v_m = 1.312 v_b,
and from this the following values of the mean velocity are
obtained:--
+-----------------------+---------------+-------------+
| |Bottom Velocity|Mean Velocity|
| | = v_b. | = v_m. |
+-----------------------+---------------+-------------+
| 1. Soft earth | 0.25 | .33 |
| 2. Loam | 0.50 | .65 |
| 3. Sand | 1.00 | 1.30 |
| 4. Gravel | 2.00 | 2.62 |
| 5. Pebbles | 3.40 | 4.46 |
| 6. Broken stone, flint| 4.00 | 5.25 |
| 7. Chalk, soft shale | 5.00 | 6.56 |
| 8. Rock in beds | 6.00 | 7.87 |
| 9. Hard rock. | 10.00 | 13.12 |
+-----------------------+---------------+-------------+
The following table of velocities which should not be exceeded in
channels is given in the _Ingenieurs Taschenbuch_ of the Verein
"Hütte":--
+--------------------------------+---------+---------+---------+
| | Surface | Mean | Bottom |
| |Velocity.|Velocity.|Velocity.|
+--------------------------------+---------+---------+---------+
| Slimy earth or brown clay | .49 | .36 | .26 |
| Clay | .98 | .75 | .52 |
| Firm sand | 1.97 | 1.51 | 1.02 |
| Pebbly bed | 4.00 | 3.15 | 2.30 |
| Boulder bed | 5.00 | 4.03 | 3.08 |
| Conglomerate of slaty fragments| 7.28 | 6.10 | 4.90 |
| Stratified rocks | 8.00 | 7.45 | 6.00 |
| Hard rocks | 14.00 | 12.15 | 10.36 |
+--------------------------------+---------+---------+---------+
§ 127. _Regime of a River Channel._--A river channel is said to be in
a state of regime, or stability, when it changes little in draught or
form in a series of years. In some rivers the deepest part of the
channel changes its position perpetually, and is seldom found in the
same place in two successive years. The sinuousness of the river also
changes by the erosion of the banks, so that in time the position of
the river is completely altered. In other rivers the change from year
to year is very small, but probably the regime is never perfectly
stable except where the rivers flow over a rocky bed.
[Illustration: FIG. 131.]
If a river had a constant discharge it would gradually modify its bed
till a permanent regime was established. But as the volume discharged
is constantly changing, and therefore the velocity, silt is deposited
when the velocity decreases, and scour goes on when the velocity
increases in the same place. When the scouring and silting are
considerable, a perfect balance between the two is rarely established,
and hence continual variations occur in the form of the river and the
direction of its currents. In other cases, where the action is less
violent, a tolerable balance may be established, and the deepening of
the bed by scour at one time is compensated by the silting at another.
In that case the general regime is permanent, though alteration is
constantly going on. This is more likely to happen if by artificial
means the erosion of the banks is prevented. If a river flows in soil
incapable of resisting its tendency to scour it is necessarily sinuous
(§ 107), for the slightest deflection of the current to either side
begins an erosion which increases progressively till a considerable
bend is formed. If such a river is straightened it becomes sinuous
again unless its banks are protected from scour.
§ 128. _Longitudinal Section of River Bed._--The declivity of rivers
decreases from source to mouth. In their higher parts rapid and
torrential, flowing over beds of gravel or boulders, they enlarge in
volume by receiving affluent streams, their slope diminishes, their
bed consists of smaller materials, and finally they reach the sea.
Fig. 131 shows the length in miles, and the surface fall in feet per
mile, of the Tyne and its tributaries.
The decrease of the slope is due to two causes. (1) The action of the
transporting power of the water, carrying the smallest debris the
greatest distance, causes the bed to be less stable near the mouth
than in the higher parts of the river; and, as the river adjusts its
slope to the stability of the bed by scouring or increasing its
sinuousness when the slope is too great, and by silting or
straightening its course if the slope is too small, the decreasing
stability of the bed would coincide with a decreasing slope. (2) The
increase of volume and section of the river leads to a decrease of
slope; for the larger the section the less slope is necessary to
ensure a given velocity.
The following investigation, though it relates to a purely arbitrary
case, is not without interest. Let it be assumed, to make the
conditions definite--(1) that a river flows over a bed of uniform
resistance to scour, and let it be further assumed that to maintain
stability the velocity of the river in these circumstances is constant
from source to mouth; (2) suppose the sections of the river at all
points are similar, so that, b being the breadth of the river at any
point, its hydraulic mean depth is ab and its section is cb², where a
and c are constants applicable to all parts of the river; (3) let us
further assume that the discharge increases uniformly in consequence
of the supply from affluents, so that, if l is the length of the river
from its source to any given point, the discharge there will be kl,
where k is another constant applicable to all points in the course of
the river.
[Illustration: FIG. 132.]
Let AB (fig. 132) be the longitudinal section of the river, whose
source is at A; and take A for the origin of vertical and horizontal
coordinates. Let C be a point whose ordinates are x and y, and let the
river at C have the breadth b, the slope i, and the velocity v. Since
velocity × area of section = discharge, vcb² = kl, or b =
[root](kl/cv).
Hydraulic mean depth = ab = a [root](kl/cv).
But, by the ordinary formula for the flow of rivers, mi = [zeta]v²;
.: i = [zeta]v²/m = ([zeta]v^(5/2)/a) [root](c/kl).
But i is the tangent of the angle which the curve at C makes with the
axis of X, and is therefore = dy/dx. Also, as the slope is small, l =
AC = AD = x nearly.
.: dy/dx = ([zeta]v^(5/2)/a) [root](c/kx);
and, remembering that v is constant,
y = (2[zeta]v^(5/2)/a) [root](cx/k);
or
y² = constant × x;
so that the curve is a common parabola, of which the axis is
horizontal and the vertex at the source. This may be considered an
ideal longitudinal section, to which actual rivers approximate more or
less, with exceptions due to the varying hardness of their beds, and
the irregular manner in which their volume increases.
§ 129. _Surface Level of River._--The surface level of a river is a
plane changing constantly in position from changes in the volume of
water discharged, and more slowly from changes in the river bed, and
the circumstances affecting the drainage into the river.
For the purposes of the engineer, it is important to determine (1) the
extreme low water level, (2) the extreme high water or flood level,
and (3) the highest navigable level.
1. _Low Water Level_ cannot be absolutely known, because a river
reaches its lowest level only at rare intervals, and because
alterations in the cultivation of the land, the drainage, the removal
of forests, the removal or erection of obstructions in the river bed,
&c., gradually alter the conditions of discharge. The lowest level of
which records can be found is taken as the conventional or approximate
low water level, and allowance is made for possible changes.
2. _High Water or Flood Level._--The engineer assumes as the highest
flood level the highest level of which records can be obtained. In
forming a judgment of the data available, it must be remembered that
the highest level at one point of a river is not always simultaneous
with the attainment of the highest level at other points, and that
the rise of a river in flood is very different in different parts of
its course. In temperate regions, the floods of rivers seldom rise
more than 20 ft. above low-water level, but in the tropics the rise of
floods is greater.
3. _Highest Navigable Level._--When the river rises above a certain
level, navigation becomes difficult from the increase of the velocity
of the current, or from submersion of the tow paths, or from the
headway under bridges becoming insufficient. Ordinarily the highest
navigable level may be taken to be that at which the river begins to
overflow its banks.
§ 130. _Relative Value of Different Materials for Submerged
Works._--That the power of water to remove and transport different
materials depends on their density has an important bearing on the
selection of materials for submerged works. In many cases, as in the
aprons or floorings beneath bridges, or in front of locks or falls,
and in the formation of training walls and breakwaters by _pierres
perdus_, which have to resist a violent current, the materials of
which the structures are composed should be of such a size and weight
as to be able individually to resist the scouring action of the water.
The heaviest materials will therefore be the best; and the different
value of materials in this respect will appear much more striking, if
it is remembered that all materials lose part of their weight in
water. A block whose volume is V cubic feet, and whose density in air
is w lb. per cubic foot, weighs in air wV lb., but in water only
(w--62.4) V lb.
+----------------------+-----------------------------+
| | Weight of a Cub. Ft. in lb. |
| +--------------+--------------+
| | In Air. | In Water. |
+----------------------+--------------+--------------+
| Basalt | 187.3 | 124.9 |
| Brick | 130.0 | 67.6 |
| Brickwork | 112.0 | 49.6 |
| Granite and limestone| 170.0 | 107.6 |
| Sandstone | 144.0 | 81.6 |
| Masonry | 116-144 | 53.6-81.6 |
+----------------------+--------------+--------------+
§ 131. _Inundation Deposits from a River._--When a river carrying silt
periodically overflows its banks, it deposits silt over the area
flooded, and gradually raises the surface of the country. The silt is
deposited in greatest abundance where the water first leaves the
river. It hence results that the section of the country assumes a
peculiar form, the river flowing in a trough along the crest of a
ridge, from which the land slopes downwards on both sides. The silt
deposited from the water forms two wedges, having their thick ends
towards the river (fig. 133).
[Illustration: FIG. 133.]
This is strikingly the case with the Mississippi, and that river is
now kept from flooding immense areas by artificial embankments or
levees. In India, the term _deltaic segment_ is sometimes applied to
that portion of a river running through deposits formed by inundation,
and having this characteristic section. The irrigation of the country
in this case is very easy; a comparatively slight raising of the river
surface by a weir or annicut gives a command of level which permits
the water to be conveyed to any part of the district.
§ 132. _Deltas._--The name delta was originally given to the [Greek:
Delta]-shaped portion of Lower Egypt, included between seven branches
of the Nile. It is now given to the whole of the alluvial tracts round
river mouths formed by deposition of sediment from the river, where
its velocity is checked on its entrance to the sea. The characteristic
feature of these alluvial deltas is that the river traverses them, not
in a single channel, but in two or many bifurcating branches. Each
branch has a tract of the delta under its influence, and gradually
raises the surface of that tract, and extends it seaward. As the delta
extends itself seaward, the conditions of discharge through the
different branches change. The water finds the passage through one of
the branches less obstructed than through the others; the velocity and
scouring action in that branch are increased; in the others they
diminish. The one channel gradually absorbs the whole of the water
supply, while the other branches silt up. But as the mouth of the new
main channel extends seaward the resistance increases both from the
greater length of the channel and the formation of shoals at its
mouth, and the river tends to form new bifurcations AC or AD (fig.
134), and one of these may in time become the main channel of the
river.
§ 133. _Field Operations preliminary to a Study of River
Improvement._--There are required (1) a plan of the river, on which
the positions of lines of levelling and cross sections are marked; (2)
a longitudinal section and numerous cross sections of the river; (3) a
series of gaugings of the discharge at different points and in
different conditions of the river.
_Longitudinal Section._--This requires to be carried out with great
accuracy. A line of stakes is planted, following the sinuosities of
the river, and chained and levelled. The cross sections are referred
to the line of stakes, both as to position and direction. The
determination of the surface slope is very difficult, partly from its
extreme smallness, partly from oscillation of the water. Cunningham
recommends that the slope be taken in a length of 2000 ft. by four
simultaneous observations, two on each side of the river.
[Illustration: FIG. 134.]
§ 134. _Cross Sections_--A stake is planted flush with the water, and
its level relatively to some point on the line of levels is
determined. Then the depth of the water is determined at a series of
points (if possible at uniform distances) in a line starting from the
stake and perpendicular to the thread of the stream. To obtain these,
a wire may be stretched across with equal distances marked on it by
hanging tags. The depth at each of these tags may be obtained by a
light wooden staff, with a disk-shaped shoe 4 to 6 in. in diameter. If
the depth is great, soundings may be taken by a chain and weight. To
ensure the wire being perpendicular to the thread of the stream, it is
desirable to stretch two other wires similarly graduated, one above
and the other below, at a distance of 20 to 40 yds. A number of floats
being then thrown in, it is observed whether they pass the same
graduation on each wire.
[Illustration: FIG. 135.]
For large and rapid rivers the cross section is obtained by sounding
in the following way. Let AC (fig. 135) be the line on which soundings
are required. A base line AB is measured out at right angles to AC,
and ranging staves are set up at AB and at D in line with AC. A boat
is allowed to drop down stream, and, at the moment it comes in line
with AD, the lead is dropped, and an observer in the boat takes, with
a box sextant, the angle AEB subtended by AB. The sounding line may
have a weight of 14 lb. of lead, and, if the boat drops down stream
slowly, it may hang near the bottom, so that the observation is made
instantly. In extensive surveys of the Mississippi observers with
theodolites were stationed at A and B. The theodolite at A was
directed towards C, that at B was kept on the boat. When the boat came
on the line AC, the observer at A signalled, the sounding line was
dropped, and the observer at B read off the angle ABE. By repeating
observations a number of soundings are obtained, which can be plotted
in their proper position, and the form of the river bed drawn by
connecting the extremities of the lines. From the section can be
measured the sectional area of the stream [Omega] and its wetted
perimeter [chi]; and from these the hydraulic mean depth m can be
calculated.
§ 135. _Measurement of the Discharge of Rivers._--The area of cross
section multiplied by the mean velocity gives the discharge of the
stream. The height of the river with reference to some fixed mark
should be noted whenever the velocity is observed, as the velocity and
area of cross section are different in different states of the river.
To determine the mean velocity various methods may be adopted; and,
since no method is free from liability to error, either from the
difficulty of the observations or from uncertainty as to the ratio of
the mean velocity to the velocity observed, it is desirable that more
than one method should be used.