drainage basin, two types of hydraulic

at-a-station; and (2) relating these variables to the discharges of a given frequency of

used Qs, the suspended sediment transport rate. The former are called **at-a-station**

relationships and the latter **downstream **relationships. The distinction between at-a-station

and downstream hydraulic geometry relations is illustrated in Figures 5.19 and 5.20.

Figures 5.19 and 5.20 illustrate how the hydraulic relations at-a-station and in the downstream

direction may be different from one basin to another. For example, the width and depth at-a-

station do not change very much in Basin A. The width to depth ratio is almost constant but the

velocity increases, as it must, as the discharge increases at-a-station. In Basin B the width to

depth ratio decreases with an increase in discharge. That is, the width changes very little but

the depth increases significantly with discharge at-a-station. Note, that in both basins, width

5.24

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