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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