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A STUDY'OF HYDRAULIC JUMP

The protection of stream beds below spillways and weirs
against erosion.has long been a problem for the hydraulic
engineer to solve. Large sums have been spent in constructing
aprons, weirs, cushion pools and other structures below dams
to protect the stream beds against damaging erosion. In
1838 the phenomena of the hydraulic Jump was noticed and
study was made or the mathematical formulas which.would
gsvern it. It remained for the Miami Conservancy District
engineers in 1915 to make the first real study for the con-
trol of the hydraulic Jump and use of it to prevent stream
erosion. Since that time more interest has been aroused in
the subject and today the control of the standing wave or
jump below a epillway is considered to be one of the essen-
tial parts of the design of dams. The authors of this
thesis took up the study for two reasons. The first was to
improve their knowledge of the control of stream erosion
and the mathematics or hydraulic Jump. The second was to
build a model which could be used for this study and for
future study of dam construction and design in the
hydraulic laboratory of Michigan State College.

The first step was to build a model in the form of s
trough with glass sides in which.a model dam or weir could
bezmounted and the action of the water observed while pass-
ing over this dam. In addition a study was made of the
available data and experiments that had been made in regard

 

 

to stream erosion. It was interesting to note that one of
the first laws, The conservation of Momentum, is still used
and is one of the basic laws regarding hydraulic Jump. With
the aid of the model the results obtained were compared with
the existing laws and compared very closely with experiments
whieh had been conducted in years past along this line.

The Hydraulic Jump or Standing Wave gets its nae from
the wave which firms below a dam under certain conditions.
It remains at the same heigit and stationary. This forma-
tion translated from a German article is called a surface
roller. this is due to the rolling appearance of the stand-
ing wave. It is difficult to discuss this formation without
bringing in the mthenstieal discussion. The mthmtics of
the forsation will be discussed later. It has been noticed
thst as the water reaches the toe of a spillway it is
traveling at a high velocity. If this velocity is main-
tained ever the tee and out on the stream bed a large amount
of material would be carried away, resulting in time in the
undermining of the toe of the sea and the failure of the
structure.

is the water strikes the tee and the standing wave is
formed the kinetic energ of the water is taku up by the
impact with the wave. The velocity of the stream is
reduced and water flows away at the normal velocity of the
stream. In other words, the energy of the falling water is
dissipated by the impact with the water telow and is used

 

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in piling up the wave into a frothy, foamy fornnti on that
cuts down the velocity or absorbs it and dissipates it in
the form of heat.

One of the most effective ways of dissipating energy
is to introduce friction. In some structures a weir is
constructed below the toe to introduce a tumble bay or a
cushion pool so as to increase the resistance to flow. is
water is usually considered incompressible this body of
water offers considerable friction to the mass flowing into
it at a high velocity. The friction of the bottom and the
friction introduced by the use of water directly in its
path absorbs the velocity of the stream and causes the
water to pile up. he cushion pool in the model was intro-
dueed by raising the end gate (see drawing of model). With
this end gate down the high velocity of the stress continued
throughout the length of the Mel and no standing wave was
introduced. By raising and lowering the end gate the posi-
tion of the jump could be varied from the tee of the dam
out to the end gate. When gravel and sand were placed in
the stream the wave would form with the end gate down.
his would indicate that there must be friction in some
form or other in order to overcome the velocity of stream
enough to allow the water to pile up. Also a higher eleva-
tion of the and gate was needed to start the Jump than to
nintain it. After the Jump was once started the end gate

could be lowered considerably and the wave seemed to main-

tain itself.

 

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The time limit on this thesis doesnot allow much study
on erosion of the stream bed. A few runs were made. some
using fine sand for a bed and others using coarse gravel.
The main deduction from these was that the Jump itself
should occur on an apron of material that would not be
eroded by the action of water. Where the Jump occurred
below the apron it hollowed out the bed to a considerable
depth below the apron. This shows that the position of the
Jump is a necessary part of the design of a spillway. How-
ever. the uterial eroded by the Jump will be redepcsited
a short distance below the point of occurrence of the Jump.
It is therefore necessary to have the Jump occur as near
the toe of the dam as possible to have economical design
of structure. In the February 8, 1927 issue of Engineering
lewe Record, page 190 is a very good'article on the costly
erosion below the Wilson Dam of the Muscle Shoals proJect.
led the engineers in charge of this mode“ designed the
spillway such that the Jump had occurred on a concrete
apron the costly repairs to the structure could have been
eliminated. The only way in which the authors of this
discussion can recomend that the position of the Jump be
determined before the structure is built is by the use of
a model. This method has been used throughout Europe and
is being used more and more in the United States. By its

use the toe of the dam and the apron can be so arranged

that with the laws of similitude and the assumed flow the

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designer can be reasonably sure that costly erosion will
not occur. So far no one seems to have been able to place
the point of occurrence of the Jump in a mathematical
formula which will allow the designer to predator-ins the
standing wave and design his apron accordingly. The mathe-
matics of the standing wave or surface roller will now be

discussed.

 

 

 

 

“TREATICS OI NIP.

Unwins formula for conservation of momentum [see
Encyclopedia Brittanica, 9th Edition, Vol. 12, p. 499.]
[Belonger 1830].

D"¢¥W
TT’TL

d - depth above the Jump
D 't depth below the Jump
'1'. velocity above the Jump
g -. acceleration of gravity
1 Derivation of the above formula is as follows: (see

sketch)

      
  

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'1 '- velocity of stream entering hydraulic Jump
1) ' depth of stream leaving hydraulic Jump
73" Velocity of stream leaving hydraulic Jump ‘
11,- depth of same stream at critical flow
Y0. velocity at critical flow
. rim v5 - g D.
Q - quantity in second feet of flos per unit width of

stream.

Qe'ld'YgD'Yfig'W (13)

 

 

 

Haas of water flowing/sec. "w Q/g

' ‘w - unit wt. of water
Change of velocity if 71 - 72
change" of‘momentum/sec. - i9“ (VI-Y2)
Substituting _v+L -V vs

'_‘év_9._(71 - rid

- 33;; 3 (11-4)

Static pressure acting on the face (a -'b)

- wd2
.-g-

Static pressure acting on the face (e - f)

- :33

Hence:

D-d)‘ (Dz-d2)
#5 ‘ Ji-
Dividing both sides'by w(D - d)

lultiplying thru

1324.1» - 29,13
8

Substituting (Q - us)
I" + m - 1123.6.
‘ 8

7.

(1b)
(10)

(1d)
(1e)
(11')

(1)

(2)

(5)
(4)
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s

i/,* 3' T In ":3"—
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3: substituting + for 11 in (e)

2‘ z '
n dDd‘ s (it)
fi— .
en n+1 - ‘ - 3 (see la) (15)
1—24,..3...‘ ”0
dD D-l-d -. n93 (is)

Dividing thru by n.” and substituting a: and y (lb and lo)
for their equivalents

r: (x + y) - z (17)
mutation (1d) shows that tan d .- D. D also equals D,
Substituting: in (it)

Dannd-Dnl - ’95 (is)
2.25 £ 13.31) - n.” (19)

Substituting 3 - n.

133‘ d- n.’ - 13,3 (80)
D,3 e D‘,3 ‘1213

e no. equation (ld) is symmetrical in d and D. If d is
less than D.. then D must consequently be greater. If d
is greater than Dg ( impossible) than D would have to be

 

 

9.

less than D.. As there seems to be no physical phenomenon
that would cause a reversal of the Jump, the conclusion is
that the Jump can occur only when (d) is less than (D6) and
(D) is greater than (Do).

When the conditions are right, so that the Jump
occurs, it will always take place at the critical depth.

The action of the hydraulic Jump is a continuous
violent inelastic impact internally. by which the kinetic
energ of the swiftly moving stream entering the Jump is

converted into heat.

To plot the results of the experiment the formula
D " f E V 33 1' I: - d
17 '71—'- 1— r
is reduced to the following form
Dot J - D

hl - velocity head of stream entering Jump.

i'hen nt%_-‘ gym (at)
n3+w+_i_2_-u%§g_+_%§ (es)
Dividing thru by (dz)
hate-fists ‘2"
Substituting :-_ 3‘; and hl- 3&1

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LAWS Ol' HYDRAULIC EIHILITUDE

11.

The relnti on between the dimension: for etrnotnrea

in nature and for models or such structures follow:

‘Subceript n in used to denote model.

11 in need to denote scale ratio

 

amromt 3411931 on

Linen: L z :2 n : 1
Are; A : :2 I:2 x 1
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Weight I z 'n a: n5 z 1
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18.

In the construction of the model, the principle object
was to obtain a nodel which would be satisfactory in deter-
mining the reliability of the law of conservation of
mentun. lo effort was made to correlate the model to
existing conditions at some particular den site. The ogee
spillwey was made to scale, however, representing type I

of spillway shown on page 697 of the Engineering lows-Record,

of Kay 3, 1928. for all the purposes of this experinnt
no consideration of the laws of sinilitude was necessary.
If mrther experiments had bem mn, however, with the
purpose of determining the location md effect of the
hydraulic :up for some proposed dan, the node]. would have
been erected so as to closely resemble the conditions at
the partimlar site, and in order to correlate the results
from such a model to the dam, the laws of sinilitude would
be used in all cases.

The model as constructed is shown by the drawings,
Plates 1, 2, and 8. The inside width of the trough is
1.00 feet, the heigxt 2.00 feet, the overall length 5.00
feet. i'he sides and floor are of wood construction with
Joints waterproofed with asphalt paint. The Joints so
waterproofed were satisfactory during the period the
experiments were run. lo perceptible leakage occurred
mer the first day. The greatest difficulty encountered
was to obtain a watertight spillway. After the first day

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13.

the ogee spillway. made of white pine lumber and paraffined,
began to swell. The gasket used between the spillway and
the glass sides consisted of old rubber inner tubing. This
gasket did not effectively check water seeping thru from
the back of the dam. It did not compress sufficiently to
allow for the expat-ion of the den. The consequences of the
expansion of the dam was that the glass sides were put .
under too great a strain. One side which had been nrked
for cross-sectioning had several long cracks which followed
these markings. The glass sides, 1/4” x 2' x 8-1/2' plate
glass, were the most costly part of the model. It is to
be reeouended that no scratching or cutting of the surface
of the glass be done in cross-sectioning. The ogee spill-
way 1: made of wood should have sufficient clearance to
allow for swelling of wood. The wood. waterproofed with
paraffin, swelled considerably. Oreosote is recomended
for waterproofing a spillway built of wood. felt. impreg-
nated in oil is to be recommended for a gasket. rubber
does not compress sufficiently. The spillway should be
made of concrete, steel or some material which will not
expend when imaersed in water.

The coke baffle, (coke broken up into small sises)
worked very efficiently in quieting any turbulence above

the spillwq.
The results of this experiment are best shown by

fig. 1.

 

 

 

 

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14.

Bidone made a rectangular trough 1.066 feet wide an!
conducted a series of experiments to verin the law.
(hensactione. Royal society of Turin. 161’ pp. 22-80).
His greatest difficulty was in the magnitude of the Jump
occurring, it being too small to be or value in verifying
eny law. Following is a table of values from Bidone'e
experiments.

“BIB I
Bilone's Observations on Hydraulic Jump

 

Series Observeti one i D 1 hi 111
in series feet 22/445 feet a

1 4 0.154 4.47 0.431 2.30 0.510 2.02

11 5 .209 5.57 0.550 5.01 .481 2.30

111 5 .245 5.55 .749 5.03 .557 2.58

17 4 .150 4.57 .414 2.75 .524 2.15

Darcy and Basin made some observations with a timber
trough 6.63 feet wide. Their experiments were made in
1866. They are of no particular value in verifying the
law. Observations were made at the Lehigh University in
1894 by Robert Icrridey. Hie eXperinents were conducted
with a trough 0.66 foot wide. His results are plotted in
113. 1. His experiments verify the law quite closely.
(See Trsnesctions. Am. Soc. 0.3.. Vol. 80, p. 386.)

mm 11
let-riders Observations on the Hydreulio Jun).

Series Observation 6 D J h
in series feet ftjieo feet isit ['1'

I 3 0.050 2.18 0.143 2.88 0.074 1.48
11 8 .044 2.98 .150 8.41 .138 3.14

 

V.

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...-.-....

Esble II (Continued)

15.

 

Series Observation 4 71 D 6‘ hl h

in series fist ft/see feet feet I;
III 6 .066 6.66 .166 4.26 .196 6.46
17 6 .066 6.66 .168 6.10 .207 6.28
7 6 .096 4.69 .267 2.81 .298 6.14
71 4 .086 6.02 .286 6.44 .690 4.70
711 6 .071 6.02 .290 4.08 .690 6.60
11. 4 .066 6.60 .162 2.96 .190 6.46
x 4 .046 6.96 .176 6.80 .242 6.26
11 6 .042 4.06 .188 4.48 .666 6.08
111 4 .068 4.66 .206 6.40

Professor 1. 8. Gibson (see Proceeding Inst. 0.6..

701. 197, p. 266.

Also rrene. 1s. Soc. 0.]..‘701. 80,

p. 416), of Dundee Scotland conducted a series of experi-

nente in a trough three feet wide. His results check the

law very closely.

in Iabll III.

24826 III

Gibson's Observetions on the Hydrsnlic Julp

i'he results of his experiments are given

 

Series 4 7' D h h
1- feet ft[%ec #gbet J ggit; ‘1
.4 0.0766 4.60 0.266 6.61 0.288 6.92

.0761 6.82 .660 4.79 0.626 7.20
.0760 7.20 .466 6.26 0.806 11.06
.0729 8.66 .660 7.28 1.079 14.82
.0728 8.96 .670 7.86 1.246 17.11
.0760 9.74 .612 8.69 1.474 20.22
.0726 11.66 .760 10.44 2.110 29.00
.0727 16.12 .860 11.69 2.676 66.80
8 0.1466 6.46 0.267 1.82 0.186 1.26
.1696 6.68 .467 6.66 0.601 6.69
.1690 A 6.82 .687 4.22 0.726 6.20
61390 3039 e723 6.22 1.096 7.86‘
.1690 9.40 .808 6.81 1.672 9.88
.1690 10.66 .910 6.66 1.726 12.41

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16.

 

Series 4 7 D I h h
19.5 ft/iee :gg.t p.12 11

5 0.2075 4.45 0.419 2.02 0.505 1.47
.2070 4.55 .440 5.15 .515 1.54

.2045 5.54 .550 2.59 .445 2.15

.2045 5.09 .575 5.51 .575 2.52

.2040 5.55 . .575 5.52 .554 5.55

.2045 7.05 _ .554 5.54 .775 5.77

.2045 7.54 .755 5.59 .515 5.95

the researches of the Maui Conservancy District (see
The Hydraulic Juap as a leans of Dissipating Energy. by
Rose 1!. Diesel and John C. Deebe also. Theory of the
Hydraulic Jun .16 Backwater Curves. by Sherman l. Ioodward,
reehnieal reports. Part III. Dayton, Ohio, 1917).

run 17
he lie-i Conservancy Districts? Observations on the
Hydraulic Julp, 5.21.. 5 '

 

 

 

In 21.135 d 1 D .1 h h
0 e
n... W 15}: I“

100 5.5 0.52 5.50 0.84 2.52 0.675 2.11
101 11.0 .58 5.75 0.50 2.75 0.510 2.53
102 9.0 .34 5.15 0.75 5.04 0.585 2.44
111 14.2 .17 7.95 0.70 4.11 0.980 5.75
112 10.0 .24 7.57 0.85 5.54 0.890 5.71
115 7.5 .54 5.88 1.00 2.94 0.591 2.05
114 5.4 .59 7.27 1.05 2.72 0.818 2.10
119 4.5 .48 8.55 1.55 2.82 1.080 2.25
130 9.5 .51 7.75 1.05 5.59 0.952 5.01
121 15.5 .21 8.52 .87 4.14 1.125 5.55
122 12.5 .24 9.45 1.07 4.45 1.585 5.78
124 9.0 .52 _ 9.25 1.27 5.97 1.550 4.15
125 8.5 .58 10.10 1.58 5.55 1.580 4.18
129 5.7 .55 11.05 1.77 5.22 1.890 5.44
150 5.2 .42 10.85 1.59 5.78 1.820 4.54
151 8.8 .55 10.72 1.45 4.55 1.780 5.40
152 11.7 .27 10.28 1.50 4.81 1.550 5.07
155 11.8 .29 10.95 1.45 4.94 1.850 5.58
154 9.0 .55 11125 1.59 4.54 1.985 5.88
157 5.5 .40 12.50 1.70 4.25, 2.420 5.04

 

 

fable IV ( Continued

17.

 

Position

Inn of Jump. d 71 D J hl h

r..t. r..t :51... 1..9 15.5 11
15s 4.5 0.49 12.55 1.90 5.55 5.550 5.20
159 2.0 .55 10.57 2.07 2.44 1.550 1.95
142 2.0 .57 11.15 2.24 2.55 1.950 2.25
145 4.5 .52 15.52 2.07 5.95 2.575 5.54
144 5.5 .55 12.55 1.74 4.55 2.550 7.12
145 11.0 .54 11.12 1.54 4.55 1.920 5.55
145 15.5 .50 11.75 1.54 5.47 2.150 7.17
147 9.5 .55 14.50 1.55 5.55 5.250 9.55
145 5.0 .54 15.70 2.10 5.59 5.910 5.59
149 5.5 .54 15.52 2.25 5.52 2.950 4.55
150 5.0 .59 15.77 2.52 5.55 2.940 4.25

The plotted results of the different experilents will
be found in Pig. 1.

The experiments were run on the model in the follow-
ing manner. from Series A to JP (see graphs) a smooth
apron was run from the toe to the and gate. The coefficient

of friction of this apron was very low. The wave was con-

trolled and mintained by raising or lowering the and gate.
In all of the observations taken the wave was maintained
in a stable for: directly at the point where the toe Joined

the apron. Once it was spotted at this point it would

maintain itself for that flow.
decreasing the flow the position of the wave would immedi-

But upon increasing or

ately change. The quantity of water flowing was found by

weighing the water and timing the flow for a quantity of

100 pounds. he area of the water at D and d was found

by measuring with a thin rule calibrated to .01 feet.

Several trials were taken on each series in order to insure

..___

 

 

 

-..

18 e

a mean value ac near correct‘ae poeeible. The drawings on
the eeriee of rune are calf-explanatory. i'he aurfaoe of
water waa plotted over the entire epillway, and the poeition
of D or “can depth relative to the toe in ehown on each
graph. following ie a eample computation:

Series A
71 - _g_ ' 100 - 2.73' per sec.

1 e
hl - v; - [2.1322 9' .115

1" D ' .0895 ‘5.97
T 613' '

The result: obtained from the eeriee of rune do not check
exactly with the theoretical formula. l‘hie would aeem to
indicate that an emperical value ehould be given thia
machine. Other experiments which have been carried on in
the pact seemed to bear thie up. Whether or not a large
etructure euch ac a dam would require thia empirical coeffi-
eient it ie not known. If euch a coefficient in needed it
might be found that each change in ehape of epillway would
effect the valuee obtained in a different dunner. It wae
noted that a very small change in head had a very marked
effect upon the standing wave. In come caaea a alight
increase would cauae the jump to diaappear ntirely‘. In
other canoe a slight decrease would cauae the jump to
extend up the face of the epillwag. It wae deducted frcn
thin that the coefficient of friction of the bed waa eo

 

 

1’.

low that any slight change in the velocity would overcome
the friction or be overcome by it so that the character
of the hydraulic jump was changed very.much.

When the sand was substituted for the smooth stream
bed the jump was much more stable. d.grsater change in
head was necessary to cause it to vary. The jump occurred
after the stream came into contact with the sand bed.
This caused the wave to hollow out the bed into an oval
shaped pool. After this pool had been formed and the flow
continued there was very little change in the shape of the
bottom. .Alcng‘the‘bottom there was noticed considerable
backrcll which.tendcd to carry the sediment upstream in?
stead of down. Only the larger particles were left in the
pool the snller ones having been carried down onto a bar
directly below the jump. This showed that an.artificial
pool formed by placing a sill across the apron would prob-
ably solve the problem of locating the jump. This has been
done on.various dams and works very well. The position
and size of the sill has to be designed very carefully to
insure proper'results. A description of the dentated sill
is given in freeman's Hydraulic Laboratory Practice on
page 18‘. With.the gravel used as a bed the jump was much
more pronounced. This condition was very similar to rip
rep or broken rock used below a spillway. The sand and
gravel bed observation bears up the statement previously
made that friction plays a large part in the position and

stability of the jump and is a necessary part of its formation.

 

20.

001013810]

i'he results of the experiments run in this thesis
were very satisfactory. A more accurate method of obtain-
ing the quantities and uasuring the depths is .to be
recs-mded.

the hydrulic jump my be used as a means for destroy-
ing the energy of the discharge of a spillway and thereby
preventing stream bed erosion.

A rough surface below the spillway which will not
produce vibration is to be recommended. rhis will prevent
the usual wide variation in the jump under conditions of
slight friction or a smooth apron.

{the quantities assured are very sun. A slight
difference in the value of one measurement will cause a
large variation in the calculations based upon these
measurements.

as length of the juap is approximately 4 to 5 times
the depth of the streamleaving the qup.

s: drOppina a colored nus in the stream above the
spillway, the diminishing velocity of the stream W-
ont the jump was observed and traced.

fhe model as now built and with a few minor correc-
tions may be mad for further interesting study of stream

bed erosion, spillway discharge, and other hydraulic

studies .

 

21.

3131-1061“!!! .

I'heory of the Hydraulic Jump and Backwater Curves and the

Hydraulic Jump as a leans of Dissipating Energy. i'he
liami Conservancy District Technical Reports. Part III.

lygzaulic Laboratory Practice, freeman, p. lot, 494, I“,

 
   
     

e P0
of Pit River Due.
Vol. 80, 1916, p. 356, rhe Hydraulic Jump in Open
Channel flow at High Velocity.

eri laws-Record
c . , . p. 242, Planning a Huge Diversion

Fair 30 c7 Oarre illway.

vol. 105, s 14 29, p. 2 4, Progress of Tests and Design
of Bonnet Carre Spillway.

vol. 101. c/z/zc. p. 17s. Prevention of Scour Below
Overfall .

vol. 100. 5/5 :8. p. soc. Studies en Prevention of Scour
Below Overiall Dams.

Vol. 100, 5/1 S, p. 369. Ice lrosion Below Dams.

Vol. 99. 12/15 2‘1, p. 974. the fission Below Overfall

Me

vel. 93, 2/3/27. p. 190, Concentrated rm Erodes Beck
Below Wi so Dem.

701. 91, l l 726, p. 000, Do Baffle I'iers Dispel Inergy?

Vol. 9‘, 7 23 25. p. 154, Preventing Undercutting of Dams.

vol. 94, 5/27/25, p. ace, Preventing Dam Undercutting.

vei. n, 5/7 25, p. vac, Scour Below Dam Prevented by
Special 'l‘oe Construction.

701. 94, 6/4/25. p. 945, Ogec Section Defendsd.

vol. 94. 4/9/25, p. 597. High Velocity Discharge of
Overlal one and Forms of Spillway Profile.

Vol. 88, d 4 22, p. MS. Early Report on the Hydralic

Jump.

Vol. SS, a/i/zs, p. 928, Early Studies of the Hydrano
Jump.

vei. as, elm/so, p. s55. rieeae Iash Out pm of Apron
of Granite eef Dem.

101. 81, 7/25 18. p. 186. Repair Washout Under Dam by
Sheet Pile Cutoff Embedded in Concrete.

nscin New .
*eI. 73 ‘72 5516. p. 974, hilure of Diversion lair on

Salt'River Project.

 

88.

Bibliography ( Continued)

p. 109, Slope of’Dam Apron Changed to
Prevent Backlash.

 

  

ugus , an 24, 1928, The Standing Wave.

Transactions A. 8.1!. E.
Se eptembcr- 8 Vol. 60, No. 36. Computation

of the Tailwater Depth of the Hydraulic Jump in
sloping Plumes.

 

 

 

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