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7rf 

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THE SUSPENSION OF SOLIDS IN FLOWING 

WATER. 

By ELON HUNTINGTON HOOKER. 



A Thesis presented to the Faculty of Cornell University for 
the Degree of Doctor of Philosophy. 


ITHACA, N. Y. 

MAY, 189G. 












I 




< < 
<■ < < 


Cornell Uuiv. Lib. Bxehsng, 

*AR 28 »*i)8 


6 


C 


















v 



1896. 


AUGUST. 


No. 6. 


AMERICAN SOCIETY OF CIVIL ENGINEERS. 

INSTITUTED 18 52. 


PAPERS. 

Note.—T his Society is not responsible, as a body, for the facts and opinions advanced 

in any of its publications. 


THE SUSPENSION OF SOLIDS IN FLOWING 

WATER.* 

--;-— 4 »’*• 

By Elon Huntington Hooker. 


Considerable sjiace in this paper is devoted to the historical side of 
the subject because the sources of information are widely scattered, 
and it is desired to indicate, so far as possible, the origin of the ideas 
and observations upon sedimentary movements which have become 
common knowledge. In the second part of the paper a comparison of 
particular facts and observations leads to certain general conclusions 
with reference to the manifestation of the phenomena studied. The 
concluding portion is devoted to an analysis of the different explana¬ 
tions of the cause of suspension, for the purpose of building ujj a satis¬ 
factory theory. 

To conduce to uniformity and clearness, the following symbols will 
be used throughout this discussion : 

* Printed by the American Society of Civil Engineers in advance of its publication in 

Proceedings. 










4 


HOOKER OK SUSPENSION OF SOLIDS IN RIVERS. 


g — 32.2 ft. per second = acceleration of gravity. 

F = area of right section of body considered. 
h = head of water corresponding to the velocity v. 
d z=. mean velocity of the stream in the vertical considered. 
v = surface velocity at the vertical considered. 
x> n = bottom velocity at the vertical considered. 
i = inclination of the water surface. 

/' = tangent of the angle of sliding friction. 

/ = tangent of the angle of rolling friction (properly a distance). 

y = heaviness of the liquid considered. For the purposes of 

this article, y = 62.5 lbs. per cubic foot = heaviness of 

% 

fresh water. 

y' = heaviness of the solid considered. 

P = resultant thrust, in the stream direction, exerted by a 
moving liquid upon a solid. 

V = volume. 

G = weight. 

M = mass. 

Jc = constant determined by experiment. 

Z = mean depth of stream. 
z = variable depth below surface. 
b = width of stream. 
r = radius of sphere. 

q =z liquid discharge per unit width of stream. 
cl = solid discharge per unit of width. 

Part I. —Historical Development of the Problem 1 . 

The varied phenomena incident to the flow of rivers have demanded 
the consideration and even the anxiety of riparian owners from a time 
far antedating the development of the modern science of hydraulics. 
Wherever rugged slopes discharge their melted snows or heavy rains 
are gathered from steep, impervious water-slieds, a mountain torrent 
has its birth, and the householder in the valley early learned to study 
its varying humors. The rapid descent of the Apennines to the sea 
and the consequent turbulent character of the streams of northern 
Italy, made this a fruitful field of study to a people w T hose scientific 

1 It is proposed here to select, from the mass of literature touching upon this problem, 
only those discussions and observations which seem to mark a distiuct step toward ils final 
scientific solution. 



HOOKER OK SUSPEKSIOK OF SOLIDS IK RIVERS. 


5 


spirit had already gone far toward establishing the fundamental laws 
of fluid motion. 

During the latter part of the seventeenth century means were sought 
for the amelioration of these mountain rivers. Dominique Gruglielinini, 
physician and hydraulician, was employed by Venice and other Italian 
cities to prepare plans looking toward the prevention of their ravages. 
His greatest work was the building of the levees on the Po above 
Plaisance, and his writings 1 gave the first impetus to a scientific study 
of fluvial phenomena. 

In 1773 Johann Silbersclilag produced his comprehensive treatise 
on hydraulics 2 , covering this field to some extent, but it was left to 
Dubuat ’ in 1786 to publish the first experimental studies which can 
be considered authoritative. These experiments were made at Paris by 
order of the French government. His determination of the different 
velocities at which solid particles begin to be moved by flowing water 
has been accepted by subsequent writers, and with his work begins the 
scientific knowledge of the movement of alluvions in sedimentary rivers. 

Dubuat’s artificial canal was- formed of planks 12 pieds 4 in length, 
3 pouces thick and 18 pouces wide, so fitted that the form could be 
altered from rectangular to trapezoidal by the addition of supple¬ 
mentary bracing. Its total length was 132 pieds, and in the trapezoidal 
form it had a clear bottom width of 5f pouces to a surface width of 3 
pieds. Water was discharged into the canal under a maximum head 
of 7 to 8 pieds. 

Surface velocities were measured with floats, noting the time of 
< . • • 

traversing 10 toises . For determining bottom velocities a small ball 
of mastic was first adopted. Its specific gravity was such that it lost 
yf of its weight in water, and w'as thus very easily moved. Later, 
more satisfactory results were obtained with red currants. These were 
smoother, moving with less friction, and could be more easily seen. 
The time of passing through a distance of 60 pieds was noted. 

1 “ Della natura di fiumi trattato fisico matematico. ” Guglielmini, Bologna, 1697. 

2 “ Hydrotechnik oder des Wasserbaues.” Johann Silbersclilag, Leipzig, 1773. Copy at 
Zurich Polytechnikum. 

3 “Traite d’Hydraulique.” Dubuat, Paris, 1786. Page 57, Volume I. Third edition 
published in 1816. Copy at Zurich Polytechnikum and at Ecole des Ponts et Chaussees, Paris. 

\ “ Systeme ancien ” of France— 

1 pouce = 1.066 ins. 

1 pied = 1.066 ft. 

1 toise = 6.395 ft. 

These units were in use previous to 1812. Between 1812 and 1840 the “systeme usuel ” 
was in vogue. Its values are slightly larger.—1 pouce = 1.093 ins. 






6 


HOOKER OH SUSPENSION OF SOLIDS IN RIVERS. 


The bottom velocities at which various materials began to be 
moved by the current were as follows: 

Potter’s clay (beginning -with a velocity of 
45 pouces, it continued to be carried 
away as the velocity was gradually de¬ 
creased to 7 pouces. At 7 pouces a 
deposit of fine sand took place, which 
continued on down through a velocity 
of 4 pouces, until at 3 pouces per 

second the clay ceased to show action) 3 pouces per second. 


Gravel (size of anise seed). 4 pouces. 

Gravel (size of peas).. 7 


Coarse sand (sand remained stable while bottom 
velocity was increased from 3 up to 7 pouces. At 
8 pouces it began to be entrained and for veloci¬ 
ties of 12 to 45 pouces per second it continued to 

be entrained and suspended). 8 “ 

Sea pebbles (1 pouce diameter).24 “ 

Dubuat’s experiments also showed that a current velocity of 10 or 
12 pouces per second was sufficient to produce sand waves in a bot¬ 
tom whose grains were large enough to be easily visible. He describes 
these furrows as perpendicular to the longitudinal axis of the cur¬ 
rent with a short steep down-stream face and a long gentle posterior 

slope. Each sand grain was slowly rolled along the up-stream incline 

» 

and fell of its own weight down the crest, thus advancing the wave 
by steps equal to the diameter of the grains. 1 2 He computes a velocity 
under these circumstances which requires two years to cover a length 
of 2 400 toises. 

The expression now universally used to represent the thrust exerted 
by a current against a solid of any form was deduced by Dubuat' and 
the coefficients experimentally determined. 

He argued that the pressure on the up-stream face (pj) would be 
greater and that on the down-stream face (p 2 ) would be less than the 

1 For a similar description see “ Report of Chief of Engineers, U. S. A.,” 1875,11, pp. 502- 

504. 

Also “Handbuch der Wasserbaukunst.” G. Hagen, 1871,'Zweiter Tlieil, “ Die Strome,” 
S. 161-162. 

2 See Flamant, “ Hydraulique,” 1891, p. 561. 


I 








HOOKER OH SUSPENSION OF SOLIDS IN RIVERS. 


<*• 

t 


pressure ( p ) at the corresponding points if the solid were removed. 
Therefore, the total impelling force would be 

P = F ( Pl —p 2 ) = F [( Pl — p) + (p —p 2 )]. 

These pressures can be written as functions of the velocity height 
h, w r hence 


v 


2 9 9 


P\ p XT p —p 2 

7 2 g y 

when m and n are experimentally determined. 

By substitution, 

9 

/ C~‘ 

P = (in -f- n) y F-— or, as usually written, 

*9 

ft 

P = k y F 

2p 

The fact that floating solids move with a velocity superior to that 
of the current which bears them was noted by Dubuat 1 . His expla¬ 
nation of it was inaccurate, but the phenomenon itself has an import¬ 
ant bearing on the present discussion. 

Of interest in this connection is the experimenter’s statement with 
reference to the theoretical form of bed best adopted for flowing 
streams. He rejects the rectangle and semicircle as being unable to 
sustain their own weight in soft soils and chooses the trapezoidal cross- 
section as offering a proper talus. These right lines will be rounded 
by the stream itself, the slope being proportioned to the diminishing 
velocity from center to sides. 2 

Dubuat’s work further includes various studies into the regime of 
rivers and the development of rules for that radius of curvature at 
bends which will best conduce to stability. To him belongs the honor 
of inception along these lines. 

J. A. Fabre was the next to publish systematic studies 3 on the 
movements of solids in torrents and rivers followed in 1811 by the 
voluminous encyclopedia of Wiebeking. 4 These men extended the 
range of observed data without making material additions to the 
theory of fluvial action. 

In the year 1845, Bouniceau 5 discussed at considerable length the 

i“Principes d’Hydraulique,” Dubuat, No. 220. Quoted by Durand-Claye, Annales des 
Fonts et Chausst^s, 1886, 1, 530. 

2 See "Principes d’Hydraulique,” Dubuat, 1786, Vol. I, p. 119. 

3 “ Essai sur la tbeorie des torrents et des rivieres,” J. A. Fabre, Paris, 1797, Premiere 
Partie. 

4 “ Wasserbau.” C - F. Wiebeking, Munich, 1811-1817, 4 volumes. 

» “ Etude sur la navigation des rivieres a marees et la conquete de lois et relais de leur 
embouchure.” Bouniceau, Paris, 1845. 








8 


HOOKER OK SUSPEKSIOK OF SOLIDS IK RIVERS. 


shifting of sands in tidal estuaries, and showed himself a close student 
of the laws of erosion by water action. His excellent little volume 
brings to light and discusses a number of anomalies in this form of 
action. 

He gives a set of values of the bottom velocities at which erosion 
begins to take place with different materials : 1 

Clay.0.08 — 0.15 m. per second. 

Coarse sand. 0.22 — 0.30 “ 

Coarse gravel.0.11 — 0.61 “ 

Ordinary pebbles . 0.65 — 1.00 “ 

Stones (size of an egg). 1.00 — 1.20 “ 

Conglomerates.1.52 “ •' 

Sedimentary rock.1.83 “ 

Solid rock.3.00 

The River Garonne for a length of 45 miles below the embouchure 
of the Lot was made the object of a series of observations covering 11 
years by M. Baumgarten. 2 These measurements deal with the varying 
discharges from month to month, with the constant changes in form of 
cross-section and maximum depths, determinations of the fall and 
heights of water as well as with geological and meteorological studies of 
the valley. It has formed the model for later fluviatile studies. Here are 
given the first measurements of discharge of detritus 3 which the author 
has been able to find. Daily samples were taken at Marmande from 
the surface of the river in a vessel containing 4.6 liters. This w r as 
allowed to stand for nine or ten days, the clear water decanted and 
the sediment filtered until thoroughly dry. After weighing, a simple 
calculation gave the weight in grams of mud per cubic meter of water. 
These measurements were continued from 1839 to 1846 continuously, 
and the average monthly solid discharge of the river at this point 
computed. 4 M. Baumgarten distinguishes three different methods 
of movement common to these solids. 

1 “Etude sur la Navigation/' etc., p. 19. 

2 “ Navigation fluviale, Garonne.” M. Baumgarten, Ingenieur ordinaire. Annales des 
Fonts et Chausstcs, 1848. 2, 1-157. 

3 Same, pp. 47, 140. 

4 In order to see if the water contained the same amount of suspended matter at all 
depths, Baumgarten made a series of tests of specimens from different depths and taken from 
points where the velocity was different. 

From the results given in the table in the continuation of this note on the opposite 
- page he decided that a surface specimen gave a fair average. 











HOOKER ON SUSPENSION OF SOLIDS IN RIVERS. 


9 


First. —A discontinuous rolling motion along tlie bed of the stream 
which takes place when the velocity of the current is limited or the 
materials large. 

Second .—TVith greater velocities or smaller particles, a discontinuous 
suspension in the lower laminae of the current. 

Third .—Movement in continuous suspension when the particles are 
carried throughout the entire length considered. 

The sand waves which M. Dubuat had observed on a small scale 
are reproduced in the Garonne on a large scale in gravel shoals, and 
M. Baumgarten made careful measurements of the yearly progress of 
one of these crests. 

In 1840 the talus down stream had a vertical height of 1.3 m., a 
base of 2.8 m., while the length of the crest was 180 m. In 1841, 
the form was nearly the same, but the crest had moved down stream 
parallel to itself about 30 m. In 1842 the forward motion was 20 m. 
These gravels were of about the size of a walnut, and the velocity 
of the water averaged 2.25 m. per second. 

Thus far attention had been especially directed towards the phe¬ 
nomenon of dragging, and the laws it follows had been, to some extent, 
investigated. Inspecteur-General Dupuit, 1 in 1848, emphasized the 
true imjjortance of suspension in the movements of soft river bottoms, 
and to him is due the first scientific study of the causes which pro¬ 
duce this action. 


Dates. 

Depth at which the water 
was taken. 

Weight of Filtered Sediment. 

In dead water of a bv idge 
or in a gentle current. 

In a stroi g current. 


rat the botiom at 7.0 m_ 

Grams. 

0.72 

Grams. 

March 25th, 1847.. 

| at 3 5 m . 

0.75 

• • • • 


( at the surface. 

0.82 

• • • • 


(at the bottom at 8.0 m.... 

0.34 

• • • • 

March 27th, 1847.. 

| at 4.0 m. 

0.32 

. • • . 

( at the surface . 

0.19 

• • • • 


(at the bottom at 8.75 m... 

1.29 

0.93 

April 9th, 1847- 


1.43 

1.18 

(at the surface. 

1.13 

1.22 


(at the bottom at 9.0 m_ 

1.89 

1.90 

April 15th, 1847.... 

jat 4.50 m . 

1.78 

2.15 

( at the surface. 

• • • • 

1.60 


(at the bottom at 8.0 m.... 

0.94 

0.90 

April 18th, 1847.... 

| at 4.00 m. 

0.87 

0.68 

(at the surface. 

1.33 

0.87 


\ 


1 “ Etudes tlieoriques et pratiques sur le mouvement des eaux." Paris, 1848. Second 
edition, Paris, 18C3, pp. 214-229. J. Dupuit, Inspecteur-General. 

































10 


HOOKER ON SUSPENSION OF SOLIOS IN RIVERS. 


Dupuit calls attention to the exjDeriment of revolving rapidly a 
glass of water containing sand grains. He notes that there is a direct 
relation between the velocity of the water and the amount of sand in 


suspension, and that the grains tend to arrange themselves in succes¬ 
sive laminae according to the order of their size; as the velocity is 
decreased, they descend successively to the lower strata. These facts 
had all been observed before his time, but Dupuit goes farther than 
his predecessors in noting that the maximum amount of suspension, 
i. e., that in the lower layers, corresponds, not to the greatest absolute 
velocity of the current, but to the maximum relative velocity of con¬ 
tiguous molecules. This is a distinct step in advance. Dupuit finds 
here his explanation of the phenomenon of suspension. 

Starting with the fact first noted by Dubuat that the velocity of a 
float exceeds that of the current, 1 he calls attention to the tendency of 
such bodies to move toward the filaments of greatest velocity and ex¬ 
plains this upon the principle of least work. Assuming the resistance 
to its motion to vary with the direction of its path, this direction will 
necessarily be that which offers the least resistance. Therefore an 
oblique path toward the most rapid current in the stream line will 
result, since this will offer the least difference in velocity between the 
solid and the fluid and so the least frictional work. 

Dupuit derives a law for this lateral movement as follows: 

Let v = the absolute longitudinal velocity of the body. 

u = the absolute transverse velocity of the body. 

w = absolute velocity of filament at shore side of body. 

✓ 

w’ = absolute velocity of neighboring filament toward center 
of stream. 

The relative velocity of the body as regards the liquid surrounding 
it may be expressed by: 


y/ u 2 v' 2 


w -f- w 1 


(1) 


Considering the resultant of the resistances which the body suffers 
as approximately proportional to this relative velocity, the value of u 
may be found for which this resultant is a minimum. 

Calling Q the angle of inclination of the tangent to the curve of 
velocities at the point considered, 

w' = w -(- u tan. Q .(2) 

1 Dupuit, in common with Dubuat, ascribes this excess of velocity to the accelerating 
force represented by the component of the bodies’ weight parallel to the surface of the cur¬ 
rent. M. Du Boys, Annales des Fonts et Chaussees, 1886, 1, 199-242, has clearly demonstrated 

he incorrectness of this explanation. 


t 







HOOKER ON SUSPENSION OF SOLIDS IN RIVERS. 


11 


Substituting (2) in (1) and putting the first differential coefficient of 
the expression equal to zero, he finds that the resistance will be a mimi- 
mum for 

tan. 0 , - 

« = •— 


i. c., the transverse velocity should decrease with the tangent of 
the curve of surface velocities, or, in other words, from the banks to 
the center of the stream. This is equivalent to saying that the maxi¬ 
mum lateral velocity will correspond to the maximum relative velocity 
of the filaments. 1 

Applying this same law to the velocities considered in a longitudi¬ 
nal section, he finds a resultant force acting obliquely upward, which 
produces the phenomenon of suspension. As this force will be greatest 
where the relative velocities are greatest, i. e., near the bottom, the 
lower laminae will carry the heavier load of particles. Solids of 
equal density will arrange themselves from bed to surface in the order 
of their volume. Suppose, now, the relation of a solid to neighboring 
ones is considered. The presence of another will tend to decrease the 
relative velocities of the filaments, and so the two will be obliged to 
descend to a lov r er lamina than would the one alone. Descent or 
ascent will follow according as the bodies approach each other or 
separate. 

Dupuit formulates these laws as follows: 


“ First .—A w r ater current can suspend solids of a density superior 
to its own. 

“Second .—The power of suspension depends upon the relative 
velocity of the filaments and is greater according as this relative 

(l V 

velocity is greater. In general, it is proportional to the quantity —— 

Cv z 

(where v == velocity of current and z = depth below surface) so that 
lover layers can carry either more solids or those of greater volume. 

“ Third .—The power of suspension of a bed is limited, i. e., a square 
meter of cross-section can only carry a certain number of solids of 
a definite volume. Thus each lamina has a different degree of 
saturation.” 


1 Observations made by Major CunniDgham in the Ganges Canal seemed to indicate a 
current from the shore to mid-stream whose intensity followed this same law (see Proceed¬ 
ings of the Institution of Civil Engineers, Vol. LXXI, p. 66.) As this current was indicated only 
by the behavior ol certain floats, it is more in consonance with present knowledge to believe 
their action due to the cause given here by Dupuit than to suppose au actual lateral motion 
of the water. 






12 


HOOKER OX SUSPENSION OP SOLIDS IN RIVERS. 


Dupuit assumes a river flowing with section and fall unchanged, 
and saturated with sediment. The entire load will be carried to the 
embouchure. Suppose the section to vary. At each change will come 
a change in the curve of velocities and a consequent change in the 

# 4 

power of suspension. 

When this power is reduced, there will follow a deposit, and 
when it is increased, erosion will take place. He makes it clear 
also that these results are dependent, not only upon the section 
at the point where the change takes place, but also upon the anterior 
portion of the river as affecting the state of saturation in which the 
river reaches the section in question. These effects can be brought 
about at any point whatever by suitably changing the up-stream 
section. 

When a deposit occurs the material comes wholly from the lower 
laminae, and they, in turn, receive from the upper ones the material in 
excess of their power of suspension. This exjdains the lamination of 
river-beds in materials increasing in size with depth below the bottom. 
The frequent presence of beds of finer particles interrupting this 
structure he explains by the principle that saturation may be 
obtained either by the size of the particles or by their nearness 
together. 

The numerous variations to which this laminated movement is 

subject is noticed, and explains the constant rising and falling of 

particles from one lamina to another, while the nature of the horizontal 

curve of velocities is such as to cause a constant movement of particles 

from the banks toward the center. To this mav be attributed the 

%/ 

tendency of a river to form islands in the middle of its bed at the 
expense of its banks. 

Among the German writers of this period, the discussion of the 
transportation of stones by torrents was especially taken up by Joseph 
von Gumppenberg Pottmes 1 , but no further experiments were pub¬ 
lished until 1857, when Blackwell 2 , in England, extended the investi¬ 
gations of Dubuat to solids of larger dimensions. 

The velocities given in the table on the next page are those at 
which movement began: 

1 “ Der Wasserbau an Gebirgsfllisten,” Joseph Freiherrn von Gumppenberg Pbttmes, 
Augsburg, 1834. 

2 See “Report of tbe Referees upon the Main Drainage of the Metropolis,’’ July 31st, 
1857, Appendix IV. Also, fur table here quoted, see rroceedivgs of the Institution of 
Civil Engineers, Vol. 82, p. 48. 




HOOKER OK SUSPENSION - OF SOLIDS IN RIVERS. 


13 


Description 

of 

substance. 

Cubic Contents. 

1 . 

Cubic 

inches. 

a. 

Cubic 

inches. 

Brickbats. 

2.59 

18.5 

Brickbats. 

4.76 

18.5 

Oolites. 

2.39 

17.68 

Flints. 

1.95 

10.37 

Slate. 

2.38 

9.06 


Velocities. 


1. 

Feet per 
second. 

3. 

Feet per 
second. 

( 1.75 

2.75 

! to 

to 

( 2.00 

3.00 

( 2.25 

2.75 

to 

to 

( 2.50 

3.00 

( 2.00 

2.75 

to 

to 

( 2.25 

3.00 

( 2.50 

3.00 

to 

to 

( 2.75 

3.25 

( 2.00 

2.75 

to 

to 

( 2 25 

3.00 


Increase of 
contents 
of 

substances 

moved. 


7.14 

3.97 

7.40 

5.32 

3.81 


Increase 

of 

velocities. 


Sixth root 
ot increase 
of contents. 


1.37 

to 

1.70 

1.10 

to 

1.33 

1.22 

to 

1.50 

1.09 

to 

1.30 

1.22 

to 

1.50 


1.38 

1.26 

1.39 

1.32 

1.25 


The inqiortant idea of saturation with solid material is definitely 
stated 1 by M. Scipion Gras in a valuable paper 2 3 published at this 


time. 


He defines saturation in a stream as that state at which the least 
addition to the solid material already carried will cause a deposit, and 
its power of entrainment as the total weight of material which a given 
stream in a state of saturation can carry. He assumes this power of 
transport to vary directly with the velocity, density and depth of the 
water, and, these quantities remaining constant, to vary with the 
volume, density and form of the solids submitted to its action. Upon 
these principles he explains erosion as a necessary consequence, when 
the saturation corresponding to the actual velocity is incomplete, un¬ 
less the bed offers too great a resistance. 

Measurements of the advance of the crests of shoals, similar to 
those undertaken by Baumgarten in 1840, were made by Hiibbe 4 in 
1861 on sand bars. His observations show the same wave form on a 
large scale, which Dubuat had noticed in the minute form, and con¬ 
firm Baumgarten’s statement of the forward motion of the crests. 

The results of the exhaustive study of the Mississippi Biver 4 and 


1 Probably first stated by Frisi, “ On Rivers and Torrents,” 1732. See Report on Missis¬ 
sippi River. Humphreys & Abbot, pjj. 190, 415. 

s “Etudes sur le torrents des Alpes.” M. Scipion Gras , Annales des Ponts et Chaussees, 
1857, 2, pp. 1-96. 

3 Zeitschrift fur Bauweseu, Jahrgang xi, 1861. Abstracted in Zeitschrift des Architekten 

und Ingenieur- Vereins, Hauover, 1863, p. 518. . 

4 “Report on the Mississippi River.” Humphreys and Abbot, 1861. Reprinted with 
additions, Washington, 1876. 





































14 


HOOKER OK SUSPENSION OF SOLIDS IN RIVERS. 


its delta were published in 1861, and contain a wide range of data on 
the distribution of sediment. Observations along the same lines as 
those of Baumgarten were instituted at Carrollton in 1851, and lasted 
throughout two years. They were conducted by Prof. Forshey. Sam¬ 
ples were taken from a point near the east bank, where the high-water 
depth was 100 ft., from the middle of the river, and from a point near 
the west bank, where the high-water depths were 100 and 40 ft., respect¬ 
ively. These tests were made daily (except Sundays) and samples taken 
from surface, mid-depth and bottom by means of a small weighted keg, 
with valves opening upward, which was designed to allow free passage 
to the water until it reached the desired depth. At the station near 
the west bank only surface and bottom samples were taken. An 
average value was obtained for the weight in grams of sediment to 
600 grs. of water at each of the positions, 100 grs. of the water being 
measured out into its proper precipitating bottle for each of the six 
working days of the week, and corresponding to each of the eight 
positions. 

During the second year samples were taken only from the surface 
and at the position near the east bank. The tabulated results of these 
measurements are given in Humphreys and Abbot’s Report (edition 
of 1876, pp. 134, 417). 

From the study of these results, Humphreys and Abbot drew the 
following conclusions: 

“ This table is fruitful in results. It establishes that the Mississippi 
water is not charged to its maximum capacity with sediment, because 
the distribution of the material is different from what must have place 
were this the case. Dupuit demonstrates that the power of sus¬ 
pension is due to the fact that the different layers of water are actuated 
by different velocities, and thus exert different pressures upon the dif¬ 
ferent sides of the suspended atoms. Hence, the greater the difference 
in the velocity of consecutive layers, the greater will be the power of 
suspension. Now, it is conclusively proved in Chapter IV 1 that the 
change of velocity from layer to layer is in horizontal planes, greatest 
near the banks and the least near the thread of the current; and in 
vertical planes, parallel to the current, the greatest near the bottom 
and surface, and the least at a point about 0.3 of the depth below the 
surface, where the absolute velocity has its maximum value. If, then, 
the water be either charged to its maximum capacity or overcharged 
with sediment, we must find the greatest amount near the banks and 


1 “ Report on the Mississippi River.” Humphreys and Abbot. 




hooker ok suspension of solids in rivers. 


15 


neai the surface and bottom, and the least amount near the thread of 
the current and near the layer 0.3 of the depth below the surface. If the 
water be undercharged, on the contrary, the distribution of sediment 
will folloAv no law, the amount at any point being fixed by the acci¬ 
dental circumstances of whirls, boils, etc., although, of course, there 
'will be an accumulation of material near the bottom, where the sus¬ 
pending power is very much greater than elsewhere. Bearing these 
well-established principles in mind, an inspection of the preceding 
tab±e must convince any one that the Mississippi water is under¬ 
charged with sediment, even in the low-water stage. A most im¬ 
portant practical deduction may be drawn from this fact, namely, the 
error of the popular idea that a slight artificial retardation of the cur¬ 
rent, that caused by a crevasse, for instance, must produce a deposit 
in the channel of the river below it.” 

Sediment observations were also made at Columbus bv the Mis- 

«/ 

sissippi survey from March to November, 1858, but, as only surface 
specimens were taken and no tabulated results give a means of com¬ 
parison between the amounts in suspension near the banks and at the 
thread of the current, they can be of little service in the scientific 
study of the distribution of sediment in the cross-section of the river. 
Curves are shown, however, on Plates XII and XIII of their Report, 1 2 
from which the relation between the mean velocity of the river and 
the corresponding mean amount of suspended matter at Carrollton 
and Columbus may be seen.“ The values from which these curves 
are plotted are given at page 417 of the same report (edition of 
1876). 

From them Humphreys and Abbot are led to the same conclusion 
as before, i. e., that the Mississippi water is not saturated with sedi¬ 
ment, using the term in the sense in which it is used by M. Scipion 
Gras. 3 

Their line of reasoning is as follows: If the water be at all times - 
charged with sediment to the maximum capacity allowed by its veloc¬ 
ity, then the amount of sediment at different stages must vary pro- 
jmrtionately with the mean velocity. 

“ At the date of highest water, both in 1851 and 1858, the river held 
in suspension but little more sediment per cubic foot than at dead low 
water. * * * Moreover, it will be seen that an analysis of the dis¬ 

tribution of sedimentary matter held in suspension leads to the same 

1 “ Report on the Mississippi River.” Humphreys and Abbot. 

2 See page Gl. 

3 See page 13. 




I 


1G HOOKER OK SUSPEKSIOK OF SOLIDS IK RIVERS. 

conclusion, by establishing that the river is never charged to its maxi¬ 
mum capacity of suspension.” 

Extreme care was taken in all these measurements determining the 
amount of sediment in the sample obtained. It was shown that deter¬ 
minations of sediment must be made by weight and not by volume, 
as the latter method introduced discrepancies. These were due to the 
difference in density of the sediment, resulting from different methods 
of manipulation by various observers. 

An extended series of measurements had been in progress on the 
Elbe, at Harburg, during the years 1837 to 1855, by Baurath Blohm. 1 
The data obtained were minutely examined and formed the nucleus 
for a work treating of the subject in all its bearings. The early 
death of Herr Blohm prevented the publication of anything but the 
introductory part of the proposed book. Reference will be made to 
these observations later. 

In 1871, M. Partiot published studies 2 on the movement of sands 
in the Loire, and enriched the knowledge of the subject by minute 
observations and extensive measurements. This monograph deserves 
especial mention, as it brings out strongly the importance of vortices 
and eddies in the suspending power of water. 

Attention is called to the interaction of the suspended particles 
in changing their forms by friction, to the suspension and disin¬ 
tegration of the clays in the higher layers and their mixture with 
vegetable matter to form the rich alluvial deposits which settle on the 
summit of the shoals at the embouchure of the Loire. The slower 
moving sands are carried only intermittently in suspension. 

Measurements are given to show that the quantities of sediment de¬ 
crease toward the river mouth, and the interesting point is determined 
by exjjeriment that the amount of silt varies not only with the height 
of the flood, with reference to others, but also with its own relative 
state. Increasing as the flood crest approaches, it reaches a maxi¬ 
mum at its summit and descends to a lower point at the middle of 
the posterior slope than at the corresponding point of the anterior 
slope. 

Partiot emphasizes the idea that the sands are only sustained by 

1 “ Ueber die in fliessenden Wasser suspendirt erhaltenen Sinkstoffe.” Zeitschrift dtt 
Architekten und Ingenieur Vereins, Hanover, 1867, pp. 240-297. 

5 “ M6moire sur les Sables de la Loire.” M. Partiot. Annales des Ponts et Chaussies, I., 1871. 




HOOKER OH SUSPENSION OF SOLIDS IN RIVERS. 


17 


eddies and vortices. He refers to experiments made at Nantes in 1869. 
Samples were taken at different depths at a point where the river was 
straight and free from eddies. Sand was not found in suspension, 
though when introduced 60 ft. above, in a surface velocity of 1.4 ft. 
per second, its presence was readily detected. At another point, where 
there was a marked eddy, grains of sand and mica were seen to surge 
to the surface and glitter in the sunlight, while grains of quartz were 
brought up in the receptacle from all depths. When the vortices were 
rapid, grains could even be taken in the hand. 

The production of these vortices and eddies is attributed to the in¬ 
equalities of the bottom, the solids deposited there, the deflecting ac¬ 
tion of concave banks and the action of floods. As the flood moves 
down a river in the form of an attenuated wave, the water flows 
down the front incline with an accelerated velocity. It overtakes the 
surface water down stream and flows over it, causing eddies. These 
grow greater as the crest of the wave is approached, since the fall 
increases, reach a maximum at the summit and decrease on the 
posterior portion, where the fall is decreased. This view is cor¬ 
roborated by the corresponding measurements of sediment in sus¬ 
pension. 

The great velocities which these vortices reach in time of flood ex¬ 
plain the movement of boulders, which could not be taken up by 
ordinary waters. M. Partiot calls to mind the lifting strength of whirl¬ 
winds as a parallel case. An interesting point is brought up in the ref¬ 
erence to the action of ice in the movement of these solids. The sand 
grains and pebbles, as w’ell as large stones, at times become frozen 
into the ice forming at the bed and banks of streams. With the least 
rise in the water this may become detached and carried to great dis¬ 
tances. 

It is to dragging rather than suspension that Partiot attributes the 
motion of sands in the Loire, and quotes some valuable researches 
made by M. Sainjon in this connection. 

A body immersed in a moving liquid is subjected by the current to 
a thrust which may be expressed by 

k y Fli — k y F -f —. 

2 g 

M. Sainjon takes the constant k = 1.46 for a prism, and k = .60 for a 
sphere, or as an average k = 1 for the particles making up gravels. 



18 


HOOKER OK SUSPENSION OF SOLIDS IN RIVERS. 


The action of gravity upon this immersed body tending to roll it 
down stream is put equal to 



Since i rarely reaches the value it is neglected and the approximate 
expression becomes 

— V O' — r)f, 

whence the resultant force in the direction of the current becomes 

P = yF v O' — r)f 

assuming k = 1. 

To determine the value of f, M. Sainjon uses the results of the ex¬ 
periments of Dubuat. In this case the bottom velocities were determ¬ 
ined at which the various materials ceased to be moved by the current 1 2 
and at this point he considers that the approximate resultant force 
obtained above may be put = o, i. e., 

2 

r F — v O' — r)f= 

whence 

V v 2 

f-pir'-r) = r ¥ -, 

1 This value is inexact. The correct value is derived as follows: 

Represent the resultant weight in water of the body rolling down the inclined river-bed 


by W — V (y' — y). Let 0 = the angle of the inclination of the bed. 

The gravity component parallel to the bed = IVsin . 0. (1) 

The normal component of W = N = W cos. 0. 

The rolling friction = F = N tan. Q — W cos. 0 tan. Q . (2) 

Therefore the resultant force acting is: 

W sin. 0 — W cos. 0 tan. Q. 

or since tan. 0 = i and tan. Q =f, this becomes 


v (V — y) (--- P q - — cos. 0 tan. q) = V(y' — y ) ( — — D ‘ — cos. 0 tan. 

sec - P ' Vyi + tan.2 0 J 

= r<y - v) (7f?W - /CO8 '0- 

By neglectiDg i, however, thereby virtually putting 0 = o, this final expression reduces to 
the same form as M. Sainjon’s approximate expression — V (y'— y) f which is the one 
used. Therefore no results are vitiated. 

2 Sainjon is so quoted by Partiot (“ Sables de la Loire,” p. 32). In one case, at least, that 
of large sand, Dubuat states (” Traite d’Hj draulique,” Paris, 178G, p. 04) that the velocity 
given is that at which the sand began to be moved as Ihe bottom velocities were increastd 
gradually from 3 up to 8 polices per second. 












HOOKER ON SUSPENSION OF SOLIDS IN RIVERS. 


19 


These results are tabulated as follows: 


Kind of material. 

Bottom velocity 
of current. 
Meters per 
second. 

v 2 

2 9 

y’ — y 

V 

F 

-jrb’-y) 

Y 2 ~g 

/ 

Dark potter's clay. 

0.C81 

0.0003 

1.64 





Sand deposited by this 








clay. 

0.102 

0.0013 






Coarse sharp yellow sand. 

0.216 

0.0024 

1.36 

0.002 

0.0027 

0.0024 

0.88 

{Size of anise 








Seine j seed.. 

0.108 

0.0006 

1.545 

0.001 

0.0015 

0.0006 

0.40 

gravels Size of peas. 
b j Size of small 

0 189 

0 0018 

1.545 

0.003 

0.0046 

0.0019 

0.41 

[ beans. 

0.325 

0.0054 

1.545 

0.0045 

0.0069 

0.0054 

0.78 

Rounded sea pebbles ul 








an inch or more diaine- 








ter. 

0.650 

0.0215 

1.614 

0.018 

0.0291 

0.0215 

0.74 

Angular flints (size of a 








hen’s egg). 

0.975 

0.04847 

1.250 

0.045 

0.0562 

0.0484 

0.86 


The mean value of f is 0.68. Eliminating the two values, 0.40 and 
0.41, so widely different from the others, the mean would be 0.80. 
Since 0.68 is approximately the tangent of the slope of a natural talus 
of ordinary earth, while wet sand and earth should have a greater co¬ 
hesion, M. Sainjon chooses to use the value f = 0.80. 

Taking the ratio of the bottom velocity in the Loire to the mean 
velocity at 0.7 (determined by measurements with a Woltmann’s wheel), 

V 

and assuming in general y' — y = 1.50, while equals two-thirds of 


the diameter for round forms and equals the diameter for angular 
ones, he computes the following table of velocity limits above which 
gravels will begin to be dragged. 


Size of gravel. Diameter in 
meters. 

Velocity at bottom. Meters 
per second. 

Mean velocity. Meters per 
second. 

0.0025 

0.25 

0.36 

0.01 

0.50 

0.70 

0.04 

1.00 

1.43 

0.10 

1.50 

2.14 

0.17 

2.00 

2.86 

0.38 

3.00 

4.29 

0.67 

4.00 

6.21 


M. C. Lechalas published a memoir, 1 also in 1871, in which he takes 
exception to the theory attributing suspension to the phenomenon of 
relative velocities. He urges that this assumes flow in parallel fila- 


i •• Les rivRres a fond de sable.” Annales des Ponts et Chaussecs, 1871. Published also, 
after revision, as an annex to Guillemain’s “Navigation Interieure—RivRres et Canaux.” 
Tome I. Paris, 1885. 


















































20 


HOOKER OK SUSPENSION OF SOLIDS IN RIVERS. 


ments which corresponds in no wise to movements under great veloc¬ 
ities. His explanation attributes suspension to repeated shocks from 
the molecules of water moving more rapidly than the suspended body 
and to the action of eddies caused by the banks and bottom. He also 
calls attention to the fact that the variations in velocity in large rivers 
are much more rapid in the vertical than in the horizontal direction. 
To these rapid vertical variations he attributes the formation of some 
horizontal vortices. 

The body of this valuable paper is devoted to an attempt to derive 
numerical results for the values of the mean depth, mean velocity and 
fall, in alluvial rivers, which will follow the contraction of its width, 
throughout a given length, by training walls. Certain parts, however, 
of M. Lechalas’ work have a direct bearing on the relations between 
velocity and movement of sedimentary matter. It will be seen that he 
lays stress on the distinction between transportation by dragging and 
by suspension. 

Referring to Dubuat’s experiments, he expresses the excess of 
pressure on the up-stream face of an immersed body as proportional 
to the square of the velocity of the water surrounding the body, i. e., 
for sand grains on a river bed, 

Thrust of the moving water = a 1 v 0 2 , 

where a is a constant which varies with the dimensions, form and posi¬ 
tion of the grains of sand. 

The resisting force of the sands of the Loire is put equal to a 0.25 2 
since they are not transported until the bottom velocity reaches 
0.25 meter. 2 

The resultant force— 

P — a ( v 0 2 — 0.25 2 ) 

is put equal to the mass of the particle multiplied by its acceleration 3 
parallel to the direction of v 0 . Measuring the velocity v 0 in the 
direction of the axis of the river, M. Lechalas considers this resultant 


a v o* corresponds to the expression 


k 


2 g 


used by M. Partiot, quoted on page 17. 


2 Compare table quoted from Sainjon at page 19. This refers to sands not already com¬ 
pacted by the continued action of currents of velocity too slight to transport the grainp, but 
yet sufficient to increase the resisting power of the surface lamina. 


2 M. Lechalas has used the word vitesse here. It must, however, be meant for accelera¬ 
tion.-See “Navigation Interieure,” Guillemain, Annexes—Rivieres a fond de sable, 
p. 489. 





HOOKER OK SUSPENSION - OF SOLIDS IN RIVERS. 


21 


force proportional to tlie discharge of sand in the river and puts— 

a (v 2 — 0.25 2 ) — bd 
or 

d = y K, 2 — 0.25 2 ) = m [v 2 — 0.25 2 ) - m (v 2 - 0.06) 

where d represents the discharge of sand per unit of width. 

The value of in is to be determined by observation, and in this 
way a correction made for the use of v 0 , the absolute velocity of the 
water at the bed, instead of the relative velocity of the water and the 
solid particle. When the velocity v 0 becomes greater than a certain 
value, the particles are lifted and cease to roll on the bottom. The 
term — in 0.06 then disappears and 

d = m v 2 . 

for particles suspended immediately above the bottom. 

The advancement of the crests of the sand bars measured by M. 
Sainjon in the Loire gives a method of determining the value of d for 
corresponding values of surface velocity v^, and also a means of com¬ 
paring this advancement with the corresponding velocities of the 
current v 0 at the bottom. 

Table of Observations on Advancement of Crests of Band Bars. 


Observed surface 
velocity in meters 
per second. 

Height in meters of the 
crest above the down¬ 
stream bed. 

Displacement of the crest in hundred- 
thousandths OF A meter per second. 

Observed for a lapse of 
many days. 

Computed. 

0 58 • 

0.000 

3.0 

3.0 

0.64 

0.300 

3.3 

3.9 

0.73 


5.1 

5.5 

0.75 

0.782 

6.3 

5.9 

0.81 

0.967 

6.7 

7.1 

0.81 


7.5 

7.1 

0.83 

0.760 

7.6 

7.5 

1.00 

0.953 

10.5 

11.6 

1.016 

0.920 

12.4 

12.0 

1.016 

0.580 

12.0 

12.0 

1 03* 

0.487 

6.2 

12.35 

1.05 

0.612 

7.0 

12.9 

1 11 

1.198 

5.8 

14.6 

1.13 

0.650 

8.7 

15.2 

1.33 

0.950 

5.6 

21.6 


* In the table quoted above by Lechalas from Sainjon the following note appears: •' It is 
wrong to suppose the co-existence of rolling and suspension for the velocities 1.03 m. * * * * 
The absolute lack of accord with the law of advancement up tj Uj =1 016 can only corre¬ 
spond to a complete transformation of the method ot transport. If there was a mixed period 
extending between the surface velocities 1.03 and 1.33, the calculated velocities would only 
differ gradually from the observed ones. Instead of that, we see that beyond = 1.03 the 
observed movements are not more than half the computed ones. This remaining advance¬ 
ment is explained by the deposit of sands held in suspension—a deposit caused by the 
sudden diminution of velocity below the crest. No trace of the mixed period remaining 
when the surface velocity reaches 1.03 m., it is probable that it was about over when the 
value reached 1.02 m.” 



















22 


HOOKER OK SUSPEKSIOK OF SOLIDS IK RIVERS. 


The computed values of the displacement of the crest given in the 
last table are derived from the formula— 

Displacement = 0.00013 (v x 2 — 0.11). 
in which M. Sainjon expresses the rate of advancement of the crest 
so long as the surface velocity does not exceed 1.016 m. per second. 

In discussing this formula M. Lechalas calls attention to the fact 
that this displacement becomes o for v x = -\/0.11, which virtually 1 
corresponds to a bottom velocity of 0.25 m., and that one could write 

d = m {v 2 — 0.11). 

by giving to m the value 0.00013 x the mean height of the crests in the 
last table corresponding to the surface velocities from 0.58 m. up to 
1.016 m. This mean height is equal to 0 77 m. and the product gives 
m = 0.0001. 


Whence 

d = 0.0001 (V —0-H) .(0) 

He objects, however, to this form because v t , the surface velocity, 
will vary with the depth of the stream and so introduce another 
variable. 

Since the bottom velocity v 0 ought not to vary widely, he prefers 
his own equation 

d = m ( v 0 2 — 0.06).(1) 

to equation (o), preceding, as given by M. Sain j on. 

Referring to the preceding table, he adopts the value 1.016 m. 
as the upper limit for surface velocities at which dragging occurs. Up 
to this velocity the equation (1) will be used, and beyond this limit 
the equation 

d =m v 2 .(2) 

will be used to express the relation between solid discharge and 
velocity at the bottom. 

It remains to find the value of v () 'which corresponds to r 1 = 1.016 m. 
M. Lechalas does this by using the formulas of Darcy and Bazin 


Zi 


v‘ 


0.00028 + 


0.00035 


and v 1 — v + 14 y/ Z i .... 
By combining these two equations, 


= 1 + 14 

» M 


Zi 


V 


= 1+14 


s|°- 


00028 + 


0.00035 

Z 


(a) 

■(*) 

(c) 


1 M. Sainjon (see page 
— 0.23 m. per second. 


) considers v 0 = .7 v,; t>, = >/ 0.11 = 0.331; v e = 0.7 X 0.33 A 


















HOOKER OK SUSPENSION OF SOLIDS IN RIVERS. 


23 


Substituting the value = 1.016 m. and the values 
Z = mean depth = 0.5 m. 

= 1.0 m. 

= 2.0 m. 

he derives the corresponding values 

v = 0.71, 0.75 and 0.78. 

By combining Darcy and Bazin’s equations 

v \ — v + 14 V Z i and 
©i = v 0 -}- 24 V Z i 

he obtains 

v 0 = v — 10 's/ Z i . 

which combined with (a) gives 

4 


(d) 




= 1 — 10 I 0.00028 -f 


0.00035 


(«) 


v \J 1 Z . 

Substituting the values of v which correspond to the assumed 
values of Z, the values of v 0 are obtained— 

v Q = 0.49, 0.56 and 0.61. 

M. Lechalas adopts a mean of these values, v 0 = 0.55 m., as the 
upper limit of the bottom velocity corresponding to transport by 
rolling on the bed of the Loire. Since the range of velocities for 
which the table on page 21 gives indications of a combined mode of 
transport occupies such a small part of the velocity scale (from = 
1.016 m. to < 1.03 m.), he assumes the same value, v 0 = 0.55, as the 
lower limit of the bottom velocity corresponding to transport by sus¬ 
pension. 

To show more clearly the actual relation of these velocity limits to 
the variables of the current, an ideal canal is assumed, of constant 
width and of a flow equal to 3 cu. m. per unit of width, so that 

5 = 1 and q = 3. 

Equations (a) and (d) preceding combined with the equation 

q = Z v, 

which expresses the definition of liquid discharge when b is put equal 
to 1, give, for v 0 = 0.55 in., 

i = 0.000035, Z = 4.50 m., and v = 0.67 m. 
and, for v 0 = 0.25 m., 

i = 0.000003, Z = 10.00 m. and v = 0.30 m. 

To express these results in the words of M. Lechalas: 

“A bed of regular width, filled with sand which is not renewed, 
and which lies at an inclination exceeding a certain limit, receives a 










24 


HOOKER OK SUSPENSION" OF SOLIDS IN RIVERS. 


discharge of 3 m. of water per second per unit of width. After a 
length of time, greater or less, according to the fall and the length of 
the canal, a state of unstable equilibrium establishes itself. The mean 
depth is then 4.50 m., the mean velocity 0.67 m., the fall 3.5 cm. per 
kilometer, and the bottom velocity 0.55 m. 

“ The sand, however, is still transported, but in quantities smaller 
and smaller each second. After a considerable time a new state of 
equilibrium is established. This is final; it corresponds to a mean 
depth of 10 m., a mean velocity of 0.30 m., a fall of 3 mm. per kilo¬ 
meter, and a bottom velocity of 0.25 m. Although these computations 
apply only to an ideal channel, yet they are of interest as showing what 
an important role is played by the consideration of these velocity 


limits in the study of alluvial rivers.” 

Returning to the equations— 

d = in ( v 0 2 — 0.06) for 0.25 m. < v 0 < 0.55 m.(1) 

and d = m v 0 2 for v 0 >> 0.55 m.(2) 


M. Lechalas uses the following method to determine the value of 
m. By combining equations (c) and (e) of pages 22 and 23. 


v 

V 


1 


0 


1 + 14 


1 — 10 


0.00028 + 



0.00028 + 


0.00035 

z 

000035' 

z 



which gives the ratio between the surface and bottom velocities in the 
artificial canal used by Darcy and Bazin. 

The bottom velocity ought to be less dependent upon the depth than 
that at the surface. If a formula is expressed in terms of the bottom 
velocity, it may properly be transformed into terms of the surface 
velocity and mean depth, or of mean velocity and mean depth. On 
the other hand, when the formula is in terms of the surface velocity, 
and it is desired to express it in terms of the bottom velocity, it is 

necessary to assume the ratio a constant for all values of the mean 

depth. This introduces an approximation unavoidable without a new 
series of observations. 

Assuming Z = 1 meter 1 in equation (/) 


- 1 = 1.80 2 . 


1 The mean of the values of Z used to obtain the critical value v 0 — 0.55 m., and hence 
the most consistent value to use in determining m. 

2 For Z = 3 m., — = 1.60. 

v 












HOOKER OK SUSPENSION OF SOLIDS IK RIVERS. 


25 


Equation (o) page 22, which is based upon M. Sainjon’s empirical 
foimula, may be considered reasonably accurate for the range of 
velocities for which it is intended, as can be seen from a study of the 
computed results in the table of page 21. 

Equation (o) and equation (1) of page 22 may now be written 
d == 0.0001 (v 2 — 0.11) = m {v 2 — 0.0G) 
and, by introducing the approximate value v l = 1.80 v 0 from page 24, 
the value found 

0.0001 \1.8 v 0 — 0.11/ 
v,/ — 0.06 

For v 0 = 0.50 m. per second. 

m = 0.00037 1 

and M. Lechalas’ equation (1) becomes 


d = 0.00037 (v 2 — 0.06).(3) 

and (2) becomes 

d = 0.00037 v 2 .(4) 


The objections to the introduction of the uncertain value of the 

'V 

ratio — in obtaining these final equations are all admitted, but M. 

v o 

Lechalas maintains that if a numerical coefficient can be used when 
the discharge d is expressed in terms of v y at the surface, 2 one can be 
much more reasonably used when the equation is in terms of the bot¬ 
tom velocity v 0 . 

The years 1874 to 1879 marked the arousal of a great popular inter¬ 
est in the United States in the question of silt movements in the 
Mississippi. The bitter controversy between the Government engineers 
and Captain James B. Eads with his associates over the improvement 
of the mouth of the river need not be entered into here. Suffice it to 
say that the many spirited articles written on the subject during those 
years were not of great scientific value and left the knowledge of the dis¬ 
tribution of the sediment in the river in the same state of incompleteness 
in which it was left by the report of Humphreys and Abbot in 1861. 

Mr. Eads states his views in a letter 3 of March 15, 1874, with refer¬ 
ence to these sediment movements in the following words: 

1 For values of Vo between 0.40 m. and 0.55 m., the corresponding values of m range 
between 0.00041 and 0.00036. 

2 As is done in the equation (o) based on M. Sainjon’s formula; displacement = 0.00013 X 
(u 2 —0.11). 

3 To William Windom, United States Senate, Chairman of Committee on Transportation 
Routes to the Seaboard. 

See “The Mississippi .Jetties,” p. 28, E. L. Corthell, New York, 1881. 







26 


HOOKER ON SUSPENSION OF SOLIDS IN RIVERS. 


“ By far the greatest portion is, however, transported in suspen¬ 
sion. The amount of this matter and the size and weight of the 
particles which the stream is enabled to hold up and carry forward 
depend wholly upon the rapidity of the stream, modified, however, 
by its depth. * * * A certain velocity gives to the stream 

the ability of holding in suspense a proportionate quantity of solid 
matter and when it is thus charged it can sustain no more. 

The fact that a given current will keep in suspension a corresponding 
quantity of solid matter; that at a less velocity a portion of it will 
be deposited and taken up again at a greater, is fully recognized 
in experimental science and has been extensively made use of for 
analysis of soils. An eminent investigator of this subject, Prof. E. 
W. Hilgard, of the University of Michigan, now of the University 
of California, Oakland, Cal., has classified silts according to the 
different velocities at which they deposit. 1 This independent line 
of research fully confirms the view herein advanced in explanation of 
the phenomena presented through the alluvial bed of the Mississippi.” 

Gen. A. A. Humphreys, Chief of Engineers, expresses his views in a 
report 2 to the Secretary of War, dated April 15, 1874, in the following 
words: 

“It has been recently stated by a civil engineer, 3 in a pam¬ 
phlet concerning the improvement of the mouths of the Mississippi 
River by jetties, that the amount of sedimentary matter carried in 
suspension by the Mississippi River is in exact proportion to the 
velocity of its current; and that, as a given velocity of current will 
keep in suspension a corresponding quantity of solid matter at a less 
velocity a certain portion of it will be dropped. * * * The first 

statement is in direct conflict with the results of the long-continued 
measurements made upon the quantity of earthy matter held in sus¬ 
pension by the Mississippi River at Carrollton, near New Orleans, and 
at Columbus, 20 miles below the mouth of the Ohio, one of the chief 
objects of which was to determine this very question, whether any 
relation existed between the velocity and the quantity of earthy 
matter held in suspension. These results prove that the greatest 
velocity does not correspond to the greatest quantity of earthy matter 

1 American Journal of Science III, VI, 337. 

“ The classified table of Prof. Hilgard gives the relative velocities created in a mechanical 
contrivauce made for test purposes in a laboratory in which coarse sand is dropped at a cer¬ 
tain velocity of the machine, which may be represented in nature as a current of about 2.5 
ins per second; the finest sand when the current is 0.3 of an inch per second; the coarsest 
silt when the velocity is 0.14 of an iuch per second; the finest silt when the velocity is 0.02 of 
an iuch per second.” 

“ Report of Chief of E lgineors, 0. S. Army,” 1874, Part I, p. 865. 

* “ Report of Chief of Engineers, U. S. Army,” 1874. Part I, p. 863. 

3 James B. Eads. 





HOOKER ON SUSPENSION OF SOLIDS IN RIVERS. 27 

held in suspension; on the contrary, at the time of the greatest veloc¬ 
ity of the current at Carrollton, the river held in suspension but little 
inoie sediment per cubic foot than when the velocity was least. When 
the quantity of earthy matter held in suspension was greatest the 
velocity was 2 ft. per second less than the greatest velocity, the quan¬ 
tity of earthy matter in the one case being three times as great as in the 
othei. We find at another time, when the velocity was one-half the 
greatest velocity the quantity of earthy matter held in suspension 
was double the amount. Again, we find the quantity of earthy matter 
in susjmnsion the same, the velocity in the one case being 6.75 ft. per 
second and in the other, 1.5 ft. per second. 

I. —Carrollton, 1851. 


Date. 

Weight in grains 
of sediment in 
1 cu. ft. of 
water. 

Mean velocity 
of river in 
feet, per sec¬ 
ond. 

Remakes. 

February 20th. 

450 

6.5 


March 20th. 

200 

6.2 


April 15th. 

150 

5.6 


May (last week of). 

100 

3.75 


June 20th. 

650 

4.3 


July 10th to 30th. 

450 

4.8 


August 1st to 20th. 

450 

From 4.8 to 
3.5 

Change in velocity regularly 
decreasing, while suspended 
matter remains the same. 

September 8th . 

300 

3.0 


October and November. 

100 

1.75 


December. 

175 

1.85 


January 20th, 1852. 

400 

2.75 



II.— Columbus. Twenty Miles below the Mouth of the Ohio, 1858. 


Date. 

Weight in grains 
of sediment in 
1 cu. ft. of 
water. 

Mean velocity 
of river in 
feet per sec¬ 
ond. 

Remarks. 

April 1st . 

300 

7.00 


April 10th . 

300 

5.25 


April 25th. 

May 1st. 

450 

300 

7.25 

7.50 


Mav 10th. 

300 

5.75 


Mav 22d. 

160 

6.75 


June 16th. 

330 

8.25 


July 16tb-17th. 

650 

3.75 


August, 2d. 

350 

4.75 


Aupust. 9th ... 

250 

4.00 


fipntftmlier 2d. 

600 

2.50 


9t,h to 93d. 

200 

2.25 


October (all of). 

200 to 100 

1.50 

t Uniform decrease in amount 
| of sediment, the velocity 
( remaining the same. 





































































28 


HOOKER OK SUSPENSION OF SOLIDS IN RIVERS. 


“ The tables (on page 27) illustrating what has just been said, have 
been prepared from the report on the Mississippi River. The figures 
given express the conditions not only on the day noted but on several 
successive days. 

“It is to be remarked that the investigations respecting the sedi¬ 
ment in suspension show that the quantity depended on the river 
from which the volume of discharge was at the time chiefly derived. 

“The cross-sections, both at Carrollton and Columbus, remained 
unchanged during the above observations.” 

In order to define still more clearly the position of General Hum¬ 
phreys on this question, the following quotation is made from his re¬ 
port of 1875 1 1 

“It has been sometimes stated that every velocity of current is 
capable of carrying in suspension a certain fixed quantity of earthy 
matter, and that the water of a muddy river is always thus charged with 
the maximum quantity of earthy matter it can carry. * * * But this 
assumption as to the carrying power of currents is utterly disproved 
by long series of exact measurements upon the Mississippi River. 
* * * These measurements upon the quantity of earthy matter 

suspended in the Mississippi River show that at no time has the w r ater 
been so heavily charged with it that the current could not carry it 
along in suspension to the same extent as it did when the quantity of 
earthy matter was least; and they further show that the current of the 
Mississippi River, when most feeble, can carry in suspension the 
greatest quantity of suspended earthy matter found in it to the same 
extent that it can carry the least quantity found in it. 

“It was undoubtedly the observation of facts similar to these that 
led to the conclusion, entertained by some, that the suspending power 
of the current of a river did not depend upon its absolute rate of mo¬ 
tion, but upon the difference of velocity between the adjoining fillets 
of water. There is good reason to conclude that this is one of the 
causes or sources of the suspending power of a stream. 

“This proposition, therefore, respecting certain velocities of cur¬ 
rent always carrying certain fixed quantities of earthy matter, and 
always adjusting those quantities according to its own variations of 
strength is so entirely disproved by facts that it will not be considered 
again.” 2 

1 Annual Report of Chief of Engineers. U. S. A..” 1875, Part I., pp. 959-975. Reprinted in 
Humphreys and Abbot’s “Report on the Mississippi River.” Edition of 1876. Appendix M. 
p C84 

2 Those readers who wish to go farther into the details «>f this somewhat amusing cou- 
troveisy are referred to Humphreys and Abbot's Report on the Mississippi River, Edition of 
1876. Appendices. 

Review of same by James B. Eads, M. Am. Soc. C. E. in Van JVostrand’s Engineer inq Maq 
azine, Vol. XIX. 1878, pp. 211-229. 

Answer to Mr. Eads’ attack by General Henry L. Abbjt, Van Nostrand’s Engineerinq Maq 
azin“, Januaiy, 1879, Vol. XX, pp. 1 6. 

Auswer tj General Abbot by Mr Eadi, Van Nostrand's Engineering Magazine, Vol. XXI, 
1879, p. 154. 




HOOKER ON SUSPENSION OF SOLIDS IN RIVERS. 


29 


An article by Mr. G. K. Gilbert, 1 upon the erosion of the Colorado 
canons, appeared in the American Journal of Science, July and August, 
1876. While subject to some criticism, it may be regarded as a most 
valuable contribution to the knowledge of the laws of transport of 
solid bodies by water currents. It is believed that Mr. Gilbert is the 
only writer who has called attention to the fact that the same expendi¬ 
ture of energy will transport a greater weight of tine particles than of 
coarse ones of the same density. 

A series of observations was conducted by Assistant Engineer J. B. 
Johnson at Helena, on the Mississippi River, in 1879. 2 Longitudinal 
and transverse soundings were made to determine the existence and 
movement of sand waves in the river bed, and the results plotted 3 4 so 
as to show clearly the presence of these undulations. From the obser¬ 
vations Mr. Johnson deduces the following facts: 

“ Average length of waves from crest to crest, about 100 m. 

“Extreme length of waves from crest to crest, about 150 m. 

“ Average height of waves from crest to valley, about 5 ft. 

“ Extreme height of waves from crest to valley, about 8 ft. 

. “ Average velocity of motion of crest, 5.41 m. per day. 

“ These results were obtained in a depth of water varying from 13 
to 30 ft. The stage of the river varied from 12 to 18 ft. above low water 
at Helena. The waves decreased in size for a falling river and vice versa. 

Their rate of motion down stream is a function of the velocitv of the 

%} 

water. They do not extend from bank to bank at Helena but disap¬ 
pear about 200 m. from each shore, covering about 1 000 m. of the 
cross-section of the river.” 

Sediment measurements were made by the same party from March 
1st to June 18th, 1879, and deserve special mention because of the in¬ 
troduction of an improved sediment can for bringing up specimens 
from the bottom. 

Samples were taken each day from the surface and 1 ft. above the 
bottom at points one-fourth and three-fourths of the distance across 
the river. Proportions of sediment were determined by weight in the 
later experiments and the mean velocity of the river was determined 
by floats upon five occasions during the extent of the observations. 5 

1 See digest in Engineering Ne.ws, August 19, 1870. 

2 See “ Report of Chief of Engineers, United States Army,” 1879, Part III, pp. 1963- 
1970. 

3 See Plate I. p. 1966. “Report of Chief of Engineers,” 1879, Part III. 

4 For sketch, see “Report of Chief of Eugineers, United States Army,” 1879, Part III, p. 
1965. 

5 The tabular result of these observations is given at p. 1969 of above Report. 




30 


HOOKER ON SUSPENSION OF SOLIDS IN RIVERS. 


Simultaneous observations of a like nature were conducted at St. 
Louis 1 by R. E. McMath. Tbev are more satisfactory in that they 
offer a slight opportunity for study of transverse distribution of sedi¬ 
ment. Both sets give velocity measurements. 

The most extended set of observations published upon sedi¬ 
ment movements and sand waves are those instituted by the Missis¬ 
sippi 2 and Missouri 3 Commissions in 1879-1881. These were made at 
St. Louis, Carrollton, Prescott, Winona, Clayton, Hannibal, Grafton 
and St. Charles. They are wide enough to put at rest certain debated 
questions, but yet fail in several points to be completely satisfactory— 
notably in failing to give data on horizontal distribution. These 
measurements will be again referred to. 

Major Allan Cunningham made a series of observations on the 
Ganges Canal 1 to determine the amount of sediment carried and its 
distribution in the cross-section. 

A tube 12 ft. long, open at both ends, was thrust down vertically 
from a floating boat until the bottom was reached. It was then closed 
at the bottom, by a lid worked by a spring, and the column of water, 
extending from bed to surface, carefully separated from its sediment 
by decantation and filtration. 

This sediment, when weighed, gave a result, called silt-density, 
which represented the average density in the vertical examined. 

To determine the distribution of silt, two collections were made, 
by the method indicated, at each of nine points in the width of the 
canal at two different cross-sections. Each set was completed as 
rapidly as possible. The mean silt velocity past each vertical was 
computed by multiplying this silt density by the corresponding mean 
velocity. Cajitain Cunningham then plotted three transverse curves 
on a common base, using as ordinates the silt density, the mean silt 
velocity, and the mean velocity past each of the nine verticals. From 
the want of relative connection between these curves he concludes that 
in the Ganges Canal there is no close relation between the silt and 
the velocity at different parts of the channel, and that the silt density 
at any point varies from instant to instant. 

1 Van Nostrand’s Engineering Magazine, 1883, p. 33. 

2 “ Report of Chief of Engineers United States Army,” 1883, Ilf. 

3 “ Report of Chief of Engineers, United States Army,” 1887, IV. 

■i " Roorkee Hydraulic Experiments,” Roorkee, 1881, Chap. XXIV. Abstracted and discussed 
in Proceedings of the Institute of Civil Engineers, 1882, Vol. LXXI, pp. 1-94. Same reproduced 
in Van Nostrand's Engineering Magazine, April and May, 1883. 




HOOKER ON SUSPENSION OE SOLIDS IN RIVERS. 


31 


In continuation of these measurements, 73 collections were made at 
four of the cross-sections, the depth and velocity at two of them being 
very different. These results led to the conclusion that the mean silt 
density in no way depended upon the depth or velocity in this canal, 
but rather upon the state of the supply water from the Ganges. 

The best known formula for the determination of the size of parti¬ 
cles dragged by a current of a given velocity is that proposed by Mr. 
Wilfred Airy, and derived by him as follows: 1 

Let a = the length of the largest cube the current could move. 
Then weight of cube = y' a 3 * (y' const.). 

Friction of cube on bed of river =/' y< a 3 (p const.). 

Total pressure of current on exposed face of cube = k a 2 v * (k 
const.) 

For equilibrium— 

/' y' « 3 = k v 0 2 a 2 

whence 




therefore the weight of the largest cube which a current with a bottom 
velocitv v„ could move would be 

«/ it 


y' a 3 — y 



k 




v. 


f> 


If G' and G" were the weights of cubes of silt, etc., which could 
just be moved by currents of bottom velocities v 0 ' and v 0 " respectively, 
then 


G’ 

G" 



or, numerically: 

If v 0 is increased by £ of itself, it will move particles of twice the 

/ 1 V 5 1 

weight, since / an( l ^ velocity v 0 is doubled, it will 

z 1 \ 6 1 

move particles of 64 times the weight, since ( —- j = — . 


1 See condensed description in Proceedings of the Institution of Civil Engineers, Vol. 
82, p. 25. Notation changed. 

See Church’s “ Mechanics of Engineering,” p. 831. 

A formula, showing that the scouring power of a natural stream is proportional to the 
seventh power of the velocity, is said to have been proposed about 1855, by W. Hopkins, of 

Cambridge, England. 

See Baldwin Latham in Proceedings of the Institution of Civil Engineers, Vol. 82, p. 43. 











32 


HOOKER OH SUSPEHSlOH OF SOLIDS IH RIVERS. 


Mr. Henry Law shows this formula to be also applicable to the 
case of a cube rolled along instead of sliding, and to be true for a 
sphere as well as a cube. 1 His proof follows: 

The moment of resistance of the cube to turning about its edge is 


, 3 a t a 
y a ’ = = r 


The turning moment of the thrust of the current is 


7 2 2 

k a v 0 


a k a 3 v, 


At the instant of turning the equation of equilibrium gives 


, a 


k a 3 v 0 2 


whence 


k o 

a =y v ' 


Following the same process used by Airy above, this leads to his 
formula (1). 

In the case of spheres, assume each one to be resting upon three 
others. 

Weight of sphere = n r 3 y'. 

6 

Let r sin. ft — lever arm of weight about point of turning. 

Then moment of resistance to turning is 

4 4 

— tc r s y’. r sin. ft = — tc y ] r 4 sin. ft 
o o 

Thrust of the current = k it r 2 v 2 . 

Its lever arm about the i^oint of turning would be r cos. ft. 

Then the turning moment due to the thrust would be 

k it r 3 v 2 cos. ft. 

For impending motion, the equation of equilibrium gives 

4 

k 7t r 3 v 2 cos. ft = — 7r y' r A sin. ft 

o 

whence 

3 k n - „ 

r = —;-- v n cot. ft. * * 

4 y' 

This again leads to Mr. Airy’s equation (1) as above, ft being con¬ 
stant as r varies. 


1 See Proceedings of the Institution of Civil Engineers, Vol. 82, pp. 29-30. Notation 
changed. 

* Through an oversight, Mr. Shaw has obtained an incorrect numerical coefficient for 
this last equation in having used, for the value of the section of the sphere normal to the 
current, % it d 2 instead of j n d 2 . It is corrected here. 









HOOKER, OH SUSPENSION OF SOLIDS IN RIVERS. 


33 


Mr. Shaw then concludes that the weight of particles moved by a 
current, whether cubes or spheres, and whether the action be sliding 
or rolling, will vary as the sixth power of the mean velocity of the 
current impinging on them, if cohesion between the particles be dis¬ 
regarded. 

t 

M. J. Thoulet, Professor of Science, at Nancy, published in 1884 
the results 1 of some experiments made to determine the force required 
to keep particles of different sizes and densities suspended in water. 

The apparatus used consisted of a glass tube placed in a vertical 
position and connected at its lower end by a rubber tube with a stop¬ 
cock to regulate the velocity of a water-current ascending through the 
glass tube. The water was led away by a waste-pipe connected near 
the top, and the velocity for each experiment determined from the 
weight of water flowing. The details of the experiments were carried 
out with scientific exactitude. 2 

M. Thoulet computed the mean velocity of the current required 
in tubes of four different diameters (2.2, 4.775, 6.75 and 8.0 mm.) to 
hold unmoved, at a fixed point, spheres of different sizes and densi¬ 
ties. These spheres were lead bullets of different calibers and balls 
of wax containing, in their interior, grains of tin, lead or copper. 
Their sphericity was tested under the microscope, and in all cases they 
were kept at the specified height for a length of time not less than 30 
seconds. 

From his results M. Thoulet has computed a table giving in milli¬ 
meters per second the velocities of vertical currents of water capable of 
holding in suspension, at a fixed height in the tube, spherical grains 
of known radii and of given densities. These radii vary from 1 to 2.5 
mm. and the densities from 1.5 to 4. The table 3 also gives, in milli¬ 
grams, the thrust of the current against the grain. 

This thrust is equal to the resultant weight of the grain immersed 
in water, i. e., 

-f it r 3 (y' — 1) = thrust 

and M. Thoulet has computed the values corresponding to the differ¬ 
ent values of r and y' from this formula. 

Making the assumption of spherical grains in a stream bed, he con¬ 
siders that each one may be regarded as resting on three others, and 

1 Ann ales des Mines, 1884, I, pp. 507-530. For digest, see Annates des Fonts et Chaussees, 
1885. I. pp. 492-500. 

2 See description in Annalesdes Mines, 1884, I, pp. 507-530. 

3 The same, p. 621. See p. 76 of this paper. 





34 


HOOKER OH SUSPENSION OF SOLIDS IN RIVERS. 


shows graphically that for a horizontal movement a force will be 
required sufficient to move the grain up a slope of about 37°. 1 

This force = $ 7t r 3 (y r — 1) sin. 37°. 

By referring to his table M. Thoulet determines the bottom veloc¬ 
ities required to exert a force equal to that demanded by this formula 
for the three cases given below. 


Material. 

Diameter of grains in 
millimeters. 

Velocity required in 
millimeters per 
second. 

Coarse mud.. 

0.40 

40 00 

Fine sand. . . 

0.70 

69.68 

lliver sand. 

1.70 

109.58 




M. Vautliier, in a valuable paper 2 before the French Association 
for the Advancement of Science, in 1884, developed mathematical ex¬ 
pressions for the velocity, at any instant, of a solid body falling 
through a liquid, and for the path described in a given time. 

His method consists in writing the accelerating force equal to the 
mass multiplied by the acceleration of the body, assumed to be a sphere. 

Accelerating force = weight of body — - resisting force 

= - 3 -nr (y* — 1 ) — it r k 

When the motion has become uniform the accelerating force will be 
zero, and one may write for this case. 

-L. X r* (y’— 1) — n r 2 k 0 . (1) 

«L/ 


whence, 




r g 


(r'~ i) 


( 2 ) 


where v' represents the limiting velocity, after which motion is uni¬ 
form. 

For any stage of the motion 


-Q- Tt r 3 (y 1 — 1) — 7 r r 2 k 


v 2 4 n r 3 t civ 

2 g 3 g ^ dt 
= mass X acceleration. 


(3) 


1 This is the value of the angle /3 iu Mr. Shaw’s analysis preceding. 

2 “ Do l’entrainement et du transport, par les eaux courantes, des vases, sables et gra- 
viers." 

L. L. Vauthier, “ Memoires de 1’ association fran^aise pour 1* avancement des sciences,” 
Blois, September 8, 1884. 

Abstracted at length in the “ Memoires de la Sqciete des Ingeuieurs Civil-* de Franre,” 
1885, 2, pp. 29-35. 

General results also given in Engineering News, November 1, 1884, p. 211. 





























HOOKER OK SUSPENSION - OF SOLIDS IN' RIVERS. 


35 


Subtracting (1) from (3) and simplifying 

k (V*- V 2 )= 8 dv 

4 (it 

Separating the two variables 

A ^ ^ * d v 

'8 r y' r —v 2 * . ( 4 ) 

Since v = o for t = o one may write: 

C —~ ( n = r dv 

J o 8 r y' J 0 v ,a — V * 

Integrating by partial fractions 

jl * = _Lf r _£*_ n -dv 

8 r r' 2 v 1 \J 0 v' + v J „ 

— 2 *' og ' e \zrz~i) 

whence, 

e 4- ~ V < = - +P . 

4 )7' — -y 

Putting 

Ar 3 it 

N= ~ivy v .( 5 ) 

and transforming 

, — 1 

* = * e «T"l .. (6) 

and, since cl s — v cl t 

e Nt _ 1 

ds = v 'wr+-i dt .. (?) 

whence, by integration, since when s — o, t = o : 

s = *' [* - w Iog -« 2 + w Iog -« ( 1 + -^ r )] 1 .(« 

1 Put eNt — 1 = u and e^t l — u + 2. 

Then eNt — u -f- 1. Differentiating, du = eNt Ndl, 

du 1 du 

or ’ ~ nTm - -jt ir+i • 

Substituting these values in (7) above 

, u 1 du 

ds = v 1 __ i~\ 

u + 2 N u -f 1 . ( a ) 

Separating into partial fractions by the method of indeterminate coefficients (cf. 

Osborne’s “ Differential and Integral Calculus,” p. 189). 

u 2 1 

(u + 1) (u + 2) ~ u -J- 2 — u -f 1 

Substituting this value in (a) and integrating 

N r du r du 

VT s = 2 J r+2~-> JT+1 + C ’ or 

N 

— s = 2 log., (u + 2) — log., ( t + 1) + C . (6) 

(Foot note continued on next page.) 








































36 


HOOKER ON SUSPENSION OF SOLIDS IN RIVERS. 


Assuming k = 0.5 and y' = 2.0 for a mean of the particles moved 
in river beds, and g — 9.8088 = 10, approximately, M. Yauthier 
obtains from (2) and (5) (for meter measure). 

v’ = 5.16393 \/ 2~r". (9) 


N = 


1.93648 


\/ 2 


( 10 ) 


This last equation shows that for particles of slight diameter, the 
value of AT, and consequently of e , is very large. 

Writing equation (6) in the form 

1 

1 — 

v = v’ - 

1 + 






it is at once seen that the fraction in the second member rapidly ap¬ 
proaches the value 1 as the diameter of the particle is decreased, i. e ., 
as N approaches oc. Therefore v approaches v' asymptotically, and 
at the limit the two will be equal. 

For the same reason given above the transcendental term in equation 
(8) will be so small as to be negligible for particles of slight diameter. 

By numerical substitution in equations (9), (10), and (8). M. 
Yauthier shows that, for a block as large as 1 m. in diameter, at the 
end of the first second the velocity will be only about $ of the velocity 
limit v', and the transcendental term in (8) will be too large to be 
neglected, but that, with succeeding seconds, it tends rapidly to 
approach the value v', while the transcendental term tends rapidly 
toward o. 


When s = o, t = o, and hence u — o, since u = e*t — 1. Substituting these values in (b) 
and solving for C 

C — — 2 log.e 2. 

Substituting this value in (b) and introducing the values of (u -f 1) and (u -f 2) 


s = 


2 

w 


log.e (e Nt 4- 1) — 


it' 08 * eN - — 


2 log.e 2 


(«) 


To the second number of (c) adding and subtracting 


2 tb 
~N 


log.e eNt f 


and collecting similar terms 

5 = 1)1 [^JL (log.e (eM + 11 — log e eNl) -|_ (-JL-L-) log.e eV<-l og . e 2 ] 

Since log.e — Nt, and 

log.e (eV< -f 1) — log.e eNt — log.e 
there results finally 

j = »> [« log., 2 4 los -* ( 1 + -Jm) ] 


ern -f 1 


= l08 -* ( 1 + -Jin) 


Compare Riihlmann’s “ Hydromechanik,” p. 699, for a similar solution. 






















HOOKER OK SUSPEKSIOK OE SOLIDS IK RIVERS. 


37 


To determine the length of the period of time required before the 
velocity of particles of different sizes becomes practically uniform, M. 
Yauthier has expanded equation (6) by division into 



For all but very large bodies all but the first two terms may be neglected. 
To find the length of time before the actual velocity will differ from 
the velocity limit by T - 0 V 0 -, one may put 


2 _ _ 1 _ 

eNt ~ 1000 ’ 

where e Nt = 2 000, and 

_ log. 2 000 
N log. e 


(ii) 


From equations (5) and (11) M. Vauthier has prepared a table for 
particles of different diameters, showing the length of the path de¬ 
scribed in the first four seconds, and the length of time elapsing before 
the actual velocity lacks only twoo °f the velocity limit v'. 

This table shows how rapidly the velocity of fall approaches the 
velocity limit in each case, especially with the smaller particles. It is 
not until the diameter of the particle becomes as great as 1 m. that 
there is an appreciable difference between the two at the end of the 
third second, and even then the difference is slight. 

Assuming a particle free to descend with a vertical velocity v in a 
current of water whose mean horizontal velocity is u, its vertical 
velocity, except in the case of very large bodies, may be put equal to 
the corresponding velocity limit v'. 

The direction of its path, while not a straight line because of the 

relative velocities of the filaments of water, will yet, in general, form 

v' 

an angle with the horizon whose tangent is —. 

XL 

If the height above the bottom at starting was Z it will reach the 
bed of the stream after a time 


2 



and at a distance down stream 



With numerical values M. Yauthier obtains the following results: 










38 


HOOKER OH SUSPENSION OP SOLIDS IN RIVERS. 


Diameter of particle 
2 r (in meters). 

l 

Velocity of 
current 
u (in meters). 

Original heiglit 
above bed 
z (in meters). 

Time in sinking 
to bottom 
t (in seconds). 

Distance traversed 
down stream 
l (in meters). 

0.0001 (mud) 

1.00 

1.00 

19.38 

19.38 

0.001 (sand) 

1.00 

1.00 

6.12 

6.12 

0.01 (gravel) 

1.00 

1.00 

1.94 

1.94 


Suppose this body, falling through the water w r ith a vertical 
velocity v\ meets an upward accidental current with a vertical com¬ 
ponent equal to v'. It would be kept in suspension so long as the 
current endured. This is held by M. Vauthier to explain the phenom¬ 
enon of suspension. 

He draws the following conclusions from his study: 

“ (a.) Water does not possess a special property by virtue of which 
it holds in suspension minute particles of a density superior to its own. 

“ ( b .) These particles always move toward the bottom with a velocity 
which depends upon their density and wdiicli is inversely as the square 
root of their transversal dimensions. 

“ (c.) From the value of these velocities, for materials of a density 
similar to those which form the surfaces of the beds of water courses, 
the effects of displacement and of transport observed in streams and 
rivers is very well explained by the single fact of accidental or per. 
manent currents which act upon the bottom.” 

In his “ Hydraulique” 1 M. Flamant has brought together the most 
valuable parts of the theories advanced by Dupuit, Vauthier, Partiot, 
Sainjon and Leclialas. His work is of especial interest in that he calls 
attention to the bearing upon this question of an article by M. Du 
Boys 2 intended to complete Dupuit’s explanation of the increased 
velocity of a surface float over that of the mean of the surrounding 
filaments of water. M. Du Boys has completed the explanation of 
Dupuit by adding that, while the displaced water and the floating 
body are alike subjected to the accelerating force due to gravity, yet 
the resistances, to which they are subjected are different. In the dis¬ 
placed water, a portion of the gravity work is lost from the non¬ 
parallelism of the filaments and the consequent internal frictions while 
in the floating body all the accelerating force due to gravity is used in 
overcoming the friction on the sides and in producing the increased 
velocity. 

■ 1 “ Mecanique Appliquee, Hydraulique” pp. 290-311. M. A. Flamant, Paris, 1891. Baudry 

k Cie. J 

2 Annales des Fonts et Chans sties, 188G, I, p. 199. 






















HOOKER OK SUSPEKSIOK OF SOLIDS IK RIVERS. 


39 


In summing up tlie lesults of liis study, M. Flamant expresses the 
belief that the power of suspension increases with the quantity — 

with the mean or bottom velocity and with the depth. 

Experiments reported by Mr. G. F. Deacon in connection with 
studies for the Manchester Ship Canal give an accurate description of 
the detailed action of flowing water upon a bed of sand. His sum¬ 
mary of results 1 will be reproduced here. 

“ The observations were made in a long flat-bottomed trough with 
glass sides by means of which the behavior of the sand could be 
accurately observed. The sand was from the estuary of the Mersey, 
the quantities moved were weighed and the surface velocities of the 
water carefully measured. When water flowed with a steadily increas¬ 
ing velocity over a surface of such sand, fine pieces of broken shell 
were first moved, and the surface velocity required to produce such 
movements was considerably less than 1 ft. per second. At such 
velocities, however, the sand proper was perfectly stable, and however 
long the flow continued it remained undisturbed; but the fine pieces 
of shells at the surface of the sand moved in spasmodic leaps, accumu¬ 
lating wherever the velocity was somewhat less. 

“ The first movement of sand began at a surface velocity of 1.3 ft. 
per second. This movement was confined to the smaller isolated 
grains; and if the same velocity was maintained, the grains so moved 
ranged themselves in parallel bands perpendicular to the direction of 
the current, each band taking the form of the well-known sand rip¬ 
ples of the sea shore or sand-bottomed stream, with its flat slope up¬ 
wards and its steep slope downwards in the direction of the current. 
At this velocity the profile of each sand ripple had a very slow motion 
of translation, caused by particles running up the flatter slope and 
toppling over the crest. The steep downward slope was, therefore, 
being constantly advanced at the expense of the denudation of the less 
steep upward slope. At a surface velocity of 1.5 ft. per second the 
sand ripples Avere very perfect and traveled with the stream at a speed 
of about tito of the surface velocity. At a surface velocity of 1.75, 
the ratio was reduced to about -robot and at a surface velocity of 2 ft. 
to tsw- A critical velocity was reached when the surface of the water 
moved at 2.125 ft. per second, when the sand ripples became very 
irregular, indicating greatly increased unsteadiness of motion of the 
water. Uii to this point the whole amount of scour was represented 
by the volume of the sand waves multiplied by an exceedingly low 
velocity, always less than the 4 ^ part of the surface velocity of the 
water. At about this critical velocity of 2.1 ft. per second, the particles 
rolled by the water up the flat slope, instead of toppling over the steep 


1 Proceedings of the Institution of Civil Engineers, 1894, Vol. 118, pp. 93-95. 





40 


HOOKER ON SUSPENSION OF SOLIDS IN RIVERS. 


slope, were occasionally carried by the water direct to the next crest; 
and as the velocity of the water was gradually increased, an increasing 
bombardment of each crest by the crest behind it took place. 

“ At about 2.5 ft. per second, another critical velocity was reached 
and many of the little projectiles cleared the toj) of the first or 
even of the second crest ahead of that from which they were fired. At 
surface velocities of 2.6 to 2.8 ft. per second, the sand ripples became 
more and more ghost-like, until, at 2.9 ft. per second, they were wholly 
merged in particles of sand rushing along with the water in suspen¬ 
sion. After this the scour was of a totally different character ; the 
sand and water became mixed, and a constant process of lifting, car¬ 
rying and depositing of individual particles ensued, the sand being 
stirred to a depth and lifted to a height dependent upon the velocity.” 

Mr. Deacon refers to the theory that the weight of sand moved is 
proportional to the sixth power of the velocity of the water and believes 
the method of determination of this law to be fallacious. His observa¬ 
tions showed that, within the limits of the experiments, the weight of 
material transported was proportional to the fifth power of the surface 
velocity or possibly a little more. Two curves are given expressing 
the results. One shows the relation between the surface velocity and 
the solid discharge in pounds of sand; the other, the ratio between 
the surface velocity of the current and the velocity of translation of 
the crests of the sand ripples. 

M. Gallois has described 1 a method of experiment, 2 which throws 
light upon the problem of suspension. A glass bottle 3 ins. in 
diameter is used and its flat bottom covered to a depth of 0.2 in. 
with clean sand. By corking so as to exclude all air and rotating 
rapidly by means of a twisted cord or a turn-table, the sand is 
thrown by centrifugal force against the sides of the bottle. The 
motion of the bottle is communicated to the water progressively 
from the sides to the center, the sand remaining at the outside. If 
the bottle be suddenly stopped when the velocity of the water has 
come to equal its own, the sand will at once project itself from the 
sides to the center in a cloud, gradually subsiding to form a cone at 
the bottom, with a vertical axis, whose length increases with the 
velocity of rotation. 

This cone flattens with decrease in velocity, until in still water it 
assumes the corresponding slope of equilibrium for sand. M. Gallois 

1 Le Genie Civil. See Engineering News, March 23, 1893, for a brief digest. 

9 Suggested by Dupuit in 1848. See page-, and also “ Etudes sur le Mouvement des 

Eaux,” Dupuit, pp. 216-217; Elamant “ Hydraulique,” 1891, p. 302. Foot note. 






HOOKER OK SUSPEHSIOK OE SOLIDS IK RIVERS. 41 

explains this phenomenon as follows : When the rotation of the bottle 
is stopped, the water continues to revolve, but is gradually brought to 
rest by the fiiction from the sides of the bottle. Since this retarding 
force communicates its action progressively from the outside inward, 
the inteiioi filaments soon attain a relative velocity with reference to 
the outer ones which increases toward the axis of rotation. The sand 
is pushed toward the center with a force which is proportional to the 
velocity of the fluid. Consequently the cone flattens as the velocity 
of rotation decreases. 

M. Fargue has recently described 1 some experiments of a similar 
nature started by him in 1872 and repeated lately at Rouen and 
Langon. The apparatus used consisted of a circular disc upon which 
a zinc annular ring, about 0.80 m. high, was fixed. The internal and 
external radii were respectively 0.50 m. and 1.0 m. The disc was so 
mounted upon a vertical axis as to be given any desired rate of rotation. 

By partly filling the ring with water and carefully increasing the 
rate of rotation a paraboloid of revolution was soon formed by the 
water surface. 

If a uniform bed of sand and gravel was placed on the bottom be¬ 
fore rotation began, and a number of floats at the surface, certain 
phenomena were seen to occur. 

Up to a velocity of 1 rotation in 4 seconds the solids remained un¬ 
moved on the bottom. When the time of revolution had decreased to 
8.5 seconds, isolated grains of sand and gravel moved to the concave 
side. This radial movement increased with the speed until, at a velo¬ 
city of 1 turn in 2£ seconds, the entire mass of gravel was collected on 
the concave side and showed a somewhat regular surface. The floats, 
on the other hand, gradually descended the surface of the paraboloid 
until certain ones became stranded on the convex side or the bottom. 

After the conditions had become fixed, the disc was suddenly 
stopped. The water continued its motion in a state of agitation cor¬ 
responding to the angular velocity at the moment of arrest. The 
hollow formed toward the axis was at once filled and the surface be¬ 
came horizontal. The gravel was carried toward the center, with a 
rapidity corresponding to the angular velocity at which the disc was 
stopped. 

1 “Experiences relatives a, l’Action de l’Eau Courante sur un Fond de Sable.” Paris. 
M. Fargue, Inspecteur-General des Ponts et Chaussees, Annales des Ponts et Chaussees, 
March, 1894. 




42 


HOOKER OH SUSPENSION OF SOLIDS IN RIVERS. 


When this velocity was 0.74 (time of rotation, 8.5 seconds), the 
materials covered the bottom almost uniformly. The disc was only 
bare for a discontinuous strip afc the outer edge. When the velocity 
of stoppage was 1.11 (time of rotation, 5f seconds), the sand and fine 
gravel moved rapidly to the convex wall and the average gravel spread 
itself almost uniformly, except that only a few of the large particles 
remained at the concave wall. When it was 3.14 (time, 2 seconds), all 
the gravel was violently thrown toward the center and the fine sands 
followed in spirals of varying lengths. There was little regularity in 
the motion of the floats though they generally kept to the concave 
bank. 

Part II.— Discussion of Observed Data. 

The extent of the erosive action of water courses marks it as the 
greatest factor in that definite movement of the materials of the earth’s 
surface from the high toward the low latitudes, which the modern 
“ Doctrine of Isostacy ” has sought to explain by a reverse movement 
underneath and a subsequent elevation 1 . 

The study of the torrents of Switzerland and Italy suffices to show 
the size of individual blocks which may be moved along the bed of a 
stream 2 3 or even carried freely in suspension. The burden of detritus 
brought down in the middle and side moraines of the Unteraargletscher 
in the Bernese Oberland is spread over a wide area by the headwaters 
of the Aar Biver, forming a waste of heavy boulders and coarse gravel 
covering \ square mile. Through this wilderness of stone, the milky 
waters of the river find a tortuous path, carrying in suspension to the 
Lake of Brienz below, the particles of powdered rock ground from the 
sides of the valley by the daily motion:! of the glacier. It is the pres¬ 
ence of this so-called “ gletschermilch,” which gives to the Swiss lakes 
a part of their peculiar and beautiful coloring. 

These short but destructive torrents divide themselves naturally 
into a “ sammelgebiet or erosionsgebiet,” where the water and solid 
material is gathered, the “ gebiet des murgangs ” 4 forming a canal 

1 Compare “Theory of the Earth’s Rotation and its Interior Heat,” pp. 26-32, Eton 
Huntington. Rochester, N. Y., 1895. 

2 For a graphic description of the descent of material in a mountain torrent, see 
Lechalas, “Hydraulique Ftuviale,” Annexes, pp. 424-428. 

3 The Unteraargletscher has a velocity down the valley of 0.50 m. per day. It was 
here that Agassiz made his glacier measurements. The Rhone Glacier, separated from the 
valley of the Aar only by the Niigelisgriitli divide, has a daily velocity of 1.0 m. 

4 For examples of dangerous “ murgiinge,” read Riedel’s “ Ueber Geschiebe Fuhrung 
und Murgange der Wildbiiche.” Zeitschrift des Oesterridi-Ingen- und Arch.- Vereins, 1871, 
pp. 113 and 151. 



HOOKER OH SUSPENSION OF SOLIDS IN RIVERS. 43 

through which the semi-fluid mass passes at considerable velocity and 
with little deposit, the “ ablagerungsgebiet,” where the solid material 
is deposited in the main valley, forming a clearly defined cone, with 
its apex at the point where the torrent issues from the mountain. 
Lastly, the “ablauf ” l or bed through which the water, relieved of the 
mass of its burden, finds its way to the main water course. 

A photograph, taken during the summer of 1895, shows clearly 
these lines of demarcation in the two torrents close above Guttannen 
on the west side of the lower Haslithal. The axis of each is approx¬ 
imately at right angles to the Aar, into which they discharge. At 
the foot of this same valley on the eastern slope, above Brienz, 
lies the small Swiss village of Neuschwanden. At its edge, through 
an abrupt chasm in the mountain side, and so close as to render 
the danger to the village an imminent one, issues the cone of de¬ 
jection of the Lammbach, probably the most destructive in Switzer¬ 
land. The huge mass of stone and boulders covers a fan-shaped area 
of approximately a square mile and is largely devoid of vegetation. 
The slope is nearly uniform from the apex to the banks of the Aar, 
which it has forced against the further side of the Haslithal. An ap¬ 
proximate measurement, made by the author in August, 1895, showed 
this slope to be about 8 degrees. At that time the side toward Neu¬ 
schwanden was overlaid with the fresh “ murgang ” or lava-like mass of 
gravel and boulders of the preceding autumn which had formed a semi¬ 
circular cordon about the village and was only deflected from the 
houses by heavy guide walls. The upper surface is nearly plane and 
the stream does not, as might be expected, thin out gradually to the 
edges. It forms a bed of nearly uniform thickness, forking into various 
divisions at the apex of the cone, while each edge is sharp and clearly 
defined, marking an abrupt descent to the bottom. In general one may 
liken the form of these streams to that of those beds of broken stone 
carefully arranged in prismoidal form one sees in American cities. 

That something of an analogous nature takes place in all larger 
water courses is certain. The difference is one of degree and not of 
kind. The variance of opinion among authorities now hinges on the 
ratio between the total amount so moving, in a given river, and the 
amount carried in intermittent or permanent suspension. 

1 The French writers use only the first three divisions and the corresponding terms 
“ bassin de reception,” “canal d’ecoulement ” and “cone de dejection.”—See Surell, 

“ Etude sur les Torrents des Hautes-Alpes.” Paris, 1870-72, Dunod. 





44 


HOOKER OK SUSPENSION OF SOLIDS IN RIVERS. 


Piles driven up stream from a caisson of the St. Charles Bridge over 
the Missouri are said to have been found under the caisson when it 
reached bed rock. 1 Jas. B. Eads describes the sand on the bed of the 
Mississippi at the St. Louis Bridge as moving for at least 3 ft. in depth, 
with a velocity decreasing below the surface. 2 A pile embedded up¬ 
right in the sand has been seen to move bodily down stream. 3 The 
velocity of movement in this sense has been determined by means 
of stakes driven in the bed of the Loire. 4 Other recorded cases are 
numerous and need not be multiplied here. 

The transportation of coarse gravel in free suspension is but an¬ 
other order of the same phenomenon. It has been observed in the 
Garonne, when dikes have been broken through, and gravel, borne in 
the upper laminae of the current, has been carried over the breach 
and deposited in the fields beyond. 5 In a similar case, masses of 
gravel were carried over a dike below Pittsburgh 1 ’ and deposited down 
stream, filling up hollows which had previously existed there. 

The law of decrease in mean velocity from the rise to the em¬ 
bouchure of rivers is closely followed by the steady decrease in size of 
the particles forming its bed and strewn along its banks. The ratio 
betrveen the amount of solid matter entrained and that of the liquid 
at any point in a stream has been called by M. Fargue its torrential 
coefficient. This should decrease from source to mouth with the fall 
and the mean velocity. M. E. Charlon 1 has made use of the larv in the 
deduction of a formula, by means of which he computes the velocity 
of a stream from the size of the materials transported by it. The 
question of corresponding decrease in amount of suspended matter, 
per cubic foot of water, is a disputed one. M. Fargue 8 holds the view 
that rivers become more and more muddy toward their embouchures, 
due to the accumulated transformation of the coarse into fine materials 
by frictio n. M. I artiot," on the contrary, states that 300 measure- 

1 “ The Mississippi as a Silt Bearer.” K. E. McMath. Van Aostrand's Enqineerinq Ma¬ 
gazine, Vol. XX, 1879, p. 227. y 

2 “ Report of Chief Engineer of St. Louis Bridge.” J. B. Eads, June, 1808, p. 21. Quoted 
by It. E. McMath in above paper 

Engineering News, Feb. 9, 1881, p. 65. 

4 See Partiot. “ Les Sables de la Loire,” p. 43. 

fi The same. p. 23. 

c “ Report of Chief of Engineers, United States Army,” 1870. IT. p. 5. 

1 See Le Genie Civil, Vol X\ II, 189J, p. 170. Note giving formula in Proceedings of the 
Institution of Civil Eugiueers, Vol. 102, p. 350. 

8 “ Etude sur la Largeur du Lit Moyen de la Garonne,” pp. 12, 13. M. Fargue Annales 
des Ponts et ChaussCes, October, 1882. 

o “ Memoire sur les Sables de la Loire.” M. Partiot, p. 21. 









HOOKER OK SUSPEKSIOK OF SOLIDS IK RIVERS. 


45 


ments in the Loire, during the floods of 1856, showed the turbidity to 
deeiease regularly toward tlie sea. Measurements continued through¬ 
out the flood showed, j)roceeding down stream, the weights of sedi¬ 
ment per cubic meter of water to be— 

At Feurs ... .300 grs. At Nevers .. .210 grs. At Tours 212 grs. 

Roanne... 242 “ Gien 223 “ Saumur...l77 “ 

Digouin . . 191 “ Orleans ..237 “ Nantes.... 150 “ 

M. Partiot explains the anomalies shown between Nevers and 
Orleans by the entry, between those points, of tributaries heavily 
charged with silt. 

A comparison of various measurements in the Mississippi was un¬ 
dertaken by the author. The results would seem to bear out M. Par- 
tiot’s view. 

The available data are the Carrollton and Columbus measurements 
of 1851 and 1858, the measurements of 1879, at Helena and at St. Louis. 1 
The two former were taken during floods. The two latter at a medium 
stage. The Columbus measurements represent only surface speci¬ 
mens. The Carrollton samples were taken with a defective apparatus. 
These facts render it impossible to draw any conclusions from a com¬ 
parison of the 1851 and 1858 results with those of 1879, which seem 
more reliable. 

However that may be, a mean of the results at Columbus and at 
Carrollton, from the second week of March to the second week of No¬ 
vember of their respective years (1851, 1858), shows the proportion of 
sediment to water, by weight, to be— 

Columbus.000749 

Carrollton.000601 

an evident decrease in sediment at the lower station. 2 

There is an apparent co-ordination between the two 1879 measure¬ 
ments which gives more weight to the results obtained. Taking the 
mean only of the top and bottom measurements, as no mid-depth 
quantities were taken at Helena, and covering the iieriod from April 

1 See a valuable article by R. E. McMatb iu Van Nostrand’s Engineering Magazine, Vol. 28, 
18S3, p. 33. 

2 During these measurement the mean velocity of the river ranged at Carrollton from 6 
to 1.7 ft. per second and at Columbus from 8 to 1.5 ft. per second, being, as a whole, con¬ 
siderably higher at Columbus. 







46 


HOOKER OK SUSPENSION OF SOLIDS IN RIVERS. 


10th to June 18th, 1879, at Helena, and April 14th to June 25tli, at St. 
Louis, the proportions of sediment to water, by weight, are — 

St. Louis.002046 

Helena.001079 

a much greater decrease toward the river mouth. 1 

The law cannot be demonstrated from the measurements now avail¬ 
able. That the effect of interaction among the solids moved on the 
bed should show a cumulative effect in the increased number of fine 
suspended particles down stream is to be expected, and is shown in 
Nature by the increased fineness of the deposits. That it should man¬ 
ifest itself in an increased weight of suspended matter per cubic foot, 
as held by M. Fargue, 2 does not appear to be substantiated by the 
limited number of observations at hand. 

The question is: Will a stream moving at a given velocity sustain a 
greater weight of fine particles per cubic unit of water than of large 
ones of the same density? If so, then a heavier load per cubic foot 
may be carried at the embouchure of rivers with the same expenditure 
of energy than in their higher reaches. 

Mr. G. K. Gilbert 3 has shown that the same consumption of energy 
will hold in susiiension a greater load of fine than of coarse material of 
like density. 

This may be shown as follows: Assume a stretch in the lower 
course of a river bounded by the cross-sections A and B. Assume the 
kinetic energy at the two points to be the same, so that the whole 
gravity work done by the weight of the stream in its descent is used 
up in external and internal frictional resistance. Suppose an inch 
cube of stone introduced at the surface at A. The total energy of the 
stream has now been increased. The cube reaches the bottom at B. 
It can only act on the bed between the two points by pressure, but as 
the friction on the bottom is independent of fluid pressure, this fric¬ 
tion is not increased. If the cube sinks at the same rate it would have 
chosen in quiescent water, it makes no demand upon the energy of the 

1 At Helena, the mean velocity ranged from 4.26 to 3.23 ft. per second, while at St. Louis 
it varied from 7.21 to 4.0 ft. per second, averaging considerably higher than at Helena. 

2 M. Fargue says (“La Largeur du Lit Moyen de la Garonne,” p. 13): “II s'op&re done, 
de l’amont vers d’aval, une transformation dans la qnalite et dans le mode de transport du 
debit solide: le debit eu gros materiaux traines sur le fond, qui n’a lieu que sous l’influence 
de vitesses notables, va en diminuant; celui des materiaux tenus, en suspension dans l’eau 
et obeissant aux faibles vitesses, va au contraire en augmentant. * * * les eaux devien- 
nent en effet de plus en plus vaseuses a mesure qu'ou se rapproche de la mer.” 

3 American Journal of Science, July and August, 1876, Part II; also abstract in Enaineerinn 
Neivs, August 19th, 1876. 












HOOKER OK SUSPENSION OE SOLIDS IN RIVERS. 


47 . 


stream. If it sinks more slowly, tlie difference between the distance 
actually sunk and the distance which would have been covered in quiet 
water during the time of transit from A to B, multiplied by the 

weight of the cube in water, measures the demand upon the stream’s 
energy. 

Suppose the same stone to be pulverized into small cubes and 
again introduced at A. The weight in w r ater is unchanged. The 
draft upon the stream s energy between A and B is computed in the 
same w r av as before and found to be less, because the difference be¬ 
tween the distance sunk in quiet water and in the actual case is less. 
The reason why the small particles sink more slowly is because the 
collective area at right angles to the motion is greater, and so requires 
a smaller value of the velocity of sinking in order to keep the total 
resistance to motion, 

p = *r F Tg 

a constant in the two cases. This is required because the work done 
by the cube in sinking to the bottom must be the same as that done by 
its component parts in covering the same distance. 

It is shown, then, that a less consumption of the stream’s energy 
between A and B is required to suspend the same weight of small par¬ 
ticles than where the grains are of the same density and larger size. 
It follows that the same expenditure of energy will suspend a greater 
weight of the small particles per cubic foot of water. 

This may offer an explanation of the phenomenon observed at Col¬ 
umbus and advanced by General Abbot 1 to prove that no relation ex¬ 
ists between velocity and weight of sediment per cubic foot of water. 
He states that the Ohio and Missouri Rivers move side by side at 
Columbus in the bed of the Mississippi without mingling their waters, 
and that, while their velocity is common, the Ohio water has only 
three-fourtlis as much sediment per cubic unit as the Missouri water. 

If they are actuated by the same velocity it may be assumed that there 
is the same amount of energy per cubic foot in each case diverted to the 
suspension of sediment. The sediment of the Missouri is much more 
comminuted than that of the Ohio, as shown by the appearance of the 
two streams. An excess in weight of sediment per cubic foot is to 
be expected, then, in the Missouri water, as the measurements showed, 
even though the velocity is the same in both. 


1 Van Nostrand’s Engineering Magazine, Vol. XX, 1879, p. 3. 





48 


HOOKER OK SUSPEKSIOK OF SOLIDS IK RIVERS. 


The problem now becomes one dependent upon the circumstances of 
each case. If the energy available for the work of suspension per 
cubic unit of water is the same at the mouth as at the head waters of 
a stream, the weight of sediment per cubic unit will be greater. This 
excess will diminish and finally become negative as the ratio of the 
available energy at the mouth and head waters becomes less. 

In general, it would seem that the available energy should decrease 
rapidly toward the embouchure and be accompanied by some slight 
decrease in the weight of sediment carried per cubic foot of water. 

1. Minor Agents Influencing Sedimentation. 

Temperature. —Chemical precipitation, in general, takes place more 
easily at higher temperatures. The same law appears to obtain in the 
case of matter in mechanical suspension. The author’s attention was 
first directed to the question by Mr. Allen Hazen, of Boston, who had 
noticed an appreciable increase in deposition of suspended matter at 
higher temperatures in the sewage at the Lawrence, Mass., Experiment 
Station. Bouniceau 1 and Partiot 2 are agreed that river deposits are 
greater in summer than in winter. The sediment observations of Prof. 
Riddell 3 , and those of Prof. Forshey 4 , at Carrollton, on Mississippi 
water both give corresponding temperatures. The former lasted from 
May 21st to August 13th, the temperature gradually rising from 72° to 
84° Fahr. The corresponding amounts of suspended matter show an 
irregular but still perceptible decrease. They follow, however, much 
more closely the fluctuations in the river surface above low water, so 
that the element of decreased depth and consequent decrease in velocity 
of flow enters in as a more potent factor in producing the same result. 

The sediment curves at both Columbus and Carrollton 5 seem to show 
an increase in suspended matter during the summer months of June, 
July, August and September, the temperature of the river water at 
Carrollton reaching a maximum of 86° Fahr. in August and descend¬ 
ing very regularly to a minimum of 39° in February. 

1 " Etude sur la navigation des rivieres ;i marees.” M. Bouniceau, 1845. Quoted in Pro¬ 
ceedings of the Institution of Civil Engineers, Vol. 66, p. 5. 

2 “Memoire sur les Sables de la Loire," p. 22; M. Partiot. Annales des Fonts et Chaus - 
sets, I, 1871. 

a “ Report to American Association of Geologists and Naturalists, 1846. Quoted by 
Humphreys and Abbot, “ Report on Mississippi River," p. 142. 

4 “ Report on Mississippi River," Humphreys and Abbot, pp. 134, 148. 

5 The same, Plates XII and XIII. 





hooker ok suspension oe solids in rivers. 49 

It is evident that river observations are little fitted for the study of 
this question because of the complexity of the elements involved in 
fluvial motion. A simple laboratory experiment on the length of time 
acquired foi mechanical precipitation in quiescent water under differ¬ 
ent temperatures would determine the matter. The yearly range in 
temperatuie in rivers is not great, and its influence on sedimentation 
will be veiy limited. In the case of flow in sewers it may assume more 
of practical importance. 

A laige number of different measurements in the Elbe 1 , made under 
various conditions, failed to show any change in temperature with 
depth. In the Mississippi River the difference between surface and 
bottom temperatures is usually too small to be registered by an ordi¬ 
nary thermometer. The maximum difference is a small fraction of a 
degree. 2 

Herr Blohm 1 has assigned an important place among the causes of 
suspension of finer particles to a system of circulation set up by the 
differences in temperature. He calls attention to the fact that water 
reaches its greatest density at 3.5° R. (39° Falir.) and that laminte at a 
less temperature than this would tend to rise to the surface, as well as 
those at higher temperatures. The result would be a mixture tendiDg 
to produce the uniformity actually observed at all depths. The tend¬ 
ency of the warmer laminae to rise would be equal, in his opinion, to 
the tendency of the finer particles of sediment to sink in obedience to 
gravity. 

The Mississippi may be taken as an index of the rivers of the tem¬ 
perate zones. Its waters at Carrollton during two years’ observations 
never exceeded the temperature of maximum density, so that, in this 
case at least, there could have been no circulation, from this cause, 
of colder water from below to the surface as an equalizer of temper¬ 
atures. 

Light .—The slight molecular agitation caused by the penetration 
of light has been shown to be sufficient to affect the rate of sedimenta¬ 
tion in quiescent water. Mr. Andrew Brown 3 found that a phial of 
turbid water had a uniform tendency to deposit its sediment most 

1 “Ueber die in fliessenden Wasser suspendirt entlialtenen Sinkstoffe,” Blohm. Zeit- 
schrift des Architekten und Ingenieur-Vereins, Hanover, 186/, pp. 277-278. 

2 Lieutenant Marr in “Report on Mississippi River,” Humphreys and Abbot, 1876, 
p. 149. 

3 Proceedings American Association for the Advancement of Science, 1848; also, Hum- 
phrey and Abbot's "Report on the Mississippi,” 1876, p. 144. 




50 


HOOKER OH SUSPENSION OF SOLIDS IN RIVERS. 


rapidly in the portions protected from the light, the surface of the de¬ 
posit showing a corresponding inclination. 

Viscosity. —At low water, under a hot sun, M. Partiot 1 has seen 
the rising tide, at the embouchure of the Loire, float off patches of 
sand from the bars and carry them upon its surface so long as it re¬ 
mained undisturbed by waves. A similar phenomenon is noted in the 
American Journal of Science, December, 1890 . 1 3 Blotches of sand, 1 in. 
in diameter at first, which later joined themselves into 6-in. squares, 
were eroded from a bank forming an angle of 150° with the water sur¬ 
face. These were seen floating on the surface half a mile down stream, 
and, if disturbed, would rapidly sink to the bottom. An oiled needle 
will float on the surface of water if carefully placed in position. 
Phenomena of this nature are due to what may be called superficial 
viscosity, which has a greater intensity than the viscosity in the in¬ 
terior of the fluid. 

That this latter influence has a part in the suspension of sediment 
is shown by the length of time required for quiescent turbid water to 
clear itself. Experiments showed turbidity in water taken from the 
Garonne after eight days and muddy water from the Elbe made no 
perceptible deposit until after a lapse of 24 hours.' 1 In water taken 
from the Mississippi at St. Louis in 1865, Mr. Flad 4 5 6 found that for a 
total of 1 000 parts in suspension at the beginning of the experiment, 

944.50 parts had settled during the first 24 hours. 

22.35 “ “ “ “ “ second 24 “ 

2.92 “ “ “ “ “ “ 48 “ 

30.23 “ were still in suspension after 96 “ 

Changes in viscosity and consequent suspending power are a prob¬ 
able concomitant of changes in temperature. 

Salt Water. —Observations made in 1839 by Sidel!’ at the mouths of 
the Mississippi showed that the river water alone required from 10 to 
14 days to settle. The admixture of salt in any form reduced the time 
of settling to between 14 and 18 hours. Mr. Gould 1 ’ found that a few 

1 “Sables de la Loire,” p. 36. 

2 Noted in Engineering Record, December 27, 1890, p. 65. 

3 Blohtn in Zeitschrift des Arcldtekten- und Ingenieur- Vereins, Hanover, 1867, p. 245. 

4 “ Silt Movement by the Mississippi,” R. E. McMath. Tan JVoslrand’s Engineering 
Magazine, 1883, p. 33. 

5 “ Report on the Mississippi River,” 1876. Appendix A, p. 500. 

6 ** Report of Chief of Engineers, United States Army,” 1875, II, p. 36. 







HOOKER OK SIJSPEKSIOK OE SOLIDS IK RIVERS. 


51 


pinches of salt thrown into a tumbler containing muddy water from 
the bottom of the Savannah River caused a much more rapid deposit. 
Mr. Fargue 1 has found that the same amount of mud introduced into 
a glass of fresh water and into a glass of salt water shows a difference 
in the period of settling. The salt water is clear after six hours of re¬ 
pose. To attain the same result requires eighteen hours in the fresh 
water. 

This property of saline solutions has an important bearing on the 
formation of bars at mouths of rivers discharging into salt seas. The 
load of detritus will be dropped sooner than if the receiving body were 
fresh water. 

Action of Waves .—The formation of bars in deep water at the 
mouths of tidal estuaries has in late years come to be attributed to 
wave action upon the detritus discharged by the river rather than to 
the simple process of deposition itself. It is the dynamic effect of the 
waves which heaps up the bars. 2 All sea beaches show this action so 
clearly that there can be little doubt as to its influence, in a lesser de¬ 
gree, on the movement of detritus in rivers. Observation has shown 
that sands are often moved when the bottom velocity is such as to be 
an insufficient cause. Mr. P. O’Meara 3 has observed this motion by 
diving to the bed of a tidal channel where the bottom velocity was too 
slight, unaided, to move the sands. He found that the sand at and 
near the bottom, under a depth of 10 ft., had an oscillatory motion 
corresponding to the 6 and 8-in. waves passing above. At the center 
of the wave passage the sand reached a considerable velocity; at its end 
the motion ceased and even seemed to be reversed. He holds that this 
action may be perceptible to depths of 40 or 50 ft. Waves of transla¬ 
tion stir the water to an infinite depth, theoretically, their velocity of 
translation being dependent upon the depth. Such waves will be 
confined to tidal estuaries. Waves of oscillation, such as the wind 
ripples in rivers, are felt, however, to considerable depths. 

At Cherbourg 4 these waves cease to act on the piers at a depth of 

1 “ Etude sur la Largeur du Lit Moyen de la Garonne.” Annales des Ponts et Chaussees, 
Oct., 1882, p. 21 (foot note). 

2 A formula for the scouring power of waves, giving relation between height of wave and 
size of particle moved, is developed by Mr. W. Smith in Proceedings of the Institution of 
Civil Engineers, Vol. 100, p. 201. 

3 Proceedings of the Institution of Civil Engineers, Vol. 118, pp. 81, 85. Noted, also, in 
Engineering Record, Feb. 23, 1895, p. 219. 

4 “ Les Marees Fluviales,” M. Comoy, 1881, p. 23. 



52 


HOOKER OK SUSPENSION OF SOLIDS IN RIVERS. 


about 25 ft. At Algiers tlie limit is 35 ft. There the sands cease to be 
moved at depths between 50 and 100 ft., while the limit for muds is 
450 ft. A visit to the harbor of Algiers during a heavy blow showed its 
peculiarly exposed position so that these figures may be reasonably 
considered maxima. 

Action of Ice .— The removal to great distances of boulders which 
neither ordinary nor flood velocities could move has been attributed to 
the transporting power of ice. There is a tendency in rivers to form 
what is called anchor ice at the bed and sides when the water is 
shallow. This ice attaches itself to the solids in its vicinity, and, 
because of its slight specific gravity, is easily detached by flood 
velocities and carried with its load down stream. M. Partiot 1 calls 
attention to this action on sand shoals barely covered by water 
from which the surface layer is detached by the floating away of 
the ice. 

Action of Sediment in Diminishing Velocity. —Mr. Baldwin Latham 
reports 2 3 observations covering a series of years which seem to show 
that the velocity of turbid water for the same depth and fall is less 
than that of clear water. He holds that this difference bears a ratio 
to the amount of turbidity, and is caused by the work used in trans¬ 
porting the material. The discharge of clear water multiplied by its 
velocity corresponded closely to the combined weight of sediment and 
water in the corresponding case, multiplied by its mean velocity. Mr. 
G. K. Gilbert has reached the same conclusion/ He states that the 
total energy of a clear stream is used up in friction on its bottom; that 
this friction is directly proportional to its velocity. When detritus is 
carried a certain amount of the energy of the stream is used to keep 
it in suspension, and this takes place at the expense of friction and 
consequently of velocity. It is to be remembered, however, that the 
total energy of the stream has, in the meantime, been increased by the 
addition of the energy represented by the vertical fall of the solid 
particles. 

The law of the conservation of energy will not admit of any other 


1 “ Les Sables de la Loire,” p. 36. 

2 Proceedings of the Institute of Civil Engineers, Vol. 71, p. 46. 

3 “ The Colorado Plateau Province as a Field for Geological Study,” American Journal 
of Science, July and August, 1876. Part II. Abstracted in Engineering Neivs, August 19th, 1876. 







HOOKER OH SUSPENSION OF SOLIDS IN RIYERS. 


53 


decision in this matter, though the statement has sometimes been 
made that such a retardation of velocity does not exist 1 . 

Mr. Gilbert’s statement that the work done by a clear stream is 
entirely used up in friction on the bed is somewhat at variance with 
the attitude of the best science of the present day. 2 The energy con¬ 
sumed by intermolecular resistances caused by the complex motion in 
the interior of the liquid is much greater than that actually used at the 
earth and air profiles. It should be added, however, that these intri¬ 
cate movements are induced by the bed’s rugosities. In general, it 
may be said, that the total energy is used in friction, through which it 
is transformed into heat energy. 

Assume a portion of a clear stream between the sections A and B. 
Suppose no difference of kinetic energy between the two stations, then 
the total energy of the stream expended is used in work done on fric¬ 
tion. Introduce a mass of sediment in suspension at A and a demand 

1 See “ Silt Movement by the Mississippi,” R. E. McMath. Van Nostrand’s Engineering Maga¬ 
zine, 1883, p. 36, Mr. McMath says: * We have seen that transportation of silt (up to the point 
of impaired fluidity) is not at the expense of the stream’s motion. The work of erosion and 
suspension is done by the stream, whose velocity must be diminished compared with flow 
under a like head in a smooth channel, but if the now-yielding bed should suddenly become 
rigid, the same or eveu greater force would be expended upon the obstructing roughness. 
Therefore though suspension consumes a part of the stream’s force the velocity is not 
necessarily lessened beyond what it would be in the ODly alternative condition that can be 
considered, a rigid bed equally rough.” This line of reasoning would seem to hold, so far as 
the actual work done upon the bed of the river between any two points is concerned. 
The work which would have been expended upon a rigid bed equally rough is now in part 
expended upon the mobile bed in the same way as before, while the residue is free to be used 
in carrying into suspension whatever is eroded from the bed between the sections considered. 
But this theory fails to take account of those external forces of Nature which are continually 
wearing away cliffs, disintegrating hillsides and introducing at the surface of the stream a 
mass of debris to be carried, for which the stream’s own mechanical action is not account¬ 
able. Gravity acts as an external force where bauks cave in and throw upon the stream’s 
energy an additional burden. The burden already in suspension at the entrance to the 
stretch considered must be carried in addition to that considered by this theory. It is to 
this additional burden that a consumption of energy and consequent retardation of velocity 
may be attributed. 

2 See Boussinesq “Theorie des Eanx Courantes,” Paris, 1872, introductory chapter. M. 
Boussinesq has shown that neither the friction, rightly called, upon the bed nor the added 
internal friction due to relative velocities of parallel filaments following stream lines is suf¬ 
ficient to explain the transformation of the energy of the stream, in its descent, into heat 
energy. He shows that if the velocity at the walls were assumed to be zero so as to attribute 
the whole work to friction between parallel filaments, the coefficient of interior friction is so 
small that the central filament, in a semicircular conduit of 1 m. radius and a fall of 1 in 10 000, 
would acquire a velocity of 187 m. per second before equilibrium was established between 
the accelerating force and the fluid resistances. It is, then, to the vortices that must be at¬ 
tributed the largest share in this transformation. They largely increase the total inteiior 
friction. 

See “ Journal de M Lionville,” t. XIII, 1868. Also “ Theorie des Eaux Courantes,” B jus- 
sinesq, pp. 2-6. 

Compare, also, Prof. Uuwin in “Encyclopedia Brittannica,” article on Hydromechanics. 







54 


HOOKER OH SUSPEHSIOH OF SOLIDS IH RIVERS. 


is made on tlie stream’s energy to keep it suspended to B. The 
thought at once suggests itself that the total energy has, in the mean¬ 
time, been increased. In answer, it may be said that the addition has 

also increased the friction at the bed since the formula 

v 2 

P = kFy 

2 g 

show's this friction to be a function of the heaviness of the fluid, which 
in its new compound state has been increased. These two changes 
tend to counteract each other and to leave still an increased demand 
upon the energy originally used in the passage from A to B. In con¬ 
sequence, there will result a retardation of velocity at B accompanied 
by an increase of depth if the supply be constant. For particles of a 
uniform size and density, this decrease in velocity will increase w T ith 
the w r eiglit of the load per cubic unit of w r ater. The decrease will be 
less for a given weight of fine particles than for the same weight of 
large ones, other conditions remaining the same. 

The presence of silt, then, retards velocity in tw r o ways: 

First .—It uses an amount of the stream’s enel’gv in suspending it. 

Second .—It increases the heaviness of the composite fluid and so in¬ 
creases friction. 

2. Influence of Depth on Transporting Power. 

It was once believed by hydraulicians that the adhesion between a 
liquid and its bed was stronger than the internal cohesion of the fluid 
itself. It seemed the natural deduction from the decrease in velocity 
observed near the banks and bed. The hypothesis then took form that 
the particles next the banks remained stationary, w r hile the main current 
flow r ed by in a fluid bed of its own consistency. 1 The experiments of 
Darcy on deteriorated pipes showed that velocity w r as a function of 
roughness, and the incorrectness of the former assumption. Even the 
outermost particles of the fluid substance have a motion relative to the 
bed. Is this velocity influenced by the depth ? Increased depth 
means increased weight per square unit of bed, and consequently in¬ 
creased pressure, but experiments have shown that not only the co¬ 
efficient of fluid friction, but also the friction itself, is independent of 
the pressure. 2 3 

1 This has been shown to be true for capillary tubes by M. Duclaux, of Clermont. 8ee 
Annalesde Chemie et de Physique, 4e Serie, t. XXV, i872. For flow in streams and large pipes 
it appears inadmissible. 

See also “ Theorie des Eaux Courantes.” J. Boussinesq, pp. 1-2. 

3 “ Encyclopedia Brittannica,” Article Hydromechanios. 













HOOKER OH StTSPEHSIOH OF SOLIDS IH RIVERS. 


55 


In the case of a homogeneous solid sliding down an inclined plane 
the coefficient of sliding friction is independent of the normal pressure, 
but the friction itself 

P=fN .( 1 ) 

is a function of both quantities, and remains unchanged at all veloc¬ 
ities. The liquid prism, sliding down an inclined bed, acts according 
to other laws so far as frictional resistance is concerned. Here again, 

p = (kr) ( p £) 

or 

P =/, N, .( 2 ) 

an equation of the same form as before, but made up of quantities 
formed in a different way. In equation (1) N represents the normal 
component of the body’s weight, is proportional to depth when the 
prism is homogeneous, and is independent of velocity. In (2) again 
represents weight, but this weight is directly proportional to a velocity 
height, and is independent of depth. Increased depth, velocities re¬ 
maining constant throughout, will have no effect on friction and hence 
produce no change in scouring action. 

The case is sometimes cited, as a substantiation of the view that 
depth increases transporting power for the same velocity, of the in¬ 
creased difficulty in wading a deep stream. The example is not a good 
one. A man’s foothold is lost sooner in this case than in a shallow ford 
because of his increased loss of weight rather than from an increase of 
transporting power, properly called. 

The statement has been made 1 that in practice observation shows 
the scouring power of a shallow stream at a high velocity to be much 
less than that of a deeper river running at a slower velocity. The ex¬ 
planation offered by Prof. Unwin 2 3 would seem to account for a portion 
of this difference. He attributes to the deep stream the advantage 
that the particles of its bed may be thrown up to a greater height, and, 
since the velocity of descent again to the bed should be the same in 
both cases, will be thus carried farther down stream before being de¬ 
posited. 

Flood waters offer great variations of depth which may be used in 
the determination of comparative amounts of sediment per unit volume. 

1 See Proceedings of the Institution of Civil Engineers. Vol. 82, 1884, p. 31. 

Mr. Law’s explanation of the phenomenon is based on the incorrect assumption that fluid 
friction increases with pressure. 

3 “ Encyclopedia Brittannica,” article Hydromechanics. 






56 HOOKER OH SUSPEHSIOH OF SOLIDS IH RIVERS. 

These weights will vary for the same depth with the relative stage of 
the flood, and so complications are introduced. The earlier stages of a 
heavy rainfall wash down the surface particles loosened by weather¬ 
ing. The later portion of the storm finds a more resisting surface. 
M. Partiot found traces of this fact in the relative weight of sedi¬ 
ment at different stages of the same flood. A safe basis of comparison 
would seem to be the same relative stages of floods of different heights 
when the same tributaries are discharging high water. 

M. Partiot 1 has been able to detect only a slight increase in the sedi¬ 
ment per unit volume in the Loire, with the importance of the flood. 
M. Fargue 2 3 states that the suspended sediment is very feeble at low 
water and reaches its maximum intensity at the flood crest. 

Major Allan Cunningham 1 bases the statement that no relation 
exists between depth and silt intensity, upon a series of observations 
on the Ganges Canal. 

A study of the data at hand will throw the most light on the sub¬ 
ject. Fig. 1 gives a graphic representation of data bearing upon this 
point. Further curves might be added from the extensive measure¬ 
ments of the Mississippi Commission 4 made in 1879-81 at Carrollton, 
Prescott, Winona, Clayton, Hannibal and St. Louis. The Carrollton 
and Columbus sediment ordinates are taken from the “Report on the 
Mississippi ” by Humphreys and Abbot, page 417—one for each week 
of the year. The corresponding mean gauge readings were taken from 
Plates XII and XIII of the same volume. The St. Louis co-ordinates 
are from Mr. McMath’s paper in Van Nostrand’s Engineering Magazine, 
1883, page 33. The Helena co-ordinates are from the “ Report of the 
Chief of Engineers, United States Army,” 1879, Part III, page 1968. 
The Elbe and Maas measurements are taken from the Zeitschrifi ties 
Architekten unci Ingenieur Vereins, Hanover, 1867, pages 290 and 291. 

The method of plotting these curves must be distinctly understood. 
They start from no common origin. They have no quantitative relation 
to each other. Each represents only the general trend of direction of 
the number of points from which it is constructed. These are in most 
cases widely scattered, but their general direction is clearly defined and 

1 “ Sables de la Loire,” p. 22, 1871. 

2 " La Largeur du Lit Moyen de la Garonne, p. 14, 1882. 

3 See Proceedings of the Institution of Civil Eugineers, Vol. 71, 1882, p. 35. " Roorkee 

Hydraulic Experiments.” 

4 “ Report of Chief of Engineers, U. S. Army,” 1883, III, pp. 2209 and 2266. 










HOOKER OH SUSPENSION OF SOLIDS IN RIVERS. 57 

is fairly represented by the lines shown. It is seen at once that their 
direction, in each case, is such as to show an unmistakable increase in 
sediment per cubic foot with higher stages. 1 * 3 Each is independently 
constructed and all show the same thing." The author can find no 
other values to contradict them. ' This seems to show satisfactorily 
that weight of sediment per cubic foot increases as the river rises 
and depth increases. That it proves that transporting power in¬ 



creases with depth is not claimed, for velocity increases with stage 
also, and either one or both together may be the cause of increased 

1 In the curves shown on Plate VIII of the “ Report of the Mississippi River Commis¬ 
sion,” 1879-81 (see " Report of the Chief of Engineers, United States Army,” 1883, III), this 
relation is only slightly traceable in the Carrollton measurements of 1879-80. 

9 Since the above curves were plotted the author has found a plate showing a series of 
points plotted in an analogous manner from the extensive observations of Assistant Engineer 
Seddon at St. Charles, Mo., 1879 (see “ Report of the Chief of Engineers, United States Army,” 
1887, Part IV, pp. 3090-96. For description of apparatus used see same report, pp. 3121-23). 
These points show how a curve similar to those in Fig. 1 could be drawn and are clearly 
confirmative of the conclusions here deduced. 

3 The results of Prof. Riddell’s measurements are confirmatory. See Humphreys and 
Abbot’s “ Report on the Mississippi,” 1876, p. 142. 
































































































































































58 


HOOKER OK SUSPENSION OF SOLIDS IK RIVERS. 




transport. If, however, one were to join with Humphreys and Abbot 
in denying any fixed connection between velocity and suspending 
power, then this might be looked upon as j3roof. 

The influence of a gradual or sudden decrease in depth upon trans¬ 
porting power offers a range of experiment and observation which is 
yet to be made. There is little definitely known from measurements 
about the influence of such shoaling upon the curve of velocities. 
For unchanged width, decrease in depth means decrease in sectional 
area and consequent increase in mean velocity for constant discharge. 
Such a shoaling may be likened, in its action, to a submerged weir. 
When the change is sudden there will be a mass of dead water above 
the weir, at the bottom, forming a fluid bed upon which the discharged 
water flows. This would seem to indicate conditions favorable for a 
deposit above the obstruction, upon the same principle upon which 
Humphreys and Abbot explained the formation of delta bars by de¬ 
posits in the dead angle caused by the fresh water flowing over the 
heavier salt water. The case of movable dams in many of the conti¬ 
nental rivers is in point. 

Experience, however, fails to show any shoaling of consequence 
above these structures. 1 The natural conclusion is that this dead 
water, so called, must be in a lively state of agitation, its eddies and 
vortices carrying up deposited material to the higher laminae whose 
movement of translation carries it over the obstruction. 2 

In a gradual shoaling, with banks widening to form a pool, there is 
no sudden accession of vortex motion and the decrease in velocity due 
to enlargement of section is followed by deposits until equilibrium is 
established between the velocity of the stream, the resisting power of 
the banks and that of the bed. 

The question of changes in the form of the curve of velocities, as 
affected by obstructions, is still in need of experimental research by 
measurements made at varying distances above the obstruction to de¬ 
termine the velocities throughout the range of the back-water. 

That the friction of the air has some effect in influencing the form 
of this curve has been generally accepted since the Mississippi meas¬ 
urements were made. It is only the extent of the influence which was 
then claimed for this factor which has since been called in question by 

1 For substantiation of this statement see Flamaut, Annates des Fonts et Chaussees, 1882, 
quoted by Lechalas, “ Hydraulique Fluviale,” 1884, p. 64. 

* See “ Report of the Chief of Engineers, U. S. Army,” 1876, II, p. 6. 



HOOKER OK SUSPEKSIOK OF SOLIDS IK RIVERS. 


59 


those who have other theories for the cause of the depression of the 
maximum velocity below the surface. 1 That its retarding effect is less 
than that of the friction on the bed is usually conceded. 

That both are greater than the internal friction due to laminated 
flow is a reasonable conclusion from the actual form of the vertical 
curve of velocities, substantiated by Boussinesq’s demonstration 2 of 
the slight value of this friction of laminated flow. 

Suppose any cause produces a sudden increase of roughness in the 
bed and so increased bottom friction. If the hypothesis of parallel 
filaments 3 is adopted as representing the general trend of flow, the 
natural conclusion is that the parabolic curve of velocities in a vertical 
will be tipped down stream. In other words, the lower filaments will 
be more retarded than the upper ones. That a gradual shoaling with 
a corresponding increased surface fall has the opposite effect on the 
curve of velocities has been shown by Dupuit 4 for the case of un¬ 
changed width. 

There, by a simple numerical calculation, the bottom velocity is 

shown to take on a much more rapid increase than the surface velocity, 

so that, at the crest of the shoaling, they have become more nearly 

equal than before. In the contracted section below the crest the curve 

will have been tipped uji stream, though the rapid increase of bottom 

friction (varying as the square of the velocity) will soon force it back to 

the normal position for steady flow at that velocity. He attributes to 

this disproportionate increase in bottom velocity with the mean 

velocity, the larger part of the scouring action seen immediately down 

stream from hydraulic constructions. This view would seem to throw 

additional light on the cause of non-slioaling above submerged weirs. 

Experiments are needed on this subject. 

% 

3. Influence of Retardation of Velocity. 

What, if any, is the relation between velocity and suspension? This 
is a vital question in the discussion. Assume a sedimentary stream 

1 See Prof. James Thomson, “Encyclopedia Brittannica,” Article Hydrodynamics. Also, 
P. P. Stearns. Transactions American Society of Civil Engineers, Vol. 12, p. 331, aud Vol. 7, 
pp. 122-130. 

2 Seepage 53, foot note. 

3 In spite of the statement often seen, that this in no wise corresponds to the complex 
motion seen in rivers, especially the Mississippi, an ordinary inspection of streams, even in 
times of flood, shows it to be more reasonable than any other supposition yet advanced, and 
to represent, in an average sense, the phenomena observed. 

4 “Etudes sur le mouvement des eaux,” 2d Edition’ 1863, pp. 58-68. 



I 


60 HOOKER ON SUSPENSION OF SOLIDS IN RIVERS. 

flowing between regular banks. Does any fixed connection exist 
between the velocity of the mean of all the cubic feet passing a given 
section per second and the weight of sediment contained in that ideal 
cubic foot of water? If so, is the relation one of cause and effect? 
To the first of these questions Humphreys and Abbot give an emphatic 
negative, 1 based upon a comparison between the Carrollton and Colum¬ 
bus sediment and velocity curves. Captain Ead’s criticism of this 
view was founded on a misconception. 

He proves conclusively in his article that the total weight of sedi¬ 
ment passing a given point in the river is proportional to the velocity 
of the current. This, however, was not the question at issue, and is 
settled by a moment’s reflection. The debatable problem is: Does an 
increase of velocity increase the transporting power per cubic foot of 
water passing a given cross-section? The vital question in the problem 
of jetties is not as to the ability of the contracted stream to carry 
throughout their length and beyond the sediment contained per cubic 
foot in the water of the river above. Their success hinges upon the 
capacity of the stream to take an additional load per cubic foot from 
its increased velocity until the increasing depth has again estab¬ 
lished equilibrium. The consensus of opinion of writers seems to 
answer in the affirmative as opposed to the position of Humphreys 
and Abbot. 

Partiot says 2 * that sediment in floods follows the same law as the 
velocities, increasing up to the highest stage and decreasing after¬ 
wards. 

Referring to the Missouri, Major Ruffner 2 says: 

“The water is so heavily charged with sediment that decrease in 
velocity is immediately followed by a deposit, but the converse of 
scour following an increase of velocity, although apparent, is not so 
well marked nor so extensive. * * * When from any cause the 
velocity of the current is suddenly increased, the most rapid erosion 
takes place; and the greatest deposit occurs when the velocity is sud¬ 
denly checked.” 

Captain Eads and Mr. Corthell 4 state that the current of the Mis¬ 
sissippi cannot be checked in the slightest degree in flood time, when 
its waters are heavily charged, without causing a deposit; that an 

1 See p. 26. 

2 “ Sables de la Loire,” p. 22. 

s “Improvement of Non-Tidal Rivers,” 1886, pp. 78, 138. 

-» Transactions American Society of Civil Engineers, Vol. xi, pp. 262, 263. 







HOOKER OH SUSPENSION OF SOLIDS IN RIVERS. 


6 


iron netting, with meshes 1 ft. square, set in a shoal on the Missouri, 
caused a deposit of 16 ft. in one flood. Mr. Ockerson 1 claims that the 
river is not always fully charged with sediment and so may at times 
be retarded to some extent without deposit. 

Major Allan Cunningham - ’ states that, in the Ganges canal, measure¬ 
ments have shown that silt density is independent of velocity. 

Mr. R. E. McMatli" says the cause of suspension commonly varies 
with the velocity. 



Fig. 2. 

These views represent all phases of opinion. General statements 
are not convincing so it is advisable to analyze the available measure¬ 
ments. 

Fig. 2 gives a graphic representation of five different series of 
measurements in the Mississippi. Other data at hand are unaccom¬ 
panied by velocity determinations. These curves are plotted with 
mean velocities as abscissas and amounts of sediment as ordinates. As 

1 Transactions, American Society of Civil Engineers, Vol. xi, p. 273. 

2 “ Roorkee Hydraulic Experiments.” See Proceedings of the Institution of Civil Engi¬ 
neers ; Vol. 71, p. 35. 

3 “ Silt Movement by the Mississippi.” Van Nostrand’s Engineering Magazine, 1883. p. 38. 





































































































































62 


HOOKER OK SUSPENSION OF SOLIDS IN RIVERS. 


in Fig. 1 they are not suitable for quantitative comparison. There 
is no co-ordination in vertical scales. Each represents only the general 
trend of direction of the points from which it is drawn. The points 
for the Columbus and Carrollton curves were the most scattering, as 
would be expected from an examination of the same values plotted in 
Plates XII and XIII of Humphreys and Abbot’s “Report on the 
. Mississippi. ” There, time is introduced so as to form two curves. 
Here, a general relation is expressed by one curve, omitting the time 
element and plotting each observation as a separate point fixed by its 
sediment and velocity co-ordinates. 

An examination of the two curves as plotted by Humphreys and 
Abbot seems to indicate a relation, especially in the case of Carrollton 
where samples were taken from three depths and averaged. The rela¬ 
tion is not, however, invariable. From the points as plotted here there 
can be no doubt of the existence of a law. 

Another set of measurements was instituted at Carrollton by the 
Mississippi River Commission in 1879-80 which remove all doubt as to 
the existence of a relation between velocity and sediment at that point. 1 

The investigations made by this commission at Prescott, Winona, 
Clayton, Grafton, Hannibal and St. Louis in 1880 and 1881 are the 
most extensive ever conducted and offer the data for plotting addi¬ 
tional curves. 2 3 Those made by the Missouri Commission at St. 
Charles 1 in 1879-1880 offer similar facilities. 

Of the curves shown in Fig. 2, the most reliable are those repre¬ 
senting the Helena and St. Louis measurements because of the im¬ 
proved methods used. In both these cases the law is clearly marked. 
The curve marked Ockerson is plotted from some measurements pub¬ 
lished by J. A. Ockerson, 4 M. Am. Soc. C. E. 

In Fig. 2 it might seem reasonable to have started each curve 
from the origin on the ground that stagnant water would carry no sedi¬ 
ment. On the other hand, such a proceeding would have been open 
to objection as an cirgumentum in circulo , and so was avoided. The 
curves, however, place themselves in a significant arrangement. 

1 See “ Report of the Chief of Engineers, United States Army,” 1883, III, pp.2209-2266 
and Plate VIII. 

* These curves have all been plotted by the author and show the same result as those 
drawn on Fig. 2 

3 “Report of the Chief of Engineers, United States Army,” 1887, IV, pp. 3090-3096. 

4 Transactions of the American Society of Civil Engineers, Vol. xi, p. 273. Mr. Ockerson’s 
values for sediment as given above show a marked departure from other determinations in 
the Mississippi. However, as only the relative quantities are required here, they are used 
without question. 





HOOKER OK SUSPEKSIOK OF SOLIDS IK RIVERS. 


63 


The curves are believed to show that the sediment by weight in a 
cubic foot of water does obey a general law of increase with the 
velocity in the Mississippi River. As such a relation has been largely 
conceded to exist in other rivers it is believed that it is general. That 
instant deposit always follow the least retardation is not proved or 
claimed. 

4. Distribution of Sediment in the Cross-Section. 

Distribution of Sediment in the Vertical .—Sufficient data is at hand to 
settle this question in its general bearing. That it has been a matter 
of dispute is shown by the different opinions expressed. 

M. Baumgarten, 1 from measurements in the Garonne, came to the 
decision that surface measurements were a fair index of the amount of 
sediment at all depths. 

Herr Blohm 2 quotes a number of varying opinions from English, 
German and Italian engineers ; but, from his own measurements in 
the Elbe at Harburg, finds somewhat of an excess in the surface 
quantities. He, however, gives it as his opinion that the distribution 
of sediment is about equal throughout. 

Andrew Brown" decided, after repeated trials, that sediment in the 
Mississippi was equally distributed at all depths, provided the samples 
were taken in the main current. 

M. Partiot 1 4 * takes the other view, stating that measurements in the 
Loire have shown the ratio of surface, middle and bottom quantities 
to the mean of all, to be represented by the numbers 90, 100 and 110. 

M. SurelP found that the silt intensity in the Rhone increased 
rapidly with distance from the surface. He expressed the ratio be¬ 
tween surface and bottom amounts by the relation of the numbers 100 
and 188. 

In the measurements on the Mississippi at Columbus, 6 only surface 
specimens were used, the ratio 100 to 120 being used to reduce the sur¬ 
face values to the mean for all depths as determined at Carrollton in 

1 “ Navigation Fluviale, Garonne.” M. Baumgarten. Annates des Fonts et Chausstes, 1842, 
2, p. 49, See also page 8 of this paper. 

a Ueber die in fliessenden Wasser suspendirt enthaltenen Sinfcstoffe.” Baurath Blohm. 
Zeitschnft des Aichitekten und Ingenieur Vereins, Hanover, 1867, p. 276. 

3 Humphreys and Abbot’s “ Report on the Mississippi,’' 1876, p. 143. 

4 “Sables de la Loire,” 1871, p. 24. 

8 M Guerard on “Mouth of the River Rhone.” See Proceedings of the Institution of 
Civil Engineers, Vol. 82, p. 309. 

8 See Humphreys and Abbot’s “Report on the Mississippi,” 1876, p. 136. 



64 


HOOKER OH SUSPEHSIOH OF SOLIDS IH RIVERS. 


1851-1852, The following table has been prepared from all the data 
which could be collected upon this matter : 


Distribution of Sediment with Regard to Depth. 


Place of observation and 
observer. 



Parts of Sediment in 1 000 000 
Parts of Water at the Depths 
Indicated. 

Date. 

Reference. 

Surface. 

(Mean.) 

Mid depth. 
(Mean ) 

Bottom 
(usually 1 ft. 
above). 
(Mean.) 

Mississippi at St. Louis— 
McMath. 

1879 

Van Nostrand Eng. Mag., 
1883. 

1 847 

2 009 

2 117 

Mississippi at Helena— 
Johnson.... 

1879 

Rep’t Chf. of EDg’rs U. 
S. A , 1879, III. 

799 

1 266 

Mississippi at Carrollton 
—Forshey. 

1851-52 

Rep’t on Miss. Hum¬ 
phreys and Abbot, 1861 

648 

802 

842 

Sacramento at Kersche¬ 
val’s—Le Conte.. 

1879 

Rep’t Chf. of Eng’rs U. 
S. A 1879, II. 

5 525 

9 051 

5 618 

Mississippi at Prescott— 
Miss. River Com. 

1880-81 

Rep’t Chf. of Eng’rs U. 
S. A., 1883, III. 

123 

167 

159 

Mississippi at Winona— 
Miss. River Com. 

1880-81 

Rep’t Chf. of Eng’rs U. 
S. A., 1883, III. 

34 

32 

36 

Mississippi at Clayton— 
Miss. River Com. 

1880-81 

Rep’t Chf. of Eng’rs U. 
S. A 1883, III . 

40 

42 

41 

Mississippi at Hannibal— 
Miss. River Com. 

1880-81 

Rep’t Chf. of Eng’rs U. 
S. A., 1883 III . 

165 

208 

224 

Mississippi at Grafton — 
Miss. River Com .... .... 

1880-81 

Rep’t Chf. of Eng'rs U. 
S.A 1883, III . 

319 

322 

345 

Mississippi at 8t. Louis— 
Miss. River Com . 

1880-81 

Rep’t Chf. of Eng’rs U. 
S. A., 1883, III . 

686 

906 

995 

Mississippi at Carrollton 

Rep’t Chf. of Eng'rs U. 

“ The means of surface, mid-depth 

— Miss. River Com . 

Missouri at 8t. Charles — 
Missouri River Com. ... 

1879-80 

1879 

S. A., 1883, III . 

Rep’t Chf of Eng’rs U 
S. A., 1887, IV . 

and bottom observations are in 
the ratio of 100, 144 and 183, 
respectively.” Vide Rep’t Chf. 
Eng'r, 1883, III, p. 2216. 

2 418 | 2 473 | 2 548 

Garonne—Baumgarten _ 

1847 

Anna],, des Ponts et 
Chauss6es, 1848, II. ... 

The surface, mid-depth and bot¬ 
tom quantities are in the ratio 

Elbe at Harburg — Blohm.. 

1837-54 

( Zeitschrift des Arch. 
} und Ing. Ver., Han- 

of the numbers 100,141 and 125. 
Surface mid-depth and bottom 
quantities are in the ratio of the 

, 


( over, 1867 . 

numbers 100, 93.8 and 98.2. 


A study of the table shows only one case in which the sediment per 
cubic unit is not greater at the bottom than at the surface. This is 
that of the Elbe at Harburg. There are two cases where the mid¬ 
depth amounts are less than the surface values, the Mississippi at 
Winona and the Elbe at Harburg. There are three cases where the 
bottom values are less than those at mid-depth, the Sacramento at 
Kerscheval’s, the Mississippi at Clayton, and the Garonne. 

In all other cases there is a marked increase from surface to bottom. 
When it is considered how extensive a range of measurements is repre¬ 
sented in these means and due weight is given to the careful and far- 












































HOOKER OH SUSPEHSIOH OF SOLIDS IH RITERS. 


65 


reaching observations of the Mississippi and Missouri commissions, it 
is thought that the law of increase from surface to bottom may be con¬ 
sidered as established. 

The table shows two other facts. First, that surface observations 
are not an accurate index of the mean amount of sediment. Second, 
that no general coefficient should be used to reduce surface observa¬ 
tions to mean \alues for all depths, since this coefficient will vary in 
different livers and for different stages of the same river. 

It was hoped that data might be found from which a relation could 
be established between the form of the velocity and sediment curve in 
the same vertical. Measurements at all points of the vertical are 
needed with simultaneous velocity observations at the same depths. 
The only ones obtained at all partaking of this nature are those in the 
Sacramento Eiver, which are too limited to be of service. 1 The 
observations at Carrollton in 1880 showed little change in the surface 
sediment with change of stage. 

Law of Distribution in the Horizontal .—In this study the data avail¬ 
able are still more limited. Of the extensive measurements carried on 
by the Mississippi and Missouri commissions, the author can find no 
single case where the samples from the eight positions in the trans¬ 
verse sense were kept separate and the weights published. Mr. 
Seddon 2 states that nothing of special interest was shown by these 
measurements at St. Charles, so the data were not published. The 
Carrollton observations 3 of 1880 are said to have shown a uniform dis¬ 
tribution from side to side of the channel and so are not published. 
Major Cunningham 4 obtained no data from which a law could be pre¬ 
dicated in his measurements on the Ganges Canal. 

Those at Columbus in 1858 are not printed in detail, but the Car¬ 
rollton observations of 1851-52 give the single satisfactory set. The 
measurements made in 1879 by Mr. McMath at St. Louis are not pub¬ 
lished in detail, but give some facts of interest. The last two sets are 
represented in the table on the next page by their means for such 
positions as are indicated. 

1 An article by C. C. Babb, Jun. Am. Soc. C. E , abstracted in Engineering News, 
August 10th, 1893, is said to give curves representing sediment at different depths in the 
Potomac River. 

a “Report of the Chief of Engineers, United States Army," 1887, IV, p. 3090. 

3 “Report of the Chief of Engineers, United States Army,” 1883, III, p. 2216. 

4 Proceedings of the Institution of Civil Engineers, Vol. 71, p. 34, “ Roorkee Hydraulic 
Experiments.” 



6G 


HOOKER OK SUSPENSION OF SOLIDS IN RIVERS. 


Distribution of Sediment in the Horizontal. 


Place of observation and 

Date. 

Reference. 

Parts of Sediment in 1 000 000 
Parts of Water at the 
Positions Indicated. 

observer. 



Bank. 

Middle. 

Bank. 

Mississippi at Carrollton 
—Forshey. 

1851-52 

Rep’t on Miss, Hum¬ 
phreys and Abbot,1861 

542 (300 ft. 
from e. 

573 

543 (400 ft. 
from w. 

Mississippi at St. Louis - 
Me Math. 

1879 

Van Nostrand Eng. Mag., 
1883. 

bank). 

Mean ofsedii 
parts; me 
from Mo. s 
of sedimen 
1 523 parts 

Dentin t 
an of set] 

bank). 

be river, 2 062 
liment, 281 ft. 


ide, 2 736 parts; mean 
11291 ft. from Ill. side. 


These meager measurements are offered rather to show the need of 
attention to this matter than as a proof that the maximum of sediment 
is near the thread of the stream. Mr. McMatli 1 found the maximum 
in nearly every case several hundred feet on the Missouri side of the 
line of maximum velocity. This could, perhaps, be accounted for by 
the fact of the Missouri water being more highly charged since its 
sediment is finer. If the Ohio and Mississippi flow side by side with¬ 
out miugling their waters 2 it may be that a similar phenomenon occurs 
at St. Louis. 

Surface Convexity and the Lateral Movement of Suspended Matter .— 
It has been stated 3 that crevasses in the Mississippi show a marked 
swelling or convexity at the thread of the current, where it crosses the 
levee, and that this convexity, which is due to the excess of velocity, 
has the effect of drawing floats to a narrow line at the filament of max¬ 
imum velocity. 

Major Cunningham 4 could measure no sensible curvature at the 
center of the Ganges Canal. He had expected to find such a swelling 
on the ground that increased velocity would be accompanied by de¬ 
creased pressure. He quotes the statement of General Rundall that 
the surface of the Godavery and Malianuddy Rivers was obviously con- 

1 " Silt Movement by the Mississippi,” R. E. McMatli. Van Nostrand's Engineering 

Magazine, Vol. 28, 1883, p. 36. 

3 See “Report on the Mississippi,” Humphreys and Abbot, 1876, p. 136. Compare, 
also, the statement made by McMatli in “ The Mississippi as a Silt-Bearer,” Van Nostrand’s 
Engineering Magazine, 1879, p. 222. The Missouri water was found to contain 2£ times the 
amount of sediment carried by the water of the upper Mississippi where the two streams 
were running side by side, with a common velocity, past Bissell’s Point. 

3 “ Report on the Mississippi,” Humphreys and Abbot, 1876, p. 284. 

4 Proceedings of the Institution of Civil Engineers, Vol. 71, pp. 11 and 12. 




























HOOKER OH SUSPENSION OF SOLIDS IN RIVERS. 67 

cave, plane or convex, according as the rivers were falling, stationary 
or rising. 

Mr. Flamant 1 shows that there should be no such difference in 
pressure since a fluid transmits pressure equally in all directions and 
the surface should be level for permanent motidn. He explains the 
convexity measured by Baumgarten 2 3 4 in the Garonne in one case on the 
ground that it did not correspond to a permanent state, but to a con¬ 
dition of rise when the center would show the increase before the 
sides. 

M. Baumgarten' 5 6 7 8 9 io concludes from his two measurements on the 
Garonne that the changes are scarcely sensible. 

The experiments of Darcy and Bazin* showed no results which 
could be said to express a law. 

Prof. De V olson Wood 5 states that the water is highest where the 
velocity is greatest, but gives no substantiation for the statement. 

M. Debauve 5 accepts the idea as true and explains its apparent 
disagreement with the hydrostatic law on the ground of viscosity, 
the water tending to the form which will offer the least resistance at 
the banks. 

Major Cunningham' found that surface floats placed near the banks 
were uniformly drawn out to the thread of the stream, while subsur¬ 
face floats maintained a direction sensibly parallel to the banks. 

Mr. McMatlC holds that suspended material is borne from the 
sides toward the center. The same statement is made by Dupuit'* with 
reference to bodies floating at the surface. 

M. Lagrene"* considers the upper surface in straight sections to be 
sensibly horizontal so far as present knowledge reaches. 

In general it may be said that the movement of surface floats shows 
a tendency toward the line of maximum velocity. That the difference 

1 See Annales des Ponts et Chausstes, 1882, IV, p. 56. Also Proceedings of the Institution 
of Civil Engineers,*Vol. 71, p. 66. 

2 See Proceedings of the Institution of Civil Engineers, Vol. 71, p. 12, or Annales des Ponts 
et Chausstes, 1848, 2, pp. 28-30. 

3 Annales des Ponts et Chausstes, 1848, 2, p. 30. 

4 See “ Recherches Experimentales.” Darcy and Bazin. Plates XIX to XXVI. Also Pro¬ 
ceedings of the Institution of Civil Engineers, Vol. 71, p. 12. 

5 Van Nostrand’s Engineering Magazine, 1879, p. 370. He claims that the cause lies in the 
reduction of pressure with increase of velocity. 

6 Navigation Fluviale et Maritime.” V. Debauve. 

7 Proceedings of the Institution of Civil Engineers, Vol. 71, p. 23. ‘‘Roorltee Hydraulic 
Experiments.” 

8 Van Nostrand’s Engineering Magazine, Vol. 28, p. 35. 

9 “ Etudes sur le Mouvement des Eaux,” p. 218. 

io ** Cours de Navigation Interieure,” p. 53. 




68 


HOOKER ON SUSPENSION OF SOLIDS IN RIVERS. 


in height of surface between center and sides is a concomitant phe¬ 
nomenon cannot be said to be proved. There are many statements 
that the fact exists, but such measurements as are available fail to 
substantiate them. 

Does Suspended Matter Move Faster Than the Current? —The ques¬ 
tion of the existence of a relative velocity between surface floats 
and the surface current may be said to be settled. It has been 
considered a source of error in the measurement of surface velocities 
by floats for some time past. The velocity of the float is greater than 
the mean velocity of the displaced water. For the same float, this rel¬ 
ative velocity will increase with increase in depth of flotation. 1 

Major Cunningham 2 3 takes the contrary view, claiming that the 
velocity of a submerged rod is somewhat less than the mean velocity 
past its immersed length, so that it should only extend -jVo °f the 
total depth to the bottom in order to give a true indication of the 
mean velocity in a vertical. 

The experiments of M. Du Bovs" upon the Rhone may be said to 
settle the matter beyond dispute. He found from experiments upon 
boats that they moved sensibly faster than the current, and that this 
relative velocity varied with the form of the boat and increased with 
its size and with the velocity of the current. 

Experiments by M. Berard, 4 in 1886, on an artificial canal, showed 
that a float moved faster than the surface velocity, but almost identi¬ 
cally at the same rate as the mean velocity of the displaced water. 

M. Du Boys, 5 however, gives the case of a boat which had a 
velocity of 4.46 m. in a current whose surface velocity was only 2.75 m. 
These differences are too great to be attributed to the fact that the 
maximum velocity is below the surface. 

A block of wood and a floating canal-boat will not remain side by 
side in a river, but gradually draw away from each other, the boat 
taking the lead. 

1 Noted and incorrectly explained in Zeitschrift des Architekten- und Ingenieur- Vereins, 
Hanover, 1887, p. 628. 

2 “ Roorkee Hydraulic Experiments.'' Proceedings of the Institution of Civil Engineers, 
Vol. 71, p. 22. 

3 “ La Marche des Bateaux dans les Courauts Rapides.” Annulet des Fonts et Chausseet 
1886, I, pp. 199-242. 

* “ Marche des FJotteurs dans les Courauts.” Anniles des Ponts et Chausstes 1886 II p 

830. 

s “ Hydraulique.” Flamant, p. 299. 






HOOKER OH SUSPEHSIOH OF SOLIDS IH RIVERS. 


69 


The Question of Lateral and Vertical Flow in Rivers. —Islands are 
said to be formed at the center of rivers at the expense of the banks, 
indicating a transverse flow of the particles toward the middle. Is 
there a regular lateral flow of the water obeying a law as fixed as that 
which determines the general law of translation ? If so, it will enter 
as an important factor in governing sedimentary movements. Prof. 
James Thomson 1 has demonstrated the presence of such a flow at curves 
where the motion at the surface is outward and at the bed is inward. 
This offers a suitable explanation of the cause of shoals opposite sharp 
bends. In a regular channel centrifugal force does not enter in to pro¬ 
duce these effects. Are they present ? Mr. McMath says 2 that obser¬ 
vation has failed to detect any division of a stream into ascending or 
descending areas other than local motions due to eddies. Major Cun¬ 
ningham 3 claims to have detected a surface flow toward the center in 
the Ganges Canal as indicated by the motion of the floats near the 
banks, while a sub-surface flow in the contrary sense was indirectly 
shown by the fact that the deeply immersed floats showed no general 
transverse tendency. 

Experiments made by J. B. Francis 4 in regular canals with white¬ 
wash discharged through a tube opening near the bed, seemed to show 
a vertical movement of the water from the bed to the surface, the 
whitewash appearing at a distance down-stream varying from 10 to 30 
times the depth. Mr. Francis offers this vertical movement as the 
cause of suspension of sediment. 

Prof. De Volson Wood, 5 upon the ground of these observations of 
Mr. Francis, adopts the idea of a lateral surface movement toward 
the banks, with a corresponding bottom current toward the center. 

F. P. Stearns, M. Am. Soc. C. E.,° on the other hand, holds the 
view that there is a motion of bottom water to the surface at the sides 
and toward the center at the surface. 

The observations of Mr. Francis might seem to be corroborated by 
the positive statement 7 that the Mississippi water is constantly rising 
from the bed to the surface. __ 

1 “ Encyclopedia Britannica ” Article Hydr 'dynamics, p. 498. 

2 “ Silt Movement by the Mississippi,” Van Nostrand’s Engineering Magazine, Vol. XXVIII, 
p. 35 

3 Proceedings of the Institution of Civil Engineers, Vol. LXXXII, pp. 23, 24. " Roorkee 
Hydraulic Experiments.” 

4 Transactions of the American Society of Civil Engineers, Vol. VII, p. 109. 

6 Van Nostrand’s Engineering Magazine, Vol. XXI, 1879, p. 369. 

Also Transactions of the American Society of Civil Engineers, June 17th, 1879. 

6 Transactions of the American Society of Civil Engineers, Vol. XII. p. 331. 

7 "Report of the Chief of Engineers, United States Army,” 1883, III, p. 2 218, 




70 


HOOKER ON SUSPENSION OF SOLIDS IN RIVERS. 


5. Solid Discharge of Rivers. 

In many articles on river correction and in most of the treatises on 
river hydraulics will he found statements of the amount of silt carried 
by streams in various parts of the earth. The amounts are unreliable 
in some cases and in many are expressive of only suspended matter to 
the exclusion of the more or less extensive movement along the bottom. 
A collection of such non-homogeneous data is not deemed of value for 
the purposes of the present paper. 1 

Proportionate Amounts Suspended and Dragged. —The quantitative 
relation existing between the amounts moved at or near the bottom 
and those carried in free suspension has offered another opportunity 
for diversity of opinion. 

Dupuit 2 looks upon transportation in suspension as the most 
important element, on the ground of the slight velocity of the ma¬ 
terials dragged. Prof. Forshey 3 concludes that the matter pushed 
along the bed of the Mississippi forms about three-fourths of its total 
solid discharge. 

M. Guerard 4 is persuaded that the greater portion of the solid 
matter discharged by the Rhone is pushed along its bed. The Missis¬ 
sippi River Commission, 5 6 after a series of careful measurements of 
sand waves at Carrollton, decided that not more than .08 of 1 per cent, 
of the total solid discharge was moved along the bed of the river at 
this point. Major Ruffner *’ states that the movement of material 
at the lower laminae of the Missouri at St. Charles was believed to be 
as great as that in all the rest of the river. He refers 1 also to observa- 

1 The following list gives a few sources of information upon this point: 

“Report on the Mississippi,” Humphreys and Abbot, 1876, p. 146. 

ZeUschrift des Arch.- und Ing - Verexns, Hanover, 1867, pp. 245-50. 

Proceedingt of the Institution of Civil Engineers, Vol. XXI, pp. 15, 27, 459; Vol. LIU, p.18; 
Vol. LI, pp. 216, 217; Vol. LVII, pp. 272-4. 

The Engineer, October 25th, 1889, p. 343. 

Annates des Ponts et Chaussees, 1848, II, pp. 46-48; 1860, I, p. 137; 1860, II, p. 374; 1869, I, 
p. 588; 1871, I, p. 15. 

“Canal and River Engineering,” Stevenson, p. 318. 

“ Improvement of Non-Tidal Rivers,” Ruffner. 

“Cours d’Hydraulique Agricole et Urbaine.” M. Bechmann, 1895, pp. 109, 120. Also 
treatises on general geology. 

2 “ Etudes sur les Mouvements des Eaux,” p. 216. 

3 Proceedings of the American Association for the Advancement of Science, Nashville, 
1877. See also Van Nostrand's Engineering Magazine, Vol. XX, p. 227. 

4 Proceedings of the Institution of Civil Engineers, Vol. 82, p. 309. 

s “ Report of the Chief of Engineers, United States Army,” 1883, III, p. 2218. 

6 “ Improvement of Non-Tidal Rivers,” p. 78. 

" The same, p. 138. * 



/ 


HOOKER OH SUSPEHSIOH OF SOLIDS IH RIVERS. 71 

tions at Lake Providence, La., which indicated that a large amount 
was moving along the bed of the Mississippi at that point of which the 
moving sand waves represented only a small portion. 

Captain Eads and Mr. Corthell 1 join with the Mississippi River 
Commission in the statement that nearly all the solid matter in the 
Mississippi is carried in suspension, while but a small proportion is 
dragged on the bottom. 

It is believed that this last statement represents the actual case in 
most sedimentary rivers. 

6. Tabulation of Observed Data on Dragging. 

A tabulation of the best known results of experiment on velocities at 
at which dragging begins is given on the next page. Those given by 
Bouniceau appear to be taken from the results of Dubuat’s and Tel¬ 
ford’s experiments. The others are believed to represent original 
measurements or computations. In some cases the published results 
do not state whether the velocity measured was that at surface, mid¬ 
depth or bottom. 

Further statements of velocities are given by Weisbach, Unwin, 
Church, Bechmann and others, but they are all based upon the measure¬ 
ments detailed in this table. A limited collection of similar data is 
given in the Report of the Chief of Engineers, U. S. Army, 1885, I, 
pages 569-570. 

The detail of Dubuat’s and Sainjon’s measurements has been given 
in a preceding part of this paper. The measurements in the Upper 
Rhine were made near Alt-Breisach, with a smooth river-bed. In the 
cases so marked, motion did not take place until the particles were 
subjected to a slight disturbance from the outside. 

Login’s experiments were made with a stream averaging $ in. in 
depth. Telford’s velocities are those at which erosion begins. Black¬ 
well’s measurements have been referred to on page 12. 

The materials are given as described by the experimenter. A careful 
set of measurements of these velocities, made with improved appara¬ 
tus, is much needed. 

Displacement of Crests of Sand Waves .—For purposes of comparison, a 
collection has been made on page 74 of measurements on sand waves. 


1 Transactions of the American Society of Civil Engineers, Vol. XI, p. 262. 




Table giving Velocities of Current at which Dragging Begins. 


72 


HOOKER OH SUSPENSION - OF SOLIDS IN' RIVERS. 




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74 


HOOKER OH SUSPENSION OF SOLIDS IN RIVERS. 


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HOOKER OH SUSPENSION OF SOLIDS IN RIVERS. 


75 


Additional measurements, made by M. Sainjon, are given at page 
21 of this paper. 

In general, it may be said that the displacement of these wave 
crests increases with the stage and with the velocity, at least up to a 
certain point of bottom velocity. Their form and motion is most reg¬ 
ular at times when the river is neither rising nor falling. Their motion 
is most rapid when the river is rising. They are largest in deep water. 


7. Velocities of Vertical Current Required to Keep Particles Suspended. 


A particle whose specific gravity is greater than 1 will sink in quies¬ 
cent water when viscosity is overcome. It will, when falling freely from 
rest, after a very short period of acceleration, attain a state of velocity 
practically uniform. The same particle should be kept suspended in¬ 
definitely by a constant current, whose vertical component has a ve¬ 
locity equal to that asymptotically approached by the falling particle. 

Experiments have been made to determine the velocities of currents 
necessary to keep particles of various sizes and specific gravities 
suspended when acting in a direction opposed to gravity. 

The table 1 on page 7G gives the results of experiments made by M. 
Thoulet. It represents the maximum velocities in millimeters per 
second of, and the thrusts in milligrams exercised by, currents capable 
of holding suspended, at a fixed point in a tube, spherical grains of 
radius varying from 0.1 to 2.5 mm., and of densities, in air, between 1.5 
and 4.0. The velocities are taken from diagrams constructed from 
results of actual experiment. The thrusts P are computed on the sup¬ 
position that a current with a velocity v, in millimeters, which holds in 
suspension a body of density y\ exercises against the body a thrust 


P — - 7r r 3 (y' —1) (in milligrams, where r is 
3 


expressed in millimeters). 

The range of values for (r) and (y') is sufficient to cover all ordi¬ 
nary mineral grains. 

There can be little doubt that the velocity of vertical current or the 
amount of vertical thrust required for the suspension of ordinary 
grains is very slight. Col. Mansfield reports 2 observations on grains 
of sand allowed to fall from rest in water. The maximum velocity 

1 For further details see p. 33; also Annales des Mines, 1884, I, p. 521. 

“ Report of Chief of Engineers, United States Army,” 1886, pp. 1298-1299. 







Velocity ( v ) of the Current in Millimeters per Second Thrust (Pi in Milligrams Exercised Against the Spherical 

Grain of Density (?'). 


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HOOKER ON" SUSPENSION OF SOLIDS IN RIVERS. 


77 


readied at tlie bottom showed that a vertical current of 1 or 2 ins. per 
second would keep them suspended. 

Mr. T. E. Login, 1 found that the following materials had a rate 
of sinking in water of, and consequently would be sustained by a ver¬ 
tical current of, a velocity equal to the numbers given below: 

Velocity required. 

Materials. (Feet per second.) 

Brick clay (mixed with water and allowed half 

hour to settle). 0.009 

Fresh water sand. 0.166 

Sea sand. 0.196 

Rounded pebbles (size of peas). 1.000 

Diagrams have been prepared by Richards 2 and Woodward to show 
the relations existing between the specific gravity, diameter and veloc¬ 
ity of fall in water of various minerals. They are constructed from a 
formula given by Rittinger/' 

V= 2.44 VD{8 — 1) 

where V = velocity of fall in millimeters per second. 

D = diameter of particles in meters. 

8 = specific gravity of the mineral. 

These diagrams are extended to cover specific gravities between 1 
and 15, diameters between 0 and 0.06 m., and velocities of fall up to 
2.3 m. per second. 

8. Results Expressed in the Form of a Series of Propositions. 

The following propositions are believed to express the main facts, 
so far as they are known at present, which must be recognized by any 
broad theory of the cause of the suspension of sediment: 

1. The movement of solids by water currents may take place by 
dragging, by intermittent suspension or by continuous suspension. 

2. Motion in each of the three ways is increased with increase of 
depth; yet the depth itself can only affect the intermittant motion. 

3. Motion by each of the three ways is increased by increase in 
mean velocity. 

1 Proceedings of tlie Royal Society of Edinburgh, Vol. Ilf, p. 475. 

Also “Canal and River Engineering,” Stevenson, p. 315. 

2 “ The Velocity of Bodies of Different Specific Gravity Falling in Water.” Richards and 
Woodward. Transactions of the Amei’ican Institute of Mining Engineers, 1890, Vol. 18, pp. 
644-648. 

s Rittinger’s “ Aufbereitungskunde,” 1867, p. 195.. 


LQF& 








78 


HOOKER OH SUSPENSION - OF SOLIDS IN RIVERS. 


4. The presence of sediment in a stream decreases its mean velocity. 

5. Dragging as well as suspending power increases with the heavi¬ 
ness of the liquid and with its coefficient of viscosity. 

G. By far the greater amount of solid material transported by sedi¬ 
mentary rivers is carried in continuous suspension. 

7. Sand waves of considerable size are formed in the larger rivers. 
Their motion down stream is slow, increasing in rate up to a certain 
critical value of bottom velocity and then decreasing with further 
increase. 

8. The fineness of material in suspension increases from the rise to 
the embouchure of rivers. 

9. The weight of suspended material per cubic unit of water de¬ 
creases from rise to embouchure. 

10. Increase in vortex motion increases power of transport. 

11. Continuous vertical or lateral currents which could not be 
explained by local causes in rivers have not yet been proved. 

12. The phenomenon of suspension requires for its explanation a 
continuous upward force. Intermittent forces are not sufficient, 
although the intensity of the force may vary. 

13. Bodies suspended in flowing water, either intermittently or 
continuously, tend to acquire a velocity greater than that of the water 
surrounding them. 

14. The transportation of material consumes part of the energy of 
any silt-bearing stream. A greater load of fine particles than of coarse 
can be carried with the same expenditure of energy. 

Part III.— Discussion of Theories of Suspension. 

The explanations offered as to the cause of this phenomenon, as 
detailed in the preceding pages, fall naturally into four categories. It 
will be well to examine each in the light of the facts at hand before 
drawing conclusions as to their relative value. 

(a) Theory of a Continuous Upward Flow. —J. B. Francis 1 Past- 
President Am. Soc. C. E., sought to explain the suspension of sedi¬ 
ment in streams by the presence of a continuous upward flow of the 
water at the bed. This flow was believed to be proved by experi¬ 
ments made by liberating whitewash at the bed of two different 
channels and noting its appearance at the surface at distances down- 


1 Transactions of the American Society of Civil Engineers, May and June, 1878. 








HOOKER OH SUSPEHSIOH OE SOLIDS IH RIVERS. 


79 


stream varying from 10 to 30 times the depth. The experiments of 
M. Tlioulet 1 2 3 * have shown how slight an upward current is required 
to produce this effect. Were such a resultant upward current proved, 
it might ofter a satisfactory explanation of the jihenomenon. There 
are, however, certain objections to this view which seem to render it 
inadmissible. Considering a long stretch of river, a steady flow from 
bed to surface at the center must be accompanied by a corresponding 
flow outward to the sides and a downward reverse flow along the 
bed from the sides to the center. Such a flow has not been observed. 
On the contrary several writers” have advanced the idea of a flow in 
the contrary sense, though this, too, must be regarded as yet un¬ 
proved. 

Again, even were the flow from bed to surface at center and out¬ 
ward to the sides proved, it would constitute no explanation of 
continuous suspension, unless the velocities of flow at center and 
sides were unequal, since it would give no resultant upward thrust. 
The algebraic sum of the vertical comjmnents of all the internal move¬ 
ments of the liquid should reduce to zero in the stretch considered. 

Proofs of flow from bed to surface or from sides to center based 
upon indications offered by floating substances—the case in both Mr. 
Francis’ and Major Cunningham’s experiments—can not be considered 
conclusive. In the one case the movement might be attributed to 
local currents induced by the presence of the experimental apparatus. 
In both cases, the fact that a floating body tends to move faster than 
the surrounding medium and would follow the line of least resistance 
would cause floating substances to indicate currents which did not 
actually exist. This matter will be referred to later. 

(b) Discontinuous Upward Flow or Eddy Theory .—There are certain 
writers who attribute the entire suspending power of streams to ver¬ 
tical currents incident to the complex eddying motion induced by the 
asperities of the bed. The numerous valuable articles 5 by R. E. 
McMath, M. Am. Soc. C. E., have taken the strongest position on 

1 See p. 33. 

2 See Proceedings of the Institution of Civil Engineers, Vol. 71, p. 23 and discussion. Also 
Transactions American Society of Civil Engineers, Vol. 12, p. 333. 

3 " Silt Movement by the Mississippi.” Van Nostrand’s Engineering Magazine, Vol. 28, 
l 883, p. 32. 

“ The Mississippi as a Silt Bearer.” Van Nostrand’s Engineering Magazine, Vol. 20, 1879, 
p. 218. 

“ Theory and application of the Permeable System of Works for the Improvement of 
Silt-Bearing Hirers.'’ Engineering News, November 1st, 1879. 



80 


HOOKER OK SUSPENSION OF SOLIDS IN RIVERS. 


this line. Upon the basis that the fact of suspension itself is an evi¬ 
dence of a resultant upward thrust, Mr. McMatli states his position 
in these words •} 

“ This third hypothesis goes but one step farther in ascribing to the 
irregular movements the whole work of suspension upon the ground 
that a cause known to exist wherever the fact to be accounted 
for occurs, and admitted to be efficient, must be considered the sole 
cause unless a co-working agency is known, or the cause is insufficient 
to produce the observed result. Observation readily detects whirls, 
boils and eddies in the act of bringing water and suspended material 
from the bottom to the surface and laterally from the sides to the 
center of the river. Observation has never detected any other cause 
incident to the flow of streams which produced these effects.” 1 2 

The efficiency of vortices in producing local results has passed be¬ 
yond the range of controversy. M. Boussinesq 3 divides them into two 
classes; first, those in which the constituent water is constantly 
changing as the result of an axial flow which increases in velocity as 
the radius of curvature decreases ; second, those in which the con¬ 
stituent water remains the same, being formed during an interval of 
time by a definite mass of fluid and manifesting themselves in a com¬ 
plex rotation. 

A case of the first kind is the common vortex over the discharge 
pipe of a hand basin. One of the second is seen in the experiment of 
the revolving glass or basin of water described earlier in this paper in 
the experiments of M. Fargue and M. Gallois. 

In both forms of vortex the tangential velocity increases with, ap¬ 
proach to the axis in contradistinction to the case of a rotating solid. 
In each case solid material obeys an impulse toward the axis whether 
there be a flow of liquid in that direction or not. This is due to the 
tendency of the solid to move faster than the liquid and in so doing to 
follow the line of least resistance, i. e., the line where differences of 
velocity are least. In the first form an additional impulse toward the 
axis is received from the current setting in that direction. These 
phenomena are clearly shown in the experiments referred to, though 

1 “ Silt Movement by the Mississippi.” Van Nostrand’s Engineering Magazine, Vol. 28, 
p. 85. 

“ “ The suspension of matter involves an upward motion of the water or medium in 
which such matter is suspended, if in no other way, by the law of adhesion. * * * Upward 
motion of the suspending medium involves of necessity a downward movement of equal 
volume, but not necessarily of equal velocity.” McMatli in Engineering News, November 1st, 
1879. 

s “ Essai sur la Theorie dos Eaux Courantes,” p. 616. 


0 





HOOKER OK SUSPEKSIOK OF SOLIDS IK RIVERS. 


81 


M. Gallois’ 1 explanation is at variance with that given above. He takes 
the position that the particles are impelled toward the center because 
the relative velocities are greatest at that point. This is incompatible 
with the theory that the standard form of a vertical velocity curve in 
streams is a parabola. If the friction initiated at the sides exercises 
its retarding effect progressively toward the center, as is believed to be 
the case in rivers, the absolute velocity near the axis of the vortex 
should be greater, but the relative velocities of the fluid filaments 
should be a maximum near the sides. In other words the solids would 
move toward the axis of the vortex with radial velocities varying 
directly with the distance from the axis. According to M. Gallois’ ex¬ 
planation this relation would be an inverse one. 

In a flowing stream these eddying movements are present in the 
forms described and in numerous intermediate states. Irregularities 
in the bottom sometimes send currents to the surface in the form of 
boils w T here there appears to be only a swift vertical velocity without 
much rotation. In the complex motion of the stream these visible dis¬ 
turbances are but the type of an infinite variety of similar movements 
taking place between small groups of molecules. These are not so 
evident to the eye, but consume in the aggregate a large proportion of 
the stream’s energy. The asperities of the bed and banks send off 
these rotary groups with axes inclined at all angles and with velocities 
of translation having a resultant in the stream direction. Whatever 
resistance is offered along the air profile will manifest itself in com¬ 
plicating the direction of these movements of translation. The vital 
question is to know the direction of the resultant motion. Has it a 
component acting upward ? 

It is seen at once that vertical motion of the water itself can only 
be a local phenomenon. The resultant of all the external forces act¬ 
ing upon a given stretch of river will have a direction down stream, 
and will approach zero as the motion approaches uniformity. Its 
vertical component will be downward, not upward. The same thing 
will be true of all the complex motions in the interior of the liquid 
prism considered in the aggregate. It is true, however, that the 
reverse downward current need not equal in velocity the original 
upward one. 2 The diminution in velocity may be counterbalanced 

1 See Engineering Nevjs, March 23d, 1893. 

2 This fact is noted, without comment, by Mr. McMath in Engineering News, November 
1st, 1879. It is thought that he was the first to suggest it in connection with suspension. 



82 


HOOKER OH SUSPENSION OF SOLIDS IN RIVERS. 


bj an increase in the moving area. This is an important point affect¬ 
ing the power of suspension. While there is no resultant vertical 
motion of the water, there is a resultant vertical thrust exercised upon 
sediment in the aggregate and acting as an efficient cause of suspension. 

The thrust exercised by a current upon an immersed solid is 


P = k F y 



and obviously varies with the square of the velocity. An eddy, having 
an upward component of velocity represented by 4, would exert an 
upward thrust, available for suspension, proportional to 16. In 
descending, the same water might cover twice the area, and have a 
downward velocity of 2. Then the corresponding downward thrust 
would be proportional to 8. The resultant of such a local system 
would be an upward thrust available for continuous suspension. In 
addition to this, particles carried by the vortex to the surface, and 
especially when freed at that point from its influence, would exhibit 
the phenomenon of temporary suspension, before gravity had again 
brought them back to the river bed. Cover the stream -with systems 
of this character, and there results a modified form of w r hat is 
actually seen. Eddies are induced by the rugosities met w r ith in 
the earth and air profiles. The abrupt changes in the bottom, and its 

relatively unyielding character, cause the upward currents to have 

% 

the greater velocity in general, though they must of necessity be 
restricted in relative extent. 

The fact that the descending areas must be much greater than the 
ascending ones in order to give rise to this force shows that, in the 
major part of the stream, an active force is at work tending to carry 
material to the bottom, aside from the force of gravity acting upon the 
particles. This suggests, though it does not prove, the existence of 
another cause of suspension, and such a cause is actually found, as 
shown later. 

Here, then, in obedience to the law which governs the thrust exer¬ 
cised by water impinging upon a solid, there is a resultant force 
acting upward in flowing streams, and capable of performing an 
efficient part in the continuous suspension of solid matter. While the 
aggregate of all the interior motions of the fluid can have no upward 
resultant, still the thrusts exercised upon solid suspended matter by 
these motions can and do have such a resultant as shown. 








HOOKER OH SUSPEHSIOH OF SOLIDS IN RIVERS. 


83 


It is evident that this force would disappear in circular conduits, 
flowing full, since a uniform profile may be assumed. The eddies 
would be thrown off with like intensity at all points toward the 
center. 

M. Boussinesq 1 attributes increase in vortical motion at the walls 
to any one of these causes; increased mean local velocity, increased 
roughness of lining or increase in hydraulic mean radius. The effect 
of the first two causes is evident. The influence of the increase of 
area per unit of wetted perimeter is thought to be shown by the 
greater field of action allowed for oscillatory movements perpendicu¬ 
lar to the walls. These vibrations help in detaching groups of mole¬ 
cules and cause such groups to suffer sudden changes in intensity of 
tangential friction at the walls which, in turn, favors their formation 
into vortices. 

In a circular conduit running full this would indicate an increase 
in vortical movements at the walls with increase in diameter. In 
passing from the walls to the center of the current, M. Boussinesq 
thinks the vortical agitation should increase where the wetted peri¬ 
meter is concave, as in the case of closed conduits of rectangular or 
rounded forms, running full, because the field of aotion for each vortex 
is constantly narrowing and interference increases. On the other 
hand, this agitation should be less and be practically uniform for a 
rectangular cross-section of indefinite length of base. The agitation 
should be of about the same intensity in a large circular or rectangular 
conduit running full as when running half full. The action of the 
free surface in reflecting these movements ought not to differ widely 
from the action of the upper half of the section upon the lower half 
when running full. 

Considered as a whole, the vortex theory, 2 as advocated by Mr. Mc- 
Math, is entitled to a position as one of the major causes of suspen¬ 
sion. The statement that it is the sole cause, and that observation has 
never detected any other cause incident to the flow of streams which 
has produced these effects would seem too broad a claim. 

(c.) Theory Based upon Eddies and Relative Velocities .—Partiot was 
probably the first to give prominence to the suspending power of 

1 “Theorie (lea Eaux Courantes,” p. 47. 

t A valuable paper by William Starling, M. Am. Soc. C. E., Chief Engineer of the Missis¬ 
sippi Levee Commission, is in press at the present time, which brings this theory promi¬ 
nently forward. The term “vortex theory ” is here used in a restricted sense, as indicated by 
the context. 





84 


HOOKER OK SUSPENSION OF SOLIDS IN RIVERS. 


eddies. He bases bis explanation of suspension upon the presence, in 
these vortical movements, of exaggerated relative velocities which, in 
conjunction with the motion of the body, cause a thrust toward the 
most rapid filaments. The particles are then carried along with the 
motion of translation of the eddy. 

Where eddies are not the acting force, he follows Dupuit’s explana¬ 
tion as given hereafter. 

( d'.) Theory Based upon Relative Velocities .—This theory was brought 
forward by Dupuit. It utilizes the fact that all partially or wholly 
submerged bodies tend to move faster than the mean velocity of the 
displaced water. This makes a choice of path for relative motion 
necessary. The path chosen will be that which offers the least re¬ 
sistance, i. e., diagonally toward the most rapid filaments. 

Take the case of a floating homogeneous sphere. Let it have a 
velocity of translation in the stream direction equal to that of the 
mean of the displaced filaments. Assume uniform stream motion. 
Place it near the left-hand shore. The filaments of water at the right 
of the body are moving faster than its center of gravity, and by their 
friction tend to revolve it in an anti-clockwise direction. On the op¬ 
posite side of the ball the friction is reversed in direction, and aids in 
producing the same rotation. Wliat is the result upon transverse motion ? 

The ball will tend to place its center of gravity in the line of the 
resultant resistance of the filaments ahead of it. But in the theoreti¬ 
cal case considered there is no resultant resistance upon the sphere, 
since it moves with the same mean velocity as the displaced water. 
The law stated above, then, does not call for any transverse motion 
under the conditions named. 

Introduce the fact that the body has a velocity greater than that of 
the mean of the displaced filaments. A new element is introduced, 
for there is now a resultant resistance from the filaments ahead which 
is proportioned to the square of the relative velocity. This relative 
velocity, and consequently the resistance, increases until the resistance 
is equal to the accelerating force, when the motion becomes uniform. 
It is now incumbent upon the body to choose a path relatively to the 
fluid. It will tend to place its center of gravity in the line of action 
of the resultant resistance. This line of action is to the right of the 
center of gravity of the body. It will then take a diagonal path to 
the right, or toward the most rapid filaments. 


HOOKER OKI SUSPENSION OF SOLIDS IN RIVERS. 


85 


The line of application of the mean resistance will coincide with 
the line of action of the mean velocity of the water immediately 
ahead. At any point of its path the mean resistance offered will be 
least, i. e., the velocity of the body will be most nearly equal to the 
mean velocity ahead, if the center of the body is in the line of action 
of this mean velocity. 

In summing up, then, Dupuit’s theory of suspension rests upon 
the fact that submerged bodies tend to move faster than the mean 
velocity of the displaced water, and in so doing choose the line of 
least resistance. 

It is the only satisfactory explanation yet offered of the phenomena 
shown in the experiments with rotating vessels, which are described 
by M. Fargue and M. Gallois. It offers an explanation of the increase 
in suspending power, with increase in depth and velocity. In the “Re¬ 
port on the Mississippi River ” by Humphreys and Abbot (page 254) 
it is shown that the parameter of the horizontal parabola of velocities 
5 ft. below the surface, at any stage, is proportional to the square 
root of the corresponding mean velocity. The same should be true of 
the vertical curve. 

The reasonableness and value of this theory seem evident. Its in¬ 
completeness as an explanation of the whole phenomenon is shown by 
the fact that it takes no account of the suspending power of eddies 
and offers no explanation of the presence of sediment above the line of 
maximum velocity. 

Before stating the conclusions to which this analysis of theories 
leads, it will be well to place emphasis upon a cause of suspension 
which has only been indicated in a general way. 

Dupuit 1 suggested that the resultant of all the pressures upon a 
submerged body was greater in flowing or agitated than in quiescent 
water. He explains it on the ground of relative velocities of the fila¬ 
ments as detailed above. 

M. Flamant 2 has adopted the same idea as an explanation of the ten¬ 
dency of surface floats to move faster than the current. His argument 
is that the floating body weighs more in agitated water than the 
weight of the water displaced, and that the excess shows its presence 
in the increased velocity. The explanation implied is that there is a re¬ 
sultant upward thrust in the surrounding fluid due to its agitation. 

1 " Etudes sur Le Mouveinent des Eaux, ” 1818, p. 217. 

3 “ Hydraulique,” 1891, p. 298. 



86 


HOOKER ON SUSPENSION OF SOLIDS IN RIVERS. 


The explanation offered by Du Boys 1 for the increased velocity of 
surface floats has indicated the presence of a forward thrust, but has 
not called attention to the fact that this thrust has a direction above 
the horizontal. This follows from the discussion under (&) preceding. 
Experimental demonstration is wanting, but the fact that continuous 
suspension exists above the line of maximum velocity is an evidence 
that Nature shows the phenomenon indicated by theory. 

It should be remembered that the line of maximum velocity has 
often been found at the surface. In such cases the suspending power 
due to the body’s excess of velocity and tendency toward the line of 
least resistance acts from bed to surface in conjunction with the 
upward thrust due to the eddies. In the majority of cases it is found 
below the surface, and here a force must be acting downward to assist 
gravity in opposing suspension above the line of maximum velocity. 
This would indicate for such cases a decrease in suspended matter 
near the surface. The burden of the work then falls upon the up- 
ward thrust of the vortices. The fact that agitation increases from 
the line of maximum velocity to the surface tends to make the effect 
of this downward thrust disappear, since the latter decreases in inten¬ 
sity toward the vertex of the vertical curve. In this way marked dif¬ 
ferences in the intensity of sediment charge at the surface and at the 
line of maximum velocity are prevented. A difference in the amount 
carried at the surface ought to be shown according as the vertex of the 
curve of vertical velocities is raised or lowered—for instance, bv the 
wind. The surface quantities should be greater when the maximum 
velocity is at the surface. 


Conclusions. 

It is believed that the suspension of sediment in flowing water may 
be attributed to three causes: 

First .—The resultant upward thrust due to eddies, conditioned 
upon the facts that the earth profile offers more rugosities than the air 
profile and the effort exerted by a current upon a solid varies as the 
square of the relative velocity. 

Second .—The resultant upward motion of solids due to the fact that 
an immersed body tends to move faster than the mean velocity of the 

1 “ Sur la Marche des Bateaux,” Annales des Fonts et Chausstts, January, 1886, pp. 199- 

242 . 



HOOKER OH SUSPENSION OF SOLIDS IN RIVERS. 87 

displaced water and in such motion tends to follow the line of least re¬ 
sistance. 

Third. —The viscosity of the water. 

All of these causes will be present in every stream flowing under 
natural conditions. The first tw T o causes will alternate in efficiency 
with the complex motion of the stream. At certain points of the 
vertical curves of velocities the second cause may entirely disappear as 
the curve becomes irregular, but such conditions are abnormal. 

Experimental research is needed for further progress along these 
lines. Some of the more important questions awaiting investigation 
will be indicated. 

Can it be shown experimentally that the power of suspension in 
flowing water increases with the increase of relative velocities in the 
vertical ? 

Can a measurable difference in volume of displacement be detected 
for the same body in quiescent and in flowing water ? 

What is the normal form of the vertical and horizontal sediment 
curve ? 

Can it be shown by experiment that suspending power increases 
with depth, the mean velocity remaining unchanged ? 

What is the influence of a contraction in width upon the form of 
the vertical curves of velocity and sediment ? 

What is the form of the vertical curve of velocities corresponding 
to incipient dragging for stream beds composed of the range of 
materials usually found in practice ? 

Can it be shown by experiment 1 that a heavier load of fine than of 
course particles can be carried with the same expenditure of stream 
energy? 

What is the effect of ten^erature upon viscosity and mechanical 
suspension? 

An apparatus has been designed for the purpose of studying the 
first of these questions. It consists of a long, narrow trough, fitted 
with glass sides for nearly its whole length, which is joined to a 
wooden receiving reservoir. A steady flow is set up from the reservoir 
through the experimental channel. 

Two submerged jets, each covering the entire width of the channel, 

1 A general method of experiment has been indicated by Ashbel Welsh, Past-President 
Am. Soc. C. E., Transactions, American Society of Civil Engineers, Vol. xi, p. 162. 



88 


HOOKER OK SUSPENSION OE SOLIDS IN RIVERS. 


are introduced near the reservoir through suitable copper guides, 
arranged so as to create the minimum amount of agitation in the jets 
and in the steady flow progressing independently. 

These jets discharge under a head of 40 ft. One is placed near the 
bed, another near the surface of the channel. The object sought in 
the design was to obtain a means of controlling the form of the vertical 
velocity curves by varying the discharge from each of the jets. This 
is controlled by stop-cocks. The velocities at different parts of the 
vertical are measured by a cluster of Pitot’s tubes. 

The experiments have not yet been made. Preliminary trials have 
shown, however, that the control of the curve of velocities is ap¬ 
preciable for a sufficient distance from the jets to afford ample field for 
experimentation. 

The author takes pleasure in expressing his deep sense of obliga¬ 
tion to Prof. E. A. Fuertes for constant assistance and suggestions, 
and to Prof. I. P. Church for kindly criticism, which has led to modi¬ 
fications in some of the views expressed. This opportunity is gladly 
taken to mention the kindness of M. Edouard Collignon, Inspecteur 
General des Ponts et Chaussees, in affording, among other courtesies, 
unusual library privileges at the Ecole Nationale des Ponts et Chaus¬ 
sees, Paris; and of Prof. Conrad Zscliokke, of Zurich, in assistance 
rendered in various ways. He wishes, also, to express his apprecia¬ 
tion of the continued courtesy shown him by M. Cordier, the librarian 
of the Ecole des Ponts et Chaussees at Paris, and by Prof. Rudio, 
the librarian of the Zurich Polytechnikum, during his use of those 
libraries. 














LB S ’08 















































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