Water-Supply and Irrigation Paper No. 150 Series M, General Eydrographie Investigations, 16 

DEPARTMENT OF THE INTERIOR

UNITED STATES GEOLOGICAL SURVEY

CHARLES D. WALCOTT, DIRECTOR

WEIR EXPERIMENTS, COEFFICIENTS, 

MD FORMULAS

BY

ROBERT E. HORTOTST

WASHINGTON

GOVERNMENT PRINTING OFFICE

1906
Water-Supply and Irrigation Paper No. 150 Series M, General Hydrographic Investigations, 16

DEPARTMENT OF THE INTEEIOE 

UNITED STATES GEOLOGICAL SURVEY

CHARLES D. WALCOTT, DIRECTOK

WEIR EXPERIMENTS, COEFFICIENTS, 

AND FORMULAS

BY

ROBERT E. HORTO1NT

WASHINGTON

GOVERNMENT PRINTING OFFICE

1906
CONTENTS AND ILLUSTRATIONS.

Page.

Introduction.............................................................. 7

Definitions of terms ................................................... 7

Notation............................................................. 8

Base formulas........................................................ 9

Equivalent coefficients..........................'...................... 9

Approximate relative discharge over weirs............................. 9

Eeferences........................................................... 10

Theory of weir measurements ............................................. 10

Development of the weir ............................................. fc 10

Theorem of Torricelli................................................. 10

Elementary deduction of the weir formula ............................. 11

Application of the parabolic law of velocity to weirs ................ 12

General formula for weirs and orifices ................................. 12

Vertical contraction .................................................. 13

Velocity of approach ..................................................... 14

Theoretical formulas.................................................. 14

Distribution of velocity in channel of approach......................... 16

Distribution of energy in channel of approach.......................... 17

The thin-edged weir...................................................... 20

Earlier experiments and formulas .............^....................... 20

Castel...........................................:............... 20

Poncelet and Lesbros............................................. 21

Boileau............... .......................................... 21

East Indian engineers' formula.................................... 22

Experiments and formula of James B. Francis ......................... 23

Experiments and formula of Fteley and Stearns........................ 26

Bazin's experiments..-.-.......................'...................... 29

Bazin's formulas for thin-edged weirs. ^................................ 31

Derived formulas for thin-edged rectangular weirs...................... 34

Fteley and Steams-Francis formula................................ 34

Hamilton Smith's formula ........................................ 34

Smith-Francis formula............................................ 37

Parmley's formula ............................................... 37

Extension of the weir formulas to higher heads ........................ 39

Comparison of weir formulas........................................... 40

Comparison of various velocity of approach corrections............. 40

End contractions incomplete contraction ............................. 44

Compound weir..................... ................................ 46

Triangular weir...................................................... 46

General formula ................................................. 46

Thomson's experiments .......................................... 46

3
4 CONTENTS.

The thin-edged weir Continued. Page.

Trapezoidal weir ..................................................... 47

The Cippoletti trapezoidal weir ................................... 47

Cippoletti's formula .............................................. 48

Requirements and accuracy of weir gagings ................................ 49

Precautions for standard weir gaging.__......__._...__-__-..._.._._.... 49

Plank and beam weirs of sensible crest width ..-......-._.._.-..--.__.. 52

Eeduction of the mean of several observations of head.................. . 52

Effect of error in determining the head on weirs.--......-...-..-....... 53

Error of the mean where the head varies ...-..--.......-..-......-..-. 54

Weir not level ........................................................ 57

Convexity of water surface in leading channel........-.-.--,....-..-.... 58

Results of experiments on various forms of weir cross sections...--....--.... 59

The use of weirs of irregular section ................................... 59

Modifications of the nappe form .....-..-..-..-..--..-..--..-....-..... 60

Experimental data for weirs of irregular cross section .-...._............ 61

Base formula for discharge over weirs of irregular cross section ...... 62

Bazin's experiments on weirs of irregular cross section.............. 63

Bazin's correction for velocity of approach ..................... 63

Recomputation of coefficients in Bazin's experiments........... 66

Cornell University hydraulic laboratory ........................... 85

Experiments of United States Board of Engineers on Deep Waterways. 86 

Experiments at Cornell University hydraulic laboratory on models of

old Croton dam ..-..........-.-.-.--..-.-.--.....--...-...--... 90

Experiments of United States Geological Survey at Cornell University

hydraulic laboratory......-........-.-..-....--.--..-........... 95

Experiments on model of Merrimac River dam at Lawrence, Mass... 107

Flow over weirs with broad crests .....--...--.........--..--.......... 110

Theoretical formula of Unwin and Frizell.......................... 110

Blackwell's experiments on discharge over broad-crested weirs...... 112

East Indian engineers' formula for broad-crested weirs.. -...-....-.. 114

Fteley and Stearns experiments on broad- crested weirs.............. 116

Bazin's formula and experiments on broad-crested weirs ............ 117

Experiments of the United States Geological Survey on broad-crested

weirs..-...-...--.----.-...-.-...-..-.....--...-..-........-... 119

Table of discharge over broad-crested weirs with stable nappe....... 121

Effect of rounding upstream crest edge...........-.----...--..--....-.. 122

Experiments on weirs with downstream slope or apron of varying inclination ............................................................... 124

Triangular weirs with vertical upstream face and sloping aprons..... 124

Triangular weirs with upstream batter 1:1 and varying slope of apron. 126 

Experiments on weirs of trapezoidal section with upstream slope of

i:l, horizontal crest, and varying downstream slopes.. ............ 127

Combination of coefficients for weirs with compound slopes ............. 127

Weirs with varying slope of upstream face ........---..-........-.-.... 128

Dams of ogee cross section, Plattsburg-Chambly type ....-.----..--.--.- 130

Experiments on discharge over actual dams...........-..-.-.---.---..- 131

Blackstone River at Albion, Mass ................................. 132

Muskingum River, Ohio ........-.........."...--.......-.--....,.. 132

Ottawa River dam, Canada........................................ 132

Austin, Tex., dam..-...-..--.-......-..-.....-.....- --........--- 133

Roughness of crest............. ...................................... 133

Falls ................................................................ 135

Weir curved in plan.................--...-.....-..-...--..-.---.--.-. 136
CONTENTS AND ILLUSTKATIONS. 5

Page.

Submerged weirs......................................................... 137

Theoretical formula ..-......_._........._..---...---......--.--..---- 137

Fteley and Stearns submerged-weir formula.........._..--........ ---. 138

Clemens Herschel's formula .......................................... 139

The Chanoine and Mary formula.-............-...........---.---.-..- 140

R. H. Rhind's formula ............................................... 141

Bazin's formulas. .__.._......__....._.._......_._.....-...--.---.---- 141

Increase of head by submerged weirs .................................. 142

Eankine's formulas............-............-......-...-..---.---. 142

Colonel Dyas's formula ........................................... 143

Submerged weirs of irregular section .................................. 143

Bazin's experiments. .............. .............................. 143

Data concerning East Indian weirs .................--..---.-.---.. 144

United States Deep Waterways experiments ................--.---. 146

Weir discharge under varying head ............^ .......................... 146

Prismatic reservoir, no inflow.....-......-...--.-----...--.-.-.------- 147

Approximate time of lowering prismatic or nonprismatic reservoir....... 147

Reservoir prismatic, with uniform inflow ........-...--.------..-..---- 148

General formulas................................................. 148

Formulas for time of rise to any head H, prismatic reservoir with uniform inflow..-..--..-.----..-----.---.------..----.----.------- 149

Nonprismatic reservoir, uniform inflow......................---.-.---- 153

Variable inflow, nonprismatic reservoir................................ 154

Tables for calculations of weir discharge ....-......-..........-.---.---.... 156

Table 1. Head due to various velocities..........-....-..---...--..-.-. 157

Table 2. Percentage increase in discharge by, various rates of velocity of

approach.........._..._........._..__.._............-..-...---..-. 159

Tables 3, 4. Discharge over a thin-edged weir by the Francis formula.... 162

Tables 5, 6. Three-halves powers...................................... 171

Table 7. Flow over broad-crest weirs with stable nappe................. 177

Table 8. Backwater caused by a dam or weir..._....................... 180

Index..._............................................................. 187

PLATE I. Bazin's coefficients..................................... 32

II. Effect of errors in weir experiments ..................... 54

III. Modifications of nappe form ..--.-........-.......-..--. 60

IV-XII. Bazin's experiments-.....-.....----..---...-.---...---. 66

XIII, XIV. Cornell hydraulic laboratory experiments..--.........-.. 86

XV-X VIII. United States Deep Waterways experiments ............. 90

XIX-XXII. Croton dam experiments................................ 94

XXIII-XXXII. United States Geological Survey experiments ............ 106

XXXIII. Merrimac River dam experiments..--.---.--........---. 108

XXXIV-XXXV. Cross sections of ogee dams ............................. 130

XXXVI. Coefficient diagram for ogee dams ....................... 130

XXXVII. Experiments on actual dams............................ 132

XXXVIII. Diagram of variable discharge........................... 150

FIG. 1. Torricellian theorem applied to a weir ............................. 11

2. Rectangular orifice ............................................... 12

3. Distribution of velocities .......................................... 16

4. Triangular weir ............-...........-......................_.. 46

5. Trapezoidal weir ................................................. 47
ILLUSTRATIONS.

FIG. 6. Sections of the Francis weir ____.._..:.._....-................-.-.- 51

7. Inclined weir .................................................... 57

8. Broad-crested weir................................................ 110

9. Coefficient curve for triangular weirs............................... 125

10. Fall ............................................................. 135

11. Weir curved or angular in plan.................................... 136

12. Submerged weir ..................................!...-........... 137

13. East Indian weir section .......................................... 145

14. East Indian weir section .......................................... 145

15. Concave backwater surface ........................................ 180

16. Convex backwater surface ..-........---.----.-..--.---.---.....-. 181
WEIR EXPERIMENTS, COEFFICIENTS, AND 

FORMULAS.

By EGBERT E. HORTON.

INTRODUCTION.

DEFINITIONS OF TERMS.

The word '' weir" will be used to describe any structure used to 

determine the volume of flow of water from measurements of its 

depth on a crest or sill of known length and form. In this general 

sense timber and masonry dams having various shapes of section, 

reservoir overflows, and the like ma}7 be weirs. Terms, more or less 

synonymous, used to describe such weirs are "comb," "wasteway," 

" spillway," "overwash," "rollway," and "overfall."

The French term "nappe," suggesting the curved surface of a cloth 

hanging over the edge of a table, has been fittingly used to designate 

the overfalling sheet of water.

The expression "wetted underneath" has been used to describe the 

condition of the nappe designated by Bazin as "noye"es en dessous," 

signifying that the water level between the nappe and the toe of the 

weir is raised by vacuum above the general water level below the 

weir.

"Thin-edged weir "and "sharp-crested weir" are used to designate 

a weir in which the nappe, or overfalling sheet, touches only the 

smooth, sharp upstream corner or edge of the crest, the thickness of 

which is probably immaterial so long as this condition is fulfilled.

A "suppressed weir" has a channel of approach whose width is the 

length of the weir crest.

A "contracted weir" has a crest length that is less than the width 

of the channel of approach.

The term "channel of approach," or "leading channel," defines the 

body of water immediately upstream from the weir, in which is 

located the gage by which the depth of overflow is measured.

"Section of approach" may refer to the cross section of the leading 

channel, if the depth and width of the leading channel are uniform; 

otherwise it will, in general, apply to the cross section of the channel 

of approach in which the gage is located.

7
8 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

" Weir section 11 refers to the cross section of the overflowing stream 

in the plane of the weir crest.

"Crest contraction" refers to the diminished cross section of the 

overflowing stream resulting from the upward curvature of the lower 

water filaments in passing the crest edge. It does not include the 

downward curvature of the water surface near the weir crest.

The u vertical contraction of the nappe' 1 includes both the crest 

contraction and the surface contraction.

"Incomplete contraction 11 may take place either at the crest or at 

the ends of a weir, and will occur when the bottom or side walls of 

the channel of approach are ^o near the weir as to prevent the complete curvature of the water filaments as they pass the contracting 

edge.

Dimensions are uniformly expressed in feet and decimals, velocities 

in feet per second, and quantities of flow in cubic feet per second, 

unless otherwise stated in the text.

In the preparation of this paper much computation has been involved 

and it is expected that errors will appear, which, if attention is called 

to them, may be corrected in the future. Information concerning 

such errors will be gratefully received.

NOTATION.

The symbols given below are used in the values indicated. The 

meaning of additional symbols as used and special uses of those that 

follow are given in the text:

D=Measured or actual depth on the crest of weir, usually determined as the difference of elevation of the weir crest and the water level, taken at a point 

sufficiently far upstream from the weir to avoid the surface curve.

H=The head corrected for the effect of velocity of approach, or the observed head 

where there is no velocity of approach. As will be explained, D is applied 

in formulas like Bazin's, in which the correction for velocity of approach is 

included in the coefficient. H is applied in formulas where it is eliminated. 

<z>=Mean velocity of approach in the leading channel, usually taken in a cross section opposite which D is determined.

h=Velocity head=a^-.

<;=Acceleration by gravity. Value here used 32.16. 

P= Height of weir crest above bottom of channel of approach, where channel is

rectangular.

W= Width of channel of approach where D is measured. 

A=Area of cross section of channel of approach.

(r=Area of channel section where D is measured, per unit length of crest. 

a Area of weir section of discharge=D L. 

L Actual length of weir crest for a suppressed weir, or length corrected for end

contractions, if any.

I/=Actual length of crest of a weir with end contractions. 

N= Number of complete end contractions. 

£=Breadth of crest of a broad-crested weir. 

S=Batter or slope of crest, feet horizontal to one vertical.
INTRODUCTION. 9

d= Depth of crest submergence in a drowned or submerged weir. 

Q= Volume of discharge per unit of time. 

C, M, m, //, a, f, etc., empirical coefficients.

BASE FORMULAS.

The following formulas have been adopted by the engineers named:

2 __ 

(?= MLH^gH. Hamilton Smith (theoretical).

Bazin, with no velocity of approach.

Bazin, with velocity of approach. 

Francis a (used here). 

= CLE^+fL. Fteley arid Stearns.

EQUIVALENT COEFFICIENTS.

The relations between the several coefficients, so far as they can be 

given here, are as follows:

yw= g M.

M is a direct measure of the relation of the actual to the theoretical weir discharge.

8.02 =5.35 M

"=>"= r0- 18706-

APPROXIMATE RELATIVE DISCHARGE OVER WEIRS.

For a thin-edged weir, the coefficient C in the Francis formula is 

3.33=- . Let C' be the coefficient for any other weir, and x the 

relative discharge as compared with the thin-edged weir, then

 =»£ ..

or, as a percentage,

^ = 100 0=30

.... (1)

«The coefficient C of Francis includes all the constant or empirical factors appearing in the 

formula, which is thus thrown into the simplest form for computation.
10 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

This expression will be found convenient in comparing the effect on 

discharge of various modifications of the weir cross section. For a 

broad-crested weir with stable nappe, ^=2.64, see p. 121. The discharge over such a weir is thus seen to be 79.2 per cent of that for a 

thin-edged weir by the Francis formula.

REFERENCES.

The following authorities are referred to by page wherever cited 

in the text:

BAZIN, H., Recent experiments on flow of water over weirs. Translated by Arthur 

Marichal and J. C. Trautwine, jr. Proc. Engineers' Club Philadelphia, vol. 7, 

No. 5, January, 1890, pp. 259-310; vol. 9, No. 3, July, 1892. pp. 231-244; No. 4, 

October, 1892, pp. 287-319; vol. 10, No. 2, April, 1893, pp. 121-164.

BAZIN, H., Experiences nouvelles sur 1'ecoulement en deversoir, 6me art., Annales 

des Fonts et Chaussees, Memoires et Documents, 1898, 2me trimestre, pp. 121-264. 

This paper gives the results of experiments on weirs of irregular section. 

Bazin's earlier papers, published in Annales des Ponts et Chaussees, 1888, 1890, 

1891, 1894, and 1896, giving results of experiments chiefly relating to thin-edged 

weirs and velocity of approach, have been translated by Marichal and Trautwine.

BELLASIS, E. S., Hydraulics.

BOVEY, H. T., Hydraulics.

FRANCIS, JAMES B., Lowell hydraulic experiments.

FEIZELL, JAMES P;, Water power.

FTELEY, A., and STEARNS, F. P., Experiments on the flow of water, etc. Trans. Am. 

Soc. Civil Engineers, January, February, March, 1883, vol. 12, pp. 1-118.

MEEEIMAN, MANSFIELD, Hydraulics.

RAFTER, GEORGE W., On the flow of water over dams. Trans. Am. Soc. Civil Engineers, vol. 44, pp. 220-398, including discussion.

SMITH, HAMILTON, Hydraulics.

THEORY OF WEIR MEASUREMENTS.

DEVELOPMENT OF THE WEIR.

The weir as applied to stream gaging is a special adaptation of mill 

dam, to which the term weir, meaning a hindrance or obstruction, has 

been applied from early times. The knowledge of a definite relation 

between the length and depth of overflow and the quantity also probably antedates considerably the scientific determination of the relation 

between these elements.

In theory a weir or notch a is closely related to the orifice; in fact, 

an orifice becomes a notch when the water level falls below its upper 

boundary.

THEOREM OF TORRICELLI.

The theorem of Torricelli, enunciated in his De Motu Gravium 

Naturaliter Accelerate, 1643, states that the velocity of a fluid passing 

through an orifice in the side of a reservoir is the same as that which 

would he acquired hy a heavy hody falling freely through the vertical

a Commonly applied to a deep, narrow weir.
THEOEY OF WEIR MEASUREMENTS. 11

height measured from the surface of the fluid in the reservoir to the 

center of the orifice.

This theorem forms the basis of hydrokinetics and renders the weir 

and orifice applicable to stream measurement. The truth of this proposition was confirmed by the experiments of Mariotte, published in 

1685. It can also be demonstrated from the laws of dynamics and the 

principles of energy. a

ELEMENTARY DEDUCTION OF THE WEIR FORMULA.

Iii deducing- a theoretical expression for flow over a weir it is 

assumed that each filament or horizontal lamina of the nappe is actuated by gravity acting- through the head above it as if it were flowing 

through an independent orifice. In fig. 1 the head on the successive 

orifices being H^ H^ H^ etc., and their respective areas A^ Az, A3 , 

etc., the total discharge would be

. . . . (2)

FIG. 1. Torricellian theorem, applied to a weir.

If the small orifices A be considered as successive increments of head 

Jf, the weir formula may be derived by the summation of the quantities 

in parentheses. H comprises n elementary strips, the breadth of each

77" 77^ 2 77"

is . The heads on successive strips are , , etc., and the total

n ^ ft- n ' '

becomes

y l2H~, IZH , A ^+V^r+V^r+J ....

T ff where =A-\-Al , etc., for a rectangular weir. The sum of the

LH , IE 2 n

seres

Hence the discharge is

The above summation is more readily accomplished by calculus.

a See Wood, Elementary Mechanics, p. 167, also p. 291.
12 WEIE EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

APPLICATION OF THE PARABOLIC LAW OF VELOCITY TO WEIRS.

The following elementary demonstration clearly illustrates the character of the weir :

r. According to Torricelli's theorem (see fig. 1), the velocity (v) of a

filament at any depth (so) below surface will be v This is the

equation of a parabola having its axis OX vertical and its origin 0 

at water surface. Replacing the series of jets by a weir with crest at 

.X, the mean velocity of all the filaments will be the average ordinate 

of the parabola OPQ. The average ordinate is the area divided by 

the height, but the area of a parabola is two-thirds that of the circumscribed rectangle; hence the mean velocity of flow through the weir 

is two-thirds the velocity at the crest, i. e., two-thirds the velocity 

due to the total head If on the crest. The discharge for unit length 

of crest is the head H, or area of opening per unit length, multiplied 

by the mean velocity. This quantity also represents the area of the 

parabolic velocity curve OPQX. The mean velocity of flow in the 

nappe occurs, theoretically, at two-thirds the depth on the crest.

The modification of the theoretical discharge by velocity of approach, 

the surface curve, the vertical contraction at the crest, and the various 

forms that the nappe may assume under different conditions of aeration, form of weir section, and head control the practical utility of the 

weir as a device for gaging streams.

GENERAL FORMULA FOR WEIRS AND ORIFICES.a

Consider first a rectangular opening in the side of a retaining vessel. 

The velocity Of flow through an elementary layer whose area is Ldy 

will be from Torricelli's theorem:

FIG. 2. Rectangular orifice.

The discharge through the entire opening will be, per unit of time, 

neglecting contractions,

Q-- Jfyy.Ldy. ...... (4)

"The correlation of the weir and orifice has been given by Merriman. See Hydraulics, pp. 42-4S.
VERTICAL CONTRACTION. 13

This is a general equation for the flow through any weir or oriiice, 

rectangular or otherwise, Q being expressed as a function of y. In 

the present instance L is constant. Integrating,

. ..... (5)

For a weir or notch, the upper edge will be at surface, H^ O^ and 

calling Hz = H\n equation (5),

....... (6)

In the common formula for orifices, only the head on the center of 

gravity of the opening is considered.

Expressing H^ and H^ in terms of the depth H on the center of 

gravity of the opening and the height of opening d, Merriman obtains, 

after substituting these values in and expanding equation (5) by the 

binomial theorem, the equivalent formula,

. . (7)

The sum of the infinite series in brackets expresses the error of the 

ordinary formula for orifices as given by the remainder of the equation. This error varies from 1.1 per cent when h=d to 0.1 per cent 

when h=3d.

VERTICAL CONTRACTION.

Practical weir formulas differ from the theoretical formula (6) in 

that velocity of approach must be considered and the discharge must 

be modified by a contraction coefficient to allow for diminished section of the nappe as it passes over the crest lip. Velocity of approach 

is considered on pages 14 to 20. Experiments to determine the weir 

coefficient occupy most of the remainder of the paper. The nature of 

the contraction coefficient is here described.

Vertical contraction expresses the relation of the thickness of nappe, 

s, in the plane of the weir crest, to the depth on the crest, H. If the 

ratio x II were unity, the discharge would conform closely with the 

expression

The usual coefficient in the weir forijiula expresses nearly the ratio 

slH.

The vertical contraction comprises two factors, the surface curve or 

depression of the surface of the nappe and the contraction of the 

under surface of the nappe at the crest edge. The latter factor in
14 WELR EXPERIMENTS, COEFFICIENTS, AND FOEMULAS.

particular will vary with form of the weir cross section, and in general variation in the vertical contraction is the principal source of 

variation in the discharge coefficient for various forms of weirs.

The usual base weir formula, $=2/3 LHvQgH, is elsewhere given 

for an orifice in which the upper edge is a free surface. If instead 

the depth on the upper edge of the orifice is d, the surface contraction, 

there results the formula

(8)

This is considered as the true weir formula by Merriman. a In this 

formula only the crest-lip contraction modifies the discharge, necessitating the introduction of the coeificient. The practical difficulties of 

measuring d prevent the use of this as a working formula.

Similarly a formula may be derived in which only the effective 

cross section s is considered, but even this will require some correction 

of the velocity. Such formulas are complicated by the variation of s 

and d with velocity of approach. 6 Hence, practical considerations 

included, it has commonly been preferred to adopt the convenient

base formula for weirs, Q=^ MLH^lgH, or an equivalent, and throw 

all the burden of corrections for contraction into the coefficient M.

VELOCITY OF APPROACH.

THEORETICAL FORMULAS.

Before considering the various practical weir formulas in use some 

general considerations regarding velocity of approach and its effect on 

the head and discharge may be presented.

In the general formula (4) for the efflux of water when the water 

approaches the orifice or notch with a velocity -*>, then with free discharge, writing D-\-h in place of H, for a rectangular orifice, we have

(9)

D1 and D% being the measured depth on upper and lower edges of the 

orifice, and ^=^-, the velocity head. 

To assume that Z>+A equals H is to assume that the water level is

a Hydraulics, p. 123.

&See Trautwine and Mariclial's translation of Bazin's Experiments, pp. 231-307, where may also be 

found other data, including a resume of M. Boussinesq's elaborate studies of the vertical contraction of the nappe, which appeared in Comptes Rendus de 1'Academie des Sciences for October 24,1887.
VELOCITY OF APPROACH. 15

increased by the amount A, or, as is often stated, that If is "measured 

to the surface of still water." This is not strictly correct, however, because of friction and unequal velocities, which tend to make 

H D>h, as explained below. 

For a weir, Dj equals zero; integrating,

O __ 3

Since Q = iL^gH^, we have

(da)

This is the velocity correction formula used by James B. Francis." 

Since h appears in both the superior and inferior limits of integration, it is evident that h increases the velocity only, and not the section of discharge. The criticism is sometimes made that Francis's 

equation has the form of an increase of the height of the section of 

discharge as well as the velocity.

The second general method of correcting- for velocity of approach 

consists of adding directly to the measured head some function of the 

velocity head, making

ff=D+ah 

in the formula

or

Q=CL

This is the method employed by Boileau, Fteley and Stearns, and 

Bazin. No attempt is made to follow theory, but an empirical correction is applied, affecting both the velocity and area of section.

By either method v must be determined by successive approximations unless it has been directly measured.

Boileau and Bazin modify (95) so as to include the area of section of 

channel of approach, and since the velocity of approach equals Q/A, 

a separate determination of v is unnecessary. Bazin also combines 

the factor for velocity of approach with the weir coefficient.

The various modifications of the velocity correction formulas are 

given in conjunction with the weir formulas of the several experimenters.

oBovey gives similar proof of this formula for the additional cases of (1) an orifice with free discharge, (2) a submerged orifice, (3) a partially submerged orifice or drowned weir, thus establishing 

its generality.
16 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

DISTRIBUTION OF VELOCITY IN CHANNEL OF APPROACH.

The discharge over a weir takes place by virtue of the potential 

energy of the layer of water lying above the level of the weir crest, 

which is rendered kinetic by the act of falling over the weir. If the 

water approaches the weir with an initial velocity, it is evident that 

some part of the concurrent energy will facilitate the discharge.

The theoretical correction formulas may not truly represent the 

effect of velocity of approach for various reasons:

1. The fall in the leading channel adjacent to the measuring section 

is the source of the velocity of approach, and this fall will always be 

greater than that required to produce the existing velocities, because 

some fall will be utilized in overcoming friction.

2. The velocity is seldom uniform at all parts of the leading channel and the energy of the water varies accordingly. This effect is 

discussed later (p. 17).

3. It is not certain just what portion of the energy of the water in 

the section of the leading channel goes to increase the discharge.

FIG. 3. Distribution of velocities.

In general the threads of the water in the cross section of the channel of approach to a weir have varying velocities. It follows that, as 

will be shown, the ratio of the actual energy of the approaching water 

to the energj^ due to the mean velocity will be greater than unity, and 

for this reason the correction for velocity of approach will be greater 

than if the energy were that due to a fall through a head produced by 

the mean velocity v. The more nearly uniform is the velocity of the 

water in the leading channel the smaller will be the necessary coefficient a in the velocity head formula. The velocity may be rendered 

very nearly uniform by the use of stilling racks or baffles. Where 

this was done in the experiments on which a formula was based (that 

of Francis, for example) a larger velocity of approach correction than 

that obtained by the author may be necessary in applying the formula 

to cases where there is wide variation in the velocity in the leading 

channel. To avoid such a contingency it is desirable, when practicable, to measure head to surface of still water, because more accurate 

results can be obtained and wash against instruments prevented.
VELOCITY OF APPROACH. 17

The vertical and horizontal velocity curves in an open channel usually closely resemble parabolas. A weir interposes an obstruction in 

the lower part of the channel, checking- the bottom velocities. The 

velocity is not, however, confined to the filaments in line with the section of the discharge opening of the weir. As a. result of viscosity of 

the liquid, the upper rapidly moving layers drag the filaments underneath, and the velocity may extend nearly or quite to the channel bottom. There will usually, however, be a line (A B C, fig. 3), rising 

as the weir is approached, below which there is no forward velocity.

The line A B C is the envelope of the curves of vertical velocity 

in the channel of approach.

There will be a similar area of low velocity at each side of the channel for a contracted weir. The inequality of velocities for such weirs 

being usually greater than for suppressed weirs, it follows that a 

larger coefficient in the formula for velocity of approach may be 

required. This is confirmed by experiment.

Various assumptions have been made as to what portion of the 

energy of the approaching stream goes to increase the discharge, («) 

that resulting from the mean velocity deduced from the discharge 

divided by the area of the entire section of the channel of approach; 

(5) that of the mean velocity obtained by using the sectional area of 

the moving water, above the line ABC, fig. 3; (c) that of the filaments lying in line with or nearest to the section of the weir opening-, 

determined approximately by the surface velocity."

DISTRIBUTION OF ENERGY IN CHANNEL OF APPROACH.

Consider unit width of the channel of approach: 

Let Vg = Surf ace velocity.

^=Mean velocity.

rb = Bottom velocity.

r Velocity at a height x above bottom.

JT= Depth of water in channel of approach.

 w Weight of unit volume. 

The general formula for kinetic energy is

K.E.-J^ ........ (10)

where W= weight of the moving mass.

If the velocity increases uniformly from bottom to surface, the 

velocity at heig-ht ay will be

a Smith, HamOton, Hydraulics, p. 68. 

IKB 150 06 2
18 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

Let dx be the thickness of a lamina one unit wide at height x. The 

total kinetic energy for the depth JTwill be

If the velocity is uniform, the total kinetic energy per unit width 

is found by integration to be

 Y^, 3

K

TTI _ ' ^*~ }m f-\f)\ 

 & - <> ....... (12)

Integrating for the simple case where vb =0 and the velocity increases

V-.

uniformly from the bottom to the surface so that 'Um ~cj, we have

K. E.='^^ ....... (13)

ff

Comparing this with the expression for kinetic energy of a stream 

flowing with the uniform velocity v (formula 12), we find the mass 

energy of the stream with uniformly varying velocity to be twice as 

great as for the uniform velocity.

By a similar integration the ratio of the total kinetic energy to the 

kinetic energy corresponding to the mean velocity in the channel of 

approach can be obtained for any assumption as to the distribution of 

velocities in the leading channel. The resulting ratio will depend 

upon the relative areas of section with low and high velocities which 

go to make up the mean, and in practice it will generally exceed unity.

The lowering of the water surface from the level of a still pond will 

also be greater in the case of unequal velocities than in the case of a 

uniform velocity equal to their mean. The theoretical weir formula 

indicates the same discharge in case of a uniform velocity of approach 

v as in case of varying velocities whose mean is equal to v, although 

in the former case the actual drawing down of the head if it were 

measured would be found greater. If h were the velocity head 

corresponding to the mean velocity, and if v^ v2 , va , etc., vn were the 

actual velocities in the n unit areas of cross section, the actual velocity 

head h' will be such that

:. E.

Now,-

_^V ^,2 w QJi K. E. of average velocity.

As shown above, the integral K. E. is the greater.
VELOCITY OF APPROACH. 19 

It follows that h">h. If «=*'

h 

Then

Introducing velocity of approach in the discharge formula we substitute D-\-h for H, and integrate between the limits zero and D. 

Hence, for the same discharge, the area of weir section is greater 

without velocity of approach by nearly the amount hL.

For a given measured head D, the effect of velocity of approach, 

whatever it may be, appears as an increase in the mean velocity of 

discharge in the plane of the weir. The relation of the mean velocity 

of discharge for a weir with velocity of approach to that for a weir 

without such velocity is shown by the following expression, the mean 

head being the same in both cases :

Mean velocity in the plane of the weir=-j£ 

then :&::2fi

It will be seen that the discharge over a weir with velocity of 

approach is less than that for the same total head and greater than 

that for the same measured head without velocity of approach, and 

that with a given measured head the greater the velocity of approach 

the greater will be the discharge.

In a weir section opening out of still water there is always a considerable surface velocity, the parabolic law (see fig. 3) being modified 

by fluid friction, which tends to equalize the velocities. Velocity of 

approach, being usually greater at the surface, furthers this equalization. Some of the kinetic energy of the swifter-moving filaments is 

transferred to their slower-moving neighbors, the result being that 

while the kinetic energy of the whole mass Q passing the weir per 

second remains constant, yet the average velocity is accelerated and 

the discharge rate is increased as compared with the theoretical quantities. This will be clearer if we consider two contiguous filaments, 

each having unit section «, one with a velocity of 1, the other of 2 feet 

per second. The two will discharge 2+1 units flow per second, having the total kinetic energy indicated below:

 _, lxla . 2X28 fm
20 WEIE EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

If, now, the velocities are equalized, 9 units of kinetic energy will 

be equally divided between the two filaments, so that, the new velocity 

being -y,

%awv X v*

and ^

The average velocity before equalization was 1.5.

The discharge from two filaments having equal velocities will be 

3.802 units, as compared with 3.00 for two filaments having unequal 

velocities.

THE THIN-EDGED WEIR.

EARLIER EXPERIMENTS AND FORMULAS.

Prior to 1850 the practice of weir measurement was in a somewhat 

chaotic condition, especially in England, Germany, and the United 

States. There were many experimental results, but the experiments 

were made on so small a scale that the various influences affesting the 

measurements and the lack of proper standards made the results 

erratic and untrustworthy in detail. Greater advancement had been 

made in France by such savants as Dubuat, Eytelwein, L>'Aubuisson, 

Castel, Poncelet, Lesbros, and Boileau. Some of the work of the 

early French experimenters has proved, in the light of wider experience, to be of considerable value.

EXPERIMENTS OF CASTEL.

The first experiments deserving consideration are of those of M. 

Castel, conducted at the waterworks of Toulouse in 1835 and 1836. a 

Castel erected his apparatus on a terrace in conjunction with the water 

tower, which received a continuous supply of 1.32 cubic feet per 

second, capable of being increased to 1.77 cubic feet per second. The 

weir consisted of a wooden dam, surmounted by a crest of copper 

0.001 foot in thickness, situated in the lower end of a leading channel, 

19.5 feet long, 2.428 feet wide, and 1.772 feet deep. Screens were 

placed across the upper end of the channel to reduce oscillations. 

The head was measured at a point 1.60 feet upstream from the weir by 

means of a point gage. The overflow was measured in a zinc-lined 

tank having a capacity of 113.024 cubic feet. The length of the crest 

for weirs with suppressed contractions varied from 2.393 to 2.438 

feet. Heights of weirs varying from 0.105 to 0.7382 were used, and

"Originally published in M6moires Acad. Sci. Toulouse, 1837. See D'Aubuisson's Hydraulics, Bennett's translation, pp. 74-77. Data recomputed by Hamilton Smith in his Hydraulics, pp. 80-82 and 

138-145. The recomputed coefficients 'will be found valuable in calculating discharge for very small 

and very low weirs.
THIN-EDGED WEIRS. 21

a similar series of experiments was performed on suppressed weirs 

1.1844 feet long. The head varied for the longer weirs from about 

0.1 to 0.25 foot. Additional experiments were made on contracted 

weirs having various lengths, from 0.0328 to 1.6483 feet, in a channel 

2.428 feet wide, and for lengths from 0.0328 to 0.6542 foot in a channel 1.148 feet wide. The experiments on- these narrow slit weirs 

included depths varying from 0.1 or 0.2 foot to a maximum of about 

0.8 foot.

D'Aubuisson gives the following formula, derived from the experiments of Castel for a suppressed weir:

..... (14)

where W is the measured central surface velocity of approach, ordinarily about 1.2t>.

EXPERIMENTS OF PONCELET AND LESBROS.

The experiments made by Poncelet and Lesbros, at Metz, in 1827 

and 1828, under the auspices of the French Government, were continued by Lesbros in 1836. The final results were not published, however, until some years later. a

The experiments of Poncelet and Lesbros and of Lesbros were performed chiefly on a weir in a fixed copper plate, length 5.562 feet. 

The head was measured in all cases in a reservoir 11.48 feet upstream, 

beyond the influence of velocity of approach. The crest depth varied 

from about 0.05 to 0.60 or 0.80 foot. The experiments of Lesbros 

are notable from the fact that a large number of forms of channel of 

approach were employed, including those with contracted and convergent sides, elevated bottoms, etc. The experiments of Lesbros on 

these special forms of weirs have been carefully recomputed by Hamilton Smith, and may be useful in determining the discharge through 

weirs having similar modifications. 5

EXPERIMENTS OF BOILEAU.

The experiments of Boileau c at Metz, in 1846, included 3 suppressed 

weirs, having lengths and heights as follows:

(1) Length 5.30 feet, height 1.54 feet.

(2) Length 2.94 feet, height 1.12 feet.

(3) Length 2.94 feet, height 1.60 feet.

The depth of overflow varied from 0.19 to 0.72 foot. Boileau 

obtained the following formula for a suppressed weir:

« Experiences hydrauliques sur les lois de I'e'coulement de 1'eau, Paris, 1852.

6 Smith, Hamilton, Hydraulics, pp. 96 and 97 and 104-107. Also plates 1-2 and 8.

"Gaiigeage de cours d'eau, etc., Paris, 1850.
22 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

This formula includes the correction for velocity of approach. The 

coefficient O, it will be noticed, is given as a constant. Boileau afterwards gave a table of corrections varying with the depth, indicating a 

discharge from 96 to 107 per cent of that obtained with the constant 

coefficient. Additional experiments by Boileau on suppressed weirs 

having a crest length of about 0.95 foot have been recomputed by 

Hamilton Smith. 05 The heights of weirs were, respectively, 2.028, 

2.690, 2.018, and 2.638 feet. In these experiments the discharge was 

determined by measurement through orifices.

EAST INDIAN ENGINEERS' FORMULA. 6

The East Indian engineers' formula for thin-edged weirs is

~ 2

where

Reducing,

M=lM= 0.654 -0.01 H 

(7=3.4:989-0.0535 H

(16)

. . . (17)

This formula applies to a suppressed weir. Method of correction 

for velocity of approach is not stated. Coefficient M has a maximum 

value 0.654, and decreases slowly as the head increases. Limits of 

applicability of formula are not stated. Values of O are given below:

Coefficient Cfor thin-edged weirs, East Indian engineers' formula. c

Hin

feet.

0

1

o

3

4

n

6

7

8

9

0.0

3.499

3.445

3.392

3.338

3. 285

3.221

3.178

3.124

3.071

3.017

0.1

.3.494

3.440

3.386

3.333

3.280

3. 226

3.172

3.119

3.066

3.012

0.2

3.488

3.435

3.381

3.328

3.274

3.221

3.167

3.114

3.060

3.007

0.3

3.483

3.429

3.376

3.322

3.269

3. 215

3. 162

3.108

3.055

3.001

0.4

3.478

3.424

3.370

3.317

3.264

3.210

3.156

3.103

3.050

2.996

0.5

3.472

3.419

3.365

3.312

3.258

3.205

3.151

3.098

3.044

2.991

0.6

3.467

3.413

3.360

3.306

3.253

3.199

3.146

3. 092

3.039

2.985

0.7

3.462

3.408

3.354

3,301

3.248

3.194

3.140

3.087

3.034

2.980

0.8

3. 456

3.403

3.349

3.296

3.242

3.189

3.135

3.082

3.028

2.975

0.9

3.451

3.397

3.344

3.290

3.237

3.183

3.130

3.076

3.023

2.969

a Hydraulics, pp. 133-135.

ft Given in J. Mullins's Irrigation Manual, introduced in United States by G. W. Rafter and used in 

region of upper Hudson River. Not given in Bellasis's recent East Indian work on hydraulics.

c For East Indian engineers' broad-crested weir formula, using coefficients derived from the above, 

see p. 114.
THIN-EDGED WEIBS. 23

EXPERIMENTS AND FORMULA OF JAMES B. FRANCIS.

The experiments on discharge over thin-edged weirs, a upon which 

the Francis formula is based, were made in October and November, 

1852, at the lower locks of the Pawtucket canal, leading from Concord River past the Lowell dam to slack water of Merrimac River. 

Additional experiments were made by Francis in 1848 b at the center 

vent water wheel at the Boott Cotton Mills in Lowell, with gates 

blocked open and with constant head. A uniform but unknown volume of water was thus passed through the turbine and over a weir 

having various numbers of end contractions, the effect of which was 

thus determined. Similar experiments were made in 1851 at the Tremont turbine,c where a constant volume of water was passed over 

weirs of lengths ranging from 3.5 to 16.98 feet and with from two to 

eight end contractions. These experiments were made to determine 

the exponent n in the weir formula

Q= CLHn.

Francis here found % 1.4:7, but adopted the value n=\. 5=3/2, in 

the experiments of 1852.

The Pawtucket canal lock was not in use at the time of the Lowell 

experiments in 1852 and the miter gates at the upper lock chamber 

were removed and the weir was erected in the lower hollow quoin of 

the gate chamber. The middle gates at the foot of the upper chamber were replaced by a bulkhead having a sluice for drawing off the 

water. A timber flume in the lower chamber of the lock was used as 

a measuring basin to determine the flow over the weir. Its length 

was 102 feet and its width about 11.6 feet. A swinging apron gate 

was so arranged over the crest of the weir that, when opened, the 

water flowed freely into the measuring basin below, and when closed, 

with its upper edge against the weir, the overflow passed into a 

wooden diverting channel, placed across the top of the lock chamber, 

and flowed into Concord River. An electric sounder was attached 

to the gate framework, by which a signal was given when the edge of 

the swinging gate was at the center of the nappe, when either opening 

or closing. By this means the time of starting and stopping of each 

experimental period was observed on a marine chronometer. The 

depth on the weir was observed by hook gages. The readings were 

taken in wooden stilling boxes, 11 by 18 inches square, open at the 

top, and having a 1-inch round hole through the bottom, which was 

about 4 inches below the weir crest. The weir was in the lower quoin 

of the gate recess, and the hook gage boxes were in the upper quoin, 

projecting slightly beyond the main lock walls. In weirs with end

a Francis, J. B., Lowell Hydraulic Experiments, pp. 103-135. oldem, pp. 96-102. "Idem, pp. 76-95.
24 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

contractions the full width of the channel was used. For suppressed 

weirs, a leading channel having a width equal to the length of the 

weir crest was formed by constructing vertical timber walls within 

the main canal, extending 20 feet upstream from the weir and having 

their upper ends flaring about 1 foot toward the canal walls. Water 

was freely admitted on both sides of these timber walls. The hook 

gage boxes were outside of this channel. The holes in the bottom 

were plugged, and flush piezometer pipes were used to connect the 

hook-gage boxes with the inner face of the side walls of the channel 

of approach. Observations of the head by hook gage were taken at 

intervals of about 15 seconds. Each experimental period covered 

from 190 to 900 seconds. The hook-gage readings were reduced to 

weir crest level as a datum and arranged in groups of two or three, 

which agreed closely. The mean head was determined by the correction formula (48). In one period, 18 observations of heads ranged 

from 0.6310 to O.G605 foot; their arithmetical mean was 0.6428; 

the computed correction was minus 0.0008.

The measured head was corrected for velocity of approach by using 

the theoretical formula given below. The range and character of the 

experiments, together with the general results, are shown in the following table:

Thin-edged weir experiments of J. R. Francis at the lower locks, Lowell, Mass., 1852.

Serial

numbers of

experiments.

1

3

e

i

5

11

34

36

44

51

56

62

67

72

79

85

o

EH

4

10

33

35

43

50

55

61

66

71

78

I*

£

|

d

"o

EH

4

6

23

2

8

7

5

0

5

5

7

84 fi

88 4

ft

c3 'CD

"3 ^

cd ^

d ?,

51

o S 

M- d O H

_rt CD S ^

 3-g

is

13.96

13.96

13.96

13.96

13.96

9. 992

9.992

13.%

13.96

9.992

13.96

13. 9f>

13.96

|3

,0^

 j-g

£H

3.%

<ij ' O

0-3,0

-C °

l-Sib

p

5.048

5.048

5. 048

5.048

2.014

5.048

5.048

5.048

2.014

5.048

5.048

2.014

2.014

.Range of observed head,

in feet.

Range of velocity of 

approach, in

feet per

second.

; ,1

ft

1. 52430

1.23690

. 91570

1. 01025

1. 02805

. 97450

. 99240

. 77690

. 77115

.7362

. 59190

.63135

i 1

1. 56910

1. 25490

1. 06920

1. 02625

1.07945

. 98675

1.00600

. 81860

. 88865

. 81495

. 65525

. f 6385

.66940 .68,815

fe

0.7682

.5904

.3951

. 3527

.9496

.5376

.5477

.3170

.6694

.3659

.2182

. 5193

.4382

o

a

1

0

^

*o3 *

> 0)

"8

A^s

&E

CD

H r-1

0. 7889

.6000

.4863

. 3596

1.0049

.,5455

.5589

.3405

.7963

. 4213

.2509

. 5496

. 4526

9.997

9.997

9.997

7.997

9.997

9.995

9.995

9.997

9.997

9.. 995

9.997

9.997

7.997

d

.0 

1

g

|

 o

d

CD

0)

,0

a

^

2

2

2

4

2

0

0

2

2

0

OH.:

<ll

*.s

a SH"

^

CD-O

S ^

'S-d

o aj

II

1.56

1. 'J6

1.00

1.02

1.06

0.98

1.00

0.80

0.83

0.80

2 ,0.62

2

4

0.65

0.68

Discharge coefficient 

G.

5

'R

03

^

3.3318

3.3412

3.3333

3. 3617

3.3567

3.3437

3. 3349

3. 3287

3.3435

3. 3424

3.3306

3. 3278

3. 3382

a

I

£

!§

3.3002

3. 3159

3. 3110

3. 3586

3. 3498

3. 3366

3.3243

3.3188

3. 3376

3. 3341

3. 3237

3.3244

d

03

OJ

^

3. 3181

3.3338

3. 3223

3. 3601

3. 3527

3.3409

3. 3270

3. 3246

3.3403

3.3393

3. 3275

3. 3262

3.3333 1 3.3368
FBANCIS EXPEEIMENTS. " 25

From a discussion of these experiments Francis presents the final 

formula 

> . . (18)

If there are end contractions,

_ Tl r 

 Mi '

If there is velocity of approach,

3

The mean velocity v was determined by successive approximations; 

ft was determined by the usual formula 

The Francis formula for velocity of approach correction is cumbersome, and several substitutes have been devised, some of which are 

described in the following paragraphs.

(1) Determine the approximate velocity of approach vt by a single 

trial computation of $, using D H. 

Then use

to determine the final value of Q. For a given value of v this gives 

too large a value of H, but the approximate value of i\ is somewhat 

too small, partially counterbalancing the error and usually giving a 

final value of Q sufficiently precise.

(2) By developing into series and omitting the powers hlD above 

the first, h being always relatively small, the following closely approximate equivalent of the Francis correction formula, given by Emerson,a 

is obtained:

2 , 

(3) Hunking and Hart 6 derive from the Francis correction formula 

the following equivalent expression:

}fr ...... (20)

where G is the area of channel section in which D is measured, per 

unit length of crest.

«Hydrodynamics, p. 286. &Jour. Franklin Inst, Phila., August, 1884, pp. 121-126.
26 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS. 

For a suppressed weir,

For a contracted weir, (22)

"'"Z'-O.l.VZ*.. 

Hunking and Hart have computed values of K by the 

the above formula for each 0.005 increment in D G to 

results extended by formula (23) are given below.

Velocity of approach correction, factor K, Hunking and Hart formula,

DIG

.000

.005

.010

.015

.020

.025

.030

.035

.040

.045

.050

.055

.060

.065

.070

.075

.080

.085

.090

.095

0.0

1. 00000

1. 000006

1. 000026

1. 000058

1. 000103

1. 000161

1.000231

1. 000314

1. 000409

1. 000518

1. 000638

1. 000772

1. 000917

1. 001075

1. 001246

1. 001429

1.001624

1. 001832

1. 002051

1. 002284

0.1

1. 002528

1. 002785

1. 003053

1. 003335

1.003628

1. 003933

1. 004251

1.004581

1.004923

1. 005278

1. 005644

1. 006023

1. 006414

1. 006817

1. 007232

1.007659

1. 008099

0.2

1.009980

1.010480

1. 010994

1. 011519

1. 012057

1. 012607

1.013169

1. 013744

1. 014331

1. 014931

1. 015543

1. 016167

1. 016805

1.017455

0.3

1. 022359

1. 023110

1.023875

1. 024653

1.025444

, 1.026248

1. 027065

1. 027895

1. 028739

1. 029596

1.030467

1. 031350

1. 032248

1. 033117

1.018107 1.034113

1. 018792

1. 019480

1. 035109

1.035856

1.008551 1.020180 1.036852

1. 009015

1. 009491

1. 020893

1. 021620

1. 037848

1. 038844

0.4

1. 039840

1. 040836

1. 041832

1. 042828

1. 043824

1.045069

1. 046065

1. 047061

1. 048306

1. 049302

1. 050298

1. 051543

1. 052788

1. 053784

1. 055029

1. 056274

1. 057270

1. 058515

1. 059760

1. 061005

0.5

1. 062250

1. 063495

1. 064740

1. 065985

1. 067230

1. 068724

1. 069969

1. 071214

1. 072708

1. 073953

1. 075198

1. 076692

1. 078186

1. 079431

1. 080925

1.082419

1. 083664

1.085158

1. 086652

1. 088146

0.6

1.08964

1. 091134

1.092628

1. 094122

1. 095616

1.097359

1.098853

1. 100347

1. 102090

1. 103584

1. 105078

1. 106821

.108564

1. 110058

1. 111801

1. 113544

1. 115038

.116781

1. 118524

1. 120267

olution of 

.36. The

The general formula for K is too complex for common use. The 

expressions

^=1+0.2489 0Tf ...... (23)

and

-«- TT 1-4-1 x ^l 2£r/ ) (9A\ v^*/

are stated to give results correct within one-hundredth and one-fiftieth 

of 1 per cent, respectively, for values of K less than 0.36.

EXPERIMENTS AND FORMULAS OF FTELEY AND STEARNS.

The first series of experiments by Fteley and Stearns on thin-edged 

weir discharge a were made in March and April, 1877, on a suppressed 

weir, with crest 5 feet in length, erected in Sudbury conduit below 

Farm Pond, Metropolitan waterworks of Boston.

Water from Farm Pond was let into the leading channel through

a Fteley, A., and Stearns, F. P., Experiments on the flow of water, etc.: Trans. Am. Soc. C. E., 

vol. 12, Jan., Feb., Mar., 1883, pp. 1-118.
EXPERIMENTS OF FTELEY A1STD STEARIN'S. 27

head-gates until the desired level for the experiment, as found by 

previous trial, was reached. A swinging gate was then raised from 

the crest of the weir and the water was allowed to flow over. The 

maintenance of a uniform regimen was facilitated by the large area 

and the consequent small variation of level in Farm Pond, so that the 

outflow from the gates was sensibly proportional to the height they 

were raised. The water flowed from the weir into the conduit channel below, and was measured volumetrically. For the smaller heads 

the length of the measuring basin was 22 feet, and for the larger 

heads 367 feet.

The crest depth was observed by hook gage in a pail below the weir, 

connected to the channel of approach by a rubber tube entering the 

top of the side wall, 6 feet upstream from the weir crest. Hook-gage 

readings of head were taken every half minute until uniform regimen 

was established, and every minute thereafter. The depths in the measuring basin were also taken by hook gage. The bottom of the conduit 

was concave, and was graded to a slope of 1 foot per mile. It was 

covered with water previous to each experiment, leaving a nearly 

rectangular section.

The experiments in 1877 included 31 depths on a suppressed weir 

of 5 feet crest length, 3.17 feet high. The observed heads varied 

from 0.0735 to 0.8198 foot.

In 1879 a suppressed weir, with a crest length of 19 feet, was 

erected in Farm Pond Gate House. Head-gates and screens were 

close to weir; otherwise the apparatus for measuring head and starting 

and stopping flow was similar to that used in previous experiments. 

The crest of the weir was an iron bar 3£ inches wide and one-fourth 

inch thick, planed and filed and attached to the upper weir timber with 

screws. No variation in level of the weir crest occurred. As in the 

preceding experiments, no by-pass was provided, and the entire overflow entered Sudbury conduit below the weir. The conduit was 

partly filled with water at the start, leaving a nearly rectangular section, 11,300 feet in length and about 9 feet wide. A difference of 3 

feet in water level was utilized in measuring discharge, the total capacity being 300,272 cubic feet. Seniipartitions were provided to reduce 

oscillation of the water. Many observations, covering a considerable 

period of time, were required to determine the.true water level. This 

series of experiments included 10 depths on a suppressed weir 19 feet 

long and 6.55 feet high, with measured heads varying from 0.4685 to 

1.6038 feet and velocities of approach ranging from 0.151 to 0.840 foot 

per second.

From measurements on weirs 5 and 19 feet in length, respectively, 

and from a recalculation of the experiments of James B. Francis, 

Fteley and Stearns obtained the final formula

...... (25)
28 WEIB EXPERIMENTS, COEFFICIENTS, AND FOBMULAS. 

In the above, if there is velocity of approach,

« =1.5 for suppressed weirs. 

a =2.05 for weirs with end contractions.

The value of the velocity head coefficient a was determined from 

94 additional experiments on the 5-foot weir in 1878. These involved 

measured heads ranging from 0.1884 to 0.9443 foot, heights of weir 

ranging from 0.50 to 3.47 feet, and velocities of approach reaching a 

maximum of 2.35 feet per second. Also 17 experiments were made 

on weirs 3, 3.3, and 4 feet long respectively; the first with two and 

the last two with one end contraction. These experiments included 

measured heads varying from 0.5574 to 0.8702 foot, and velocities of 

approach from 0.23 to 1.239 feet per second.

, In all experiments on velocity of approach, the head was measured 

6 feet upstream from crest. The width of channel was 5 feet. a

Fteley and Stearns found the following values of ot for suppressed 

weirs:

Fteley and Stearns's value of a for suppressed weirs.

Measured 

depth on 

weir, 

in feet.

0.2

.3

.4

.5

.6

.7

.8

.9

1.0 

1.1 

1.2 

1.3 

1.4 

1.5 

1.6 

1.7 

1.8 

1.9 

2.0

Depth of channel of approach below weir 

crest, in feet.

0.50

1.70

1.53

1.53

1.53

1.52

1.51

<-1.50

1.49

1.48

........

1.00

1.87

1.83

1.79

1.75

1.71

1. 68

"1.65

1.63

1.61 

1.59 

1.57 

1.55 

1.54 

1.52 

1.51 

1.49 

........

1,70

1.66

1.65

1.63

1.62

1.60

1.59

1.57

1.56

1.54 

1.53 

1.51 

1.49 

1.48 

1.46 

1.44 

1.43 

1.41 

1.40 

1.38

2.60 &

1.51

1.50

1.49

1.48

1.47

1.46

1.45

cl.44

1.43 

1.42 

1.41 

1.40 

1.39 

1.38 

1.37 

1.36 

1..35 

1.34 

1.33

a Fteley and Stearns, idem, pp. 5-23. 

& Applicable to greater heights of weir, 

o Limit of experiments.
BAZIN'S EXPERIMENTS. 29

Current-meter measurements showed a nearly uniform distribution 

of velocities in the channel of approach above the 19-foot weir, a fact 

to be taken account of when the formulas are applied to cases where 

the velocity of approach varies in different portions of the leading 

channel.

If there are end contractions, the net length of weir should be determined by the Francis~formula,

The head should be measured at the surface of the channel of 

approach, 6 feet upstream from the weir crest.

BAZIN'S EXPERIMENTS.

Bazin's experiments on thin-edged weirs were performed in the side 

channel of the Canal de Bourgogne, near Dijon, France, and were 

begun in 1886. Their results were published in Annales des Fonts et 

Chaussees and have been translated by Marichal and Trautwine. a

The standard weir consisted of horizontal timbers 4 inches square, 

with an iron crest plate 0.276 inch in thickness. Air chambers were 

placed at the ends of the weir on the downstream side, to insure full 

aeration of the nappe. End contractions were suppressed. The 

height of the first weir was 3.27 feet above channel bottom, and the 

head was measured in "Bazin pits," one at each side of the channel 

16.40 feet upstream from the weir crest. The pit consisted of a lat: 

eral chamber in the cement masonry forming the walls of the canal. 

The chamber was square, 1.64 feet on each side, and communicated 

with the channel of approach by a circular opening 4 inches in diameter, 

placed at the bottom of the side wall and having its mouth exactly 

flush with the face of the wall. The oscillations of the water surface 

in the lateral chamber were thus rendered much less prominent than 

in the channel of approach. The water level in the Bazin pit was 

observed by dial indicators attached to floats, the index magnifying 

the variations in water level four times, the datum for the indicators 

having been previously determined by means of hook gages placed 

above the crest of the weir and "by needle-pointed slide gages in the 

leading channel.

A drop gate was constructed on the crest of the weir to shut off the 

discharge at will. In each experiment the head-gates through which 

the water entered the leading channel were first raised and the water 

was allowed to assume the desired level. The weir gate was then 

raised, and the head-gates were manipulated to maintain a nearly con-

« Bazin, H., Recent experiments on flow of water over weirs, translated from the French by Marichal and Trautwine: Proc. Engineers' Club Phila., vol. 7, Jan., 1890, pp. 259-310; vol. 9, pp. 231-244, 

287-319; vol. 10, pp. 121-164.
30 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

stant inflow. The arithmetical mean of the observations during each 

period of uniform regimen was used as the measured head for that 

experiment.

The overflow passed into a measuring channel, 656.17 feet in length, 

whose walls were made of smooth Portland cement concrete. The 

channel was 6.56 feet wide, its side walls were 3.937 feet high, and its 

lower end was closed by water-tight masonry. Its bottom was graded 

to a slope of about 1:1,000. The volume of inflow was determined by 

first covering the channel bottom with water, then noting the change 

of level during each experimental period, the capacity of the channel 

at various heights having previously been carefully determined. A 

slight filtration occurred, necessitating a correction of about one-eighth 

of 1 per cent of the total volume. The observations for each regimen 

were continued through a period of 12 to 30 minutes.

Sixty-seven experiments were made on a weir 3.72 feet high, includ 

ing heads from the least up to 1.017 feet. Above this point the 

volumetric measuring channel filled so quickly as to require the use of 

a shorter weir. Thirty-eight experiments were made with a standard 

weir, 3.28 feet long and 3.72 feet high, with'heads varying from the 

least up to 1.34 feet. For heads exceeding 1.34 feet it was necessary 

to reduce the height of the weir in order that the depth above the weir 

should not exceed that of the channel of approach. Forty-eight 

experiments were made on a weir 1.64 feet long and 3.297 feet high, 

with heads ranging from the least up to 1.780 feet. These experiments 

sufliced to calibrate the standard weir with a degree of accuracy stated 

by Bazin as less than 1 per cent of error.

In order* to determine the effect of varying velocities of approach 

the following additional series of experiments were made on suppressed weirs 2 meters (6.56 feet) in length.

Experiments on suppressed weirs 2 meters in length.

Number 

of experiments.

28+30

29+29

27+41

44

Kange of head in feet.

From 

0.489

.314

.298

.296

To 

'1.443

1.407

1.338

1.338

Height of 

experimental weir, in 

feet.

2.46

1.64

1.15

0.79

The standard weir was 3.72 feet high, and the experimental weirs 

were placed 46 to 199 meters downstream. The discharge was not 

measured volumetrically. A uniform regimen of flow was established 

and the depths on the two weirs were simultaneously observed during 

each period of flow.
BAZIJSr'g FOBMULAS. 31

These experiments afforded data for the determination of the relative effect of^ different velocities of approach, corresponding to the 

different depths of the leading channel.

From these experiments Bazin deduces coefficients for a thin-edged 

weir 3.72 feet high, for heads up to 1.97 feet, stated to give the true 

discharge within 1 per cent. a

BAZIN'S FORMULAS FOR THIN-EDGED WEIRS.

Starting with the theoretical formula for a weir without velocity of 

approach, in the form

and substituting

for II, in the case of a weir having velocity of approach, there results,

Bazin obtained, by mathematical transformation, the equivalent 6

or

2

^a^q

Bazin writes

m=,

for which equation he obtains, by mathematical transformation, the 

approximate equivalent 0

The calculation of the factor v appearing in this formula requires 

the discharge Q to be known.

Assuming that the channel of approach has a constant depth P below 

the crest of the weir, and that its width is equa1. to the length of the.

a Bazin, H., Experiences nouvelles sur I'ecoulement en deversoir: Ann. Fonts et Chausse'es, M&m. et 

Doc., 1898, 2me trimestre. See translation by Marichal and Trautwine in Proc. Eng. Club Phila., vol. 7, 

pp. 259-310; vol. 9, pp. 231-244.

&The steps in the derivation of this formula are given by Trautwine and Marichal in their translation of Bazin's report of his experiments, in Proc. Eng. Club Phila., vol. 7, p. 280.

"The steps in detail are given by Trautwine and Marichal in their translation of Bazin, in Proc. 

Eng. Club Phila., vol. 7, No. 5, p. 281.
32 WEIE EXPERIMENTS, COEFFICIENTS, AND FOEMULAS.

weir, v may be expressed in terms of these factors, and of the discharge

=. Q L(P+D). 

Using this value of v, Bazin obtains the expression

..... (28)

where ^=0 a >nz- <® is a nearly constant factor, varying only with

m*. The value of GO as well as that of a can be determined by comparative experiments on thin-edged weirs of different heights."

From a discussion of his own experiments and those of Fteley and 

Stearns, Bazin finally obtained the formulas

i no velocity of approach; 1 

^ with velocity of approach.]

0.003X3.281 0.00984 6 , ov

 =0.405H - (pO)

g For a weir with velocity of approach a= g and &7=0.55. Substituting in equations (27) and (28),

m=,i 1+0.555 ........ (32)

These formulas give values of in agreeing with the results of the 

experiments within 1 per cent for weirs exceeding about 1 foot in 

height within the experimental range of head.

Approximately, for heads from 4 inches to 1 foot,

m=0.425+0.21 frT ..... (33)

correct within 2 to 3 per cent.

The following table gives Bazin's experimental coefficients, the head 

and height of weir (originally meters) having been reduced to feet:

"For detailed analysis see Trautwine and Marichal, Proc. Eng. Club Phila., vol. 7, pp. 282-283. 

b Experimental tabular values of ju. differing very slightly from the formula within the range of 

Bazin's experiments are also given.
U. 8. GEOLOGICAL SURVEY 

f-

# &

L£

-^

ft-f-

 8 .9 1.0 1.1 1.2 1.3 

Heads in feet.

BAZIN'S EXPERIMENTAL COEFFICIENT m FOR THIN-EDGED WEIRS OF VA

FORMULA Q=mLin'29 D.
BAZIN 8 FORMULAS. 33

3

Values of the Bazin coefficient C in the formula Q=CLH'^ for a thin-edged weir, without

end contraction.

Measured 

head D.

Feet.

0.164 

.197 

.230 

.262 

.295

.328 

.394 

.459 

.525 

.591

.656

.787 

.853 

.919

.984 

1.050 

1.116 

1.181 

1.247

1.312 

1.378 

1.444 

1.509 

1.575

1.640 

1.706 

1.772 

1.837 

1.903 

1.969

Meters.

Height of crest of weir above bed of channel of approach, in feet.

0.66

C

3.673 

3.657 

3.649 

3.657 

3.665

3.681 

3.705 

3.737 

3.777 

3.810

3.850 

3.882 

3.914 

3.946 

3.978

4.610

0.98

C

3.633 

3.609 

3.593 

3. 585 

3.585

3.585 

3.593 

3.609 

3.633 

3.657

3.681 

3.705 

3.729 

3.753 

3. 785

3.810 

3.834 

3.858 

3.874 

3.898

3.922 

3.938 

3.962 

3.978

0.20 0.30

1.31

C

3.617 

3.585 

3.569 

3.553 

3.545

3.545 

3.545 

3.553 

3.561 

3.569

3.585 

3.601 

3.625 

3.649 

3.665

3.689 

3.705 

 3. 721 

3.745 

3.761

3.785 

3.801 

3.818 

3.834 

3.850

3.866 

3.874 

3.890 

3.906 

3.922 

3.930

0.40

1.64

C

3.609 

3.569 

3.553 

3.537 

3.529

3.521 

.3.513 

3.513 

3.513 

3.521

3.529 

3.545 

3.561

3. 577 

3.593

3.609 

3.625 

3.641 

3.657 

3.673

3.681 

3.697 

3.713 

3.729 

3.745

3.753 

3.769 

3.785 

3.793 

3.810 

3.818

0.50

1.97

C

3.601 

3.569 

3.545 

3.529 

3.513

3.505 

3.497 

3.489 

3.489 

3.489

3.497 

3.505 

3.513 

3.529 

3.537

3. 553 

3.561 

3.577 

3.593 

3.601

3.617 

3.625 

3.641 

3. 657 

3.665

3.681 

3.689 

3.697 

3.713 

3.721 

3.737

0.60

2.62

C

3.601 

3.561 

3.537 

3.513 

3.497

3.489 

3.473 

3.465 

3.457 

3. 457

3.457 

3.457 

3.465 

3.465 

3.473

3.481 

3.497 

3.505 

3.513 

3.521

3.529 

3.537 

3.545 

3. 553 

3.561

3.569 

3.577 

3.585 

3.593 

3.601 

3.617

0.80

3.28 4.92

C C

3.601 

3.553 

3. 529 

3.513 

3.497

3.481 

3.465 

3.449 

3.440 

3.432

3.432 

3.432 

3.432 

3.440 

3.440

3.449 

3.449 

3.457 

3.465 

3.465

3.473 

3.481 

3.489 

3.489 

3.497

3.505 

3.513 

3.513 

3.521 

3.529 

3.537

1.00

3.593 

3.553 

3.529 

3.505 

3.489

3.473 

3.449 

3.432 

3. 424 

3.416

3.408 

3.400 

3.400 

3.400 

3.400

3.400 

3.400 

3.400 

3.400 

3.400

3.400 

3.408 

3.408 

3.408 

3.408

3.416 

3.416 

3.416 

3.424 

3.424 

3.424

1.50

6.56

C

3.593 

3. 553 

3.521 

3.505 

3.481

3.473 

3.449 

3.432 

3.416 

3.408

3.392 

3.392 

3.384 

3.384 

3.384

3.376 

3.376 

3.376 

3.376 

3.376

3.376 

3.376 

3.376 

3.376 

3.376

3.376 

3.376 

.3. 376 

3.376 

3.376 

3.376

2.00

CO

c

3.594 

3.550 

3.522 

3.499 

3.481

3.466 

3.441 

3.422 

3.405 

3.392-

3.380 

3.371 

3.364 

3.358 

3.353

3.348 

3.343 

3.338 

3.333 

3.328

3.323 

3.319 

3.316 

3.311 

3.306

3.303 

3.298 

3.294 

3.289 

3.285 

3.282

00

Measured 

head D.

 Meters.

0.05 

.06 

.07 

.08 

.09

.10 

.12 

.14 

.16 

.18

.20 

.22 

.24 

.26

.28

.30 

.32 

.34 

.36 

.38

.40 

.42 

.44 

.46 

.48

.50 

.52 

.54 

.56 

.58 

.60

This table, unfortunately, is inconvenient for interpolation in English 

units. The values also differ slightly from those computed from the 

formulas. The table illustrates the difficulty of practical application 

of a weir formula in which the coefficient varies rapidly both with 

head and height of weir.
34 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

A table has been added giving values of jw computed by formula 

(30) for a thin-edged weir without velocity of approach.

Values of u in the Bazin formula for weirs of infinite height, uith no velocity of approach.

H.

Feet.

0.0

.1

_ 2

.3

.4

.5

.6

.7

.8

.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

0.

0.5034

.4542

.4378

.4296

.4247

.4214

.4191

.4173

.4159

.4148

.4139

.4132

.4126

.4120

.4116

.4112

.4108

.4105

.4102

.4099

0.01.

1. 389

.4944

. 4518

.4367

.4290

.4243

.4211

.4189

.4171

.4158

.4147

.4139

.4131

. 4125

.4120

. 4115

.4111

.4108

.4104

.4102

0.02.

0. 8970

.4870

.4497

. 4357

.4284

.4239

. 4208

.4187

.4170

.4157

.4146

.4138

.4131

.4124

.4119

.4115

.4111

.4107

. 4104

.4101

0.03.

0. 7331

.4807

.4478

.4348

.4278

. 4236

.4206

.4185

.4168

.4156

.4146

.4137

.4130

.4124

.4119

.4114

.4110

.4107

.4104

.4101

0.04.

0. 6510

.4753

.4460

.4339

. 4273

.4232

.4204

.4183

.4167

.4154

.4145

.4136

.4129

.4123

.4118

.4114

.4110

.4107

.4103

.4101

0.05.

0. 6018

.4706

.4444

.4331

. 4268

.4229

. 4202

.4181

.4166

.4153

.4144

.4136

.4129

. 4123

.4118

.4113

.4110

.4106

.4103

.4100

0.06.

0. 5693

.4665

. 4429

. 4324

. 4264

. 4225

. 4200

.4180

.4164

. 4152

.4143

.4135

. 4128

.4122

.4117

.4113

.4109

.4106

.4103

.4100

0.07.

0. 5457

. 4628

.4414

.4316

.4260

. 4222

. 4197

.4178

.4163

. 4151

. 4142

.4134

.4127

. 4122

.4117

.4113

.4109

.4106

.4103

.4100

0.08.

0. 5280

. 4596

. 4401

.4309

. 4255

. 4219

.1195

.4176

.4162

. 4150

.4141

.4133

. 4127

. 4121

.4116

.4112

.4108

.4105

.4102

.4100

0.09.

0. 5142

. 4568

.4389

.4302

. 4251

.4216

.4193

.4174

.4160

.4149

.4140

.4133

. 4126

.4121

.4116

.4112

.4108

.4105

.4102

.4099

0.1.

0. 5034

.4542

.4378

.42%

.4247

. 4214

.4191

.4173

.4159

.4148

.4139

. 4132

. 4126

.4120

.4116

.4112

.4108

.4105

.4102

.4099

DERIVED FORMULAS FOR THIN-EDGED RECTANGULAR WEIRS.

A number of weir formulas have been derived from subsequent 

analysis or recomputation of the experiments of Francis, Fteley and 

Stearns, and Bazin, differing more or less from those given by the 

experimenters.

FTELEY AND STEARNS-FRANCIS FORMULA. 05

..... (34)

Correction for end contractions is to be made by the Francis 

formula; velocity of approach correction by the Fteley and Stearns 

formulas

H=D-\-\,5h, for suppressed weir. 

ZT=ZM-2.05A, for contracted weir.

HAMILTON SMITH'S FORMULA. b

The base formula adopted is

~ 2

(35)

aFteley and Stearns, Experiments on the flow of water, etc.: Tranis. Am. Soc. C. E., vol. 12, p. 82. 

& Smith, Hamilton, Hydraulics, pp. 123-132.
DEBITED EORMTTLAS FOR THIN-EDGED WEIRS. 35

The velocity of approach correction is made by the use of the 

formulas

Jff=D-\-1.Ah, for contracted weirs." 

for suppressed weirs.

A diagram and tables of values of the coefficient Mare given by the 

author. The correction for partial or complete contraction is included 

in the coefficient, separate values of M being given for suppressed and 

contracted weirs.

2 _ 

Making Cf= o J^v^ffi the Smith formula (35) may be written

which is directly comparable with the Francis formula.

Smith's coefficients in the above form are given in the following 

tables.

Hamilton Smith's coefficients for weirs witJi, contraction suppressed at l>oth ends, for use in

3

flu formula Q=CLH-*.

H= 

Head, 

in feet.

0.1 

.15

.2

.25

.3

.4

.5

.6

.7

.8 

.9 

1.0 

1.1 

1.2 

1.3 

1.4 

1.5 

1.6 

1.7 

2.0

7/=length of weir, in feet.

19

3. 5l5 

3.440

3.397

3.371

3.349

3.322

3.312

3.306

3.306

3.306 

3. 312 

3.312 

3.317 

3.317 

3.322 

3.328 

3.328 

3.333 

3.333

15

3. 515 

3.445

3.403

3.376

3.354

3.328

3.317

3. 312

3. 312

3.317 

3.317 

3.322 

3.328 

3.333 

3.338 

3.344 

3.344 

3.349 

3.349

10

3.520 

3.445

3. 408

3.381

3.360

3.333

3.322

3.317

3.317

3.322 

3.328 

3. 338 

3.344 

3.349 

3.360 

3. 365 

3.371 

3.376 

3.381

7

3.520 

3.451

3.408

3.386

3.365

3.344

3.338

3.333

3.338

3.344 

3.3.54 

3.360 

3.371 

3.381 

3.386 

3.392 

3.403 

3.408 

3.413

5

3.526 

3.451

3.413

3.392

3.376

3.360

3.354

3.354

3.360

3. 365 

3.375 

3.386 

3.397 

3.403 

3.413 

3.424 

3.429 

3.435

4

3.461

3.429

3.403

3. 3S6

3.371

3.371

3.371

3.376

3.386 

3.397 

3.408 

3.419 

3.429 

3.440 

3.445 

3. 456 

3. 401

3«

3.472

3. 435

3.413

3.403

3.386

3.386

3.392

3.397

3.408 

3.418 

3.429 

3.445 

3.456 

3.467

la

3.488

3. 450

3.429

3.418

3.403

3.408

3.413

3.424

3.441 

3.451 

3.467

0.666

3.611 

3. 542

3. 510

3.494

3.483

3.478

3.478

3.483

3.494

3.510

a The use of the head corresponding to central surface velocity without correction, to determine D, 

is also recommended. 

b Approximate.
36 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

Hamilton Smith's coefficients for weirs with two complete end contractions, for use in the

formula Q=

H= 

Head.

0.1

.15

.2

.25

.3

.4

.5

.6

.7 

.8 

.9 

1.0 

1.1 

1.2 

1.3 

1.4 

1.5 

1.6 

1.7 

2.0

£'=length of weir, in feet.

0.66

3.381

3. 312

3.269

3.237

3. 215

3.183

3.156

3.140

3.130

la

3.419

3.344

3.306

3.274

3.253

3. 215

3.189

3.172

3.156

2

3.456

3.392

3.349

3. 322

3.296

3.258

3.237

3.215

3.199 

3.183 

3.167

3. 156 

3.140 

3.130 

3.114 

3.103

2.6

3.478

3.408

3. 365

3.333

3.306

3.274

3.247

3.231

3.210 

3.199 

3.189 

3.172 

3.162 

3.151 

3.135 

3.124

3.103

3

3.488

3.413

3.371

3.338

3.312

3.280

3.253

3.237

3.226 

3.215 

3.199 

3.183 

3.172 

3.162 

3.151 

3.140

3.114

4

3.494

3.419

3.376

3.344

3. 322

3.285

3. 264

3.247

3. 231 

3.221 

3. 210 

3.199 

3.189 

3.178 

3.167 

3.156 

3.151 

3.140

5

3.494

3.424

3.376

3.349

3. 322

3. 290

3.269

3.253

3.242 

3.231 

3.22G 

3.215 

3. 205 

3.194 

3.199 

3.178 

3.167 

3.162

7

3.499

3. 424

3.381

3. 354

3. 333

3.301

3.280

3.269

3. 258 

3. 247 

3.242 

3.231 

3. 226 

3.215 

3.205 

3.199 

3.189 

3.183 

3.178

10

3.504

3.429

3.386

3.360

3.338

3.306

3.290

3.280

3.274 

3. 269 

3. 258 

3.253 

3.242 

3.237 

3.231 

3.221 

3. 215 

3.210 

3.205

15

3. 504

3. 435

3.392

3.360

3. 338

3.312

3. 295

3. 285

3.280 

3.274 

3.269 

3.264 

3.258 

3. 253 

3.247 

3.242 

3.237 

3.231 

3.226

19.

3.510

3. 435

3.392

3. 365

3.344

3.317

3. 301

3.290

3.285 

3.280 

3.274 

3.269 

3.264 

3.264 

3. 258 

3. 258 

3.253 

3.247 

3.247

a Approximate. 

Hamilton Smitli's coefficient C for long weirs.

H

0.00

.01

.02

.03

.04

.05

.06

.07

.08

.09

0.1

3. 5096

3.4957

3.4818

3. 4678

3.4539

3.4400

3. 4314

3.4229

3. 4143

3.4058

0.2

3. 3972

3. 3908

3. 3844

3. 3780

3.3716.

3. 3652

3.3537

3. 3512

3. 3488

3.3463

0.3

3. 3438

3. 3411

3.3384

3.3358

3.3331

3. 3304

3. 3277

3.3250

3. 3224

3. 3197

0.4

3. 3170

3. 3154

3. 3138

3. 3122

3.3106

3.3090

3. 3074

3. 3058

3. 3042

3.3026

0.5

3.3010

3. 3005

3.2999

3.2994

3. 2988

3. 2983

3. 2978

3. 2972

3. 2967

3. 2961

0.6

3. 2956

3.2945

3. 2935

3.2924

3. 2913

3. 2902

3. 2892

3.2881

3. 2870

3. 2860

0.7

3. 2849

3. 2838

3.2828

3. 2817

3. 2806

3. 2796

3. 2785

3. 2773

3. 2762

3. 2752

Hamilton Smith's formula is based on a critical discussion of the 

experiments of Lesbros, Poncelet and Lesbros, James B. Francis, 

Fteley and Stearns, and Hamilton Smith; including series with and 

.without contractions and having crest lengths from 0.66 to 19 feet.
DERIVED FORMULAS FOB THIN-EDGED WEIRS. 37 

SMITH-FRANCIS FORMULA.

The Smith-Francis formula, * based on Francis's experiments, reduced 

to the basis of correction for contractions and velocity of approach 

used with Hamilton Smith's formula, is, 

for a suppressed weir,

0=3.29 Z+JT* ...... (36)

for weir of great length or with one contraction,

0=3.297;^ ......... (37)

for weir with full contraction,

0=3.29 L-lH* ...... (38)

If there is velocity of approach,

' H D-\-\A h, for a contracted weir. 

A, for a suppressed weir.

PARMLEY'S FORMULA. b 

Parmley's formula is

Q=CKLD% ....... (39)

If there are end contractions, the correction is to be made by the 

Francis formula,

The factor ^represents the correction for velocity of approach. 

The factor has been derived by comparing the velocity correction 

factor in the Bazin formula (formula 32), written in the form

with the approximate Francis correction as deduced by Hunking and 

Hart (formula 23), written in the form

(

where a is the area of the section of discharge, for either a suppressed 

or contracted weir, and A is the section o|f the leading channel. It is 

observed that there is an approximately constant relation between the 

two corrections, that of Bazin being 2.2 times that of Francis.

« Smith, Hamilton, Hydraulics, pp. 99 and 137.

6Rafter, G. W., On the flow of water over dams: Trans. Am. Soc. C. B., vol. 44, pp. 350-359, discussion by Walter C. Pannley.
38 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

Parmley adopts the Bazin correction and gives the following table, 

which may also conveniently be applied in computing discharge by 

Bazin's formula.

The discharge coefficient C used by Parmley is that for a weir with 

no velocity of approach, as in the Francis formula. It is not, however, constant. Its values have been deduced from a mean curve representing the experiments of Francis, Fteley and Stearns, and Bazin.

Velocity of approach correction, K, Parmley and Bazin formulas.

a 

A

0.00

.01

.02

.03

.04

.05

.06

.07

.08

.09

0

1.0001

3.0002

1. 0005

1.0009

1.0014

1.0020

1.0027

1.0035

1. 0044

0.1

1.0055

1.0066

1.0079

1.0093

1. 0108

1. 0124

1.0141

1.0159

1.0178

1. 0198

0.2

1. 0220

1.0243

1.0266

1.0291

1. 0317

1.0344

1. 0372

1. 0401

1.0431

1.0463

0.3

1. 0495

1. 0529

1. 0563

1.0599

1.0636

1. 0674

1. 0713

1. 0753

1.0794

1.0837

0.4

1. 0880

1.0925

1. 0970

1. 1017

1.1065

1.1114

1.1164

1. 1215

1. 1267

1. 1321

0.5

1. 1375

1. 1431

1. 1487

1.1545

1.1604

1. 1664

1. 1725

1. 1787

1. 1850

1.1915

Parmleifs weir formula, coefficient C.

D,

feet.

0.0

.1

.2

.3

.4

.5

.6

.7

.8

.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

0.00

3.580

3.478

3.420

3.385

3.368

3.358

3.351

3.346

3.340

3.334

3.329

3. 324

3. 319

3. 313

3.307

3.301

3.296

3.290

3.285

3.280

0.01

3.568

3.471

3.416

3.383

3. 367

3.357

3. 351

3.345

3.339

3.334

3.328

3.324

3.318

3.312

3.306

3.301

3.295

3.290

3. 285

0.02

3.556

3.464

3.412

3.381

3.366

3.356

3.350

3.345

3. 339

3.333

3, 328

3.323

3.318

3. 312

3.306

3.300

3.295

3.289

3. 284

0.03

3.544

3.458

3.408

3.380

3.364

3.356

3.350

3.344

3.338

3.332

3.328

3.322

3.317

3.311

3.305

3.300

3.294

3. 288

3. 284

0.04

3.532

3.451

3.404

3.378

3.363

3.355

3.349

3.344

3.338

3.332

3.327

3. 322

3.317

3.311

3.305

3. 299

3.294

3. 288

3.283

0.05

3.520

3.444

3.400

3.376

3. 362

3.354

3.349

3.343

3. 337

3. 332

3. 326

3. 322

3. 316

3. 310

3.304

3. 298

3. 293

3.288

3. 282

0.06

3.512

3.439

3.397

3.374

3.361

3.353

3.348

3. 342

3.336

3. 331

3.326

3. 321

3. 315

3.309

3.303

3. 298

3. 292

3.287

3. 282

........

0.07

3.503

3.434

3.394

3.373

3 360

3.353

3.348

3.342

3.336

3.330

3.326

3.320

3.315

3.309

3.303

3 298

3.292

3. 286

3.282

0.08

3.495

3.430

3.391

3.371

3.360

3.352

3.347

3.341

3.335

3.330

3.325

3.320

3.314

3.308

3.302

3.297

3.291

3.286

3.281

0.09

3.486

3.425

3.388

3.370

3.359

3.352

3.347

3.341

3.335

3.329

3.324

3.320

3.314

3.308

3.302

3.296

3.291

3.285

3.280
THIN-EDGED WEIRS. 39

EXTENSION OF THE WEIR FORMULA TO HIGHER HEADS.

' It will be noticed that all the accepted formulas for discharge over 

thin-edged rectangular weirs are based on experiments in which the 

head did not exceed 2 feet above crest. It is often desirable to utilize 

the weir for stream gagings where the head is greater, especially for 

the determination of maximum discharge of streams, the head frequently being as large as 6, 8, or even 10 or 12 feet.

In the experiments at Cornell University on weirs of irregular section it was often necessary to utilize depths on the standard weir 

exceeding the known limit of the formula. A series of experiments 

was accordingly carried out in which a depth on a standard thin-edged 

weir (16 feet long) not exceeding the limit of the formula was utilized 

to determine the discharge over a similar but shorter standard thinedged weir (6.56 feet long) for depths up to approximately 5 feet." 

The results of these experiments, as recomputed, eliminating slight 

errors in the original, are given below.

It will be noted that the weir was short and the velocity of approach 

relatively large, yet, according to the results when corrected by the 

Francis method, the average value of (f for heads from 0.75 to 4.85 

feet is 3.296, or 98.88 per cent of the Francis coefficient for a thinedged weir. The average value of Cf for heads from 0.746 foot to 2 

feet is 3.266, and for heads from 2 to 4.85 feet, 3.278.

United States Deep Waterway* experiments at Cornell hydraulic laboratory for extension of

thin-edged weir formula.

Standard weir, 16

feet long, 13.13

feet high.

Cor. D, longitudinal, 

piezometer, centi1

12.28

15. 30

18.39

21.65

24.16

27. 21

 30. 16

30. 22

37.90

44. 22

59.00

74.22

81.69

Q, Bazin 

formula, 

in cubic 

feet per 

second.

.>

14.12

19.42

25. 35

32.24

37.86

45.13

52. 62

52.77

73.46

92.79

143.90

202. 37

233. 81

Lower thin-edged weir: _P=5.2, X=6.56.

Observed 

P, flush, 

piezometer, centimeters.

8

22. 744

27. 855

33. 175

39. 419

44. 000

49. 699

55. 213

55. 128

68. 238

80. 566

105. 639

130.286

142. 557

-O,

in feet.

4

0. 7462

.9139

1.0885

1.2933

1. 4436

1.6306

1.8115

1.8088

2. 2389

2. 6434

3. 4660

4. 2747

4. 6773

D 

P+J>

.">

0.1255

. 1495

.1731

.1992

. 2173

.2387

.2583

.2581

.3010

. 3370

.4000

. 4512

. 4735

K 

Hunking 

and 

Hart.

6

1.0041

1.0056

1.0075

1 . 0099

1. 0122

1. 0141

1. 0166

1. 0166

1.0225

1.0283

1.0398

1. 0504

1.0557

n*

7

0. 6469

. S7.H7

1.1434

1. 4849

1. 7564

2. 1116

2. 4787

2. 4730

3. 4254

4. 4193

0. 6095

9. 2867

10. 67S9

Q, cubic

second, 

per foot 

(corrected) .

«

2. 1066

2. 9143

3. 8183

4. 86.H5

5. 7252

6. 8333

7. 9750

7. 9977

11. 1516

14. 0960

21 . 8902

30. 8008

35. 5933

-4

m

9

3. 256

3.317

'A. 831

3. 279

3. 260

3. 1236

3. 218

3.234

3. 226

3.190

3.312

3. 317

3.333

a Rafter, C4. W., On tlie flow of water over dams: Trans. Am. Soc, C. E., vol. 44, p. 397.
40 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

If it is borne in mind that the influences which go to make up variation in the weir coefficient are more potent for low than for larger 

heads, it may be confidently asserted that the Francis formula is applicable within 2 per cent for heads as great as 5 feet, and by inference it 

is probabty applicable for much greater heads as well.

COMPARISON OF WEIR FORMULAS.

The later weir formulas all give results agreeing, for the range of 

heads covered, within the limit of accuracy of ordinary stream measurements. Which of the several formulas to use will be determined 

by convenience and by the conditions attending the measurements.

The Francis formula is applicable for weirs with perfect bottom 

contraction and for any head above 0.50 foot.

The Hamilton Smith, Fteley and Stearns, and Bazin formulas are 

more accurate for very slight heads, or where bottom contraction is 

imperfect, this element, which tends to increase discharge, being 

included in the larger velocity of approach correction. These formulas are, however, based on experiments none of which exceeded 2 

feet head, and they have not been extended.

For suppressed weirs in rectangular channels having conditions 

closely duplicating Bazin's experiments, his formula is probably most 

applicable. The head should preferably be measured in a Bazin pit, 

opening at the bottom of the channel, 16.4: feet upstream from the 

weir, in a suppressed weir, if the nappe is allowed to expand laterally after leaving the weir, the computed discharge by any of the formulas should be increased from one-fourth to one-half of 1 per cent.

Comparative discharge by various formulas over weirs of great height and length; no end 

contractions nor velocity of approach.^

Formula.

Castel............ .........

Fteley-Stearns-Francis. . . .

East Indian engineers ....

Coefficient C, for heads ranging from 

0.20 to 4 feet.

0.20

3. 4872 

3. 3455 

3.4025 

3.33 

3. 5004 

3. 642684 

3.3800 

3. 3972 

3.29 

3.478 

3.488

0.50

3. 4872 

3.3455

3.33 

3. 3269 

3. 406094 

3.3300 

3. 3010 

3.29 

3.368 

3.472

1.00

3. 4872 

3. 3455 

3. 3136 

3.33 

3.317 

3. 326696 

3.319 

3.284 

3.29 

3.334 

3.445

4.00

3. 4872 

3. 3455

3.33 

3. 3109 

3. 26783 

3" 31375 

3.284 

3.29

3.285

Per cent of discharge by Francis 

formula for heads ranging from 

0.20 to 4 feet.

0.20

104. 616 

100. 365

100.00 

105. 012 

109. 281 

101. 400 

101.916 

98.70 

104. 340 

104.640

0.50

104. 616 

100.365

100.0 

99.807 

102. 183 

99.90 

99. 030 

98.70 

101. 040 

104. 16

1.00

104. 616 

100. 365 

99.408

100.0 

99.51 

99. 801 

99. 570. 

98. 520 

98.70 

100. 020 

103. 35

4.00

104. 616 

100. 365

100.0 

99. 327 

98.035 

99.412

98. 550

a Computed by H. R. Beebe, C. E.
COMPABISON OP WEIE FOEMULA8. 41

Table showing comparative discharge per foot of crest for suppressed weirs of various 

lengths, heads, and velocities of approach. «

Height (P)..... ........... .....................

Head(D) ......................................

Castel ..............................

Boiieau .............................

Bazin ...............................

Fteley-Stearns-Francis ............... 

Hamilton Smith .....................

Smith-Francis .......................

Average. ......................

i

1.0

1.90

3. 7822

3. 8630

3. 5373

3. 7268

3. 7845

3. 7297 

3. 9220

4. 0581 

3. 7924

3.800

2

1.0

1.18

3. 6127

3. 5484

3. 4218

3. 4729

3. 3766

3. 4752 

3. 6392

3. 7109 

3. 5337

3.532

10

2

1.0

1.16

3. 6217

3. 5484

3. 4218

3. 4730

3.3766

3. 4752 

3. 4872

3. 4847 

3. 5337

3.490

10

4

1.0

.68

3. 5308

3. 4144

3. 3632

3. 3669

3, 4002

3. 3690 

3. 3878

3. 3876 

3. 3347

3. 395

10 '

4

4

2.15

30. 3037

30. 9046

28. 2983

29. 7470

29. 7555

29. 7000

31. 573

30.040

a Computed by H. R. Beebe, C. E. 

COMPARISON OF VARIOUS VELOCITY OF APPROACH CORRECTIONS.

The various modes of correction for velocity of approach used by 

different investigators can be rendered nearly identical in form, varying, however, in the value of the coefficient oc adopted.

Comparative coefficients of correction for velocity of approach for thin-edged weirs with end

contractions suppressed.

Experimenter.

Boiieau. ......................................

Lesbros ......................................

Bazin ........................................

Value of a in the formula H=D+a ga=1.8

a=1.56

a=1.5

a a-l 2-JTl

g a=1.69 or 

o

Values of w in 

the formula

6 oo =0.2489

Emerson. 6 Hunking and Hart.

The above values were all derived from experiments on thin-edged 

weirs. Bazin's experiments covered the larger range of velocities and 

were most elaborate. It may be noted that the correction applied by 

Bazin is two and two-tenths times that of Francis for a given velocity
42 WEIE EXPERIMENTS, COEFFICIENTS, AND FOEMULAS.

of approach. Bazin's correction is, in effect, an increase in the measured head of 1.69 times the velocity head, while Francis increases the

2 /7f measured head by an amount - ^J -^ less than the velocity head accordo LJ 

ing to Emerson's formula.

Ratio of tl\t> various corrections for velocitiy of approach for suppressed weirs.

Bazin ................. ,-.-...-_.....

Fteley and Stearns ...................

Hamilton Smith .....................

Bazin.

1.000

.887

.789

.454

Fteley and 

Stearns.

1.127

1.000

.887

.511

Hamilton 

Smith.

1.271

1.128

1.000

.576

Francis.

2.2

1.957

1.736

1.000

The factors in the above table are not strictly accurate, for the reason that the expressions used to deduce the equivalents from the different formulas are in some cases approximations. They serve to 

illustrate the relative magnitude of the different corrections for thinedged weirs without end contractions. For thin-edged weirs with end 

contraction, Hamilton Smith uses the coefficient a 1.4 and Fteley and 

Stearns give the coefficient <*=2.05.

There are no experiments available relative to the value of the 

velocity correction for other than thin-edged weirs. It is necessary, 

therefore, to utilize the values above given for weirs of irregular section. It will be seen that it matters little in what manner the correction for velocity of approach is applied, either by directly increasing 

the observed head, as in the formulas of Hamilton Smith and Fteley 

and Stearns, or by including the correction in the weir coefficient, as 

is done by Bazin, or by utilizing a special formula to derive the corrected head, after the manner of James B. Francis. The three methods 

can be rendered equivalent in their effect.

The important point is that the corrected result must be the same as 

that given by the author of the formula which is used to calculate the 

discharge. As to the relative value of the different modes of applying the correction, it may be said of that of Francis, that in its original 

form it is cumbersome, but it renders the correction independent of 

dimensions of the leading channel, as do also the formulas for correction used by Hamilton Smith, and Fteley and Stearns. Inasmuch as 

the velocity head is a function of the discharge, successive approximations are necessary to obtain the final corrected head by any one of 

these three formulas.

By using the Hunking and Hart formula the correction for the 

Francis weir formula becomes fairly simple, as it does not require the 

determination of the mean velocity of approach by successive approxi-
COMPAKISON OF VTEIK FOEMULAS. 43

mations, but to apply this formula it is necessary to know the dimensions of the leading1 channel and of the weir section. The approximation given by Emerson is also much simpler than the original Francis 

formula.

Bazin's method of including the velocity correction in the coefficient 

makes the weir coefficients obtained by the experiments comparable 

one with another only when both the head and velocity of approach 

are the same in both cases." His correction also involves the dimen* 

sions of the leading channel as factors. Obviously, in the case of 

many broad-crested weirs utilized for measuring flow, the dimensions 

of the leading channel can not be ascertained accurately and there is 

great variation of velocity in different portions of the section of approach. It becomes necessary that the correction should be in such a 

form that it is a function of the velocity and not of the channel 

dimensions.

It is to be noticed that where an attempt has been made in the 

weir experiments to eliminate velocity of approach effect from the 

coefficient the velocity has been nearly equalized by screens and has 

been determined by successive approximations. It is suggested that 

where the velocities vary widely they be determined by current 

meter in several subdivisions of the section, the approximate integral 

kinetic energy estimated, and a value of a selected depending on the

h' 

ratio of -r so obtained, where A is the velocity head corresponding to

the mean velocity and It! is the velocity head which would result if the 

actual velocities were equalized. Inasmuch as the surface velocity 

usually exceeds the mean velocity in the channel of approach in about 

the same ratio that h' exceeds A, the suggestion is made by Hamilton 

Smith 6 that where the velocity of approach is unavoidably variable, 

or the boundaries of the current are uncertain, the surface velocity vs 

be measured by floats and applied directly in the determination of the 

quantity k.

The variations in discharge over a thin-edged weir, by the different 

formulas, are often less than the difference in the correction for velocity 

of approach would indicate. In the formula of Fteley and Stearns, as 

compared with Francis, for example, the larger velocity correction is 

in part compensated by a smaller weir coefficient, and the same is true 

of the formulas of Hamilton Smith and Bazin for cases where the head 

is large.

"See special discussion of the point, p. 63. b Smith, Hamilton, Hydraulics, p. 84.
44 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS. 

END CONTRACTIONS INCOMPLETE CONTRACTION.

The formula for end contractions deduced by James B. Francis is 

very generally used. The correction is made to the length of weir, 

the result obtained being the length of a suppressed weir that will 

give the same discharge.

L=L'-INH .............. (40)

S =A coefficient, the value of which, deduced by Francis, is

5=0.1.

L' Actual length of weir crest. 

L Length of equivalent suppressed weir crest. 

N Number of end contractions. 

H= Effective head, feet.

The experiments of Fteley and Stearns,a while somewhat discordant, 

indicate an average value of 5 for heads from 0.3 to 1 foot, of about 

0.1. The value of 5 apparently decreases as the head increases. It 

also decreases if the end contraction piece is so near the side of the 

channel as to render the contraction incomplete.

Hamilton Smith shows that side contractions and bottom or crest 

"contraction are mutually related, and that the side width of the channel of approach should be fully three times the least dimension of the 

weir. Usually L is much greater than II, and the side width may be 

made at least as great as 3ff. The specification of Francis is, side 

width ^ H.

Smith's rule indicates that to provide complete contraction the area 

of leading section A. must bear a relation to the area of weir section a 

depending upon the relative head and length of crest.

For three weirs of equal section a, the following values of A, the 

necessary channel-section area, are given:

Z' = 12 11= 1 a=12 A= 72=6«

L'--= 4 11= 3 a=12 vl

L'= 1 #=12 «=12 J.

Hamilton Smith prefers to use separate coefficients for suppressed 

weirs from those for contracted weirs, the relation between the coefficients being expressed by the formula

^' (41)

6^,=Coefficient for partially suppressed weir, as with complete suppression on sides and full contraction at bottom.

a Fteley and Stearns, Experiments on the flow of water, etc.: Trans. Am. Soc. C. E., vol. 12, pp. 

108-113.
END CONTRACTIONS. 45

6*e= Coefficient for completely contracted weir.

JT=Least dimension of weir, whether L or H,

R Wetted perimeter of weir=Z+2ZT

T = Distance from any side of weir to the respective side of channel,

where there is partial suppression. 

xS' =Length of sides on which there is partial suppression.

Smith's values of contraction coefficient s in formula 31 are

Y/X z 

3 0.000 

2 .005 

1 .025 

4 .06 

0 .16

The ratio J7JT approximately measures the amount of contraction. a

Bazin does not give a formula for weirs with end contractions. The

Bazin formula may be applied to weirs in which the height of weir is

so small that the bottom contraction is partially suppressed. The

Bazin coefficient then includes:

1. Effect of contraction from surface curve.

2. Effect of crest contraction and its modification by both velocity 

of approach and by partial suppression, if any.

3. Effect of velocity of approach proper.

4:. Effect of distribution of velocities in channel of approach.

5. Loss of head from friction and eddies.

As the Baziu weirs were very low, and these factors go to increase 

the correction necessary, it will be seen that the relatively large 

velocity of approach correction required by Bazin's formula may be 

readily accounted for.

The experiments of Flinn and Dyer on the Cippoletti weir (see 

p. 48) indicate that the effect of end contraction may be somewhat 

greater than that indicated by the Francis formula. Any experiments 

in which similar volumes of water have been successively passed over 

weirs with and without end contractions may be utilized to determine 

the effect of such contractions.

It may be added that a more elaborate study of end contractions is 

desirable. It is to be borne in mind, however, that to secure greater 

accuracy in this regard a more complicated or variable correction than 

that of Francis must probably be used, and the result will be to greatly 

increase the labor of weir computations in the interest of what is 

usually a comparatively small matter, the better remedy being probably the use of weirs with end contractions suppressed, wherever 

practicable.

a Smith, Hamilton, Hydraulics, pp. 118-123. Smith's critical discussion of this subject will be 

found of value in calculating discharge for weirs with partially (suppressed contraction either at sides 

or bottom.
46 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

COMPOUND WEIR.

A weir with a low-water notch depressed below the general crest 

level may sometimes be used to advantage in gaging small, variable 

streams. The discharge over such a weir, constructed with end contractions on both sections, can be calculated as for two separate weirs, 

the lower short section having end contractions for all heads. The 

flow over the two upper sections is computed as for a suppressed weir.

Such a weir has been used for the determination of the low-water 

flow of very small streams, for which purpose it is well adapted, the 

entire stream when at low stages flowing in the central notch, in a 

stream relatively deep and narrow.

The measurement of very thin sheets of water on a broad weir is 

subject to peculiar difficulties, including uncertainty of coefficient, 

adhesion of nappe to weir face, dispersion by winds, and a large percentage error in the results if there is a small error in measuring the 

head.

TRIANGULAR WEIR. 

GENERAL FORMULA.

Referring to fig. 4, we may write

l\H-y\\L\H. 

 v « ...... L .......

FIG. 4. Triangular weir.

Substituting, in equation (4),

L(H-y]

THOMSON'S EXPERIMENTS.

The mean coefficient of contraction for a thin -edged triangular weir 

deduced experimentally by Prof. James Thomson, of Belfast, is M= 

0.617," the formula being

..... (43)

"British Association Report, 1858 (original not consulted). Merriman gives the mean value of Jbf 

for heads between 0.2 and 0.8 foot as 0.592.
TBAPEZOIDAL WEIKS. 47

For a right-angled notch,

L=ZIIand $=2.64^ ...... (44)

The length of the contracting edges in a triangular notch being proportional to the depth, it is believed that the coefficient of discharge 

is somewhat more constant than for a rectangular weir. a

TRAPEZOIDAL WEIR.

The discharge in this case may be determined directly from the 

integral formula (4) as for a triangular weir, by integrating between 

the limits AD and CE, fig. 5. It may also be derived as follows:

zl=f A

c. *

*-----

y

i -.1 ./._

ay\

- . !_ __-_

Ify-

-- *

FIG. 5. Trapezoidal weir.

2= slope of one side to the vertical.

By integration,

^ ...... (45)

in which coefficients of contraction for the horizontal crest and for 

the end slopes must be introduced.

THE CIPPOLETTI TRAPEZOIDAL WEIR.

* *

The discharge over a trapezoidal without contraction would be the 

sum of that for a rectangular weir added to that for two triangular 

weirs forming the ends. From the experiments of James B. Francis 6 

it appears that each end contraction reduces the effective length of the 

weir 0.1IT. The contraction decreases the discharge by the amount

If the ends of the weir, instead of being vertical, are inclined outward in such manner that the discharge through the added area counterbalances the decrease from the end contraction, then the effective

o The coefficient 2.64 is the same as that deduced for broad crest weirs with stable nappe. A table of 

values of 2.64S^ is given on page 177, which may be applied in calculating flow over triangular weirs. 

6 Lowell Hydraulic Experiments.

IBR 150 06 1
48 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

length of the weir will remain constant as the head increases, the same 

as in a suppressed weir. The discharge through the end triangle ABC 

will be, from equation (42),

Where z is the width or base of the end triangle. Equating the 

two expressions for Q, and solving for 2, we find, assuming M to have 

the same value in both cases,

z=\H ......... (46)

This condition defines the Oippoletti weir. a

CIPPOLETTl'S FORMULA.

Cippoletti derived his formula from a discussion of the experiments 

of James B. Francis, selecting a coefficient 1 per cent greater, making

Q=l X0.629Z^/2<7#=3.367Z^ .... (47)

L is the length of the crest or base of the trapezoid.

Flinn and Dyer 6 experimented at the testing flume of the Holyoke 

Water Power Company by passing the same volume of water successively over a trapezoidal experimental weir and over the gaging weir 

of the turbine testing flume 19.7 feet downstream. The latter, it is 

stated, complied in form with Francis's specifications.

The depths were observed by hook gage; eleven readings, as a rule, 

being taken and their arithmetical mean used for the determination of 

a head. The thirty-two series of valid experiments range from 0.3 

foot depth on a weir with sill length of 3 feet to a head of 1.25 feet on 

a sill 9 feet long.

The discharge over the standard weir was calculated by the formulas 

of J. B. Francis and of Hamilton Smith. The correction for velocity 

of approach at the experimental weir was made by the formula of 

Hamilton Smith, for use with contracted rectangular weirs,

Flinn and Dyer's coefficients are as follows:

Mean of 32 experiments, (7=3.283 

Mean after rejecting 5 diminished weights, (7=3.301

In general, the coefficient diminished as the head increased, suggesting that the end inclination should slightly exceed ^H in the Cippoletti

4:

weir, to provide complete compensation, and that the end contraction

"First described by Cesare Cippoletti in Canal Villoresi, Modulo per la Dispensa delle Acque, 1887. 

b Flinn, A. D., and Dyer, C. W. D., The Cippoletti trapezoidal weir; Irons, Am, goe. C. E., vol. 32, 

1894, pp. 9-33,
WEIE GAGINGS. 49

coefficient in the trapezoidal weir may be greater than QAIf, as used 

by Francis.

The question is complicated by velocity of approach. For example, 

had the Francis velocity-correction formula been used by Flinn and 

Dyer, their values of C would have been larger. As a tentative conclusion it is probable that the application of either the Francis formula with his velocity-head correction or the Flinn and Dyer coefficient 

with the Smith velocity correction will, when applied to a Cippoletti 

weir, give results as accurate as the precision of the coefficients will 

justify.

REQUIREMENTS AND ACCURACY OF WEIR GAGINGS.

PRECAUTIONS FOR STANDARD WEIR GAGING.

Certain specifications were laid down by James B. Francis as guides 

in cases where the utmost precision is desired in weir measurements. a 

The limits of applicability of the weir have been greatly extended 

since 1852, and some of the uncertainties as to the effect of various 

modifications of weir construction have been removed.

In general, for standard thin-edged weirs 

1. The upstream crest edge should be sharp and smooth.

2. The overflowing sheet should touch only the upstream crest 

corner.

3. The nappe should be perfectly aerated.

4. The upstream face of the weir should be vertical.

5. The crest should be level from end to end.

6. The measurements of head should show the true actual elevation 

of water surface above the level of the weir crest.

7. The depth of leading channel should be sufficient to provide complete crest contractions, and, if they are not suppressed, the width of 

channel should be sufficient to provide complete end contractions.

8. A weir discharging from a quiet pond is to be preferred. If this 

is not available, the velocity of approach in the leading channel should 

be rendered as uniform as possible and correction made therefor by 

the method employed by the experimenter in deriving the formula.

In order to fulfill these requirements, certain secondary conditions 

are necessary. The depth on the weir should be measured at a point 

far enough upstream from the crest to be unaffected by the surface 

curvature, caused by the discharge.

"Francis, J. B., Lowell Hydraulic Experiments, pp. 133-136.
50 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

The distance upstream to the point of measuring the head has been 

as follows:

Distance upstream from weir to gage used by various experimenters.

Experimenter.

Poncelet and Lesbros ..................................

Lesbros ..-.-.-._.-._._.__........._....-.-....----.-.

Francis ............................ ...---.-........-.

Hamilton Smith, jr.......... ..........................

Fteley and Stearns ....................................

Bazin ................................................

Date.

1828

1834

1852

1874-1876

1878

1886

Distance 

upstream, in 

feet.

11.48

11.48

6.00

7.60

6.00

16.40

Six feet upstream from crest is a distance frequently used, but this 

may be insufficient for suppressed weirs, and also for those having 

irregular cross sections or upstream slopes. Boileau considered the 

origin of the surface curvature to be at a distance from the weir equal 

to about 2.5 times the height of crest above the bottom of the channel 

of approach, indicating that for a suppressed weir the head should 

be measured at least this distance from the crest. a For a weir discharging from a still pond the head can be measured at any considerable distance from the weir. Hamilton Smith 6 states that, for weirs 

with full contraction, II can be measured at any convenient point from 

4: feet to 10 feet from the crest.

The head may be measured, directly by a graduated scale or hook 

gage, or by means of a piezometer tube having its orifice flush with 

the side wall of the leading channel, and at right angles to the direction of flow of the water.

The depth of the leading channel in Francis's experiments was 4.6 

feet below crest, and Francis lays down the rule that the depth of the 

leading canal should be at least three times the head on the weir. 

Hamilton Smith fixes the minimum depth of the leading channel below 

the crest at 2ZT.

Fteley and Stearns c state that the depth of the leading channel 

below weir crest should be at least 0.5 foot, in order that correction 

for velocity of approach may be reliably made for depths occurring in 

their measurements, and that a greater depth of leading channel is to 

be preferred.

To provide complete end contractions, Francis states that the distance from the side of the channel of approach to the end of the weir 

overflow should be at least equal to the depth on the weir. Hamilton

aTteley and Stearns, Experiments on the flow of water, etc.: Trans. Am. Soc. C. E., vol. 12, p. 47. 

& Smith, Hamilton, Hydraulics, pp. 129-131. 

c Ibid., pp 112-U4.
WEIE GAGINGS. 51

Smith considers that the distance from the end of the weir to the side 

of the channel should be at least 2^, and that the depth of channel 

below crest, also the side distance, should in no case be less than 1 

foot. Francis further specifies that the length of weir crest should 

be at least three times the depth of overflow. The nappe should not 

be allowed to expand laterally immediately below a suppressed weir. 

In order that the nappe may be perfectly aerated, Francis considers 

that the fall below crest level on the downstream side should be not 

less than \H, increasing for very long weirs or in cases where the 

downstream channel is shallow. He found, however, no perceptible 

difference in the discharge fora head of 0.85 foot, whether the water 

on the downstream side was 1.05 feet or 0.0255 foot below crest level. 

Fteley and Stearns and Hamilton Smith agree that, if the water is

' Crest 4. 6 above 

main channel

FIG. 6. Sections of the Francis weir. A, General section of weir; B, detail of crest.

deep below, it may rise to crest level on downstream side of weir 

without sensible error, and Fteley and Stearns add that a weir may be 

submerged to a depth of 15 per cent of the head without an error 

exceeding 1 per cent.

The thickness of crest lip is immaterial so long as the edge is sharp 

and square and the nappe cuts free and is freely aerated. The latter 

conditions require, however, that the crest shall be thin, especially 

where the head is slight.

Fig. 6 shows cross sections of the crest of the weir used by James B. 

Francis at the Lower Merrimac locks at Lowell, in 1852, in deriving 

his formula. The crest consisted of a cast-iron plate 13 inches wide 

and 1 inch thick, planed true and smooth on all surfaces. Its upper
52 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

edge was chamfered on the downstream side at an angle of 45° to a 

thickness of 0.25 inch at the edge. As shown, the nappe cut 

clear from the top of the crest in an unbroken sheet. The lowest 

head used by Francis was over 0.5 foot. For very low heads the crest 

lip should be thinneT. A wooden crest tends, by capillary attraction, 

to cause the nappe to adhere to the flat top surface under low heads. 

A wooden crest is cheap, easily adjusted, and convenient for temporary use, but it will, in time, tend to become somewhat rounded, 

reducing the vertical contraction of the nappe.

A cast-iron crest will usually have to be made to order. A large 

steel angle bar may often be obtained from stock sizes of the rolling 

mills more cheaply. Such a bar, with legs, say 3 and 6 inches, 

respectively, with the 6-inch flat face planed and its edge trued, will 

form a rigid and permanent crest. The 3-inch leg may be bolted to 

the top of the timbers forming the body of the weir.

It may be added that approximate corrections for rounding of 

upstream corner of the crest, inclination of the weir upstream or 

downstream, or incomplete contractions can be made from data now 

available. In constructing gaging weirs preference, however, should 

be given to those forms which render the determination of the discharge the most simple, and the extent to which the preceding specifications may be departed from judiciously will depend upon the exigencies of the case and the purposes for which the results are desired.

PLANK AND BEAM WEIRS OF SENSIBLE CREST WIDTH.

Experiments on weirs with crest boards 1, 2, or 4 inches in thickness 

were made by Blackwell, Fteley and Stearns, and Bazin. The results 

show that for depths exceeding 1.5 to 2 times the crest width the 

nappe will break free, and if properly aerated the coefficient will then 

be identical with that for a thin-edged weir.

When the nappe adheres to the crest the coefficients are very uncertain for such weirs, adhesion of nappe to downstream face of crest 

and modified aeration entering to give divergent values.

The precise stage at which the change from an adhering to a free 

nappe or the reverse occurs is not constant, but varies with velocity of 

approach and with rate of change of the head as the changing point is 

approached, being different for a sudden and for a gradual change, and 

also when the point of change is approached by an increasing as compared with a decreasing head.

REDUCTION OF THE MEAN OF SEVERAL OBSERVATIONS OF HEAD.

In measuring a constant volume of water, several observations of 

the head on the weir are desirable, the accuracy of the result, 

according to the theory of least squares, being proportional to the 

square root of the number of observations.
WEIB GAGINGS. 53

In weir experiments it is often impossible to maintain a perfectly 

uniform head or regimen. If the variations are minute the arithmetical mean may be used directly. If the variations are of wider range, 

or if the utmost precision is required, the following correction 

formula of Francis may be applied: a

Let Z>j, .Z>2 , DM etc., Dn represent the several successive observed

heads. 

Let £j, £2 , £3 , etc., tn represent the corresponding intervals of time

between the several observations. 

Let ^represent their sum, or the total time interval. 

$=the total volume of water flowing over the weir in the time T. 

Z>=the mean depth on the weir that would discharge the quantity

Q in the time T. t 

Z=the length of weir crest. 

C=ib.Q weir coefficient. 

We have, very nearly,

Also,

Equating, eliminating the common factor CL, and solving forZ>, we 

have

J)= JT (I

EFFECT OF ERROR IN DETERMINING THE HEAD ON WEIRS. 6

Consider the formula

Differentiating, we have

The error of any gaging when H-\-dH\s, taken as the head instead 

of the true head H being used will be dQ, and the ratio of this quantity to the true discharge Q will be

This formula will give nearly the correct value of the error if the 

increment dH approaches an infinitesimal.

« Francis, J. B., Lowell Hydraulic Experiments, p. 113.

ftRafter, G. W., On the flow of water over dams: Trans. Am. Soc. C. E., vol. 44, p. 686; data here 

given based on discussion by Walter C. Parmley.
54 WEIK EXPEBIMENTS, COEFFICIENTS, AND FOBMULAS.

In the following table is shown the effect of errors of one thousandth, five thousandths, one hundredth, and five hundredths foot, 

respectively, for various heads. This clearly illustrates both the 

necessity of proper care and the folly of ultra precision in measuring 

the relatively large values of H with which we are mainly concerned. 

The curves of error on PL II are equilateral hyperbolas, which have 

been reduced to straight lines by plotting on logarithmic scales.

Percentage error in discharge resulting from various errors in tlie measured head on weirs.

Head, in 

feet.

0.1 

.5

1.0

5.0

10.0

Error in measured head, in feet.

0.001

Per cent. 

1.5 

.3

.15

.03

.015

0.005

Per cent. 

7.5 

1.5

.75

.15

.075

0.01

Per cent. 

15 

3

1.5

.3

.15

0.05

Per cent

15

7.5

1.5

.75

An error of a half-tenth foot under 5 feet head causes the same 

error in the result as an error of one-half hundredth foot with a head 

of one-half foot.

In weir experiments it is important to know the effect of an error 

in head lion the resultant coeflicient of discharge C. The error in C 

is evidently equivalent to the error in Q found above, where H is 

constant.

ERROR OF THE MEAN WHERE THE HEAD VARIES.

In determining the volume of flow over dams where gaging records 

are kept, the method usually pursued has been to have readings taken 

twice daily, as at morning and evening, showing the depth flowing 

over the crest of the dam. The average of the two readings for each 

day has been found and the volume of flow corresponding to this 

average head has been taken as the mean rate of flow over the dam 

for the day.

It is evident, however, that as the discharge varies more rapidly 

than the head (usually considered to be proportional to the three-halves 

power of the head), the volume of discharge obtained as above 

described will be somewhat less than the amount which actually passes 

over the dam. The following analysis has been made to show the 

magnitude of the error introduced by using the above method.

Assuming that the initial depth on the crest of the dam is« zero, but 

increases at a uniform rate to H^ at the end of a time interval T7, the
U. S. GEOLOGICAL SURVEY 

10 

1

5

6 

7

8 

9 

[easured head

a

o ^-jbo

.3

A

.5 

so b

S1 s

^

5

v-

*

S5 g

S,

s1-

5

v 

^

£

§

>- < 

, . °. <

("f,

v .'?/>

V -

§ ss

ins^=

!

X

S

\ 

 ^

V

(O

1§

p

^

^

=

e

S

N 1.4

S

\

d 

(

s

A

ic

e 

v

\

^

\

3

d

 ^

\

^

J

«

c

^

S

^

^.

m

°o

\

1 £ S

Per cent of error in discharge. 

EFFECT OF ERRORS IN WEIR MEASUREME
WEIR GAGINGS. 55

mean head deduced from observations at the beginning and end of the 

period would be i II " the head at any time t would be

where f is a constant.

We may write the usual formula for weir discharge Q= 

then, if the head varies from zero to H^ the total volume of flow in 

the time Twill be

. . . (50)

The total discharge corresponding to the average head ~ Hv is

The ratio of the discharge is

Volume by average head <gav 

Actual volume t

=0.8840 (52)

It appears that where the initial or terminal head is zero the volume 

of flow determined by using the average head will be 11.6 per cent 

too small. This percentage of error is the same whatever may be the 

maximum head H, and whether the stream is rising or falling. It is 

also independent of the rate of change in the head.

Conditions like those above discussed occur at milldams during the 

season of low water, when the pond is allowed to fill up at night and 

the water is drawn down to crest level or below during the day when 

mills are running.

The following example will illustrate. Suppose a sharp-crested 

weir without end contractions, with crest 1 foot long, on which the 

water rises to a depth of 1 foot in a period of 10 seconds 

Mean depth on a weir with varying head.

Head, in feet, at end of each second. . .

I

.1 

.05

.05

9

.1

.15

3

.3 

.15

.25

4

.4

2

.35

5

.5 

.25

.45

6

.6 

.3

.55

7

.7 

.35

.65

8

.8 

.4

.75

9

.9

.45

.85

10

1.0

.5

.95
56 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

Using the average head during each second the volume of flow may 

be approximately integrated by finite differences, as follows, the discharge being taken from Francis's tables:

Discharge over a weir with varying head.

Time, in seconds.

0 to 1

1 to 2

2 to 3

3 to 4

4 to 5

5 to 6

6 to 7

7 to 8

8 to 9

9 to 10

Total .......

Average head, 

in feet.

0.05

.15

.25

.35

.45

.55

.65

.75

.85

.95

Discharge, in 

second-feet.

0. 037

.194

.416

.690

1.005

1.358

1. 745

2.163

2.609

3.083

13.30

The average head for the entire period, 0.5 foot, gives a discharge 

for 10 seconds of 11.773 second-feet, or 88.5 per cent of that given 

above, the numerical result agreeing closely with that obtained by 

analysis. The volume of flow from average head equals seven-eighths 

of the true integral volume of flow, approximately.

If there is an initial head IT0, then when the head varies uniformly,

= CL 

The total volume of flow in time T will be

T f*T

Qdt=CL I (H0+f$dt=

Jo

The average head during time Tis

The total volume of flow corresponding to this head is
WEIR G AGINGS. 57

The ratio of the actual or integral discharge to the discharge by the 

average head is

(

JT -\-- fTi® T 

______ _____ _______ ° % J ________ ,~o\

Integral volume ~^ \ C H + fT^ H~

The value of this ratio is independent of the coefficient or length of 

weir, but varies with the rate of change of head.

WEIR NOT LEVEL.

If the crest of a gaging weir is not truly horizontal, but is a little 

inclined, the discharge may be closely approximated by the use of the 

average crest depth If in the ordinary formula, or more precisely by 

the formula below, applicable also to weirs of any inclination.

FIG. 7. Inclined weir

The flow through the elementary width dl is 

dQ= GIF til

rL s

Total discharge^ Q- \ CH*dl=C

J(> . 

Integrating,

2<7Z

^l\dl

(54)

In this formula either the mean coefficient deduced by Thomson

2

(see p. 46) for a triangular weir, in which - 6'== 1.32, or that of Frano

o

ois, in which G =1.332, may be used. If there are end contractions, 5

the net length,

Z=Z'-0.2 

should be used.
58 WETE EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

The discharge using the average head,

«- 2

is

(55)

The extent of variation from the true discharge resulting from the 

use of formula (55) in place of the integral formula (54) is illustrated 

by the following:

Let Z=10 feet, Ha=\.Q foot, ^ 6"= 1.332.

Discharge by (55) for average head = 33.30 cubic feet per second. 

If Hz Hv 0.01 foot true discharge, $=33.30 cubic feet per second. 

If Hz Jfi^ = 0.10foot true discharge, $=33.30 cubic feet per second. 

If HZ H^=- 0.50 foot true discharge, $=33.54 cubic feet per second.

In general, since the discharge varies more rapidly than the head, 

the effect of calculating the discharge from the average head will be 

to give too small discharge, the error increasing with the variation in 

crest level.

Hence the discharge obtained by using the average crest level for a 

weir having an inclined or uneven crest will be somewhat deficient. 

The magnitude of the variations in height of the crest will determine 

whether the average profile can be used or whether the crest should be 

subdivided into sections, each comprising portions having very nearly 

the same elevation (whether adjacent or not), and the discharge over 

each section computed as for a separate weir.

In general it may be stated that the error in the value of Q1 

increases directly in proportion as the ratio of the difference in the 

limiting heads to the average head is increased.

CONVEXITY OF WATER SURFACE IN LEADING CHANNEL.

If there are wide variations in velocity in the measuring section, the 

level of the water surface may be affected, since water in motion 

exerts less pressure than when at rest.

Conditions of equilibrium cause the swift-moving current to rise 

above the level of the slower-moving portions. If the head is measured near still water at the shore, the result may be slightly too small.

The difference in height a may be expressed in the form,

.2__ 2

A-A^-^rr1 ....... (56)

The coefficient r is often assumed equal to unity, but evidently 

varies with the distribution of velocities whose resultant effect it 

measures.

a Humphreys and Abbot, Physics and Hydraulics of the Mississippi River, 1876, p. 320,
WEIK EXPERIMENTS, COEFFICIENTS, AND FOEMULAS. 59

RESULTS OF EXPERIMENTS ON VARIOUS FORMS OF WEIR

CROSS SECTIONS.

THE USE OF WEIRS OF IRREGULAR SECTION.

Many cases arise where it is desired to estimate, approximately, at 

least, the flow over dams of peculiar cross section.

The construction of so-called standard or thin-edged weirs that 

shall be permanently useful to measure the flow of large and variable 

streams is so difficult and expensive as to be frequently impracticable. 

Existing milldams often afford a convenient substitute. In the 

following pages are presented the results of the leading experiments 

to determine proper coefficients for '' irregular" weirs, followed by a 

grouping of experiments on similar models, whether all by one experimenter or not. The data are not always as complete or consistent as 

could be desired, but the need for fair working coefficients is very 

great, and, in the line of making use of all the available information, 

the several diagrams of comparison and the conclusions therefrom are 

presented, with the understanding that these are not final, although it 

is quite certain that the laws of coefficient variation are correctly outlined by the data at present available, and they form, therefore, a safe 

working hypothesis.

Weir models of irregular section are calibrated in order that existing dams of similar cross section may be used for stream gaging. It 

becomes necessary to calibrate the experimental models for a wider 

range of heads than has commonly been emploj^ed in experiments on 

standard thin-edged weirs, in order that the range of rise and fall of 

the stream from low water to high may be included.

While the recent experimental data include heads as great as from 4 

to 6 feet, j^et it is often necessary to determine the discharge for still 

greater heads, and experiments on certain forms with heads up to 10 

or 12 feet are needed.

In this connection the greater relative facility of securing accurate 

results with weirs for high than for low heads may be noted.

The proportional error resulting from variations in crest level, as 

well as uncertainties as to the nappe form and consequent value of the 

coefficient, largely disappear as the head increases. The effect of form 

of crest and friction is also relatively diminished. It is probably true 

that the coefficients for many ordinary forms of weir section would 

tend toward a common constant-value if the head were indefinitely 

increased. The above facts render milldams especially useful for the 

determination of the maximum discharge of streams. Dams can be 

used for this purpose when the presence of logs and drift carried 

down by the flood preclude the use of current meters or other gaging 

instruments.
60 WEIE, EXPERIMENTS, COEFFICIENTS, AND FORMULAS. 

MODIFICATIONS OF THE NAPPE FORM.

The elaborate investigations of Bazin relative to the physics of weir 

discharge set forth clearly the importance of taking into consideration 

the particular form assumed by the nappe. This is especially true in 

weirs of irregular section in which there is usually more opportunity 

for change of form than for a thin-edged weir. In general the nappe 

may 

1. Discharge freely, touching only the upstream crest edge.

2. Adhere to top of crest.

3. Adhere to downstream face of crest.

4:. Adhere to both top and downstream face.

5. Remain detached, but become wetted underneath.

6. Adhere to top, but remain detached from face and become wetted 

underneath.

7. In any of the cases where the nappe is " wetted underneath" this 

condition may be replaced by a depressed nappe, having air imprisoned 

underneath at less than atmospheric pressure.

The nappe may undergo several of these modifications in succession 

as the head is varied. The successive forms that appear with an 

increasing stage may differ from those pertaining to similar stages 

with a decreasing head. The head at which the changes of nappe form 

occur vary with the rate of change of head, whether increasing or 

decreasing, and with other conditions.

The law of coefficients may be greatly modified or even reversed 

when a change of form takes place in the nappe.

The effect of modifications of nappe form on various irregular weir 

sections is shown in PL III. The coefficients are those of Bazin and 

include velocity of approach. The coefficient curve for any form of 

weir having a stable nappe is a continuous, smooth line. When the 

nappe becomes depressed, detached, or wetted underneath during the 

progress of an experiment, the resulting coefficient curve may consist 

of a series of discontinuous or even disconnected arcs terminating 

abruptly in "points cFarret" where the form of nappe changes. The 

modifications of nappe form are usually confined to comparatively low 

heads, the nappe sometimes undergoing several successive changes as 

the head increases from zero until a stable condition is reached beyond 

which further increase of head produces no change. The condition 

of the nappe when depressed or wetted underneath can usually be 

restored to that of free discharge by providing adequate aeration.

The weir sections shown in PI. Ill are unusually susceptible of changes 

of nappe form. Among weirs of irregular section there is a large 

class for which, from the nature of their section, the nappe can assume 

only one form unless drowned. Such weirs, it is suggested, may, if 

properly calibrated, equal or exceed the usefulness of the thin-edged
U. 8. GEOLOGICAL SURVEY 

3.40 8.50 3.60 3.70 3.80 3

Value of the coefficient C=mV%) (m being the original Ba

VARIATIONS IN WEIR COEFFICIENT WITH CHANGE 
WEIKS OF IEEEGULAE SECTION. 61

weir for purposes of stream gaging, because of their greater stability 

of section and because the thin-edged weir is not free from modification of nappe form for low heads.

As an example, Bazin gives the following coefficients applying to 

a thin-edged weir 2.46 feet high, with a head of 0.656 foot, under 

various conditions:

Condition of nappe.

Free discharge, full aeration .....................

Nappe depressed, partial vacuum underneath .....

Nappe wetted underneath, do wr stream water

Nappe adhering to downstream face of weir, ressault at a distance .............................

Bazin coefficient m.

0.433

.460

.497

.554

C=mV2g

3.47

3.69

3.99

4.45

Per cent of 

the Francis 

coefficient.

104.1

110.7

119.7

133.5

These coefficients include velocity of approach effect, which tends 

to magnify their differences somewhat. There is, however, a range 

of 25 per cent variation in discharge between the extremes. a

The departure in the weir coefficient from that applying to a thinedged weir, for most forms of weirs of irregular section, results from 

some permanent modification of the nappe form. Weirs with sloping 

upstream faces reduce the crest contraction, broad-crested weirs cause 

adherence of the nappe to the crest, aprons cause permanent adherence 

of the nappe to the downstream face.

EXPERIMENTAL DATA FOR WEIRS OF IRREGULAR CROSS

SECTION.

The only experiments on irregular or broad-crested weirs in which 

the discharge has been determined volumetrically are those of Blackwell on weirs 3 feet broad, of Francis on the Merrimac dam, and of 

the United States Geological Survey for lower heads, on various forms 

of section. So far as the writer is aware, all other such experiments 

have been made by comparison with standard weirs.

In the following pages are included the results of the experiments 

of Bazin on 29 forms of cross section; also those of the United States 

Deep Waterwa3T s Board under the direction of George W. Rafter, and 

those of John B. Freeman at Cornell University hydraulic laboratory. 

The results of 20 series of experiments, chiefly on weirs with broad and 

ogee crest sections, made under the writer's direction at Cornell Universit}" hydraulic laboratory, are here for the first time published.

afiazin's general discussion of the above and other modifications of the coefficient has been translated by the writer, and may be found in Rafter's paper, On the flow of water over dams: Trans. 

Am. Soc. C. E., vol. 44, pp. 254-261.
62 WEIE EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

As it has been necessary to reduce the experimental data to a uniform basis for purposes of comparison the original data, together with 

the results obtained by recalculation, have been included for theBazin, 

United States Deep Waterways, and Freeman experiments.

BASE FORMULA FOR DISCHARGE OVER WEIRS OF IRREGULAR CROSS

SECTION.

Precedent to the opening of the hydraulic laboratory of Cornell 

University the most elaborate experiments on weirs of irregular cross 

section were those of Bazin. His experiments were all reduced in 

such manner as to include the velocity of approach correction in the 

discharge coefficient.

In America the formula most commonty used is that adopted by 

James B. Francis, in which velocity of approach is eliminated from 

the coefficient by correcting the head, thus reducing the conditions as 

nearly as possible to the basis of no velocity of approach before applying the formula.

In order to render Bazin's results comparable with the later experiments, it has been necessary to adopt a standard or base formula to 

which all the experiments should be reduced. The considerations leading to the adoption of the formula of Francis here used are given 

below.

In the process of gaging streams at dams the head is usually 

measured in comparatively still water in an open pond. This condition could not be duplicated in the Cornell experiments. As the 

formula of James B. Francis is most simple in form for the case of a 

weir with no velocity of approach, and as it is often convenient to 

compare the discharge over a dam with that for a thin-edged weir of 

standard form, a weir formula of the base form used by Francis has 

been adopted in reducing the experiments. In this formula,

Q= CLH*.

Z=Length of crest corrected for end contractions, if any.

^7= Head on weir crest corrected for velocity of approach by 

the Francis correction formula or an equivalent method.

C A coefficient determined from experiments on a model dam. 

In this connection it may be remarked that the formula of Bazin 

includes the correction for velocity of approach in the weir coefficient; 

hence the coefficient for a given weir is comparable only with that for 

another weir under the same head when the velocity of approach is 

the same in both cases. Bazin's formula also expresses the velocity of 

approach implicitly by means of the depth and breadth of the leading 

channel. In actual gagings the leading channel is often of irregular 

forai, hence it becomes necessary to eliminate the depth and breadth 

of the channel from the formula.
WEIRS OF IRREGULAR SECTION, 63

There is considerable variation in the magnitude of the correction 

for velocity of approach used by different experimenters. As a rule, 

the velocity of approach is negligible at gaging stations at dams. It 

became necessary, therefore, in reducing these experiments to determine from the measured discharge and observed head what the head 

would have been had the same discharge taken place over a weir 

in a still pond. To accomplish this the formula for correction for 

velocity of approach adopted by James B. Francis has been used. 

This being the case, it is to be noted that in applying the coefficients, 

which, as given, have been reduced as nearly as possible to the basis 

of no velocity of approach, the same method of velocity correction 

must be used, and if it is used no error will result where the actual 

velocity of approach is nearly the same as that which occurred in the 

experiments.

BAZIN'S EXPERIMENTS ON WEIRS OF IRREGULAR CROSS SECTION.

These include a wide variety of forms, m&ny of which will seldom 

be found in America, and the use of which for purposes of gaging 

would be ill advised.

The small size of the models used, high velocity of approach, and 

narrow range of heads covered, limit the application of these results. 

No effort has been made to present all the results in this paper/' Certain series, useful for comparison, have been recomputed as described 

below, and by grouping similar sections we ma}^ determine the general effect of various slope and crest modifications.

BAZIN'S COKRECTION FOR VELOCITY OF APPROACH.

The base formula for weir discharge adopted by Bazin and the 

method of taking into account the velocity of approach are described 

in connection with his experiments on thin-edged weirs (p. 31).

The following discussion shows the complex character of the Bazin 

coefficients, and the fact that they do not express directly the relative 

discharging capacity of weirs of irregular section.

The effect of velocity of approach is to increase the discharge at a 

given observed head, .Z), over what it would be if the same head were 

measured in still water, as in a deep, broad pond.

Bazin's coefficients in the form published are not readily applicable 

in practice to weirs of other heights, or to weirs in ponds, or otherwise 

to any but weirs in restricted channels of the depth and width of the 

weir.

a For complete original data, see Bazin, as translated by Marichal and Trautwine in Proc. Engineers 

Club Phila., vol. 7, pp. 2^9-310; vol. 9, pp. 231-244, 287-319: vol. 10, pp. 121-164; also numerous experiments reduced to English units by Rafter and others, Trans. Am. Soc. C. E., vol. 44, pp. 220-398.
64 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

The Bazin coefficients as published may be considered as comprising two principal factors. JIT being the Bazin coefficient, we may write

M=FC.

F velocity of approach effect. 

C contraction effect.

Bazin uses a correction formula for velocity of approach, derived 

from the expression

Consider a standard weir and experimental weir both of the same 

height, but of different form, the measured depth being the same, and 

the Bazin coefficients being M and m, the velocity of approach and 

discharge T^and v and Q and ^, respectively, and C and C^ the coefficients in a formula in which the velocity of approach correction is 

eliminated from the coefficient and applied to the head; then the discharge for the standard weir would be,

using the Bazin coefficients,

where M' = M< %g and Z=1.0; 

using the coefficient 6",

taking roots

C D

\M'J ~D+aK

Bazin does not give the quantities of flow in the tables of results of 

his experiments, hence to determine h it is necessary to calculate Q. 

<y, and h from the known values M and D and from JP, the height of 

weir.

D being the same for both the standard and the experimental weirs, 

we have for the experimental weir

<7

D

Ol being the coefficient for the experimental weir, and Ax the velocity 

head.
WEIKS OF IEEEGULAE SECTION. 65 

Hence, by multiplication,

D 

and

^y>+«//'

or

The velocity of approach for a given depth on a weir is proportional 

to f, hence, since h is proportional to ?'2 , we have

Hence,

M

//1' The ratio - used by Bazin is not, therefore, precisely a measure of

the relative discharging capacities of the two weirs under similar conditions of head and velocity of approach, for the reason that the 

velocity of approach will not be the same for both weirs if the Bazin 

coefficients are different. The ratio ml Mis made up of two factors, 

one of which, OJ C, expresses the absolute relative discharging capacities of the two weirs under similar conditions of head and velocity of 

approach, and the other expresses the effect of the change in discharging capacity on the velocity of approach for a given depth on a weir 

of given height.

Thus the coefficient M for any weir has, by Bazin's method of reduction, different values for every depth and for every height of weir 

that may occur.

O For reasons elsewhere stated it is preferred to express by -^ only ^i

the relative discharging capacities of the weirs where the velocity of 

approach is the same in both. It is then practically a measure of the 

vertical contraction of the nappe, and is constant for a given head for 

any height of weir, and may be sensibly constant for various depths 

on the weir.
66 WEIK EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

RECOMPUTATION OF COEFFICIENTS IN BAZIN's EXPERIMENTS.

In reporting the results of his experiments on weirs of irregular 

section, Bazin gives the observed heads on the standard weir of comparison, the absolute coefficient in applying for each depth on the 

experimental weir and the ratio ml Moi the experimental and standard 

weir coefficients.

The results give coefficients which strictly apply only to weirs having 

both the same form of section and the same heights as those of Bazin. 

Although weirs of sectional form geometrically similar to Bazin's are 

common, yet few actual weirs have the same height as his. There 

appear to be two elements which may render inaccurate the application of Bazin's absolute coefficients to weirs of varying height: (1) The 

difference in velocity of approach; (2) the difference in contraction of 

the nappe for a higher or lower weir.

In order to render the results of Bazin's experiments comparable 

one with another and with later experiments, a number of series have 

been recomputed, the velocity of approach being treated in the same 

manner as in the computation of experiments at Cornell hydraulic 

laboratory.

The method is outlined below, the references being to the tables of 

Bazin's experiments given on pages 68 to 81.

Column 2 gives the observed head reduced to feet for the experimental weir. _

Column 4 the absolute coefficient C-^ m, >/% 

(These have been reduced from Bazin's original tables.)

Column 5 gives the discharge per foot of crest over the experimental 

weir calculated by the formula

Q=mLD

quantities in column 3 being taken directly from a table of three-halves 

powers.

In column 6 the actual velocity of approach, /?'=~TT^7), is given,

v* 

and in column 7 the velocity head, h=^-.

The dischargs over the standard weir was calculated by Bazin by 

using his own formula and velocity of approach correction. He does 

not give the discharge, however, and we have been obliged to work 

back and obtain it from the data given for the experimental weir.

Having determined the actual discharge and the observed head, we 

are now at liberty to assume such a law of velocity of approach correction in deducing our new coefficients as we choose. We will therefore deduce the coefficients in such form that when applied to a weir
U. S. GEOLOGICAL SURVEY

f'flact.O.33'-

WATER-SUPPLY PAPER NO. 150 PL. IV

Round crest

.1 .8 .9 1.0 

Corrected head in feet.

1.3 1.3 1.4 1.5

EXPERIMENTS OF BAZIN ON BROAD-CRESTED WEIRS. 

Velocity-of-approach correction by the Francis method.
IJ. S. GEOLOGICAL SURVEY WATER-SUPPLY PAPER NO. 150 PL. V

Slope 10' !, 

'Series 142

0 .1 .8 .3 .4 .5 .6 .7 .8

EXPERIMENTS OF BAZIN ON WEIRS OF TRIANGULAR SECTION WITH VARYING

DOWNSTREAM SLOPE. 

Ve(ocity-of-approach correction by the Francis method.
U. S. GEOLOGICAL SURVEY WATER-SUPPLY PAPER NO. 150 PL. VI

Coefficient 

C. Corrected head In feet. 

.1 .2 .3 A .5 .6 .7 .8 .9 1.0 1.1 1.2 1.3 1.4 1.5

SERIES ©3ty

^

r&

M-S\

<^J|iZ-_

®-

&

/S/

0 .1 .2 .8 .4 .5 .6 .7 .8 .9 1.0 1.1 1.8 1.3 1.4 1.5

Corrected head in feet.

EXPERIMENTS OF BAZIN ON WEIRS OF TRAPEZOIDAL SECTION WITH VARYING

DOWNSTREAM SLOPE. 

Ve I ocity-of-approach correction by the Francis method. (See also PI. VII.)
U. 8. GEOLOGICAL SURVEY WATER-SUPPLY PAPER NO. 160 PL. VII

Coefficient 

C. Corrected head in feet.

.7 .8 .9 1.0 1.1 1.2 1.3 1.4 1.5

EXPERIMENTS OF BAZIN ON WEIRS OF TRAPEZOIDAL SECTION WITH VARYING

DOWNSTREAM SLOPE. 

Velocity-of-approach correction by the Francis method. (For cross section see Pi. VI.)
U. S. GEOLOGICAL SURVEY 

" -7.3/7-'

WATER-SUPPLY PAPER NO. 150 PL. VIII

Coefl 

cien 

C. 

0 

3.3

3.2 

3.1 

3.0 

2.9 

2.8

Jt 

Corrected head in feet. 

.1 .2 .3 .4 .5 .6 .7 .8 9. 1.0 1.1 1.2 1.3 1.4 1.5

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0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 1.1 1.2 1.3 1.4 1.5 1.6

Corrected head in feet.

EXPERIMENTS OF BAZIN ON WEIRS OF TRAPEZOIDAL SECTION WITH VARYING 

DOWNSTREAM SLOPE.

IRK 150 06 6 Velocity-of-approach correction by the Francis method.
U. S. GEOLOGICAL SURVEY WATER-SUPPLY PAPER NO. 150 PL. IX

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EXPERIMENTS OF BAZIN ON WEIRS OF TRIANGULAR SECTION WITH VARYING 

DOWNSTREAM SLOPE.

Velocity-of-approach correction by the Francis method.
U. S. GEOLOGICAL SURVEY

Coefficient 

C.

NO. 180 PL. X

3.6 

3.5

Slope of upstream face. Run for unit rise. 

MEAN CONSTANT COEFFICIENTS FOR VARYING SLOPE OF UPSTREAM FACE.

J63

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EXPERIMENTS OF BAZIN ON WEIRS OF TRIANGULAR SECTION WITH VARYING 

UPSTREAM SLOPE.

Velocity-of-approach correction by the Francis method.
U. S. GEOLOGICAL SURVEY

Coefficient

a

0 .1 .2 .3 .

WATER-SUPPLY PAPER NO. 150 PL. XI

Corrected head in feet. 

.5 .6 .7 .8 .9 1.0 1.1 Ut 1.3 1A l£

TRIES /7,

Upst -earn r/qoe

Jf.®

m

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SERIES 151

slipt

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.1 .2 .3 .5 .6 .7 .8 .9 1.0

Corrected head in feet. 

EXPERIMENTS OF BAZIN ON WEIRS OF TRAPEZOIDAL

UPSTREAM SLOPE. 

Velocity-of-approach correction by the Francis method. (For c

1.1 1.2 1.3 1.4 1.5

SECTION WITH VARYING 

ross section see PI. XII.
U. S. GEOLOGICAL SURVEY

Coefficient 

C.

0 .1 .3 .3 .4

WATER-SUPPLY PAPER NO. 150 PL. XII

Corrected head in feet. 

.5 .6 .7 .8 .9 1.0 1.1 1.2 1.3 1.4 J.5

Ups >iope

7?

4

W

L/i istres. =rt 'cat

W

.4 .5 .6 .7 .8 .9 1.0 1.1 1.2 1.3 1.4 1.5 

Corrected head in feet.

EXPERIMENTS OF BAZtN ON WEIRS OF TRAPEZOIDAL SECTION WITH VARYING 

UPSTREAM SLOPE.

Velocity-of-approach correction by the Francis method. (See also PI. XI.)
WEIRS OP IRREGULAR SECTION. 67

in which there is velocity of approach we may apply the correction 

formula of Francis,

r~

H=\

A sufficient approximation to this formula for our present purposes 

may be obtained if we simply make

v is the velocity of approach corresponding to the trial discharge for the head Z>, no successive approximations being made, as 

^ould be necessary to determine the true head If by the Francis cor- 

~ection formula. . \

For example, in an extreme case, using a thin-edged weir

whence ff=D+-= ^ 1.0431

and Q 3.547.

By the Francis correction formula we h'nd, using three successive 

approximations,

& = 3.5183 giving v= 1.7591

$8 = 3.5387 giving ^=1.7694

^3 = 3.541 as the final discharge,

4 hat the difference is 0.11 of 1 per cent. We are therefore justified 

: n using this method to determine values of Oto two places decimals, 

or to within one-fourth to one-half per cent.

We have also used V^=8.02, as in the reduction of the Cornell 

experiments.

Column 8 gives the corrected head,

H=D+h. 

Column 10 gives the final coefficient C deduced by the formula

_ Q 

Pis. IV to XII show the resulting discharge coefficients.
68 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

Bazin's experiments on weirs of irregular section.

Bazin's Series, No. 86. 

Crest length, 6.55 feet. 

Crest height, 2.46 feet.

Cross section.

Period.

]

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

Observed

head. experimental

weir I),

in feet.

2

0. 1820

.2119

.2509

.2781

.3067

.3392

.3678

.4016

.4251

. 4527

.4770

.5075

.5360

.5639

.5973

a. 5304

a. 6032

a. 6347

1$

3

0. 0777

.0976

. 1257

.1466

.1701

.1974

. 2232

. 2549

. 2771

.3049

.3294

.3616

.3924

.4235

.4613

.3858

.4683

.5060

Ci

4

2. 7829

2. 8712

2. 9674

2. 9754

3.0957

3. 0957

3. 1839

3. 2321

3. 2641

3. 3283

3. 3844

3. 4406

3. 4807

3.5368

3. 5930

3. 4566

3. 4486

3.4646

Q, flow

per foot,

experimental 

weir, in

cubic feet

per

second.

5

0. 2160

2801

3

>rrvj 

/GO

.4369

. 5273

.6119

.7098

.8233

. 9033

1. 0153

1. 1137

1.2449

1. 3656

1. 4992

1. 6561

1. 3349

1.6156

1.7508

V

6

0.082

.105

.138

.159

.190

.218

.251

.288

.303

.350

.379

.420

.455

.4£6

.542

.446

.528

.566

v= 

2ff

7

0.0006

.0008

.0010

.0013

.0014

.0019

. 0022

.0027

.0033

.0039

.0045

.0031

.0044.

.0051

H

8

0. 1820

.2119

^.2509

'.2781

.3073

.3400

.3688

.4029

.4265

.4546

.4792

. 5102

.5393

.5678

.6018

.5335

.6076

.6398

ffi

9

0. 0777

.0976

.1257

.1466

.1701

.1983

.2241

. 2539

.2786

.3069

.3315

.3642

.3957

.4280

.4671

.3897

.4740

.5120

C

10

2.78

2.87

2.97

2.98

3.10

3.09

3.17

3.24

3.24

3.31

3.36

3.42

3.45

3.50

3.54

3.42

3.41

3.42

Bazin's Series, No. 89. 

Crest length, 6.55 feet. 

Crest height, 2.46 feet.

Cross section.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

0. 2079

. 2873

.3641

.4337

.4963

.5619

.6331

.6890

.7490

.7985

.8546

.9228

.9648

1.0236

1.0784

1. 1312

1.1866

1. 2375

1. 2959

a 1. 0807

a 1.1587

0. 0948

.1538

.2196

.2859

.3494

.4213

.5036

. 57195

.6482

.7135

.7906

.8867

.9479

1. 0362

1. 1193

1. 2028

1. 2932

1. 3767

1.4754

1. 1239

1.2477

2. 7669

2. 7669

2. 76C9

2. 8230

2. 8792

2.9674

3.0476

3. 0877

3. 1679

3.2160

3. 2962

3.3524

3. 4326

3.4887

3.5288

3. 5849

3.6331

3. 6732

3. 7052

3. 5368

3.5448

0. 2626

.4260

.6083

.8062

1. 0063

1.2513

1. 5360

1. 7673

2.0548

2. 2975

2.6090

2. 9704

3. 2513

3. 6163

3. 9511

4. 3060

4.6943

5. 0525

5. 4737

3. 9786

4. 4169

0.098

.155

.216

.279

.340

.414

.498

.5CO

.640

.705

.785

.879

.949

1.038

1.118

1.201

1.292

1.364

1.456

1.122

1.219

0. 0008

.0012

.0018

.0026

.0039

.0049

.0064

.0078

.0097

.0120

.0140

.0168

.0195

.0224

.0259

.0288

.0331

.0195

.0231

0. 2079

.2873

.3649

.4349

.4981

.5635

.6370

.6939

.7554

.8063

.8643

.9348

.9788

1.0404

1. 0979

1. 1536

1. 2125

1.2663

1. 3290

1.1002

1.1818

0.0948

.1538

.2205

.2869

.3515

.4235

.5084

.5782

.6561

.7236

.8031

.9041

.9687

1. 0606

1. 1505

1. 2396

1. 3359

1. 4245

1. 5321

1. 1537

1.2850

2.77

2.77

2.76

2.81

2.86

2.95

3.02

3.06 j

3.13 |

3.18

3.25

3.28

3.36

3.41

3.43

3.47

3.51

3.54

3.57

3.45

3.44

a Nappe free from the crest.
WEIES OF IRKEGULAK SECTION.

Sazin's experiments on weirs of irregular section Continued.

69

Bazin's Series, No. 113. 

Crest height, 2.463 feet.

Cross section.

Period.

1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18 

19

Observed 

head, experimental 

weir -D, 

in feet.

2

0.208

.289

.363

.443

.518

.592

.667

.736

.805

.863

.936

.989

1.055

1.076

1.114

1.159

1.197

1.252 

1. 320

D*

3

0. 0948

. 1554

.2187

.2949

.3728

. 4555

. 5447

.6314

.7223

.8017

. 9056

.9835

1. 0530

1.1162

1. 1758

1. 2478

1. 3096

1.4009 

1. 5166

Ci

4

2.64

2.66

2.66

2.65

2.66

2.64

2.71

2. 75

2.78

2.80

2.85

2.88

2.91

2.94

2.95

3.00

3.01

3.06 

3.11

Q, flow 

per foot, 

experimental 

weir, in 

cubic feet 

per 

second.

5

0. 2503

.4134

.5817

.7815

.9916

1. 2025

1. 4761

1. 7364

2. 0080

2. 2441

2.5810

2. 8325

3. 0641

3. 2816

3. 4686

3. 7434

3. 9419

4. 2868 

4. 7166

V

6

0.206

.269

.332

.392

.472

.542

.612

.672

.758

.821

.872

.925

.968

1.032

1.075

1.154 

1.246

1

7

0. 0007

.0011

.0017

.0024

.0034

.0045

.0058

.0070

.0090

.0105

.0118

. 0132

.0143

.0165

.0179

.0206 

.0243

H

8

0.208

.289

.3637

.4441

.5197

. 5944

.6704

. 7405

.8108

.8700

.9450

.9995

1.0468

1. 0892

1. 1283

1. 1755

1.2149

1. 2726 

1. 3443

*

9

0. 09484

. 1562

. 2196

.296*

.375

.4578

.549

.6377

.7303

.8115

. 91865

.9925

1.068

1.1364

1. 1980

1.274

1.339

1.436 

1. 558

C

10

2.63

2.65

2.65

2.64

2.64

2.63

2.69

2.72

2.75

2.77

2.81

2.84

2.86

2.88

2.88

2.93

2.94

2.98 

3.03

Bazin's Series, No. 114. 

Crest height, 2.46 feet

Cross section.

1

2

4

5

6

7

8

9

10

11

12

13

14

15

16

17

0.204

.280

.352

.433

.504

.578

.657

.735

.810

.882

.958

1.034

1.112

1.171

1.243

1.301

1.384

0. 0921

.1482

.2089

.28497

.3578

.4394

.5325

.6302

.7290

.8283

.9377

1. 0514

1. 1727

1. 2672

1.3858

1. 4839

1. 6282

2.47

2.54

2.59

2.60

2.59

2.60

2.62

2.63

2.63

2.65

2.66

2.68

2.69

2.70

2.73

2.73

2.76

0. 2275

.3764

.5411

.7409

.9267

1. 1424

1. 3952

1. 6511

1. 9173

2. 1950

2. 4943

2. 8178

3. 1546

3. 4214

3. 7832

4. 0510

4. 4938

0. 056

.137

.193

.256

.313

.376

.446

.517

.587

.655

.728

.806

.883

.941

1.021

1.078

1.168

........

0. 0006

.0011

.0015

.0022

.0031

.0042

.0054

.0068

.0083

.0102

.0120

.0137

.0161

.0181

.0213

0. 204

.280

.3526

.4341

. 5055

.5802

.6601

.7392

.8154

.8888

.9663

1. 0442

1. 1240

1. 1847

1. 2591

1. 3191

1.4053

0. 0921

.1482

.2097

.2860

.3594

.4417

.5362

.6353

. 7358

.8381

.9494

1. 0667

1. 1917

1. 2899

1. 4127

1. 5149

1.6654

2.47

2.54

2.58

2.59

2.58

2.59

2.60

2.59

2.60

2.62

2.63

2.64

2.65

2.65

2.67

2.66

2.70
70 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

Bazin's, experiments on weirs of irregular section Continued.

Bazin's Series, No. 115. 

Crest height, 2.46 feet.

Cross section.

Period.

1

1

2

3

4

5

6

. 7

8

9

10

11

12

13

14

15

16

17

18

19

Observed 

head, experimental 

weir D, 

in feet.

2

0.196

.264

.342

.415

.495

.566

.638

.716

.792

.871

.948

1.023

1.097

1.178

1.260

1.330

1.388

1.424

1.467

D%

3

0. 0868

.1357

.2001

.2674

.3483

.4258

.5096

.6059

.7049

.8129

.9230

1. 0347

1.1490

1. 2786

1. 4144

1. 5338

1. 6353

1. 6993

1. 7768

Ci

4

2.25

2.41

2.45

2.51

2.50

2.55

2.54

2.56

2.60

2.60

2.60

2.61

2.63

2.64

2.65

2.68

2.69

2.70

2.70

Q, flow 

per foot, experimental 

weir, in 

cubic feet 

per 

second.

5

0. 1953

.3270

.4902

.6712

.8708

1. 0858

1. 2944

1. 5511

1. 8327

2. 1135

2. 4098

2. 7006

3. 0219

3. 3755

3. 7482

4.1106

4.3990

4.5881

4.797

V

6

0.073

.120

.175

.233

.290

.358

.418

.487

.563

.634

.706

.775

.849

.928

1.009

1.085

1.144

1.18

1.26

ift 

2<7

7

0.0005

.0008

.0013

.0020

.0027

.0037

.0049

.0062

.0078

.0095

.0112

.0134

.0159

.0181

.0202

.0216

.0247

II

8

0.196

.264

.3425

.4158

.4963

.5680

.6407

.7197

.7969

.8782

. 9558

1.0325

1. 1089

1. 1914

1.276

1.348

1.408

1.446

1.492

H§

9

0. 0868

.1357

.2005

.2683

.3494

.42^1

. 5132

.6109

.7115

.8227

.9347

1. 0491

1. 1679

1. 2997

1. 4414

1.5651

1. 6707

1. 7388

1.8225

C

10

2.25

2.41

2.44

2.50

2.49

2.54

2.52

2.54

2.58

2.57

2.58

2.57

2.58

2.59

2.58

2.62

2.62

2.64

2.63

Bazin's Series, No. 116. 

Crest height, 2.46 feet.

Cross section.

1

o

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

0.177

225

296

.367

.435

.504

.537

.639

.713

.781

.849

.917

.986

1.053

1. 120

1. 185

1.251

1.317

0. 0745

.1068

.1611

.2224

.2870

.3578

. 3935

.5108

.6021

.6902

.7823

.8781

.9791

1. 0805

1.1853

1. 28995

1. 3992

1. 5114

2.71

2 . 

00 no

2 . Oft yv

2.92

2.95

2.98

2.99

3.01

3.00

3.00

3.02

3.02

3.05

3.06

3.08

3.09

3.10

3.12

0. 2019

.3022

.4672

.6494

.8467

1. 0662

1. 1776

1. 5375

1. 8063

2. 0706

2. 3625

2. 6519

2.9863

3.3063

3.6507

3.9859

4. 3375

4. 7156

0.076

.112

169

.229

.292

.360

.392

.497

.569

.640

.713

.793

.864

.942

1.019

1.092

1.169

1.250

0.0008

.0013

.0020

.0024

.0039

.0051

.0064

.0078

.0097

.0115

.0137

.0162

. 0185

.0213

.0243

0.177

.225

.296

.3678

.4363

.5060

.5394

.6429

.7181

.7874

.8568

.9267

.9975

1. 0667

1. 1362

1. 2035

1. 2723

1. 3413

0. 0745

.1068

.1611

.2232

.2879

.3599

.3957

.5156

.6084

.6982

.7933

.8925

.9963

1. 1021

1. 2108

1. 3203

1.4346

1. 5529

2.71

2.83

2.90

2.91

2.94

2.96

2.98

2.98

2.97

2.96

2.98

2.97

3.00

3.00

3.02

3.02

3.02

3.04
WEIES OF IRREGULAR SECTION. 71

Bazin's experiments on weirs of irregular section Continued.

 * t~ ~"

Bazin's Series, No. 117. 

Crest height, 2.46 feet.

Cross section.

Period.

1

1

2

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

Observed 

head, experimental 

weir D, 

in feet.

2

0.158

.204

.289

.361

.426

.494

.662

.635

.708

.771

.834

.912

.989

1.064

1.129

1.197

1.267

1.336

D*

3

0. 06282

. 09212

. 1554

. 21691

. 27808

. 34724

. 42134

.5060

. 59578

. 67702

.76168

. 87096

. 98352

1. 0975

1.1996

I. 3096

1. 4262

1.5442

Ci

4

2. 19

2.64

2.57

2.65

2.73

2.77

2.83

2.82

2.86

2.89

2.91

2.93

2.95

2.95

2.95

2.97

2.98

'2.99

Q, flow 

per foot, experimental 

weir, in 

cubic feet 

per 

second.

5

0. 1376

2432

.3994

. 5126

. 7592

.9619

1.1924

1. 4267

1.7049

1. 9566

2. 2155

2. 5519

2. 9014

3. 2376

3.5388

3.8895

4. 2501

4. 6172

V

6

0.043

.091

.145

.182

.263

.321

.395

.461

.539

.606

.674

.758

.841

.919

.987

1.062

1.139

1.214

If2

2<7

7

0.0005

.0011

.0016

.0025

.0033

.0045

.0058

.0070

.0090

.0110

.0132

.0152

.0175

.0202

. 0228

H

8

0.158

.204

.289

.3615

.4271

. 4956

. 5645

.6383

. 7125

.7768

.8410

.9210

1.0000

1.0772

1. 1442

1. 2145

1. 2872

1. 3588

H?

9

0. 0628

.0921

. 1554

.2178

.2791

.3493

.4247

..5096

.6020

.6849

.7713

.8839

1. 0000

1. 1177

1. 2236

1.3392

1.4601

1.5842

C

10

2. 19

2 g4

2.56

2.35

2.72

2.76

2.81

2.80

2.83

2.86

2.87

2.89

2.90

2.90

2.89

2.90

2.91

2.91

±saziu s series, INU. ±00. <i Everest length, 6.519 feet. * jjjj. 

Crest height, 2.46 feet. »i%%^ 

Cross section.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

0.183

.244

.304

.364

.424

.484

.542

.597

.658

.713

.776

.830

.887

.953

1.010

1.068

1.122

1.179

1.244

1.299

1.361

0. 0783

.1206

.1676

.2196

.2761

.3367

.3990

.4613

.5338

.6021

.6836

. 7562

.8354

.9303

1. 0150

1. 1037

1. 1885

1.2802

1. 3875

1.4805

1.5878

3.90

3.86

3.85

3.86

3.88

3.87

3.88

3.89

3.91

3.92

3.93

3.97

3.96

3.98

3.97

4.00

3.99

4.01

4.01

4.01

4.03

0.305

.467

.647

.849

1.071

1.304

1.548

1.793

2. 088

2.360

2.684

3.001

3.300

3.701

4.029

4.417

4.748

5.133

5.564

5.935

6.408

0.12

.17

.23

.30

.37

.44

.52

.59

.67

.74

.83

.91

.99

1.08

1.16

1.25

1.33

1.41

1.50

1.58

1.68

0. 0002

.0004

.0008

.0014

.0021

.0030

.0042

.0054

.0070

.0085

.0107

.0129

.0152

.0181

.0209

.0243

.0275

.0309

.0350

.0388

.0439

0. 1832

.2444

.3048

.3654

.4261

.4870

.5462

.6024

.6650

. 7215

.7867

.8427

.9022

.9711

1.0309

1. 0923

1. 1495

1. 2099

1. 2790

1. 3378

1. 4049

0. 0783

.1206

.1684

.2206

.2781

.3399

.4035

.4671

.5423

. 6.135

.6982

.7740

.8567

.9568

1. 0468

1. 1411

1. 2316

1. 3310

1. 4446

1. 5477

1.6654

3.90

3.87

3.84

3.85

3.85

3.84

3.84

3.84

3.85

3.85

3.84

3.88

3.85

3.87

3.85

3.87

3.86

3.86

3.85

3.84

3.85
72 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

Eazin' s experiments on weirs of irregular section Continued.

Bazin's Series, No. 137. 

Crest length, 6.523 feet. 

Crest height, 2.46 feet.

Cross section.

Period.

Observed 

head, experimental 

weir D, 

in feet.

1 2

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

0.268

.332

.391

.451

.513

.578

.637

.700

. 765

.822

.887

.946

1.012

1.078

1.142

1.201

1. 262

1. 322

D?

3

0. 1388

.1974

.2445

.3029

.3674

.4394

.5084

. 5857

.6692

.7452

.8354

.9201

1. 0180

1. 1193

1.2204

1.3162

1. 4178

1.5200

Oi

4

3.47

3.45

3.50

3.47

3.53

3.51

8.51

3.55

3.56

3.56

3. .56

3.62

3.59

3.61

3.60

3.62

3.62

3.64

Q, flow 

per foot, experimental 

weir, in 

cubic feet 

per 

second.

5

0.482

.680

.856

1.051

1.295

1.540

1.783

2.080

2.382

2.652

2.973

3.334

3. 662

4.043

4. 392

4.778

5.140

5.533

i>

6

0.18

.24

.30

.36

.44

.51

.57

.66

.74

.81

.89

.98

1.05

1.14

1.22

1.30

1.38

1.46

V25-

7

0. 0006 ,

.0009

.0014

.0020

.0030

.0040

.0051

.0068

.0085

.0102

.0123

.0149

.0171

.0202

.0231

.0263

.0296

.0331

H

8

0. 2686

. 3329 .

.3924

.4530

.5160

.5820

.6421

.7068

.7735

.8322

.8993

.9609

1. 0291

1. 0982

1. 1651

1. 2273

1. 2916

1. 3551

Jff*

9

0. 1395

.1922

.2454

.3049

.3707

.4440

.5144

. 5945

.6797

.7589

.8524

.9420

1. 0438

1. 15Q6

1. 2575

1.3591

1.4686

1.5773

C

1)

3.46

3.54 "

3.49

3.45

3.49

3.47

3.47

3.50

3.50

3.49

3.49

3.54

3.51

3.51

3.49

3.52

3.50

3.51

Bazin's Series, No. 138. 

Crest length, 6.532 feet. 

Crest height, 1.64 feet.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

0.194

.263

.327

.391

.447

.510

.571

.626

.685

.745

.807

.873

.927

.992

1.045

1.110

1.176

1.233

1.289

1.355

1.429

0. 0854

.1349

.1870

.2445

.2989

.3642

.4314

.4953

.5670

.6431

.7250

.8157

.8926

.9880

1. 0683

1. 1695

1. 2753

1. 3691

1. 4645

1. 5773

1. 7082

3.57

3.50

3.48

3.50

3.56

3.63

3.62

3.71

3.66

3.69

3.70

3.72

3.72

3.76

3.80

3.78

3.79

3.81

3.82

3.82

3.83

0.305

.473

.651

.858

1.064

1.321

1.560

1.838

2.075

2.373

2.683

3.036

3.318

3.715

4.060

4.422

4.851

5.220

5.577

6.036

6.542

0.17

.25

.33

.42

.50

.61

.70

.81

.89

.99

1.09

1.21

1.29

1.41

1.51

1.61

1.72

1.82

1.90

2.01

2.13

0.0004

.0010

.0017

.0027

.0039

.0058

.0076

.0102

.0123

. 0152

.0185

.0228

.0259

.0309

.0354

.0403

.0460

.0515

.0561

.0628

.0705

~1itk.

Cross section.

0. 1944

.2640

.3287

.3937

.4519

.5158

.5786

.6362

.6973

.7602

.8265

.8958

.9529

1.0229

1.0804

1. 1503

1. 2220

1.2845

1.3451

1. 4178

1. 4995

0.0854

.1357

.1887

.2473

.3039

.3706

.4405

.5072

.5820

.6626

.7507

.8481

.9303

1.0347

1.1224

1. 2332

1. 3508

1.4550

1. 5599

1.6885

1. 8362

3.57

3.48

3.45

3.47

3.50

3.56

3.54

3.62

3.56

3.58

3.57

3.58

3.56

3.59

3.62

3.58

3.59

3.59

3.58

3.57

3.56
WEIRS OF-IRREGULAR SECTION. 73

Bazin's experiments on weirs of irregular section Continued.

Bazin's Series, No. 145. 

Crest length, 6.541 feet. 

Crest height, 1.64 feet.

Cross section.

I Period.'

Observed 

head, experimental 

weir D, 

in feet.

2

0.359

.424

J$

8

0. 2151

.2761

. 479 . 3315

. 547 . 4046

. 592 . 4636

.658 .5338

. 720 . 6109

.781

.835

.902

.962

1.032

1.087

1.152

1.210

1.274

1.334

1.396

1.467

.6902

.7631

.8567

.9435

1. 0484

1.1333

1. 2364

1.3310

1. 4380

1.5408

1. 6494

1. 7768

C'l

4

3.02

3.10

3.18

3.25

Q, flow 

per foot, 

experimental 

weir, in 

cubic feet 

per 

second.

5

0.649

.856

!' if 

'iff If

678

0.32

.42

1.053 | .50

1. 316 . 60

3. 35 1. 5.53 . 69

3.38

3.42

1. 805

2.090

.78

.89

0. 0016

.0027

.0039

.0056

.0074

 0095

.0123

3.47 2.394 .99 .0152

3.49 2.663 1.07 .0178

3.53 3.025 1.19

3.53

3.53

3.58

3.58

3.61

3.61

3.62

3.64

3.64

3.332 1.28

3. 707 1. 39

4.045 i 1.48

4. 403 1. 58

4. 801 1. 68

5. 198

5. 575

6.006

6.479

1.78

1.88

1.98

2.09

.0220

. 0255

. 0300

.0341

.0388

.0439

.0493

.0549

.0609

.0679

0. 3606

.4267

.4809

. 5526

. 5994

.6675

.7323

.7962

.8528

.9240

.9875

1.0620

1. 1211

1.1908

1. 2539

1. 3233

1. 3889

1.4569

1.5349

IP

9

0. 2169

.2790

.3335

.4112

.4636

.5447

.6263

.7102

.7878

.8882

.9806

1.0944

<:

10

2.99

3.07

3.16

3.20

3.33

3.31

3.34

3.37

3.38

3.41

3.40

3.39

1. 1869 3. 41

1. 2997 3. 39

1. 4042 3. 42

1.521H 3.42

1. 6370 3. 40

1. 7586 3. 42

1.9016 3.41

Bazin's Series, No. 141. 

Crest length, 6.520 feet. 

Crest height, 2.46 feet.

Cross section.

1

2

3

4

5

(i

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0.215

.281

.355

.425

.489

.561

.624

.692

.758

.822

.888

.956

1.029

0. 0997

.1490

3. 02 0. 301

3.09 .460

. 2116 3. 07

. 2771 3. 04

.3420

.650

.842

3. 08 1. 053

.4202 3.08

.4929

. 5757

.6600

.7452

.8368

3.17

3.11

3.12

3.15

3.17

. 9347 3. 19

1. 0438 3.17

1. 113 1. 1742 3. 20

1. 165 1. 2575

1.237

1.298

1.369

1.431

1.463

1.3758

1. 4788

1.6018

1. 7118

1.7696

0.11

.17

.24

0. 0002 0. 2152

. 0004 . 2814

.0009

. 29 . 0013

. 37 . 0021

1. 294 . 43 . 0029

1.562 ' .,)! .0040

1.791 .57 .0051

2.059 .64 .0064

2. 347 . 72 .0081

2. 653 . 79 . 0097

2. 983 . 87 .0118

3. 309 . 95 0140

3.757 1. 04 . 0168

3.21 4.045 1.12 .0195

3. 20 4. 416 1.19

3. 22 4. 766 1. 27

. 0220

. 0251

3.22 5.152 '' 1.34 .0279

3.24 5.540 1.42

3.25 5.752 1.47

.0313

. 0336

.3559

.4263

.4911

.5639

0.0997

.1490

. 2124

.2781

.3441

.4235

. 6280 . 4976

.6971

.7644

.8301

.8977

.9678

1.0430

1. 1298

1. 1845

1.2590

1.3231

1.3969

1.4623

1. 4966

. 5820

.6678

.7562

.8509

.9523

1. 0652

1. 2012

1.2884

1. 4127

1. 5218

1. 6511

1. 7677

1. 8307

3.02

3.09

3.06

3.02

3.06

3.06

3.14

3.08

3.08

3.10

3.12

3.13

3.12

3.13

3.14

3.13

3.13

3.12

3.13

3.14

IRR 150 06-
74 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

Bazin' s experiments on weirs of irregular section Continued.

Bazin's Series, No. 142. 

Crest length, 6.523 feet. 

Crest height, 2.46 feet.

Cross section.

Period.

-|

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

Observed 

head, experimental 

weir 1), 

in feet.

2

0. 300

.369

.447

.509

.591

.656

.727

.795

.861

.934

1.007

1.079

1.149

1.222

1.285

1.362

1.430

rf

3

0. 1643

.2242

.2989

.3631

.4544

.5435

.6199

.7089

.7989

.9027

1. 0105

1. 1208

1.2316

1.3508

1. 4567

1. 5895

1. 7100

Oi

4

2.83

2.87

2.87

2.86

2.88

2.86

2.92

2.94

2.94

2.95

2.95

2.98

2.98

2.99

3.00

3.00

3.01

Q, flow 

per foot, experimental 

weir, in 

cubic feet 

per 

second.

5

0.464

.643

.851

1.038

1.308

1.554

1.810

2.084

2.349

2.664

2.980

3.338

3.665

4.037

4.370

4.770

5.147

V

6

0.17

.23

.29

.35

.43

.50

.57

.64

.71

.78

.86

.94

1.01

1.10

1.17

1.25

1.30

V"'

ig

i

0.0004

.0008

.0013

.0019

.0029

.0039

.0051

.0064

.0078

.0095

.0115

.0137

.0159

.0188

.0213

.0243

.0263

H

8

0. 3004

.3698

.4483

.5109

.5939

.6699

.7327

.8014

.8688

.9435

1. 0185

1.0927

1. 1649

1.2408

1.3063

1. 3863

1. 4563

H$

9

0. 1643

.2251

.2999

.3652

.4578

.5484

.6263

.7155

.8101

.9158

1.0286

1. 1427

1. 2575

1.3824

1.4925

1. 6317

1. 7569

C

10

2.82

2.86

2.84

2.84

2.86

2.83

2.89

2.91

2.90

2.91

2.89

2.92

2.92

2.92

2.93

2.92

2.93

Bazin's Series, No. 139. 

Crest length, 6.532 feet.

Crest height, 1.64 feet. Qf<m gectlon

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

0.190

.253

.312

.375

.434

.500

.552

.615

.667

.733

.798

.852

.915

.969

1.023

1. 092

1. 151

1.210

1.258

1.326

1.394

0. 0828

.1273

.1743

.2297

.2860

.3536

.4101

. 4823

.5447

.6276

.7128

.7865

.8753

.9538

1. 0347

1. 1411

1.2348

1. 3310

1. 4110

1. 5269

1.6459

3.66

3.68

3.72

3.66

3.73

3.72

3.78

3.76

3.82

3.79

3.80

3.84

3.86

3.87

3.92

3.90

3.90

3.94

0.303

.467

.647

.841

1.067

1.317

1.550

1.812

2.081

2. 380

2.709

3.022

3.378

3. 692

4.038

4.446

4.816

5. 240

3. 95 5. 570

3. 93 6. 013

3.93 6.484

0.17

.25

.33

.42

.52

.62

.71

.80

.90

1.00

1.11

1.21

1.32

1.41

1.52

1.63

1.72

1.84

1.92

2.03

2.13

0.0004

.0010

.0017

.0027

.0042

.0060

.0078

.0099

.0126

. 0155

. 0192

. 0228

.0271

.0309

.0359

.0413

.0460

.0526

.0573

.0641

. 0705

0. 1904

.2540

.3137

.3777

.4382

.5060

.5598

.6249

.6796

. 7485

.8172

.8748

.9421

.9961

1.0589

1. 1333

1. 1970

1. 2626

1. 3153

1.3901

1. 4645

0. 0828

.1280

.1759

.2323

.2899

.3600

.4191

.4941

.5607

.6482

.7385

. 8185

.9143

.9940

1. 0897

1. 2060

1.3096

1. 4194

1. 5080

1. 6388

1.7714

3.66

3.65

3.68

3.62

3.68

3.66

3.70

3.67

3.71

3.67

3.67

3.69

3.68

3.71

3.71

3.68

3.68

3.69

3.69

3.67

3.66
WE1BS OF IEEEGULAE SECTION. 75

Bazin's experiments on weirs of irregular section Continued.

Bazin's Series, No. 140. ~f~.>7$jg>£;,

Crest length, 6.532 feet. ^P^^fe^

Crest height, 1.64 feet. Crosg sectlorl.

Period.

1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

Observed 

head, experimental 

weir D, 

in feet.

2

0.192

.252

.308

.371

.436

.488

.549

.604

.664

.719

.785

.837

.905

.961

1.023

1.080

1.143

1.195

1.254

1.316

1.375

D*

3

0.0835

.1265

.1709

.2260

.2879

.3399

.4068

.4694

.5411

.6096

.6956

.7658

.8610

.9421

1. 0347

1.1224

1. 2220

1.3063

1.4043

1.5097

1. 6123

Cl

4

3.77

3.74

3.75

3.71

3.77

3.74

3.81

3.82

3.83

3.84

3.88

3.88

3.92

3.90

3.95

3.93

3.97

3.96

3.97

3.99

4.01

Q, flow 

per foot, experimental 

weir, in 

cubic feet 

per 

second.

5

0.315

.473

.641

.838

1.086

1.276

1.551

1.792

2.072

2.342

2.700

2.968

3.375

3.674

4.069

4.402

4.843

5.187

5.558

6.024

6.456

l'

6

0.17

.25

.33

.42

.52

.60

.71

.80

.90

.99

1.11

1.20

1.32

1.41

1.53

1.62

1.74

1.83

1.92

2.03

2.14

ifl

2g

1

0.0004

.0010

.0017

.0027

.0042

.0056

.0078

.0099

.0126

.0152

.0192

.0224

.0271

.0309

.0364

.0408

.0471

.0521

.0573

.0641

.0712

H

8

0.1924

.2530

.3097

.3727

. 4402

.4936

.5568

.6139

.6766

.7342

.8042

.8594

.9321

.9919

1.0594

1. 1208

1.1901

1.2471

1. 3113

1.3801

1.4462

H%

9

0.0841

. 1273

.1726

.2278

.2919

.3472

.4157

.4811

.5570

.6289

.7209

.7961

.8998

.9880

1.0898

1. 1869

1. 2981

1.3925

1.5011

1. 6211

1.7388

C

10

3.74

3.72

3.71

3.68

3.72

3.66

3.73

3.72

3.72

3.72

3.74

3.73

3.75

3.72

3.73

3.71

3.73

3.72

3.70

3.72

3.71

Bazin's Series, No. 147. 

Crest length, 6.536 feet. 

Crest height, 2.46 feet.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

0.231

.308

.373

.438

.503

 .569

.637

.681

.734

.797

.845

.898

.953

1.015

1.063

1.115

1. 165

1.217

1.265

1.332

1.394

0.1110

.1709

.2278

.2899

.3568

.4292

.5084

.5620

.6289

.7115

.7768

.8510

.9303

1. 0226

1.0960

1. 1774

1. 2575

1. 3426

1. 4228

1. 5373

1.6459

2.75

2.85

2.86

2.97

3.02

3.13

3.20

3.24

3.34

3.40

3.44

3.53

3.57

3.63

3.67

3.73

3.79

3.83

3.85

3.89

3.98

0.305

.485

.652

.861

1.078

1.343

1.626

1.821

2.109

2.417

2.673

3.004

3.320

3.703

4.037

4.401

4.775

5.132

5. 467

5.991

6.567

0.11

.18

.23

.30

.37

.44

.54

.58

.66

.74

.81

.89

.97

1.06

1.15

1.22

1.31

1.40

1.47

1.58

1.68

0.0002

.0005

.0008

.0014

.0021

.0030

.0045

.0052

.0068

.0083

.0102

.0123

.0146

.0175

.0206

.0231

.0267

. 0305

.0336

.0388

.0439

Cross section.

0. 2312

.3085

.3738

.4394

.5051

.5720

.6415

.6862

.7408

.8053

.8552

.9103

.9676

1. 0325

1. 0836

1. 1381

1. 1917

1. 2475

1. 2986

1. 3708

1. 4379

0. 1110

.1709

.2287

.2909

.3589

.4326

.5132

.5682

.6378

.7223

.7906

.8681

.9523

1.0484

1. 1286

1.2140

1.3013

1.3925

1. 4805

1.6053

1. 7244

2.75

2.84

2.85

2.96

3.00

3.10

3.17

3.20

3.31

3.35

3.38

3.46

3.49

3.53

3.58

3.62

3.67

3.68

3.70

3.73

3.81
76 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

Bazin's experiments on weirs of irregular section Continued.

Bazin's Series, No. 149. 

Crest length, 6.518 feet. 

Crest height, 2.46 feet.

Cross section.

Period.

1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

Observed 

head, experimental 

weir 1), 

in feet.

;2

0.248

.317

.390

.455

.521

.585

.653

.705

.766

.818

.882

.942

.999

1.051

1.103

1.165

1.209

1.281

1.330

1.385

1.446

J*

3

0. 1235

.1785

.2436

.3070

.3761

.4475

.5277

.5920

.6705

.7398

.8283

.9143

.9985

1. 0774

1.1584

1.2575

1.3294

1.4499

1.5338

1.6300

1.7388

CL

4

2.55

2.58

2.67

2.73

2.82

2.89

2.97

3.00

3.08

3.16

3.23

3.30

3.36

3.39

3.45

3.49

3.52

3.57

3.60

3.63

3.67

Q, flow 

per foot, experimental 

weir, in 

cubic feet 

per 

second.

5

0.316

.462

.651

.838

1.060

1.295

1.568

1.776

2.067

2.338

2.674

3.016

3.356

3.627

4.002

4.397

4.682

5.177

5.508

5.917

6.386

V

6

0.12

.17

.23

.29

.36

.43

.51

.56

.64

.71

.80

.89

.97

1.03

1.12

1.20

1.27

1.38

1.45

1.54

1.64

ift 

20

7

0.0002

.0004

.0008

.0013

.0020

.0030

.0040

.0049

.0064

.0078

.0099

.0123

.0146

.0165

.0195

. 0224

.0251

.0296

.0327

.0369

.0418

H

8

0. 2482

.3174

.3908

.4563

.5230

.5850

.6570

.7099

.7724

.8258

.8979

.9543

1. 0156

1. 0665

1. 1225

1. 1874

1. 2341

1.3106

1.3627

1.4219

1.4878

fi*

9

0.1235

.1785

.2445

.3080

.3782

.4475

.5325

.5983

.6783

.7507

.8509

.9318

1.0241

1.1006

1.1885

1. 2932

1. 3708

1.5011

1. 5912

1.6956

1. 8151

C

10

2.56

2.59

2.66

2.72

2.80

2.89

2.94

2.97

3.05

3.11

3.14

3.23

3.28

3.30

3.37

3.40

3.42

3.45

3.46

3.49

3.52

Bazin's Series, No. 150. 

Crest length, 6.518 feet. 

Crest height, 2.46 feet.

Cross section.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

0.248

.323

.379

.459

.512

.586

.637

.698

.751

.814

.869

.928

.982

1.043

1.095

1. 152

1.215

1.259

1.315

1.323

1.380

1.439

0. 1235

.1836

.2333

.3110

.3664

.4486

.5084

.5832

.6508

.7344

.8101

.8940

.9732

1. 0652

1. 1459

1.2364

1.3393

1. 4127

1.50*0

1. 5218

1. G211

1.7262

2.53

2.65

2.78

'2.82

2.91

2.96

3.07

3.12

3.18

3.26

3.31

3.37

3.42

3.47

3.51

3.56

3.58

0.314

.488

.648

.877

1.065

1.329

1.560

1.819

2.070

2.393

2.681

3.013

3.328

3.713

4.037

4.414

4.797

3.63 5.118

3. 65 5. 512

3.65 ' 5.548

3.68 , 5.962

3.73 6.416

0.12

.16

.23

.30

.36

.43

.50

.57

.64

.73

.80

.89

.97

1.06

1.13

1.22

1.30

1.38

1.46

1.47

1.55

1.64

0.0002

.0004

.0008

.0014

.0020

.0029

.0039

.0051

.0064

.0083

.0099

.0123

.0146

.0175

.0199

.0231

. 0263

.0296

.0331

.0336

.0374

.0418

0.2482

.3234

. 3798

.4604

.5140

.5889

.6409

.7031

.7574

.8223

.8789

.9403

.9966

1.0605

1. 1149

1. 1751

1.2413

1.2856

1.3481

0.1235

.1836

.2342

.3120

.3685

.4521

.5132

.5895

.6587

.7452

.8241

.9114

. 9955

1.0913

1. 1774

1. 2737

1.3825

1.4584

1.5651

1.3.566 1.5807

1. 4174 1. 6868

1.480X 1.8023

2.54

2.66

2.77

2.81

2.89

2.94

3.04

3.09

3.14

3.21

3.25

3.31

3.34

3.40

3.43

3.46

3.47

3. 51

3.52

3.51

3.54

3.56
WE1ES OF IRREGULAR SECTION.

Bazin's experiments on weirs of irregular section Continued.

Bazin's Series, No. 151. 

Crest length, 6.550 feet. 

Crest height, 2.46 feet. 

77

Cross section.

Period.

1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

Observed 

head, experimental 

weir D, 

in feet.

2

0.201

.240

.307

.391

.445

.514

.537

.573

.643

.695

.756

.800

.826

.867

.921

.975

1.027

1.090

1.112

1.140

1.209

1.248

1.314

1.352

1.416

tf

3

0.0901

.1176

.1701

.2445

.2969

.3685

.3935

.4337

.5156

.5795

.6574

.7155

.7507

.8073

.8839

.9628

1.0408

1.1380

1. 1727

1. 2172

1.3294

1.3942

1.5063

1. 5721

1.6850

a

4

2.71

2.81

2.79

2.79

2.92

2.95

2.98

3.05

3.09

3.20

3.24

3.30

3.31

3.36

3.39

3.46

3.51

3.52

3.57

3.60

3.61

3.64

3.68

3.71

3.75

Q, flow 

per foot, experimental 

weir, in 

cubic feet 

per 

second.

5

0.244

.329

.474

.684

.867

1.089

1.174

1.324

1.594

1.856

2.129

2.362

2.486

2.712

2.997

3.332

3.653

4.013

4.177

4.382

4.801

5.060

5.557

6.825

6.337

V

6

0.09

.12

.17

.24

.30

.37

.39

.43

.51

.59

.66

.72

.76

.81

.89

.97

1.04

1.13

1.17

1.22

1.31

1.36

1.47

1.52

1.64

V2

2£

7

0.0001

.0002

.0004

.0009

.0014

.0021

.0024

.0030

.0040

.0054

.0068

.0081

.0090

.0102

.0123

.0146

.0168

.0199

.0218

.0231

.0267

.0288

.0336

0359

.0418

H

8

0. 2011

.2422

.3074

.3919

.4464

.5161

.5394

.5760

.6470

.7004

.7628

.8081

.8350

.8772

.9333

.9896

1.0438

3.1099

1.1333

1. 1631

1.2357

1.2768

1. 3476

1.3879

1.4578

H*

9

0.0901

.1191

.1701

.2454

.2979

.3707

.3957

.4371

.5204

.5857

.6665

.7263

.7631

.8213

.9013

.9850

1.0667

1. 1695

1.2060

1.2543

1. 3741

1.4431

1.5651

1. 6352

1.7604

C

10

2.71

2.76

2.79

2.79

2.91

2.94

2.97

3.03

3.06

3.17

3.19

3.25

3.26

3.30

3.32

3.38

3.47

3.43

3.46

3.50

3.49

3.51

3.55

3.56

3.60
78 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

Bazin's experiments on weirs of irregular section Continued.

 » » 4 O.6K

Bazin's Series, No. 153. 

Crest length, 6.515 feet. 

Crest height, 2.46 feet.

Cross section.

Period.

1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

Observed 

head, experimental 

weir D, 

in feet.

2

0.237

.301

.372

.373

.440

.505

.576

.637

.696

.701

.760

.762

.814

.879

.937

.993

1.001

1.055

1.102

1.170

1.226

1.290

1.289

1.347

1.404

1.436

I*

3

0.1154

.1651

.2269

.2278

.2919

.3589

.4371

.5084

.6807

.5870

.6626

.6652

.7344

.8241

.9071

.9895

1. 0015

1.0836

1. 1569

1. 2656

1. 3575

1.4652

1.4635

1. 5634

1.6636

1.7208

CL

4

2.73

2.77

2.79

2.83

2.90

2.93

3.00

3.07

3.10

3.10

3.15

3.16

3.20

3.25

3.29

3.34

3.33

3.40

3.41

3.46

3.48

3.51

3.52

3.53

3.58

3.58

Q, flow 

per foot, 

experimental 

weir, in 

cubic feet 

per 

second.

5

0.314

.457

.633

.645

.847

1.052

1.311

1.560

1.801

1.820

2.085

2.101

2. 349

2.678

2.984

3.307

3.330

3.672

3.956

4.394

4.733

5.159

5.139

5.507

5.943

6.158

V

6

0.12

.16

.22

.23

.29

.35

.43

.50

.57

.58

.65

.65

.72

.80

.88

.96

.96

1.05

1.11

1.21

1.28

1.38

1.37

1.45

1.54

1.58

 ya 

20

7

0. 0002

.0004

.0008

.0008

.0013

.0019

.0030

.0039

.0051

.0052

.0066

.0066

.0081

.0099

, 0120

.0143

.0143

.0171

.0192

.0228

.0255

.0296

.0292

.0327

.0369

.0388

H

8

0.2372

.3014

.3728

.3738

.4413

.5069

.5790

.6409

.7011

.7062

.7666

.7686

.'8221

.8889

.9490

1.0073

1. 0153

1. 0721

1. 1212

1.1928

1.2515

1. 3196

1. 3182

1. 3791

1.4409

1.4748

H%

9

0.1154

.1651

.2278

.2287

.2929

.3610

.4406

.5132

.5870

.5933

.6717

.6743

.7452

.8381

.9245

1. 0105

1.0226

1. 1099

1. 1869

1.3030

1. 3992

1. 5166

1. 5132

1. 6193

1. 7298

1. 7914

C

10

2.72

2.77

2.78

2.82

2.89

2.91

2.98

3.04

3.07

3.07

3.10

3.12

3.15

3.20

3.23

3.27

3.26

3.31

3.33

3.37

3.38

3.40

3.40

3.40

3.44

3.44
WEIRS OF IRREGULAR SECTION. 79

BazMs experiments on weirs of irregular section Continued.

Basin's Series, No. 154. 

Crest length, 6.516 feet. 

Crest height, 2.46 feet.

Cross section.

Period.

1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Observed 

head, experimental 

weir D, 

in feet.

2

0.236

.308

.373

.447

.508

.577

.643

.706

.760

.823

.888

.946

1.011

1.075

1.138

1.195

1.250

1.310

1.370

1.430

1*

3

0. 1147

.1709

.2278

.2989

.3621

.4382

.5156

.5933

.6626

.7466

.8368

.9201

1. 0165

1. 1146

1. 2140

1. 3063

1. 3975

1. 4994

1. 6035

1. 7100

C'i

4

2.70

2.74

2.83

2.85

2.95

2.97

3.04

3.07

3.17

3.20

3.20

3.24

3.28

3.31

3.36

3.37

3.40

3.43

3.45

3.48

Q, flow 

per foot, experimental 

weir, in 

cubic feet 

per 

second.

5

0.311

.469

.645

.852

1.068

1.301

1.569

1.821

2.102

2.389

2.678

2.981

3.334

3.674

4.066

4.415

4.760

5.145

5.520

5.951

V

6

0.12

.17

.23

.29

.36

.37

.51

.57

.65

.73

.80

.87

.96

1.03

1.13

1.20

1.28

1.36

1.44

1.53

Ift

2g

7

0.0002

.0004

.0008

.0013

.0020

.0021

.0040

.0051

.0066

.0083

.0099

.0118

.0143

.0165

.0199

.0224

.0255

.0288

.0322

.0364

H

8

0. 2632

.3084

.3738

.4483

.5100

.5791

.6470

.7111

.7666

.8313

.8979

.9578

1.0253

1. 0915

1. 1579

1. 2174

1.2755

1.3388

1.4022

1. 4664

1T»

9

0. 1349

.1709

.2287

.2999

.3642

.4406

.5204

.5996

.6717

.7576

.8509

.9376

1.0377

1. 1396

1. 2461

1.3426

1. 4397

1.5494

1. 6601

1. 7750

C

10

2.30

2.74

2.82

2.84

2.94

2.95

3.02

3.04

3.11

3.16

3.15

3.18

3.21

3.22

3.26

3.29

3.31

3.32

3.32

3.35
80 WEIR, EXPERIMENTS, COEFFICIENTS, AND FORMULA^.

Bazin's experiments on weirs of irregular section Continued.

Bazin's Series, No. 156. 

Crest height, 2.46 feet. 

Crest width, 0.66 foot. 

Upstream slope, i to 1. 

Downstream slope, 5 to 1.

Period.

1

1

2

3

4 

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Observed 

head, experimental 

weir D, 

in feet.

2

0.246

.311

.382

.446

.508

.576

.638

.703

.764

.834

.888

.956

1.018

1.074

1.139

1.203

1.267

1.341

1.394

1.457

7,3

3

0. 1220

.1734

. 2361

.2979 

.3621

.4371

. 5096

. 5895

.6678

.7617

.8368

.9347

1.0272

1. 1130

1. 2156

1. 3194

1.4262

1. 5529

1.6459

1. 7587

Oi

4

2.76

2.80

2.84

2.90 

2.91

2.95

3.01

3.06

3.10

3.13

3.17

3.24

3.22

3.30

3.29

3.31

3.34

3.36

3.36

3.39

Q, flow 

per foot, 

experimental 

weir, in 

cubic feet 

per 

second.

5

0.337

.486

.671

.864 

1.054

1.289

1.534

1.804

2.070

2.384

2. 653

3.028

3.309

3.673

3.999

4.367

4.764

5.218

5.530

5. 962

V

6

0.12

.17

.24

.30 

.36

.42

.49

.57

.64

.72

.79

.88

.95

1.04

1.11

1.19

1.26

1.37

1.43

1.52

,fl 

2<7

7

0. 0002

.0004

.0009

.0014 

.0020

.0027

.0037

.0051

.0064

.0081

.0097

.0120

.0140

.0168

.0192

.0220

.0247

.0292

.0318

. 0359

H

8

0. 2462

.3114

.3829

.4474 

.5100

. 5787

.6417

.7081

.7704

.8421

.8977

.9680

1. 0320

1.0908

1. 1582

1. 2250

1.2917

1.3702

1. 4258

1. 4929

fl*

9

0. 1220

.1734

.2370

.2989 

.3642

.4406

.5144

.5958

.6757

.7727

. 8510

.9524

1.0484

1. 1396

1. 2462

1.3558

1.4686

1.6035

1. 7028

1. 8241

C

10

2.76

2.80

2.83

2.89 

2.89

2.92

2.98

3.03

3.06

3.08

3.12

3.18

3.16

3.22

3.21

3.22

3.24

3.25

3.25

3.27.

Bazin's Series, No. 158. 

Crest length, 6.520 feet. 

Crest height, 2.46 feet. Cross section.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0.234

.312

.383

.457

.530

.600

.672

.733

.799

.860

.930

.984

1.055

1.125

1.177

1.243

1.297

1.361

1.412

1.457

0. 1132

.1743

.2370

.3090

.3858

.4648

. 5509

. 6276

.7142

.7975

.8969

.9761

1. 0836

1. 1933

1. 276g

1.3858

1.4771

1. 5878

1. 6779

1.7587

2.79

2.72

2.77

2.79

2.81

2.82

2.86

2.90

2.91

2.95

3.00

3.04

3.10

3.12

3.15

3.19

3.22

3.25

3.30

3.32

0.316

.474

.656

.862

1. 085

1.311

1.576

1.821

2. 078

2.354

2.691

2.967

3.348

3.713

4.022

4.434

4.766

5.168

5.544

5. 839

0.12

.17

.23

.29

.36

.43

.50

.57

.64

.71

.79

.86

.95

1.04

1.10

1.20

1.27

1.35

1.43

1.49

0.0002

.0004

.0008

.0013

.0020

.0030

.0039

.0051

.0064

.0078

.0097

.0115

.0140

.0168

.0188

.0224

.0251

.0283

.0348

.0345

0.2342

.3124

.3838

.4583

.5320

.6030

.6759

.7381

.8054

.8678

.9397

.9955

1.0690

1. 1418

1.1958

1.2654

1. 3221

1. 3893

1. 4468

1. 4915

0. 1132

.1743

.2379

.3100

.3880

.4683

.5557

.6340

.7223

.8087

.9114

.9925

1. 1053

1. 2204

1.3029

1.4228

1.5200

1.6370

1.7406

1. 8215

2.79

2.72

2.76

2.78

2.80

2.80

2.84

2.87

2.88

2.91

2.95

2.99

3.03

3.04

3.08

3.12

3.14

3.16

3.18

3.22
WEIKS OF IKKEGULAK SECTION. 81

Basin's experiments on 'iveirs of irregular section Continued.

Bazin's Series, No. 159. 

Crest length, 6.511 feet. 

Crest height, 2.46 feet.

Cross section.

! 

Period.

1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

Observed 

head, experimental 

weir D, 

in feet.

2

0.234

.304

.379

.387

.457

.516

.526

.599

.664

.670

.735

.797

.861

.876

.935

.994

1.068

1.126

1.145

1.198

1.261

1. 320

1.332

1.389

1.445

1.456

rf

3

0. 1132

.1676

.2333

.2408

.3090

.3707

.3815

.4636

.5411

.5484

.6302

.7115

.7989

.8199

.9042

.9910

1. 1037

1. 1948

1.2252

1. 3112

1. 4161

1.5166

1. 5373

1.6370

1. 7370

1. 7569

Ci

4

2.68

2.75

2.82

2.82

2.81

2.91

2.84

Q, flow 

per foot, experimental 

weir, in 

cubic feet 

per 

second.

5

0.303

.462

.657

.680

.868

1.079

1.085

2. 82 1. 308

2. 87 1. 553

2.83 1.552

2. 88 1. 813

2.94 ' 2.092

2.99

2.94

2.389

2.411

2. 93 2. 649

3.01 2.983

3.03 3.333

3.10

3.05

3.08

3.704

3.751

4.035

3. 11 4. 416

3.15

3.13

3.14

3.19

3.19

4.777'

4.820

5. 150

5. 551

5.614

V

6

0.11

.17

.23

.24

.30

.36

.36

.43

.50

.19

.56

.64

.72

.72

.78

.86

.94

1.03

1.04

1.10

1.19

1.27

1.27

1.33

1.42

1.43

i>2

2tf

7

0. 0002

.0004

.0008

.0009

.0014

.0020

. 0020

.0030

.0039

.0037

.0049

.0064

.0081

.0081

.0095

.0115

.0137

.0165

.0168

.0188

.0220

.0251

.0251

.0275

.0313

.0348

H

8

0. 2342

.3044

.3798

.3879

.4684

.5180

.5280

.6020

.6679

.6737

.7399

.8034

.8693

.8843

.9446

1.0055

1. 0817

1. 1425

1. 1618

1.2168

1.2830

1 3451

1.3571

1. 4165

1. 4763

1. 4908

H%

9

0. 1132

.1676

.2342

.2417

.3100

.3728

.3836

.4671

.5460

.5533

.6366

.7196

.8101

.8311

.9172

1.009

1.1255

1. 2204

1.2526

1.3425

1. 4533

1.5599

1. 5808

1. 6850

1. 7932

1. 8188

C

10

2.68

2.74

2.80

2.81

2.80

2.89

2.83

2.81

2.84

2.80

2.85

2.91

2.95

2.90

2.89

2.96

2.96

3.04

3.00

3.00

3.03

3.06

3.05

3.06

3.09

3.09
82 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

Bazin's experiments on weirs of irregular section Continued.

Bazin's Series, No. 160. 

Crest height, 2.46 feet. 

Crest width, 1.31 feet. 

Upstream slope, i to 1. 

Downstream slope, 6 to 1.

Period.

1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

Observed 

head, experimental 

weir D, 

in feet.

2

0.451

.522

.593

.663

.735

.798

.863

.930

.998

1.074

1.129

1.193

1.254

1.326

1.389

1.457

D%

3

0. 3029

.3772

.4567

. 5399

. 6302

.7128

.8017

.8969

.9970

1. 1130

1. 1996

1. 3030

1.4043

1. 5269

1. 6370

1. 7587

G

4

2.81

2.82

2.84

2.88

2.89

2.91

2.92

2.97

2.99

3.02

3.03

3.06

3.08

3.10

3.14

3.16

Q, flow 

per foot, 

experimental 

weir, in 

cubic feet 

per 

second.

5

0. 8540

1. 0637

1.2970

1. 5549

1.8213

2.0742

2.3410

2. 6638

2. 9810

3.3361

3. 6348

3. 9872

4. 3252

4.7334

5. 1402

5. 5575

V

6

0.29

.36

.42

.50

.57

.64

.70

.78

.86

.95

1.01

1.09

1.16

1.25

1.34

1.42

v- a

20

7 

0.0013

.0020

.0027

.0039

.0051

.0064

.0076

.0095

.0115

.0140

.0159

.0185

.0209

. 0243

.0279

.0313

H

8

0. 4523

.5240

.5957

.6669

.7401

.8044

.8706

.9395

1.0095

1. 0880

1. 1449

1. 2115

1. 2749

1.3503

1. 4169

1.4883

H%

9

0.3039

.3793

.4601

.5447

.6366

. 7209

.8129

.9100

1. 0135

1. 1349

1. 2252

1. 3334

1. 4397

1.5686

1. 6867

1. 8151

C

10

2.80

2.80

2.82

2.86

2.86

2.88

2.88

2.93

2.94

2.94

2.97

2.98

3.00

3.02

3.05

3.06

Bazin's Series, No. 161. 

Crest length, 6.543 feet. 

Crest height, 1.64 feet. Cross section.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

0.298

.354

.413

.472

.529

.581

.639

.693

.750

.804

.864

.919

.960

.992

1.019

1.056

1.083

1.118

1.157

1.187

1.225

1.263

1.289

1.326

1.359

0. 1627

.2107

.2654

.3243

.3847

.4429

.5108

.5770

.6495

.7209

.8031

.8810

.9406

.9880

1. 0287

1. 0851

1. 1271

1. 1821

1. 2445

1. 2932

1. 3558

1.4194

1.4635

1.5269

1. 5843

4.31

4.30

4.26

4.23

4.22

4.25

4.24

4.26

4.28

4.31

4.31

4.32

4.33

4.30

4.31

4.28

4.27

4.24

4.17

4.16

4.12

4.09

4.11

4.08

4.08

0.701

.906

1.131

1.371

1.625

1.883

2.167

2.458

2. 782

3.107

3.461

3.806

4.073

4.248

4.434

4.665

4.825

5.003

5.171

5.380

5.378

5.808

6.001

6.242

6.446

0.36

.45

.56

.65

.75

.85

.95

1.05

1.16

1.27

1.38

1.49

1.57

1.61

1.67

1.72

1.78

1.81

1.84

1.90

1.95

2.00

2.05

2.10

2.15

0.0020

.0031

.0049

.0066

.0087

.0112

.0140

.0171

.0209

.0251

.0296

.0345

.0383

.0403

.0434

.0460

.0493

.0509

.0526

.0561

.0591

.0622

.0658

.0686

.0719

0.3000

.3571

.4179

!4786

.5377

.5922

.6530

.7101

.7709

.8291

.8936

.9535

.9983

1. 0323

1. 0624

1. 1020

1. 1323

1. 1689

1.2096

1.2431

1. 2841

1. 3252

1.3543

1. 3946

1. 4309

0.1643

.2133

.2702

.3314

.3946

.4555

.5277

.5983

.6770

.7548

.8452

.9303

.9970

1.0484

1.0944

1. 1569

1.2044

1. 2640

1. 3310

1.3858

1.4550

1. 5252

1. 5756

1.6476

1. 7118

4.27

4.25

4.19

4.14

4.12

4.13

4.11

4.11

4.11

4.12

4.10

4.09

4.08

4.05

4.05

4.03

4.01

3.96

3.88

3.88

3.83

3.81

3.81

3.79

3.76
WEIRS OF IRREGULAR SECTION. 83

Bazin's experiments on weirs of irregular section Continued.

Bazin's Series, No. 163. 

Crest length, 6.535 feet. 

Crest height, 1.64 feet.

Cross section.

Period.

1

1

2 

3

4

5

6

7

8 '

9

10

11

12

13

14

15

16

17

18

19

20

21

Observed 

head, experimental 

weir D, 

in feet.

2

0.184

.244 

.303

.366

.423

.486

.536

.593

.653

.702

.769

.827

.882

.949

.998

1.056

1.114

1.171

1.231

1.285

1.339

rf

3

0.0790

.1206 

.1668

.2215

.3761

.3388

.3924

.4567

.5277

.5882

.6744

.7521

.8283

. 9245

.9970

1.0851

1. 1758

1. 2672

1. 3658

1.4567

1. 5495

f\

4

3.81

3.83 

3.84

3.83

3.83

3.82

3.86

3.94

3.91

4.04

3.98

4.02

4.02

4.04

4.06

4.06

4.05

4.07

4.07

4.12

4.17

Q, flow 

per foot, experimental 

weir, in 

cubic feet 

per 

second.

5

0.301

.463 

.641

.850

1.053

1.295

1.513

1.799

2.063

2.376

2.683

3. 023

3.329

3.735

4.048

4.425

4.779

5.169

5.576

6.014

6.464

c

6

0.17

.26 

.33

.42

.51

.61

.69

.81

.90

1.01

1.11

1.22

1.32

1.44

1.53

1.64

1.74

1.84

1.94

2.06

2.17

t;2 

2S7

0. 0004

.0009 

.00*7

.0027

.0040

.0058

H

8

0.1844

.2449 

.3047

.3687

.4270

.4915

. 0074 . 5434

.0102 .6032

. 0126 . 6656

. 0159 . 7179

.0188

.0231

.0271

.0322

.0364

.0418

.0471

.0526

.0585

.0660

.0732

.7878

.8501

.9091

.9812

1. 0344

1. 0978

1. 1611

1. 2236

1. 2895

1.3510

1. 4122

/ft

9

0.0790

.1213 

.1684

.2241

.2791

.3441

.4001

.4683

.5435

.6084

.6995

.7837

.8667

.9717

1. 0514

1. 1505

1. 2510

1. 3541

1.4635

1. 5703

1. 6779

C

10

3.81

3.82 

3.81

3.79

3.77

3.76

3.78

3.84

3.80

3.90

3.84

3.86

3.84

3.84

3.85

3.85

3.82

3.82

3.81

3.83

3.85

Bazin's Series, No. 161. 

Crest length, 6.534 feet. 

Crest height, 1.64 feet. Cross section.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0.244

.305

.367

.425

.482

.540

..592

.651

.702

.766

.817

.877

.939

.993

1.052

1.115

1.162

1.219

1.277

1.330

0. 1206

. 1685

.2224

.2771

.3346

.396«

.4565

.5252

. 5882

. 6705

. 7385

.8213

.9100

.9895

1. 0790

1. 1774

1. 2526

1 . 3459

1. 4431

1. 5338

3.86

3.91

3.87

3.90

3.87

3.87

3.94

3.94

3.97

4.00

4.03

4.05

4.07

4.10

4.09

4.12

4.13

4.15

4.18

4.19

0.467

.659

.359

1.080

1.296

1.536

1.797

2.069

2.334

2.684

2.978

3.325

3.704

4.055

4.417

4.862

5.163

5. 602

6.019

6.411

0.25

.34

.43

.52

.61

.70

.81

.90

.99

1.11

1.21

1.32

1.44

1.54

1.64

1.76

1.84

1.96

2.06

2.16

0.0009

.0018

.0030

.0042

.0058

.0076

.0102

. 0126

.0152

.0188

.0228

.0271

.0322

.0369

.0418

.0482

.0526

.0597

.0660

.0725

0. 2449

.3068

.3700

.4292

.4878

.5476

.6022

.6636

.7172

.7848

.8398

.9041

.9722

1. 0299

1. 0938

1. 1632

1. 2146

1. 2787

1.3430

1.4025

0. 1213

.1701

.2251

.2810

.3409

.4057

.4671

.5410

.6071

.6955

.7699

.8595

.9583

1.0453

1. 1442

1. 2543

1. 3392

1. 4465

1.5564

1.6601

3.85

3.88

3.82

3.84

3.80

3.79

3.85

3.82

3.84

3.86

3.87

3.87

3.86

3.88

3.86

3.88

3.86

3.87

3.87

3.86
84 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

Bazin's experiments on weirs of irregular section Continued.

Bazin's Series, No. 165. 

Crest length, 6.544 feet.

Crest height, 1.64 feet.

Cross section.

Period.

1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

Observed 

head, experimental 

weir D, 

in feet.

2

0.337

.401

.464

.i>28

.593

.656

,720

.783

.843

.904

.969

1.029

1.090

1.153

1.217

1. 279

1.341

1.401

1.448

D*

3

0. 1957

.2540

.3161

.3836

.4567

.5313

.6109

.6929

.7740

.8595

.9538

1.0438

1.1380

1.2381

1. 3426

1. 4465

1. 5529

1. 6583

1. 7424

Oi

4

3.56

3.56

3.56

3.55

3.54

3.54

3.54

3.55

3.58

3.61

3.63

3.63

3.64

3.65

3.66

3.68

3.68

3.69

3.73

Q, flow 

per foot, experimental 

weir, in 

cubic feet 

per 

second.

5

0.698

.904

1.125

1.363

1.618

1.880

2. 162

2.461

2.771

3.103

3.462

3.789

4.150

4.526

4.904

5.336

5.704

6.125

6.490

V

6

0.35

.44

.54

.63

.73

.82

.92

1.02

1.11

1.22

1.33

1.42

1.52

1.62

1.71

1.83

1.92

2.02

2.10

iff 

20

7

0. 0019

.0030

.0045

.0062

.0083

.0105

'. 0132

.0162

.0192

.0231

.0275

.0313

.0359

.0408

. 0455

.0521

.0573

.0634

.0686

II

8

0. 3389

.4040

.4685

.5342

.6013

.6665

.7332

.7992

.8622

.9271

.9965

1.0603

1. 1259

1.1938

1. 2625

1.3311

1. 3983

1.4644

1. 5166

rf

9

0. 1974

.2568

.3202

.3902

.4660

.5435

.6276

.7142

.8003

.8926

.9940

1.0913

1. 1948

1.3046

1.4178

1.5355

1.6530

1.7714

1.8680

C

10

3.54

3.52

3.51

3.49

3.47

3.46

3.44

3.45

3.46

3.47

3.48

3.47

3.47

3.47

3.46

3.48

3.45

3.46

3.47

Bazin's Series, No. 176. 

Crest length, 6.519 feet.

urexu iieigiiu, ^.^D leeu. """"""""" 

Cross section.

1

0

3

4

5

0

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

0.237

.296

.365

.439

.494

.565

.618

.682

.73H

.797

.861

.910

.974

1.027

1.088

1.139

1.196

1.248

1.303

1. 355

1.420

0. 1154

.1611

. 2206

.2909

.3472

.4247

.4858

. 5632

.6276

. 7115

.7989

.8681

.9613

1. 0408

1.1349

1. 2156

1. 3079

1. 3942

1. 4874

1. 5773

1.6921

2. ^ti 0.317

2. 74 . 441

2.92

2.95

.645

.858

3.04 ' 1.055

3.10

3.19

3.22

3.29

3.35

3.41

3.45

3.51

3.53

3.57

1.318

1.550

1.813

2. 066

2.385

2. 724

2'. 995

3.373

3.671

4.034

3. 62 4. 416

3. 65 4. 782

3.68

3.73

3.75

3.80

5.115

5.558

5.925

6.422

0.12

.16

.23

.30

.36

.43

.50

.58

.65

.73

.82

.89

.98

1.05

1.14

1.23

1.30

1.38

1.47

1.55

1.66

0.0002

.0004

.0008

.0014

.0020

.0030

.0039

0. 2372

.2964

.3658

.4404

.4960

.5680

. 6219

. 0052 . 6872

. 0066 . 7396

.0083

.0105

.0123

.0149

.0171

~ . 0202

.0235

.0263

.0296

.0336

.0374

.0428

.8073

.8715

.9223

.9889

1. 0441

1.1082

1. 1625

1.223

1. 2776

1. 3366

1.3924

1. 4628

0. 1154

.1611

. 2214

.2919

.3494

.4281

.4905

.5695

.6366

.7250

.8129

.8853

.9835

1.0667

1.1663

1.2526

1.3508

1.4448

1.5460

1. 6423

1.7695

2.75

2.74

2.91

2.94

3.02

3.08

3.16

3.18

3.24

3.29

3.35

3.38

3.43

3.44

3.46

3.52

3.54

3.54

3.60

3.61

3.63
WEIRS OF IEREGULAR SECTION. 85

Basin's experiments on weirs of irregular section Continued.

Bazin's Series, No. 178. 

Crest length, 6.518 feet. 

Crest height, 2.46 feet.

Cross section.

Period.

1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

Observed 

head experimental 

weir D, 

in feet.

2

0.222

.299

.367

.431

.491

.556

.614

.669

.732

.789

.847

.906

.966

1.028

1.083

1.142

1.195

1.259

1.314

1.366

1.424

D*

3

0.1046

.1635

.2224

.2830

.3441

.4146

.4811

.5472

.6263

.7009

.7796

.8624

.9494

1.0423

1.1271

1.2204

1. 3063

1.4127

1.5063

1. 5965

1.6992

Ci

4

2.83

2.95

2.96

3.08

3.13

3.19

3.24

3.28

3.33

3.36

3.43

3.46

3.51

3.53

3.58

3.60

3.64

3.66

3.69

3.72

3.75

Q, flow 

per foot, 

experimental 

weir, in 

cubic feet 

per 

second.

5

0.297

.482

.658

.872

1.077

1.323

1.558

1.794

2.085

2.355

2.675

V

6

0.11

.17

.23

.30

.37

.47

.51

.57

.65

.73

.81

2. 983 . 89

, 3. 331

3.671

4.045

4.392

4.755

5.170

5.572

5.952

6.375

.97

1.05

1.14

1.22

1.30

1.39

1.48

1. 55

1.65

('-' 

2<7

7

0.0002

.0004

.0008

.0014

 .0021

.0034

.0040

.0051

.0066

.0083

.0102

.0123

H

8

0.2222

.2994

.3678

.4324

.4931

.5594

.6180

.6741

.7386

.7973

.8572

.9183

. 0146 . 9806

.0171

.0202

.0231

.0263

.0300

.0341

.0374

.0423

1. 0451

1.1032

1.1651

1.2213

1. 2890

1. 3481

1.4034

1.4663

tft C

9 10

j

0.1046

.1635

.2233

.2840

.3462

.4180

.4858

. 5533

.6353

.7115

.7934

.8795

.9716

1.0683

1. 1584

1.2575

1.3492

1.4635

1.5651

1. 6618

1. 7750

2.84

2.95

2.95

3.07

3.11

3.16

3.21

3.24

3.28

3.31

3.37

3.39

3.43

3.44

3.49

3.49

3.52

3.53

3.56

3.58

3.59

CORNELL UNIVERSITY HYDRAULIC LABORATORY. 05

This laboratory, erected in 1898, includes a reservoir formed by a 

masonry dam on Fall Creek, at Ithaca, N. Y. An experimental channel is supplied with water from the pond and has, as its general 

.dimensions, length, 400 feet; breadth, 10 feet; depth, 10 feet; bottom 

grade, 1:500. Fall Creek drains an area of 117 square miles, and 

affords a minimum water supply estimated at 12 second-feet. The 

hydraulic laboratory is located at Triphammer Falls, where a descent 

of 189 feet occurs. The weirs used in the experiments here described 

were erected in the concrete-lined experimental channel. The water 

supply was regulated b}^ wooden head-gates, operated by lever, rack, 

and pinion, the outflow from the canal passing over the declivity below.

a In reducing the experiments at Cornell hydraulic laboratory the value of g for Ithaca, latitude

_ 1 2 42° 27', altitude 500 feet, has been taken as 32.16, makingl/23<=8.02,2-f=0.015547, ^V20=5.35.
86 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

EXPERIMENTS OF UNITED STATES BOARD OF ENGINEERS ON DEEP

WATERWAYS.

These experiments were performed at Cornell University hydraulic 

laboratory in May and June, 1899, for the United States Board of 

Engineers on Deep Waterways, under the immediate direction of 

George W. Rafter, engineer for water supply, in conjunction with 

Prof. Gardner S. Williams. The results of the original computations 

were published in Trans. Am. Soc. C. E., vol. 44, together with an 

extended discussion. In the experiments a closely regulated volume 

of water was passed over a standard thin-edged weir which was placed 

near the upper end of the experimental canal and had a height of 13.13 

feet and a crest length of 16 feet, end contractions suppressed. The 

nappe was aerated, but was not allowed to expand on downstream side. 

The water flowed down the experimental canal past a series of screens 

and baffles and over the experimental weir placed at the lower end of 

the channel.

The experimental weirs were about 4.5 feet high and 6.56 feet crest 

length. A leading channel of planed boards, 6.56 feet wide and 48 

feet in length, extended upstream from the experimental weir, having 

at its upper end flaring sides extending 8.3 feet upstream and meeting 

the sides of the main channel.

The head on both weirs was read by means of open manometers 

connected to galvanized-iron piezometer pipes, placed horizontally 

across the bottom of the narrow leading channel, 37 feet upstream 

from the weir. At the standard weir two piezometers were used, one 

termed the middle piezometer, placed across the leading channel, 8 

inches above the bottom and 10 feet upstream from the standard weir. 

A second or upstream piezometer was placed 25 feet upstream from 

the standard weir. Readings of both piezometers were taken. It was 

decided, however, to use the middle piezometer as the basis of calculation of discharge over the standard weir. Near the close of the 

experiments it was found that this did not give results agreeing with 

those which would have been obtained from a piezometer placed flush 

with the bottom of the channel, as is shown to be necessary from the 

experiments of H. F. Mills a and others. A correction curve was 

accordingly deduced from comparative experiments between the middle 

piezometer and the flush piezometer, and the readings of the middle 

piezometer thus corrected were applied in the Bazin formula to calculate the discharge over the standard weir for heads not exceeding the 

limit of Bazin's experiments. For depths on the standard weir greater 

than 2 feet the discharge was computed by using coefficients deduced 

for higher heads on a shorter experimental weir, on the basis of the 

Francis formula. Owing to the uncertainty as to the piezometers and

a Mills, H. F., Experiments upon piezometers used in hydraulic investigations, Boston, 1878.
HYDRAULIC LABORATORY AT CORNELL UNIVERSIT
U. S. GEOLOGICAL SURVEY WATER-SUPPLY PAPER NO. 150 PL. XIV

 S

rCORNELL HYDRAULIC LABORATORY, ARRANGED FOR WEIR EXPERIMENTS.
WEIRS OF IRREGULAR SECTION. 87

other conditions, the original results of the experiments were credited 

with a possible error of 5 or 6 per cent.

In connection with the experiments on models of the Croton dam, a 

very thorough comparison of the so-called upstream piezometer with 

other methods of obtaining the head on a standard weir was made by 

Professor Williams. It was found that the upstream piezometer gave 

the actual head on the standard weir correctly. These results were communicated to the writer, and a recomputation of the Deep Waterways 

experiments has been made, using readings of the upstream piezometer to calculate the standard weir discharge by Bazin's formula. This 

method of calculation eliminates the necessity for correcting the pie- 

.zometer readings at the standard weir, as was necessary in the previous 

reductions. The discharge over the experimental weir has been calculated from readings of a piezometer placed 38 feet upstream from 

the weir and 8 inches above channel bottom, corrected to the basis of 

a flush piezometer.

The United States Deep Waterways experiments included, for each 

experimental model, a smaller number of heads or periods than either 

the Croton or United States Geological Survey experiments. They 

were also'the first experiments of the kind conducted at the Cornell 

laboratory, and the experience gained has probably contributed to the 

securing of somewhat greater accuracy in the later experiments. It is 

believed, however, that, as recomputed, the United States Deep Waterways experiments do not differ much in accuracy from those made on 

models of the Croton dam, which are stated by John R. Freeman to 

be reliable within about 2 per cent. The coefficients obtained by 

recomputation, when compared with the original United States Deep 

Waterways coefficients, show few differences exceeding 2 per cent. 

The variations are plus and minus in about equal numbers, and it is 

believed that these experiments are entitled to greater weight than 

they have hitherto received.

In the accompanying tables a summary of the recomputation is 

given.
WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

Recompuiation of United States Deep Waterways Board experiments on flow of water over 

model dams, Cornell University hydraulic laboratory, 1899.

Weir model.

Series 1.

P=4.91. £=6.58.

Series ft.

P=4.90. £=6.58.

Series S.

P=4.91. £=6.58.

Series 4.

P=4.91. £=6.58.

Series 5.

P=4.90. £=6.58.

Series 7.

P=4.8S5. £=6.58.

P=4.90. £=6.58. 

Series 9.

P=4.94. £=6.58.

Corrected 

depth 

D, experimental 

weir, centimeters.

3

146. 70 

123.00

98.80 

75.72 

50. 65

142. 75 

119. 20 

96. 72 

74.50

144. 00 

120. 50 

97.27

74.35 

49.77

147. 10 

123. 00 

99.62 

76.22 

51.00

P, in 

feet.

4

5.05

3.35

4.812 

4.034 

3. 242 

2.484 

1.662

4.682 

3.969 

3.173 

2.444

4.723 

3. 953 

3.192 

2.439 

1.633

4.825 

4.034 

3.268 

2.500 

1.673

Q=

flow 

i per 

foot, in 

cubic 

feet per 

second.

5

8.71

4.88

42.13

.82

41.04 

31. 22 

21.49 

13. 27 

8.21

41.40 

30.72 

21.64 

14.07

41. 22 

30.50 

21. 53 

14.18 

7.63

41.16 

30.47 

21. 48 

14.24

7.85

V

6

........

........

4.23 

3.50 

2.65 

1.80 

1.24

4.30 

3.47 

2.67 

1.92

4.28 

3.45 

2.62 

1.92 

1.17

4.20 

3.38 

2.62 

1.92 

1.19

*-£

7

........

0. 2782 

. 1959 

.1092 

.0504 

.0239

.2875 

.1872 

.1108 

.0573

.2848 

.1850 

.1067 

.0573 

.0213

. 2742 

.1776 

.1067 

.0573 

.0220

H=

D+h

8

4.972

4.872

4.853

4.138

3.368

1.725

1.190

5. 050

4. 150

3.350

2. 550

1. 750

.923

.340

5.23

1.75

5.11

4.28

2.57

6. 0902 

4. 2299 

3.3512 

2.5344 

1. 6859

4. 9695 

4. 1562 

3.2838 

2.5013

5.0078 

4.1380 

3.299 

2. 4963 

1.6543

5. 0992 

4. 2116 

3. 3927 

2. 5573 

1. 6950

C'l

9

3.584

3. 668

3.940

3.690

3.844

3.759

3. 712

3. 758

3.691

4.120

3.393

3. 382

3.485

3.485

3.574 

3.593 

3.503 

3.289 

3.751

3.737 

3.626 

3.637 

3.557

3.678 

3.623 

3.591 

3. 595

3.585

3.575 

3. 525 

3.437 

3.482 

3.557

Number of 

observations 

of 

head.

10

41

23

15

21

15

18

7

21

15

21

r,l

27

27

27

23

25 

31

27 

25 

21

29 

29

27 

27

18 

26 

23

27 

15

22 

29 

25

24

f'Same as series 7, but upstream face covered with J-inch mesh galvanized wire netting.
WEIKS OF IEREGULAE SECTION. 89

Recomputation of United States Deep Waterways Board experiments onflow of water over 

model dams, Cornell University hydraulic laboratory, 1899 Continued.

Weir model.

1

Series 10.

__ ^ .*-£6g*

P=4.57. £=6.58.

Series 11.

__ ^ *-2j6Z*

Rad-OMV/M T

Hi I

/ =4.57. £=6.58.

Series 12.

__ , * 6.56-*

? ^B

P=4.56. £=6.58.

S&fies IS

k--6.36 - 

Rad=o.33'MM;w/6 T

?

P=4.56. £-6.58.

T" «'*!i^Pi- ^-i'/.o --4

P=4.63. £=6.58.

Series IB.

Series 16.

0.75-fc*fx%-l

^^^^^^^M^

P=4.57. £=6.58.

Series 17. ~* "'"jjjjjjji^

">w%W%%$( *

P=4.57. £-6.58.

SWesjW.

i

- yiiilfs

^liltiliK

P=4.65 £-6.58.

Period.

2

1 

2 

3 

4 

5

1

2

3

4

5

1

3

4

1

2

3

4

5

1

?,

4

5

6

1

?,

3

4

1

?,

3

4

1

9:

3

4

1

2

S

Corrected 

depth D, experimental 

weir, centimeters.

3

153. 82 

132. 94 

110. 98 

87.99 

61.70

149.-15

127. 70

105.60

82. 52

58.80

154. 55

26. 80

96.75

66.30

44.70

16.60

88.52

60.02

30.80

57. 05

31.60

80.25

54.25

28.05

129. 55

105.00

79.52

53.65

126. 75

102. 55

78.02

52.00

127. 30

103. 32

78.39

32.40

38. 523

149. 362

125. 693

75.427

\

D,in 

feet.

4

5.046 

4.361 

3.641

2.887 

2.024

4.892

4.189

3.464

2.707

1.930

5.069

4.160

3.174

2.174

4.747

. 3.825

2.904

1.969

1.010

5. 151

4.317

2.633

1.780

.920

4.250

3.444

2.608

1.760

 

4157

3.364

2. 559

1.706

4.175

3.389

2.571

1.691

1.063

1.264

4.899

4.123

3

QAC

2.474

Q= 

flow 

per 

foot, in 

cubic 

feet per 

second

5

42.0 

31.2 

22.33 

14.7 

8.12

42.06

31.66

22.02

14.34

8.26

30.69

21.58

14.12

7.85

30.69

21.75

14.07

7.85

2.89

41.16

30.64

21.68

14.07

7.78

2.75

31.12

21.87

14.02

7.95

30.69

21.87

14.02

7.73

31.08

21.87

14.34

7 73

3.82

5.47

40.19

30.25

91 ^ft

13.85

I

r

6

4.37 

3.50 

2.72 

1.99 

1.24

4.45

3.60

2.72

1.97

1.26

3.19

2.48

1.82

1.17

3.28

2.60

1.90

1.19

.51

4.23

3.45

2 72

1.97

1.24

.51

3.55

2

70

1.97

1.26

3.52

2.75

1.97

1.24

3.55

2.75

2.02

1.24

.68

.92

4.23

3.45

2.70

1.94

*-*

7

0.296 

.1904 

.1150 

.0616 

.0239

.3079

.2015

.1150

.0603

.0247

.1582

.0515

.1673

. 1051

.0603

AOQQ

0040

.1959

.1150

.0603

.0247

.1926

.1176

.0603

.0239

.1959

.1176

.0634

.0239

.0072

.0132

.2782

.1850

1100

.0585

H=

D+h

8

5.3429 

4. 5514 

3.7560 

2.9486 

2. 0479

5.1999

4.3905

3. 5790

2. 7673

1.9547

5.219

4.2556

3.2255

2. 1953

4.9143

3. 9301

2 9601

1. 9910

1. 0140

5. 4292

4.5020

2.6933

1.8039

9240

4.4459

2.6683

1. 7847

4. 3496

3.4816

2. 6193

1 7299

4. 3709

3. 5066

2 6344

1. 7149

1.0702

1. 2772

5. 1772

4.3080

3. 4183

2. 5325

Ci

9

3.402 

3.220 

3.068 

2.917 

2.771

3.547

3.418

3.252

3.115

3.022

2.574

2.458

2.438

2.413

2.817

2. 790

2. 763

2.859

 2.830

3.254

3.208

3 01 o

3.183

3.219

3.096

3.320

3 O&V7

3.217

3.333

3.383

3.366

3.307

3.397

3.401

3.331

3.354

3.790

3.412

3 383

3

OQO

3.437

Number of observations 

of 

head.

10

21 

21 

20 

17 

17
90 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

Recomputation of United States Deep Waterways Board experiments on flow of water over 

model dams, Cornell University hydraulic laboratory, 1899 Continued.

Weir model.

1

Series 19.

* " ~J$<ffi$*j$/yTr300

^^jjjjjh

P=5.28. £=6.58.

 a

PH

2

8

4

5

7

Corrected 

depth

perimental

weir, 

centimeters.

3

27.04

142. 128

119 442

97.858

53.42

D, in 

feet.

4

0 8869

1.685

4.662

3

<Y| Q

3. 210

2.534

1 7V>

Q= 

flow

per 

foot, in 

cubic

feet per 

second.

5

2.75

7.46

40 04

20.78

14.17

7 68

V

6

0.44

q OQ

2.45

1.82

1 09

*=?2fif

7

0 0030

.0178

.0933

.0515

.0185

H=

D+h

8

1.7028

3.3033

2.5855

1. 7705

C'i

9

3. 276

3.357

3.651

3.594

3.461

3.401

Number of

observations 

of

head.

10

Column 5 shows the discharge over the experimental weir per foot 

of crest, deduced from the readings of the upstream piezometer at the 

standard weir, by Bazin's formula, and corrected for slight leakage.

Column 3 shows the head on the experimental weir, in centimeters, 

taken by a piezometer 38 feet upstream and 8 inches above channel 

bottom, corrected to reduce it to the equivalent reading of the flush 

piezometer.

Column 4: shows the equivalent head in feet.

Column 6 shows the absolute velocity of approach.

Column 7 shows the velocity head.

Column 8 shows the head corrected for velocity of approach; the 

correction being made by the simple addition of the velocity head to 

the measured head, which is assumed to be a sufficiently precise equivalent to the Francis correction formula for this purpose.

Column 9 gives the coefficient (715 deduced from the foregoing.

The resulting coefficient diagrams are shown on Pis. XV to XVIII, 

inclusive.

EXPERIMENTS AT CORNELL UNIVERSITY HYDRAULIC LABORATORY ON 

MODELS OF OLD CROTON DAM. a

These experiments were made in November and December, 1899, 

by Prof. Gardner S. Williams, under the direction of John R. Freeman. The standard weir used was located near the head of the experimental canal, water being admitted and regulated by head-gates in the 

usual manner. The standard weir was 11.25 feet high and 16 feet 

long on the crest. The experimental weir was placed 232.5 feet farther downstream, and also occupied the full width of the experimental 

canal. The models of the Croton dam were constructed of framed 

timber and were 6 to 9 feet high.

a Report on New York's water supply, Freeman, 1900, pp. 139-141.
U. S. GEOLOGICAL SURVEY

Coefficient 

C.

OiSenes 10

Seh'es /a

WATER-SUPPLY PAPER NO. 150 PL. XV

-C:

.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 

Corrected head In feet.

4.8 5.2 5.6 6.0

Hi

Series/O

 - 6.56'--x

Series 12

Coefficient 

f.

o

A

Se

Se

 * *'

rie.

fe}

^j

- //

,3

V

- *" ^- ** >-c

-i.

^-*

 ^ ^^

*^~

A

^^ -<:r

^~

^

^^ ^v

**

A

d)

2 81

0 .4 .8 1.8 1.6 2.0 2.4 2.8 S.2 3.6 4.0 4.4 4.8 5.2 5.6 6.0 

Corrected head in feet.

K26Z-I - -6.36^- ->t

Series 13

EXPERIMENTS OF UNITED STATES DEEP WATERWAYS BOARD AT CORNELL 

UNIVERSITY, 1899.
Coefficient 

0. 

3.80

3.60 

3.40

3.20 

1

U. S. GEOLOGICAL SURVEY 

1 I r r 1

^^ss^i

S^ 

tf

r- "> 

k

! 

A

Q

7

m. 

1 1 1

 >.

ff) rsj x 

*0 w

0 O

 "

 I "

 

1 1

/ 

^

0.67'

1-.G9tO 

?.9t'

-cr

i 

MHM «=

?ft =Ef=

^

Co eff

3.70

3.60 

3.50 

3.40 

3.30

AV

.0 2.0 3.0 

Corrected head if In feet. 

EXPERIMENTS OF UNITED STATES DEEP WATERWAYS BOARD AT 
U. S. GEOLOGICAL SURVEY WATER-SUPPLY PAPER NO. 150 PL. XVII

Corrected head in feet. 

2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.8 5.2 I

Coeffl 

c'ient

3.80 

3.60 

3.40

C

iiSer/es 7 A Series 8 

P/aned boards Wire ctotfi cover

^>,§i%%*-,

SERIES 7 -8

^P i

w«

£

CD

\

'

Y

f 

G

""ib-

 o^<-t -

(3^

 C.5

~er/es 14 /\ Series /S

ERIES 14. Sharp upstream ang/e 

" /S . Upstream ang/e rounded ; 

rad.= 0.33'

_A- 5)

^i

i\

^ }

ii22.

1.2 1.6 3.0 8.4 2.8 3.2 3.6 4.0 4.4 4.8 5.3 5.6 

Corrected head -ff in feet.

EXPERIMENTS OF UNITED STATES DEEP WATERWAYS BOARD AT CORNELL 

UNIVERSITY, 1899.
U. S. GEOLOGICAL SURVEY WATER-SUPPLY PAPER NO. 150 PL. XVIII

Crest batter 6:/

0.7S' i^^7^J

0 .4 .8 1.1 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.8 5.2

3.20

r zo '--^

''ff£r$$ffi&\'

^lillfiilf *

-V////////AL

Series 18, Indian Lake

S

(5)

St Rl,

-.5

 I

18

' $-

<3>

/:

]

3

i*

42

U .4 .8 1.8 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.8 5.2 5.6 

Corrected head H in feet.

EXPERIMENTS OF UNITED STATES DEEP WATERWAYS BOARD AT CORNELL 

UNIVERSITY, 1899.
WEIRS OF IRREGULAR SECTION.

The head on the weirs was measured by means of open glass manometers connected to piezometer tubes in the channel above each weir. 

The piezometer tubes were made of 1-inch galvanized-iron pipe with 

small holes drilled along- the sides, the ends being plugged. At the 

standard weir three piezometers were used, placed parallel to the current, at about mid-depth of the channel, one being near each side and 

one at mid-width of the channel, the mid-length of the pipes being 

26.5 feet upstream from the standard weir. A hook gage in the same 

section was used to check the observed head.

Experiments on volume of flow over models of old Oroton dam, Cornell University hydraulic laboratory, 1899.

1

Series 1 Model A.

Round crest, old Croton dam, smooth pine,

crest and slope 16 feet long. Nov. 28-29,

1899.

Series la Model A.

Round crest, old Croton dam, unplaned

plank, crest 16 feet long, smooth slope.

Nov. 6, 1899.

Series 2 Model A.

Round crest, old Croton dam, 16-foot smooth

crest, rough slope formed of cleats and

stone to simulate concrete and riprap. Dec.

4, 1899.

Series 3 Model A.

Round crest, old Croton dam, 16-foot crest,

covered with wire cloth of No. 18 wire,

i-inch mesh.a rough slope, as in series 2.

Nov. 28, 1899.

Period

No.

2

1

2

3

4

5

6

1

' 2

3

4

5

6

7

1

2

3

4

5

6

7

8

9

1

2

3

4

5

Ob 

served 

depth

on 

model 

dam, 

in feet.

3

2. 7229

2. 1857

1.4388

Mean velocity of 

approach, 

in feet 

per 

.second.

4

1.685

1.283

.749

Correction 

for 

velocity of 

approach, 

in feet.

5 .

0. 0439

.0259

.0087

. 9830 . 449 . 0031

.5907

.1230

.219

.024

.0008

.0000

2. 0897 . 978 . 0149

1. 8293 . 810 . 0102

1. 5878 . 661 . 0068

1. 2562

.9929

.6301

. 4871

2. 9227

2. 8591

2. 4948

2. 1420

1.6238

1. 2597

1. 1419

.7196

.4873

2. 0030

1. 4091

.8675

. 4288

.1184

.467 .0034

. 338 . 0018

. 175 . 0004

. Ill . 0002

1.839

1.794

1.516

1.248

.880

.623

.545

.288

.166

1.124

.712

.366

.133

.020

.0526

.0500

.0357

,0241

.0120

.0060

.0046

.0013

.0004

.0197

. .0078

.0021

.0003

.0000

Corrected 

head

on 

model 

dam, 

in feet.

6

2. 7668

2. 2116

1. 4475

.9861

.5915

.1230

2. 1046

1. 8395

1. 5946

1. 2596

.9947

. 6305

.4873

2. 9753

2. 9091

2.5305

2. 1661

1. 6358

1. 2657

1. 1465

.7209

.4877

2. 0187

1. 4129

.8656

.4251

.1144

Discharge

over 

model 

dam 

per

foot of 

length, in cubic f eet^

per second.

7

14. 762

10. 562

5.604

3.154

1.451

.147

9.578

7.883

6.284

4.287

3.006

1.494

.991

16. 175

15. 969

12. 933

10. 211

6.740

4.548

3.913

1.945

1. 087

9.037

5.308

2.527

.861

.124

8

3. 208

3.211

3.218

3. 222

3.190

3.408

3.137

3.160

3.121

3.032

3.030

2.988

2.913

3.240

3.218

3. 213

3.203

3.222

3.194

3.188

3.178

3.192

3.148

3.161

3.13K

3.099

3.205

«In experiments with wire cloth over crest, 0.004 foot is deducted from observed depth to 

compensate for thickness of wire.
92 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

Experiments on volume of flow ocer models of old Croton dam, Cornell University hydraulic laboratory, 1899 Continued.

1

Series 1 Model B.

Angular crest, old Croton dam, 16-foot crest,

all unplaned plank. Nov. 15, 1S99.

Series 2 Model B.

Angular crest, old Croton dam, 16-foot uiiplaned plank, crest slope roughened with

cleats and stone. Nov. 28, 1899.

Series 2 Continued model B.

Conditions as in preceding. Nov. 16, 1899.

Series 3 Model B.

Angular crest, old Croton dam, wire cloth on

crest, rough slope. Nov. 16, 1899.

Series 1 Model C.

Round crest, old Croton dam, 12-inch timber

on crest, 16 feet long, rough slope. Dec.

1, 1899.

Series 1 Continued model C.

Conditions as in preceding. Dec. 4, 1899.

Series 1 Model D.

Angular crest, old Croton dam, 12-iiich timber, on 16-foot crest, rough slope. Nov. 16,

1899.

Period

No.

2

!

Observed 

depth 

on 

model 

dam, 

in feet.

8

1. 8635

2 1 .9246

3

 1

5

1

2

3

4

5

6

1

2

3

4

5

1

2

3

4

1

2

3

4

5

6

7

8

1

2

3

4

5

1

2

3

.6419

.3481

.1787

2.4126

1. 5251

.9611

. 5157

. 3051

.0890

1. 8930

.9605

Mean velocity of

approach, 

in feet 

per 

second.

4:

0.973

.370

.219

.090

.034

1.298

.736

.391

.162

.077

.012

.988

.391

. 7028 . 251

. 3941 . 108

. 1952 . 039

2. 0053 1. 047

. 9787 . 389

.7391 ' .259

.1785 .032

1. 9941 1. 097

1. 1817

.8832

.6873

.4986

.2992

.1177

.0846

2. 7146

'2. 4519

1.5566

1.1046

.1070

1.2390

. 7885

.4448

.512

.328

.222

.141

.071

.019

.009

1. 632

1.436

.774

.016

.0165

. 495

.249

.113

Correction 

for 

velocity of 

approach, 

in feet.

5

0. 0147

.0021

.0008

.0001

.0000

.0262

.0084

.0024

.0004

.0001

.0000

.0151

.0024

.0010

.0002

.0000

.0170

.0024

.0011

.0000

.0187

.0040

.0017

 .0008

.0003

.0001

.0000

.0080

.0414

.0320

.0093

.0000

.0000

.0038

.0010

.0002

Corrected 

head 

on 

model 

dam, 

in feet.

6

1.8782

.9267

.6427

.3482

.1787

2. 4388

1.5335

.9635

.5161

. 3052

.0890

1.9081

.9629

.7038

.3943

.1952

a2. 0183

.9771

.7362

. 1745

2. 0128

1. 1857

.8849

.6881

.4989

. 2993

.1177

.0846

2. 7560

2. 4839

1.5659

.1046

.1070

1.2428

. 7895

.4450

Discharge 

over 

model 

dam 

per 

foot of 

length, 

in cubic 

feet per

second.

7

9.506

3.272

1.870

.741

.272

13. 478

6. 945

3.466

1.369

.631

.094

9.683

3.465

2. 165

.900

.314

10.385

3.463

2. 241

.260

9.904

4.211

2.594

1.722

1. 065

.522

.139

.067

15. 917

13.629

6.660

.112

.118

5.026

2.415

1.054

Ci

8

3.693

3.668

3.630

3.606

3.601

3. 539

3. 6f>7

3.665

3.692

3.742

3.540

3.674

3.667

3.667

3.635

3.641

3. 622

3.586

3.548

3.567

3.468

3.262

3.116

3.017

3.022

3.188

3.450

2.723

3.479

3.482

3.399

3.311

3.371

3.628

3.443

3. 551

a In experiments with wire cloth over crest, 0.004 foot is deducted from observed depth to compensate for thickness of wire.
WEIES OF IKKEGULAK SECTION. 93

Experiments on volume of flow over models of old Croton dam, Cornell University hydraulic laboratory, 1899 Continued.

1

Series 1 Model E.

16-foot angular crest, old Croton dam, without timber, but with obstructed channel,

with sharp contraction. Nov. 18, 1899.

Series 1 Model E, repeated.

Conditions as in preceding. Nov. 25, 1899.

Angular crest, old Croton dam, 16 feet long

without timber, and with slope instead of

sharp edge to upstream end of obstruction.

Nov. 27, 1899.

Period

No.

2

1

2

3

4

5

6

7

8

i

2

3

4 

5

6

1

2

3

4

5

6

7

8

Observed 

depth

model 

dam, 

in feet.

3

2. 3051

1.8125

1. 2278

. 8598

.5745

.3245

.1120

.1102

1.4569

.9168

.6866

.4820 

.2811

.1416

2. 2927

2. 2914

1. 1395

1. 1403

1.1006

1. 1099

.4763

. 0233

Mean velocity of 

approach, 

in feet 

per 

second .

4

1.154

.845

.507

.313

.179

.078

.017

.017

.636

.338

.237

.148 

.071

.028

1.139

1.138

.473

.453

.448

.457

.141

.031

Correction 

for 

velocity of 

approach, 

in feet.

5

0. 0207

.0111

.0040

.0015

.0005

.0001

.0000

.0000

.0063

.0018

.0009

.0003 

.0001

.0000

.0202

.0202

.0034

.0031

.0031

.0032

.0003

.0000

Corrected 

head

model 

dam, 

in feet.

6

2. 3258

1. 8236

1.2318

.8613

.5750

. 3246

.1120

.1102

1. 4632

.9186

.6875

.4823 

.2812

.1416

2. 3129

2. 3116

1.1429

1. 1434

Discharge 

over 

model 

dam

foot of 

length, 

in cubic 

feet per

second.

7

11. 778

8.214

4.632

2.744

1.516

.641

.133

.140

5. 957

2.982

2.037

1.240 

.579

.226

all. 613

611. 606

4.278

a 4. 097

1.1037 64.034

1. 1131

.4766

.0233

4.119

1.180

.025

8

3. 321

3.336

3.388

3.433

3.477

3.466

3.548

3.827

3.366

3.387

3.573

3.702 

3.883

4.203

3.302

3.302

3.501

3.351

3.479

3.507

3.586

7.029

a Trap open. 6 Trap closed.

At the experimental weir two similar piezometers, each about one 

third of the width of the channel from the side, were used. Owing 

to the long back slope of some of the model dams, the head was 

measured 69.75 feet upstream from, the crest of the experimental 

weirs. Headings of all the piezometers were taken at half-minute 

intervals, two and sometimes three observers working at each weir. 

The mean of ten to twenty observations was used to determine the 

head for each period in the experiment. Freeman states that he 

considers the results of these experiments for heads up to 2.5 feet, 

including all sources of errors, as certainly correct within 2 per 

cent, and probably much closer. In reducing the experiments, the 

head on the experimental weir is corrected by a method comparable 

with that of Francis. Freeman does not give the resulting coefficients 

for the weir formula, but presents the results in the form of diagrams 

showing the discharge per foot of crest for the various models. In 

the accompanying tables the computations have been carried out to
94 WEIB EXPEEIMENTS, COEFFICIENTS, AND FOBMULAS.

show the coefficients, some errors in the original data having been 

omitted.

Column 3 shows the observed head on the experimental dam, in feet.

Column 7 shows the computed discharge over the experimental dam, 

per foot of crest. This was determined by calculating the discharge 

over the standard weir by means of both the Francis and Bazin formulas, the mean of the two having been used. The result corrected 

for slight leakage, divided by 16 (the length in feet of the experimental weir model), appears in column 7.

Columns 4 and 5 show the velocity of approach and the corresponding velocity head at the experimental weir. The velocity of approach 

correction was made by adding directly the velocity head as given to 

the observed depth on the model dam, this being considered a sufficiently close approximation to the Francis method of correction.

Columns 1 to T are taken from the original computations. The 

coefficient C\ has been computed from the data in columns 6 and 7 by 

the formula

Pis. XIX to XXII show the resulting coefficients applicable in the 

formula here adopted,

correction for velocity of approach being made by the Francis correction formula or an equivalent method.

These experiments were performed for the specific purpose of 

determining the discharge over the old Croton dam. They include 

two main groups: (1) Experiments on round-crested portion of the 

dam; (2) experiments on the angular-crested portion of the dam. 

Each group includes series of experiments on: (a) Model of smoothplaned pine; (5) model of unplaned plank; (c) model with cleats and 

fragments of stone on the upstream slope to simulate the natural back 

filling; (d) model with rough slope and with ^-inch-mesh wire cloth on 

crest to simulate cut stone; (e) model surmounted by 12 -inch-square 

timber on crest. Experiments were added with a construction to 

simulate a natural rock ledge lying upstream from the angular portion 

of the dam.

The experiments were abbreviated owing to lateness of season and 

trouble from air in the gage pipes.

The value of the results is limited by the narrow range of heads 

covered. The models were of unusual forms, and show some peculiar 

differences when an attempt is made to compare the results with those 

of other weirs of similar slopes. The data are of value as showing the 

effect of various degrees of roughness on the discharge.
X

Coefficient

C- Corrected head Hin feet.

0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.

3.00

X^

\

** *^

^y.

y '

ff

.5 £/\+' ,M

Si

OD

 no

EL

ott

A

^

int

s

^

3 z/

.** . -

U/:

-c 

v^ ' ^

 - - 

5E

J&

ffie.

V 

-5

U

~-~-

/a\MOD£.

np/aned

"ft

^r- L J4

plan k

D_ -(t ^,

*

Y

3

@-

JEFflES

®

2 ODEL

?ou

A

f

sM pe ^ =e.r

3 .22

&

Q=

3.40

3.20 

3".00

^Sk \, d> ^ * -

ff

r@-

Si

s/o,

EV?/

pe

ES

Han

7\ -

 c

vi r

OD

e c

EL

/ot

A

hencre

t\\

i

|

st

3.1,

1.0 1.3 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 

Corrected head Hin feet.

k- - - 15.0 - - - -»)<- 6.0 -*f-5.Z5-**- 5Z -f ------ -------X--

SEf?/ES fa, MODEL A

EXPERIMENTS ON ROUND-CRESTED MODELS OF OLD CROTON DAM.' 

IKK 150 06 9
U. S. GEOLOGICAL SURVEY WATER-SUPPLY PAPER NO. 150 PL. XX

Corrected liead Hin feet. 

1.0 1.8 1.4 1.6 1.8 2.0 2.2 2.4

Coefficient

f.

3.80 

3.60 

3.40

-G ^ 0 ^

S4

 4

-fit

j)-

ES

Ur

I,

vL

M

3/7

3

oo

;t/

EL

p^ Me:

^int <

E= . ..

,3. 70

3.80 

3.60 

3.40

A

ft

(

N

p

^

bO

A

!

^£

S£

Rl

Ftl

Ut

*

r.s

"5

r

2,

a,

5/7

^4-

MODEL

Ml.

?d

IDt

f'\

L

/7>4

B

B

'& 

cc

/fo

Nl

"t

/ft'

h e

n

(/f

J

o

>e

4i3?

S6

<4

/

W

'Oil f*J>

Si

st

TM

'o^

Ei

e.

3

w

M

re

oo

c/\

ZL

\M

B

Ol r c

f

.j.

r ^ <fz

0 .2 .4 .6 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 

Corrected head Hin feet.

SEFtKS f A.N8 Z,MQD£L &

.J, MODEL B,ROI/6H SLOPE

EXPERIMENTS ON ANGULAR-CRESTED MODELS OF OLD CROTON DAM.
U. S. GEOLOGICAL SURVEY WATER-SUPPLY PAPER NO. 150 PL. XXI

Coeffi 

cient 

C.

3.60 

3.40 

3.20 

3.00

f1

C

r s..

y }

0

A

N

,s|/p/,ks

si

b-

-/?/

ff

rs

L-*

/,

/,

(5 ^

^

uc

Y

j

h

^x

i

^

C

c,

rcc

^<-

/vr

--^

//v,

=di

.

.^ ^ U,>- 3 49

0 .2 .4 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 

C'orrected head ff In feet.

iz"timber

Coefficient

C.

3.80 

3.60 

3.40

,

\

\

^

SCR/

 *-»,

T5

h3

A 

 --

l^O

^

D£

/

 )

fY

*~~

0 .2 .1 .6 .8 1.0 1.2 1.4 1.6 1.8 

Corrected head H in. feet.

/, MODEL O ' 

OLD CROTON DAM MODELS WITH CREST TIMBER.
U. 8. GEOLOGICAL SURVEY WATER-SUPPLY PAPER NO. 150 PL. XXII

Coefficient

a

3.80

3.60

330

-,

1

-*

\

7)^. ~"~.

te> -^. ^ ^ »~~ -<35

^

r/p> £ 5 /, M 00

-"*

rz. ^

' k- J ^ 3.

 t

32

0 Ji .4 .6 .8 LO 1.2 1A 1.6 1.8 8.0 8.3 8.4 3.0 SJ&

Corrected head Hia feet.

0 .2 .4 .6 .8 1.0 1.8 1.4 1.6 1.8 

Corrected head H in feet.

Coefficient 

C.

3.60

3.40

3.20 (a) Trap ope

 «

-n

**--^.

5£

 ^ ^

"/?/

 -*.

- -

£^ 2

C

-

, M

®*

OD

 

^

-^«

_

E

 ». ^ -^.

 >. ^ ^ * »^* 

3

la30-Z

0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 8.0 2.2 

Corrected head Hin feet.

SERIES I, MODEL E. End 3 open

SEfflES Z. MODEL E End <3 with s/op/ng approach

ANGULAR CROTON DAM MODEL, WITH CONSTRUCTION .TO 

SIMULATE ROCK LEDGE.
U. S. GEOLOGICAL SURVEY 

dent 

C.

3.20

3.00

MODEL OF OGEE DAM O~ LO7IER MOTOR COMPANY, PLATTSBUR3, N. V, LENGTH O" 

3.6 2.8 3.0 3.2 3.4 

Corrected head II in feet.

MODEL SIMILAR TO PLATTSBURG DAM, WITH BACK SLOPE MODIFIED. LENGTH OF CRE

EXPERIMENTS OF UNITED STATES GEOLOGICAL SURVEY ON VOLU

CORNELL UNIVERSITY HYDRAULIC LABORA
Coeffl 

cient 

C. 

3.60

3.40 

3.20 

3.00 

2.80

Coeff 

cien

O.

3.80 

3.60 

3.40 

3.20 

3.00

f

'^C

LXT^

©

<^ XiT

**

""1/

<--

^--

1

'

, ! *. 

-<4) 

^/:/?/

 

E5 J

 " ^

3

r-"«D « 

.4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 

Corrected head Hin feet.

OGEE MODEL AS N SERIES 32. LENGTH OF CREST, 15.969 FEET. N

15

/

/

juff

)

H

Ti><c> r

(i

»- 

5)

 

I

^i ' 

2)'

l

5L." .3 -

c5£T ^T5

4-

JO

1- \q

0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 

Corrected head // in feet.

MODEL OF OGEE DAM OF LOZIEfl MOTOR COMPANY, PLATTSBURG, N. Y LENS

EXPERIMENTS OF UNITED STATES GEOLOGICAL SURVEY ON VOLUME OF FLOW OVER 

LABORATORY, 1903.
WEIRS OF IKSEOTJLAR SECTION. 95

EXPERIMENTS OF UNITED STATES GEOLOGICAL SURVEY AT CORNELL 

UNIVERSITY HYDRAULIC LABORATORY.

In April, 1903, the writer was instructed to plan and execute a series 

of experiments on models of dams similar to those in use at gaging stations of the Geological Survey in New York, Michigan, and elsewhere.

The experiments were performed at the hydraulic laboratory of 

Cornell University, mainly during the months of May and June, 1903, 

and were conducted, under the supervision of the writer, by Prof. 

Gardner S. Williams, director of the laboratory.

The various types of dams most commonly occurring were grouped 

as follows:

1. Weirs with broad horizontal or slightly inclined crests.

2. Weirs with vertical downstream faces and inclined upstream 

slopes.

3. Weirs having compound slopes, including those with inclined 

upstream faces and with either broad crests or with sloping aprons.

tL Completely or partially curved weir sections, including those of 

ogee profile.

It was found impossible to include in the experiments all the forms 

of section desired, and it was accordingly determined to limit the 

experiments to the thorough stiuty of two classes weirs with broad 

crests and weirs with ogee sections and to extend, if possible, the 

measurements to include dams with vertical downstream faces and 

sloping upstream approaches. The order of operation used in previous experiments was transposed, the experimental models being built 

on a bulkhead forming the standard weir hitherto used and located 

near the head of the experimental canal.

The quantity of water passing over the experimental weir was measured on a standard weir below, 6.65 feet high and having a crest length 

of 15.93 feet. The head on the standard weir was measured in a 

Bazin pit, 3 by 4 feet in section, reaching to the depth of the bottom 

of the canal, and communicating therewith through a pipe -1 inches in 

diameter and about 3.5 feet long, opening at the bottom of the channel 

of approach, 29.88 feet upstream from the weir. The head on the standard weir was observed in the gage pit by means of a hook gage reading to millimeters and estimated to about one-fifth millimeter. The 

conditions at the standard weir were thus closely comparable to those 

obtained in Bazin's experiments, and his formula for this height and 

length of weir was applied to determine the discharge. Observations 

to determine the leakage between the experimental and standard weirs 

were made, and corrections were applied for whatever leakage was 

indicated, the amount being usually less than 0.01 cubic foot per second 

per foot of crest. The discharge over the standard weir was com-
96 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

puted iii cubic meters per second' and has been reduced to cubic 

feet per second, the discharge table being as follows:

Discharge over standard weir at different heads.

Head, in 

meters.

0.05

.10.

.15

.20

.25

.30

.40

.50

Q in cubic 

meters, per 

second.

0. 111863

. 296230

. 53207

. 81166

1. 12871

1. 48032

2. 27850

3. 19350

Head, in 

meters.

0.60

.70

.80

.90

1.00

1.10

1.20

Q in cubic 

meters, per 

second.

4. 21730

5. 34459

6. 57096

7. 89078

9. 30650

10. 81066

12. 40420

The discharge curve for the standard weir has also been carefully 

checked by comparing the depth flowing over with that on a similar 

weir, using the formula and method of determining the head adopted 

by Fteley and Stearns; it has also been checked by float and current-meter measurements, and for lower heads by means of volumetric 

measurement of the discharge in the gaging channel, so that it is 

believed that the discharge in these experiments is known within 1 or 

2 per cent of error as a maximum.

The work of calibrating the standard weir had been accomplished by 

Professor Williams and his assistants before the experiments of the 

United States Geological Survey were taken up, so that somewhat 

more certainty attaches to the results of these later experiments than 

to earlier experiments made before the standard-weir discharge had 

been thoroughly checked.

It was the wish of the Geological Survey that the conditions at the 

experimental weirs should conform to those actually existing at dams 

which are utilized as weirs, in connection with the stream-gaging 

operations. In such cases it is often impracticable to utilize gage pits 

of the form adopted by Bazin or to use piezometer or hook gages. 

The usual method is to read the depth directly on a graduated vertical 

scale or measure the distance to water surface from a suitable bench 

mark. The method adopted in the weir experiments consisted of 

reading directly the distance to water surface from bench marks 

located above the central line of the channel. The readings were 

taken by means of a needle-pointed plumb bob attached to a steel tape 

forming a point gage, readings being taken to thousandths of a foot.

Two gages were used, one located 10.3 feet upstream from the 

crest and another 16.059 feet upstream. In series XXXV and 

following, for the higher heads, the readings of the upstream tape 

were used. For heads where no general difference was apparent the 

average of the readings of the two tapes was taken. In general, the
WEIRS OF IRREGULAR SECTION. 97

surface curve did not perceptibly affect the reading of the gage nearest 

the weir for depths below 3 feet. The readings of the tapes were 

checked from time to time by observations with hook gages, thus practically eliminating the effect of temperature on the tapes. Observations of the head were usually taken at intervals of thirty seconds. 

Great care was used to maintain a uniform regimen of flow during 

each experimental period, and the variations of head were very slight. 

The character of the observations is illustrated by the following data 

taken from the experiments:

Readings of tapes to determine head at experimental weir.

Series XL. 

Period 10. 

Date 6, 22, 03.

Time.

h. m. $.

12 37 40

12 39 00

12 52 30

12 54 30

Mean

Readings.

42. 681

.681

.680

.681

.682

42. 680

.681

.680

.679

.680

42. 6805

Series XLIII.

Period 3.

Date 6,26.03.

Time.

h. m. s.

12 36 20

12 42 10

Mean

Readings.

2.305

.302

.302

.302

.301

.302

.301

.300

.300

2. 3015

Series XLI. 

Period 5. 

Date 6, 23, 03.

Time.

h. m. s.

1 34 30

1 51 20

Mean

Readings.

43. 633

.632

.633

.630

.630

.633

.635

.635

.634

.633

.630

.635

.630

.635

.635

.633

ijori

. ooO 

.630

.630

.630

.630

.634

.630

.628

.630

.632

.628

.631

43. 6317
98 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

For the lower heads the discharge over the experimental weir was 

volumetric-ally determined by measuring the rise of water in the canal, 

as follows:

List of experimental periods for which the discliarge was volii metrically determined.

Series.

30

31

34

37

38

Periods.

1,13,14,16

10,11

1,9,10

5«

5,6

Series.

39

40

41

43

43 «

Periods.

1

1,2

13,14

1,2,3

1,2,3,4

Series.

44

45

46

47

Periods.

1,2,3

1, 2, 3

1,2,3

1,2,3

United, States Qeological Survey experiments at Cornell University hydraulic laboratory on

model of Plnttshurg dam.

[Series No. XXX. Height of weir=P, 11.25 feet; length of weir crest=Z, 15.969 feet; width of 

channel=&, 15.970 feet; height of upstream crest corner, 10.50 feet; crest width, 3 feet.]

No.

1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

Measured head on experimental weir, in feet.

Number of

obsertions.

2

92

31

19

26

29

27

18

16

34

24

15

20

32

21

24

40

26

25

Maximum.

3

0. 0993

.7993

3. 1993

2. 3993

1. 7973

1. 1803

1. 0133

.8043

.6143

. 5083

.4283

. 2943

.2933

. 1783

.1793

Minimum.

4

0. 0818

.7863

3. 1473

2. 3473

1. 7853

1. 1723

1. 0073

.8003

.6113

. 5073

. 4263

.2903

. 2903

. 1763

. 1733

. 0893 . 0893

1. 5373

2. 1033

Mean 

=/>.

5

0.096

2.790

3.187

2.384

1.793

1.174

1. 010

.802

.613

.508

.427

.291

.292

.178

.179

.089

1. 5303 1. 532

2. 0843 2. 094

,4 = 

area of 

section 

per 

foot of 

crest.

6

11. 347

14. 041

14. 438

13. 635

13.043

12. 425

12. 260

12. 053

11. 864

11. 758

11.678

11.542

11. 543

11. 427

11. 429

11. 340

12. 783

13.345

mean velocity of approach, 

in feet 

per second.

7

0.007

1. 155

1.396

.926

.636

Head corrected _

for velocity 

of approach, 

in feet.

Jji

8

0.030

4.710

5. 766

3.710

2.412

. 354 1. 274

. 280 1. 016

.200

.136

.102

.079

.046

.044

.021

H

»

0.096

2.810

3.216

2. 396

1.799

1.176

1.011

. 719 . 803

.481

. 362

.280

.157

.158

.075

. 020 . 076

. 007 . 027

.517 ' 1.904

.793 3. 051

.614

. 50S

.428

.291

.292

.178

(^discharge 

per 

foot of 

crest, in cubic feet 

per second.

10

0.084

16. 218

20. 152

12. 631

8.282

4.399

3. 435

2.409

1.608

1. 195

.921

. 526

Discharge coefficient

11

2.835

3.443

3.495

3.405

3.433

3.454

3.381

3.348

3.345

3.302

3. 296

3.344

.506 ' 3.209

' . 240

. 179 . 226

. 089 . 076

1. 535

2. 104

6.607

10. 576

3. 202

2. 976

2.841

3.471

3.466
WEIES OF IKKEGULAK SECTION.

United States Geological Surrey experiments at Cornell University hydraulic laboratory on 

model of Plattsuurg dam Continued.

[Series No. XXXI. Height of weir=P, 11.25 feet; length of weir crest=i, 7.938 feet; width of channel^, 15.970 feet; height of upstream crest corner, 10.50 feet; width of crest, 3 feet.]

No.

1

1

2

3

4

5

6

7

8

9

10

11

12

Measured head on experimental weir, in feet.

Number of 

obsertions.

tj

25

15

44

21

39

40

26

25

30

18

40

19

Maximum.

3

5.1003

4. 2873

3. 6893

2. 8213

2. 2713

1. 3903

.9673

1. 8793

.6093

.6073

.3023

.3003

Minimum.

4

4. 9273

4. 1693

3. 5253

2. 8003

2. 2023

1.3813

. 9653

1. 8723

.6083

.6063

.2993

.2993

Mean 

=Z>.

5

5. 0014

4. 2214

A = 

area of 

section 

per 

foot of 

crest.

6

259. 535

247. 078

3.6191 507.460

2. 8185

 2. 2696

1. 3861

.9663

1.8749

.6087

.6063

.3017

.3017

224. 674

215. 780

201.998

195. 094

209. 605

189. 383

189. 345

184.471

184. 481

Head corrected

mean 

velocity 

of approach, 

in feet 

per

second.

7

1. 201

.945

.356

.551

.412

.206

. 125

.326

.065

.060

.020

.021

for velocity 

of approach, 

in feet.

H%

8

11.2C7

8.714

6.890

4.743

3. 425

1.633

.950

2.570

.475

.472

.165

.106

H

9

5. 023

4.235

3.621

2.823

2. 273

1.387

.966

1.876

.609

.606

.301

Q discharge 

per 

foot of 

crest, in cubic feet 

per

second.

10

41. 925

31. 086

23. 868

16. 170

11. 545

5.338

3.120

8.814

1.562

1. 456

.469

. 302 . 490

Discharge coefficient (i.

11

3.724

3. 567

3. 464

3. 409

3.377

3.269

3. 284

3.429

3.288

3.083

2.838

2.956

^effective 

length 

of crest 

weir.

12

7.436

7.514

7.580

7.656

7.711

7.799

7.811

7.750

7.877

7.877

7.908

7.908

[Series No. XXXII. Height of weir=P, 11.25 feet; length of weir crest=i, 7.979 feet; width of channel=&, 15.970 feet; height of upstream crest corner, 9.75 feet; width of crest, 3 feet.]

1

2

3

4

5

6

7

8

9

10

11

10

23

26

34

32

27

27

28

31

28

53

2. 1193

4. 0343

2. 6693

.9953

5. 5543

4. 7343

4. 0173

3. 3053

2. 5863

1. 9913

1. 4113

2. 0943

3. 9883

2. 6333

.9923

5. 2983

4.6443

3. 9883

3. 2673

2. 5193

1.9693

1.4023

2. 1173

4. 0053

2. 6469

.9942

5. 4712

4. 6947

4. 0007

3.2890

2. 5596

1. 9801

1. 4015

213. 476

243. 627

221. 934

195. 540

267. 038

254. 637

240. 808

232. 188

220. 539

211. 285

202. 044

0.004

.928

.535

.135

1.404

1.141

.925

.701

.506

.357

.221

3.081

8.055

4.317

.992

12. 855

10. 235

8.040

5. 984

4.104

2.790

1.661

2.117

4.018

2.651

.994

5.487

4.714

4.013

3. 296

2.564

1.982

1.402

11.029

29. 846

15. 398

3. 342

50. 440

3.8. 704

29. 395

21. 278

14. 460

9.686

5.704

3. 580

3.706

3. 567

3.370

3.924

3.782

3.656

3.556

3.524

3.471

3.434

7.767

7.577

7.714

7.880

7.430

7. 508

7.578

7.649

7.723

7.781

7. OoJ

IRR 150 06 10
100 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

United States Geological Survey experiments at Cornell University hydraulic laboratory on 

model of Plattsburg dam Continued.

[Series No. XXXIII. Height of weir=P, 11.25 feet; length of weir crest=i, 15.969 feet; width of 

channel= 6,15.970 feet; height of upstream crest corner, 9.79 feet; width of crest, 3 feet.]

No.

1

1

2

3

4

5

6

7

8

9

Measured head on experimental weir, in feet.

Number of

vations.

2

15

43

29

33

23

24

52

42

12

Maximum.

3

0. 7563

.9973

1. 3963

1. 7893

2. 3733

2.6683

.5593

.6603

.6804

1

Minimum.

4

0. 7493

.9873

1. 3903

1. 7803

2. 3363

2. 6503

. 5493

.6493

.6783

Mean

= D.

6

0. 752

.992

1.394

1.784

2.352

2.660

. 553

.653

.679

A = 

area of 

section 

per 

foot of 

crest.

6

12. 003

12. 243

12. 644

13. 034

13. 603

13. 910

11. 803

11.904

11. 930

i' = 

mean 

veloc- "y of approach, 

in feet

per second.

7

0.183

.281

.468

.671

.990

1.170

1.183

.152

.156

Head corrected

for velocity 

of approach, 

in feet.

H*

8

0.653

.990

1.651

2.396

3.641

4.387

.432

. 528

.560

H

9

0.753

.993

1.397

1.790

2.367

2.680

.572

.654

.680

Q= discharge 

per 

foot of 

crest, 

in cubic 

feet per

second.

10

2.197

3.439

5.913

8.747

13. 464

16. 272

1.397

1.808

1.866

Discharge 

coefficient 

Ci.

11

3.363

3.473

3.580

3.651

3.698

3.709

3.232

3.422

3.330

[Series No. XXXIV. Height of weir=P, 11.25 feet; length of weir crest=£, 15.969 feet; width of 

channel=6, 15.970 feet; height of upstream crest corner, 8.37 feet; width of crest, 3 feet.]

1

2

3

4

5

6

7

8

9

10

11

12

13

14

16

16

21

19

19

14

20

10

32

39

14

25

31

25

0.6453

.6423

2.0253

1. 6303

1. 2333

.9523

.4013

.6243

.2243

.1103

5. 0963

2. 7763

2.4383

2. 8013

0. 6383

.6383

2. 0043

1. 6203

1. 2293

.9483

.3983

. 6243

.2203

.1093

5. 0793

2. 7263

2. 3963

2. 7963

0.641

.640

2.015

1.628

1. 232

.950

.400

.624

.222

.110

3.089

2.749

2.421

.800

11. 891

11. 891

13.266

12. 878

12.483

12. 201

11.650

11. 875

11. 473

11. 360

14. 340

14.000

13. 672

12. 050

0.137

.141

.816

.592

.391

.261

.070

.136

.029

.009

1.509

1.258

1.051

.194

0.513

.513

2.882

2.086

1.371

.927

.253

.493

.105

.036

5.515

4. 615

3.805

.716

0.641

.640

2.025

1.633

1.235

.951

1.632

1.680

10. 826

7. 627

4.877

3.185

.400 .815

.625

.222

.110

3.122

2. 772

2.437

.800

1.613

.328

.100

21. 636

17. 612

14.374

2.341

3. 182

3.276

3. 756

3.656

3.555

3.434

3.225

3.269

3.133

2.777

3.923

3.816

3.778

3.271
WEIRS OF IBEEGULAK SECTION. 101

United States Geological Survey experiments at Cornell University hydraulic laboratory

on model of Chambly dam.

[Series No. XXXV. Height of weir=P, 11.25 feet; length of weir crest=Z, 15.969 feet; width of 

channel=6,15.970 feet; height of upstream crest corner, 10.25 feet; width of crest 4.5 feet.]

No.

1

1

2

3

4

5

6

7

8

9

11

12

13

14

15

16

17

18

19

20

21

22

23

24

Measured head on experimental weir, in feet.

Number of observations.

2

64

55

40

40

38

45

41

43

20

18

17

17

18

23

22

37

25

17

24

21

24

19

20

Maximum.

3

2. 5503

1.0103

1.5643

2. 0273

1. 7513

1. 2673

. 7593

. 4453

.3043

3.7303

3. 2193

3. 0023

2. 6743

2. 3533

1. 4923

.2103

.3913

.3313

. 2523

. 2203

.1843

.1323

.0823

Minimum.

4

2.5463

1. 0063

1. 5523

2. 0133

1. 7303

1.2583

.7543

.4373

.3003

3. 7063

3. 1813

2. 9873

2. 6643

2. 3283

1. 4873

.2093

.3903

. 3313

. 2523

.2193

.1833

.1313

.0823

Mean 

=D.

5

0.549

1.008

1.559

2.021

1.739

1.262

.757

.441

. 302

3. 755

3.195

2.987

2. 642

2. 317

1.463

.184

.391

.331

.253

.221

.183

.132

.083

.4 = 

area of 

section 

per 

foot of 

crest.

0

11. 799

12. 259

12. 810

13. 272

12. 990

12. 513

12. 008

11. 692

11.553

15. 006

14. 446

14. 238

13. 892

13. 56,8

12. 713

11.435

11. 642

11. 582

11. 504

11. 472

11. 434

11. 383

11. 384

 ?,' 

mean velocity of approach, 

in feet 

per 

second.

7

0.110

.272

.511

.744

.602

.379

.176

.079

.045

1.740

1.406

1.283

1. 077

.901

.478

.023

.065

.053

.035

.028

.021

.012

.007

Head corrected 

for velocity 

of approach, 

in feet.

jji

8

0.407

1.014

1. 954

2.890

2.304

1. 422

.660

.293

.166

7.403

5.789

5.224

4.334

3. 555

1.775

.079

.245

.191

.127

.104

.078

.048

.024

H

9

0.549

1.009

1.563

2.029

1.744

1.265

. 758

.441

.302

3.798

3.224

3.011

2. 658

2.330

1.466

.184

.391

.331

.253

.221

.183

.132

.083

Q= discharge 

per 

foot of 

crest, 

in cubic 

feet per 

second.

10

1.297

3.331

6. 548

9.874

7.816

4.747

2.114

.919

.525

26. 114

20. 319

18. 267

14. 963

12. 221

6.072

.258

.758

.609

.404

.325

.238

.136

.079

Discharge 

coefficient

Ci11

3.189

3.285

3.352

3.416

3.392

3.337

3.205

3.132

3.159

3.528

3.510

3.496

3.452

3.438

3.420

3.260

3.096

3.194

3.258

3.120

3.039-

2. 826

3.321

[Series No. XXXVI. Height of weir=P, 11.25 feet; length of weir crest=Z, 15.969 feet; width of channels, 15.970 feet; height of upstream crest corner, 10.50 feet; width of crest, 4.5 feet, with 4 inches 

radius quarter round.]

1

2

3

4

18

18

30

23

5 21

b

7

19

22

2. 7653

2.3513

2. 9373

2. 6913

2. 2913

2. 8923

1.5173 ' 1.4993

1.1143

.7553

.4948

1. 1013

.7493

.4903

2.741

2.316

1. 915

1.507

1.111

.753

.492

13. 991

13. 567

13. 166

12. 7.58

12. S61

12. 004

11.743

1.185

.936

.702

.501

.320

.177

.095

4.594

3.558

2.666

1.857

1.173

.654

.345

2.764

2.330

1.923

1.511

1.112

.753

.492

16. 586

12. 698

9.243

6.390

3.950

2.123

1.113

3.610

3.569

3.467

3.441

3.367

3.248

3.221
102 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

United Slates Geological Survey experiments at Cornell hydraulic laboratory on model of 

Dolgemlle dam with injured apron.

[Series No. XXXVII. Height of weir=P, 11.-5 feet; length of weir crest=L, 15.969 feet; width of 

channel=5, 15.970 feet; height of upstream crest corner, 10.25 feet; width of crest, G feet.]

No.

1

1

2

3

4

5

5ct

Measiiredhead on experimental weir, in feet.

Number of 

obsertions.

2

30

35

28

52

59

41

Maximum.

3

0. 9203

3. 5613

2,9123

2, 3293

1.6413

1. 3973

Minimum.

4

0. 9113

o. 4233

2. 8573

2. 2903

1. 6213

1. 3923

Meaii = 

D.

5

0.916

3. 565

2. 927

2. 324

1. 635

.435

t _

area of

per foot 

of 

crest.

6

12. 166

14. 816

14. 178

13. 574

12. 885

11.666

velocity of

approach, 

in feet 

per second.

7

0. 251

1.535

1.170

.906

. 585

.069

He

f

0

J

0

6

5

3

2

Head corrected

for velocity 

of approach, 

in feet.

If*

8

0.878

6. 828

5.060

3.570

2.100

.267

H

9

0.917

3.599

2.947

2.336

1.640

.415

Q= discharge 

per 

foot of

crest,

in cubic feet 

per second.

10

3.049

22. 744

16. 593

12. 297

7.544

.810

Discharge

coefficient

c,.

11

3.474

3.331

3.279

3.444

3. 593

3.029

United States Geological Survey experiments at Cornell hydraulic Jaboratory on model oj

Dolgerille dam.

[Series No. XXXVIII. Height of weir=P, 11.25 feet; length of weir erest=i, 15.969 feet; width of 

channel=f), 15.970 feet; height of upstream crest corner, 10.25 feet; width of crest, 600 feet.]

No.

1

1

2

3

4

5

6

7

8

9

10

11

12

Measured head on experimental weir, in feet.

Number of 

obsertions.

2

32

30

25

32

34

24

27

34

29

30

28

28

Maximum.

3

0. 3973

1. 6843

1. 1143

.7513

. 5093

.2173

3.4726

3. 1176

2. 6176

2. 2096

1. 9306

1.5286

Minimum.

4

0. 3913

1. 6713

1. 1093

.7473

.5083

.2113

3.4586

3.080G

2. 5926

2. 1806

1. 8156

1. 5106

Mean= 

/).

5

 0.395

1.689

1. 112

.749

.504

.209

3.470

3.100

2. 605

2. 198

1. 9'JO

1.517

A = 

area of 

section 

per foot 

of 

crest.

6

11. 646.

12. 940

12. 362

12.000

11.754

11. 460

14. 721

14. 351

13. 856

13. 449

13. 171

12. 768

v= 

mean velocity of 

approach, 

in feet 

per second.

7

0.068

.605

.333

.185

.101

.026

1.466

1. 271

1.019

.846

.726

.514

Head corrected

for velocity 

of approach, 

in feet.

H*

8

0.249

2.206

1.375

.649

.358

.096

6. 551

5. 521

4.243

3. 283

2.677

1. 872

H

9

0.396

1. 695

1.114

.750

.504

.209

3.501

3.124

2. 621

2.209

1.928

1. 519

Q= discharge 

per 

foot of 

crest, 

in cubic feet 

per second.

10

0.790

7.826

4.113

2. 223

1.186

.294

21. 583

18. 238

14. 124

11. 381

9. 558

6.557

Discharge coefficient &.

11

3.176

3.548

3.501

3.426

3.310

3.081

3.295

3.303

3.329

3.467

3.570

3.503

es No. XXXIX. Height of weir=P, 11.25 feet; length of weir crest=i, 15.969 feet; width of 

channel=6, 15.700 feet; height of upstream crest corner, 10.25 feet; width of crest, 6 feet.]

1

2

3

4

5

6

7

8

9

10

! -

22

33

24

23

36

32

39

32

0. 6883

.4183

3. 9806

3. 1506

2. 5066

1.8986

1. 2716

.8286

.5626

0. 6793

.4153

3. 9206

3. 1366

2.. 4806

1.8756

1. 2626

.8206

. 5606

f 0.683

[ .683

.418

3.943

3. 145

 4,488

1.884

1.268

.826

.561

11. 934

11. 934

11. 668

15. 194

14. 396

13. 739

13. 135

12. 518

12.077

3 1.811

0.170

.167

.081

1. 721

1.280

.966

.699

.401

.218

.125

0.565

.565

.270

7. 956

5. 642

3.958

2.601

1.432

.752

.420

0.684

.684

.418

3.985

3.169

2.502

1.891

1.270

.827

.561

2.027

1.995

.944

26.150

18. 431

13. 268

9.186

5.025

2.639

1.481

3.586

3.530

3.497

3.287

3.266

3.352

3.532

3.509

3.494

3.526
WEIB8 Otf IRREGULAR SECTION. 103

United States Geological Survey experiments at Cornell hydraulic laboratory on model of 

Jlat'top weirs mth vertical faces.

[Series No. XL. Height of weir=P, 11.25 feet; length of weir erest=i, 15.969 feet; width of channel= 

It, 15.970 feet; width of broad crest, 0.479 foot; nappe aerated.]

No.

1

1

2

3

4

5

6

7

8

9

10

n

Measured head on experimental weir, in feet.

Number of 

observations.

2

19

17

18

30

33

36

26

21

21

10

10

Maximum.

3

0. 6343

.2643

.2613

. 1216

1. 9916

1. 6256

1.2556

.9756

. 8213

.6513

.4503

Minimum.

4

0. 6293

. 2593

. 2513

.1206

1. 9816

1. 6106

1. 2496

.9706

.8163

.6493

.4483

Mean= 

I).

5

0.631

.260

.264

.124

1.989

1.618

1.256

.977

.820

.650

.449

A = 

area of 

section 

per foot 

of 

crest.

6

11. 882

11. 511

11. 515

11. 375

13. 238

12. 868

12. 507

12. 228

12. 070

11. 900

11. 700

v= 

mean 

velocity of 

approach, 

in feet 

per second.

7

0.133

.030

.033

.011

.710

.530

.377

.262

.204

.139

.074

Head corrected

for velocity 

of approach, 

in feet.

H%

8

0. 502

.133

.136

.044

2. 821

2. 065

1.412

.967

.743

.524

.301

H

9

0. 632

.260

.264

.124

1.996

1. 622

1.259

.978

.820

.650

.449

^discharge 

per 

foot of 

crest, 

in cubic 

feet per 

second.

10

1. 580

.343

.380

.129

9.401

6.819

4.713

3. 209

2.469

1.654

.867

Discharge coefficient

a.

11

3.148

2.584

2.794

2.943

3.332

3.301

3.338

3.318

3.325

3. 154

2.881

[Series No. XLI. Height of weir = P, 11.25 feet; length of weir crest = L, 15.969 feet; width of 

channel = b, 15.970 feet; width of broad crest, 1.646 feet; nappe partly aerated.]

[Series

1

2

3

4

5

6

7

8

9

10

11 

12 

13

14

15

25

25

26

31

28

27

25

34

30

18

34

22

20

44

3. 8606

3. 1906

2. 6906

2. 0306

1.6043

1. 2373

.9443

.6733

.4893

.3303

.2113

. 1253

.4206

.4206

3. 8256

3. 1666

2. 6656

2. 0116

1. 5973

1. 2293

.9393

.6693

.4873

.3293

.2093

. 1213

.4136

.4126

3.842

3.177

2.674

2. 022

1.601

1.233

.942

.671

.488

.330

.210 

.122

.788

.417

.417

15. 092

14. 428

13. 925

13. 272

12. 852

12. 484

12. 192

11. 922

11. 739

11. 581

11. 461 

11. 373 

12. 038

11. 668

11. 668

1.692

1.317

1. 050

.680

.462

.307

.203

.123

.078

.045

.024 

.011 

.153

.064

. 065

7. 651

5.730

4.413

2.889

2.032

1.372

.915

.575

.341

.190

.096 

.043 

.699

.270

.270

3.883

3.202

2.690

2.028

1.604

1.234

.942

.692

.488

.330

.210 

.122

.788

.417

.417

25.531

18. 995

14.624

9.021

5.936

3.835

2. 476

1.472

.910

.520

. 272 

.130 

1.840

.750

.759

3.337

3.315

3.314

3.123

2.922

2.796

2.706

2.560

2.669

2. 742

2.827 

3.047 

2.631

2.782

2.815

No. XLII. Height of weir=JP, 11.25 feet; length of weir crest=i, 15.969 feet; width of channel=6, 15.970 feet; width of broad crest, 12.239 feet; nappe partly aerated.]

1

2

3

4

5

6

7

8

9

10

11

12

33

32

26

38

32

29

38

27

34

22

36

28

0. 1706

4. 3706

3. 8316

3. 0446

3.7356

3. 5856

3. 3966

2. 2506

1. 4706

1. 0986

.6406

.6126

0. 1626

4. 3416

3. 8006

3. 0256

3. 7186

3. 5776

3. 3806

2. 2406

1. 4526

1. 0906

.6376

.6106

0.168

4. 353

3.809

3.032

1. 728

2.580

3.387

2. 243

1.449

1.096

.639

.611

11. 418

15.604

15. 060

14. 283

12. 979

13. 831

14. 638

13. 494

12. 700

12. 347

11. 890

11. 862

0.016

1.584

1.321

.971

.467

.787

1.130

.657

.369

.253

.116

.109

0.069

9.196

7.510

5.317

2. 278

4.166

6.285

3.373

1.748

1.149

.511

.478

0.168

4.389

3.835

3.046

1.732

2. 589

3.406

2.249

1.451

1.097

.639

.611

0.180

24. 716

19.8%

13. 876

6.066

10. 882

16. 535

8. 87.0

4. 682

3.129

1.375

1. 290

2.611

2.688

2.649

2.610

2.663

2. 612

2. 631

2.629

2.701

2. 723

2.689

2.700
104 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

United States Geological Survey experiments at Cornell hydraulic laboratory on model of 

fat-top weirs with vertical faces Continued.

[Series No. XLIII. Ileightof weir=P, 11.25 feet; length of weir crest=£, 15.969 feet; width of channel= 

6, 15.970 feet; width of broad crest, 16.302 feet; surface somewhat rough; nappe partly aerated.]

No.

1

1

2

3

4

5

6

7

8

9

10

11

Measured head on experimental weir, in feet.

.

Number of observations.

2

12

12

10

26

33

27

34

30

23

25

26

Maximum.

3

0. 8536

.4496

.3246

.6936

4.4506

3. 6806

2. 9426

2. 3626

1.8956

1. 4826

1.2086

Minimum.

4

0. 8506

.4426

.3106

.6856

4. 4176

3. 6306

2. 9306

2. 3586

1. 8866

1. 4756

1.2026

Mean= 

I).

5

0.851

.447

.312

.689

4.432

3. 661

2. 935

2. 360

1.890

1.480

1. 206

A= 

area of 

section 

per foot 

of 

crest.

6

12. 102

11. 698

11. 563

11. 940

15. 683

14. 912

14. 186

13. 611

13. 141

12. 731

12. 457

('= 

mean 

velocity 01 

approach, 

in feet 

per second.

7

0.168

.069

.040

.129

1.595

1. 251

.930

.706

.520

.374

.2X2

Head corrected 

for velocity 

of approach, 

in feet.

H%

8

0.786

.299

.174

.573

9.449

7.076

5.062

3.644

2. 607

1.804

1.327

H

9

0. 852

.447

.312

.690

4.469

3.686

2. 948

2.368

1.894

1. 482

- 1. 208

Q=discharge 

per 

foot of 

crest, 

in cubic 

:eet per 

second.

10

2. 027

.811

.468

1. 544

25. Oil

18. 657

13.200

9.607

6.841

4.767

3.520

Discharge coefficient

<\.

11

2.579

2.713

2.684

2.696

2.647

2. 637

2. 608

2.637

2.624

2.531

2. 652

[Series No. XLIIIa. Height of weir=P, 11.25 feet* length of weir crest=i, 15.969 feet; width of chan 

nel=h, 15.970 feet; width of broad crest, 16.302 feet: smooth planed surface.]

1

2

3

4

5

6

7

8

9

10

11

19

20

44

16

28

23

31

21

25

31

28

12 1 24

0. 3626 0. 3596

. 2496 . 2426

.1686

.9906

.1646

.9846

. 9856 . 9776

. 7886 . 7826

. 6226 . 6206

.4976 .4946

.3926 .3906

. 2806 . 2786

. 1606 . 1606

.0776 ' .0756

0.361

.246

.167

.986

11. 012 0. 050 0. 217

11.497 ' .026 .122

11.418 .013

12. 236 . 214

. 981 12. 232

.786

.621

.496

.392

.280

.161

.077

.068

.980

. 210 . 973

12.036 1 .158 .697

11. 872 . 110

11. 747 . 080

11. 643 .058

11. 530 . 036

] 1.411

11. 327

.016

.490

.350

.246

.148

.064

.006 ' .021

0.361

.246

.167

.986

.982

.786

.622

.496

.392

.280

.161

.077

0.576

.305

.153

2. 618

2.568

1.847

1.309

.945

.670

.421

.184

.066

2.653

2.494

2.240

2.673

2. 638

2.651

2.670

2.704

2. 728

2.848

2.851

3.127

[Series No. XLIV. Height of weir=P, 11.25 feet; length of weir crest=L, 15.969 feet; width of channel^, 15.970 feet; width of broad crest, 8.980 feet.]

1

2

3

4

5

6

7

8

9

10

11

12

27

20

32

20

19

17

31

28

' 31

29

26

32

0. 3136

. 1596

.4196

3. 0666

2. 3306

2. 8656

1. 5166

1. 2556

1. 0396

.8996

.7326

.5046

0. 3116

.1676

.4176

3. 0506

2. 3116

2. 8486

1. 5106

1. 2506

1. 0356

.8916

.7306

.5006

0.312

.159

.419

3. 058

2.319

1.856

1.513

1. 252

1.038

.897

.732

.502

11.563

11. 410

11. 670

14. 309

13. 569

13. 107

12. 763

12. 503

12. 289

12. 148

11. 983

11. 753

0.040

.014

.058

.986

.686

.512

.386

.297

.228

.186

.140

.083

0.174

.064

.271

5.386

3.547

2. 537

1.864

1.404

1.059

.850

.627

.356

0.312

.159

.419

3.073

2.326

1.860

1. 515

1. 254

1.039

.897

.732

.502

0.469

.163

.679

14. 105

9.307

6.712

4.928

3.718

2.803

2.254

1.676

.971

2.691

2.570

2.504

2. 619

2.624

2.646

2.642

2.649

2.646

2.652

2.676

2.727
WEIES OF IRREGULAR SECTION. 105

United States Geological Survey experiments at Cornell hydraulic laboratory on model of 

flat-top weirs with, vertical faces Continued.

[Series No. XLV. Height of weir=P, 11.25 feet; length of weir crest=£, 15.969 feet; width of channel=6, 15.970 feet; width of broad crest, 5.875 feet; nappe partly aerated.]

No.

1

1

2

3

4

5

6

Measured head on experimental weir, in feet.

Number of observations.

2

24

32

38

31

31

32

7 42

8 26

9- 33

10 23

11 36

Maximum.

3

0. 1766

. 2556

.3906

.9906

1. 2456

. 9126

.7346

1.0006

.5906

.5916

. 5216

Minimum.

4

0. 1726

. 2526

.3886

.9786

1. 2396

.9066

.7306

.9916

.5806

.5896

.5116

Mean= 

D.

5

0.174

.253

.390

.982

1.242

^1 = 

area of 

section 

per foot 

of 

crest.

6

11.424

11. .504

11. 640

12. 233

c= 

mean velocity of

approach, 

111 feet 

per 

second.

7

0.018

.029

.055

.209

12. 492 . 293

. 908 12. 159 . 189

. 733 11. 983 .139

1. 996 13. 247 . 564

1. 585 12. 836 . 410

. 590 11. 841 . 103

. 520 11. 771 . 087

Head corrected

for velocity 

of approach, 

in feet.

H%

8

0.072

.127

.243

.975

1.386

.367

.627

2.830

2.000

H

9

0.174

^discharge 

per 

foot of 

crest, 

in cubic 

feet per 

second.

10

0.207

. 253 . 337

.390 .641

. 983 2. 557

1. 243 3. 666

.909

.733

2.001

1.587

. 454 . 590

.376 .521

2. 294

1.670

7.469

5.264

1.220

1.022

Discharge coefficient 

d.

11

2.857

2.647

2.635

2.624

2.645

2.646

2.663

2.639

2.632

2.689

2.722

[Series No. XLVI. Height of weir=P, 11.25 feet; length of weir crest=i, 15.969 feet; width of channel=6, 15.970 feet; width of broad crest, 3.174 feet; nappe partly aerated.]

1

2

3

4

5

6

7

8

9

10

11

12

26

41

34

23

30

32

33

32

32

31

38

33

0. 2526

.1916

.4186

2. 9686

2.4956

2. 0376

1. 6006

1.2326

.9726

.7866

.6026

. .5056

0. 2446

.18%

.4156

2. 9416

2. 4806

2. 0126

1. 5906

1.2286

.9706

.7816

.6006

. 5026

0.250

.191

.417

2.965

2.486

2. 030

1.597

1.232

.972

.784

.602

.503

11.501

11. 441

11. 668

14. 216

13. 737

13. 280

12. 847

12. 483

12. 222

12. 035

0.029

.019

.066

1.048

.803

0.125

.083

.269

5.147

3.943

.594 2.903

.417

.291

.208

.154

2. 022

1.370

.959

.695

11.852 i .106 .467

11. 754 .082 .357

0.250

.191

.417

0.333

.221

.766

2. 981 14. 901

2.496

2.035

1.599

1.234

11.032

7.895

5.360

3.628

.972 2.549

.785

.602

.503

1.856

2.665

2.660

2. 845

2.895

2.798

2.720

2.650

2.647

2.658

2.670

1.254 2.686

.967 2.706

[Series No. XLVII. Height of weir=P, 11.25 feet; length of weir crest=/>, 15.969 feet; width of channel/), 15.970 feet; width of broad crest, 0.927 foot; nappe partly aerated.]

1

2

3

4

5

6

7_8*

9

10

11

12

27

29

31

29

29

26

29

30

27

34

27

30

0.1666

.2816

. 4156

2. 9446

2. 5306

2. 0196

1. 5786

1.2296

1. 0096

.7786

.6296

.4616

0. 1636

.2726

.4106

2. 9206

2. 5116

2. 0106

1. 5706

1.2236

1. 0046

.7756

.6276

.4606

0.165

.278

.412

2.933

2. 522

2.014

1.592

1.226

1.007

.777

.629

.461

11. 416

11. 529

11. 663

14. 184

13. 772

13. 264

12.842

12. 477

12. 258

12. 027

11. 879

11.712

0.016

.033

.060

l.]87

.970

.722

. 512

.345

.248

.163

.117

.073

0.067

.147

.265

5.076

4.037

2.874

2.015

1.361

1.012

.685

.499

.313

0.165

.278

.412

2.954

2.535

2.021

1.596

1. 228

1.008

.777

.629

.461

0.180

.377

.700

16.840

13. 360

9.572

6.582

4.308

3.046

1.965

1.389

.859

2.690

2.563

2.644

3.318

3.314

3.331

3.266

3.166

3.008

2.869

2.786

2.744
106 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

United States Geological Survey experiments at Cornell hydraulic laboratory on model of 

Merrimac River dam, at Lawrence, Masx.

[Height of weir=P, 6.65 feet; length of weir crest=£, 9.999 feet; width of channel=6,15.97 feet.]

No.

1

1" 

2" 

3« 

21 

10 

11 

22 

23 

12 

^

24 

2 

3 

13 

4 

25 

5 

6 

7 

9 

19

Measured head on experimental weir, in feet.

Number of 

observations.

 2

Maximum.

3

Minimum.

4

Mean = 

D.

5

4.001 

3.930 

3.630 

3.630 

3.166 

2.815 

2. 510

2. 223 

2.130 

2.041

1

Q£A

1.746 

1. 645 

1.496 

1. 322 

1.268 

1. 089 

.764 

. 584

KO'J

.198

A = 

area of 

section 

per 

foot of 

crest.

6

10. 651 

10. 580 

10. 280 

10. 280 

9.816 

9.465 

9.160 

8.873 

8.780 

8.691

S

Kflfl

8.396 

8. 295 

8.146 

7. 972 

7.918 

7.739 

7.414 

7.234 

7. 233 

6.848

c= 

mean 

velocity of approach, 

in feet 

per second.

7

2.618 

2.563 

2.309 

2.319 

1.939 

1.654 

1.424 

1.200 

1.127 

1.066 

.932 

.860 

.802 

.691 

.600 

.556 

.462 

.284 

.195 

.192 

.039

Head corrected 

for velocity 

of approach, 

in feet.

H*

8

8.288 

8. 054 

7.029 

7.029 

5.769 

4.837 

4.067 

3.361 

3.150 

2.933 

2. 542 

2. 327 

2. 302 

1. 832 

1.528 

1.434 

1.141 

.669 

.447 

.446 

.088

H

9

4.094 

4.018 

3.670 

3.670 

3.216 

2.860 

2. 548 

2.244 

2.149 

2.049 

1.868 

1.756 

1.743 

1.497 

1.327 

1.272 

1.092 

.765 

.585 

.584 

.198

Q= discharge 

per 

foot of 

crest, 

in cubic 

feet per 

second.

10

27. 893 

27. 120 

23. 740 

23. 738 

19. 039 

15. 660 

13. 049 

10. 652 

9.898 

9.265 

7.929 

7. 227 

6. 651 

5. 631 

4.791 

4.410 

3.581 

2.108 

1. 412 

1.389 

0.270

Discharge 

coefficient G'i.

11

3. 365 

3.367 

3.377 

3.377 

3.300 

3.237 

3.208 

3.169 

3.142 

3.158 

3.113 

3.105 

2.889 

3.074 

3.135 

3.075 

3.138 

3.151 

3.158 

3.114 

3.067

In the accompanying tables (pp. 98-106), columns 2, 3, and 4 show, 

respectively, the number of observations of head and the maximum 

and minimum readings in each experimental period. In column 5 is 

given the mean head on the experimental weir deduced from the tape 

observations above described. Column 6 shows the area of cross section of the channel of approach per foot of crest. For suppressed 

weirs this quantity equals the sum of the height of weir plus the 

measured depth on crest. For weirs with one end contraction the 

quantity A is obtained by dividing the total area of the water section, 

where D is measured, by the net length of the weir crest corrected 

for the end contraction. For those series where the depth on the 

experimental weir was increased by contracting the weir to about onehalf of the channel width and introducing one end contraction, the 

net length of crest has been determined by the method of Francis, by 

deducting one tenth the head from the measured length of crest. The 

discharge per foot of crest of the experimental weir given in column 

10 has been deduced from the discharge over the standard weir,
Coefficient 

C.

4.00

U. S. GEOLOGICAL SURVEY 

S£'ff/ES 34 

.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 

Corrected head H in feet.

O3EE MODEL SIMILAR TO PLATTSBURG DAM, WITH BACK SLOPE MODIFIED. L

EXPERIMENTS OF UNITED STATES GEOLOGICAL SURVEY ON VOLU

CORNELL UNIVERSITY HYDRAULIC LABORAT
XXIV W«' "o found innnecliatety
U. 8. GEOLOGICAL SURVEY

Coefficient

O.

3.80

Efi/ES 36

3.00

3.00

2.80

.4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0 3.3 2.4 

Corrected head H in feet.

MODEL OF OGEE DAM AT CHAMBLY, QUEBEC. UPSTREAM END EXTENDED WITH 

.8 1.0 1.3 1.4 16 1.8 2.0 2.2 2.4 

MODEL OF OGEE DAM AT CHAMBLY, QUEBEC. LENGTH OF 

EXPERIMENTS OF UNITED STATES GEOLOGICAL SURVEY ON VOLUME OF FLOW OV

LABORATORY, 1903.
U. S. GEOLOGICAL SURVEY 

Coefficient 

C.

SERIES 38

(§>

1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 

Corrected head H In feet.

MODEL OF OGEE DAM AT DOLGEVILLE, N. Y. LENGTH OF CREST

Coefficient 

C. 

S.60

3.40 

3.20 

3.00

. <k

,

/

f

/

/

>^

x-*

.---

.5

£-/?

IE-

©1

5 37

 

 -»

-~~.

^__

<xdX ~.

 ~~»

^*

.4 .6 .8 1.0 1.8 1.4 1.6 1.8 2.0 2.2 2.4 2.6 

Corrected head Hin feet.

MODEL OF OGEE DAM AT DOLGEVILLE, N. Y. - PART OF APRON CARRIED AWAY IN SECOND EXPE

EXPERIMENTS OF UNITED STATES GEOLOGICAL SURVEY ON VOLUM

CORNELL UNIVERSITY HYDRAULIC LABORATO
CnoAD-CREST WCI1. V.I3T1!, 1.0'6 FEET) LENGTH. 13.COO TFFT. [>A

o .a .4 .8 .s i.o 1.2 1.4

ROAD-"9-CT WEIT. VI3TH, 0.473 TOOT: LENGTH, 15. r CO rzlT-. 

l.J l.J 1.4 1.5

Corrected head 77 in feet. 

MODEL O T DOLCEV!LLE CAM WITH APRON REMOVED, LEAVING TRAPEZOI3. LENGTH OF 

EXPERIMENTS OF UNITED STATES GEOLOGICAL SURVEY ON VOLUME OF FLOW 

HYDRAULIC LABORATORY, 1903.
_ I

°- 3 ^

- J

V

1 «

1

'*

2

'

0

1

LL

^

*

ft

$

<o'

e 

« It

 t

«!

V,

i

c

s>

1)

J

i

3

X.

\

1(j\ -1

2

M

4

"f

te

(-

"

^

)

^

i

Cp

-d

p

: S

Cs

.

o >

5°

-i o: 

O O

IRE 150 06 11
U. S. GEOLOGICAL SURVEY 

Coeffi 

fiem 

C. 

3.20

3.00 

2.80 

2.60 

2.40 

2.20

(12

©

fa

<®

-d>

-fe

©

V\~ '& Hw ^

f-2^ *

-S^

-/?/ E± 4

I*-- -

3<?

-

Fo

_

0 .3 A

BROAD-CREST WEIR. WIDTH, 16.302 FEET; LENGTH, 15.969 FEET. SMOOTH CRES

.8 1.0 1.2 1.4

Corrected head Hjn feet.

Coeffl 

cient

a

2.80 

2.60 

2.10

(S)

(

/

JL>

<?

&~

^-

5JF "~rt}-(D- k§X f7)

<s r/?.

@\

£5 4 4-

k-

0 .2 A .6 .8 1.0 1.2 1.4 1.6 1,* ?.0 2.2 2.4 2

Corrected head Kin feet.

EXPERIMENTS OF UNITED STATES GEOLOGICAL SURVEY ON VOLUM

CORNELL UNIVERSITY HYDRAULIC LABOR
U. 8. GEOLOGICAL SURVEY WA

1.0 1.2 1.4 1.6 1.8 

BROAD-CREST WEIR. WIDTH, 5.876 FEET; LENGTH, 15.969 FEET. NAPP

.6 .8 1.0 1.2 1.4 1.6 1,8 2.0 2.2 

BROAD-CREST WEIR. WIDTH, 2.174 FEET; LENGTH, 16.969 FEET. NAPP

EXPERIMENTS OF UNITED STATES GEOLOGICAL SURVEY ON VOLUME 

CORNELL UNIVERSITY HYDRAULIC LABORATO
U. S. GEOLOGICAL SURVEY

cient 

C.

SEf\/ES 1-7 

.2 A .6 .8 1.0 1.3 1 t 1.6 1.8 2.0 

Corrected head H in feet.

BROAD-CREST WEIR. WIDTH, 0.927 FOOT; LENGTH, 15.969 FEET. NA

EXPERIMENTS OF UNITED STATES GEOLOGICAL SURVEY ON VOLUME OF FLOW OVER 

LABORATORY. 1903.
WEIRS OF IRREGULAR SECTION. 107

obtained, as described above, by dividing the total discharge by the 

net length of the experimental weir. The mean velocity of approach 

v, given in column 7, has been obtained by the formula

The correction for velocity of approach has been carefully computed 

by the Francis formula

where

3

The resulting values of Hs are given in column 8. The corresponding values of H, given in column 9, have been obtained b^y interpolation from a table of three-halves powers. The discharge coefficient Cl 

given in column 11 has been obtained by the formula

This coefficient represents the discharge per linear foot of crest, if 

the head is 1 foot, with no velocity of approach, it being the coefficient 

in a weir formula of the same form as that used b,y J. B. Francis for 

a thin-edged weir.

Pis. XXIII to XXXII show the coefficient diagrams deduced from 

these experiments.

EXPERIMENTS ON MODEL OF DAM OF THE ESSEX COMPANY, MERRIMAC 

RIVER, AT LAWRENCE, MASS/'

A series of experiments covering five different depths' on crest was 

made b^y flames B. Francis at lower locks, Lowell, Mass., November, 

1852. The model had a crest length of 9.999 feet, with end contractions 

suppressed. Height of water was measured by hook gage in a chamber at one side of the channel, 6 feet upstream from crest, so arranged 

as to give substantially the height of the still-water surface above the 

crest without correction for velocity of approach. The discharge was 

volumetrically determined as in Francis's thin-edged weir experiments.

The experiments of Francis covered depths on crest ranging from 

0.5872 foot to 1.6338 feet. From these experiments he deduced tne 

formula for discharge,

a Francis, J. B., Lowell Hydraulic Experiments, pp. 136-137.
108 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

If the discharge were expressed in terms of the usual formula,

3_

Q= C^LHZ , with a varying coefficient C^ we should have a continuously increasing coefficient.

A series of experiments on a similar model dam, 6.65 feet high, with 

crest length of 15.932 feet, was made at Cornell University hydraulic 

laboratory in 1903. The model there used differed from that shown 

by Francis only in the substitution of a natter upstream slope near the 

bottom of the canal, as shown in PL XXX1IL The end contractions 

were suppressed and the depth on crest was measured with steel tape 

and plumb bob suspended over center of channel at points 14.67 feet 

and 29.82 feet, respective^, upstream from crest of experimental weir. 

Discharge was previously measured over the standard weir, calibrated 

by Bazin's and Fteley and Stearns's formulas, located at head of experimental canal.

The experiments covered a range of heads varying from 0.198 foot to 

4.94 feet. In the majority of the experiments the head was observed 

at both points. The upper point of measuring depth was at the 

upstream end of the inclined approach. The lower point was over the 

incline, where the area of the section of approach was smaller and 

the velocity larger than in the deeper channel above. The experiments have been reduced with reference to the heads measured 29.82 

feet upstream from crest. By comparison of the depths simultaneously observed at the two points correction factors have been deduced 

for the reduction of the rejmaining experiments, in which £he head was 

observed at the downstream point of observation only.

The observed head has been corrected for velocity of approach by 

the formula of Francis. The resulting mean coefficient curve, based 

on 19 valid observations, shows a larger coefficient of discharge in the

3

formula Q= C\LH^ than does that of Francis.

For a head of 1 foot the formula of Francis for the Merrimac dam 

gives a discharge of 90.3 per cent of that for a thin-edged weir. The 

Cornell experiments show 94.5 per cent of the discharge over a thinedged weir under the same head.
U. S. GEOLOGICAL SURVEY 

3.4 2.6 

EXPERIMENTS ON MODEL SIMILAR TO MERRIMAC RIVER DAM AT LAWRENCE, MASS., 

AT CORNELL UNIVERSITY HYDRAULIC LABORA
WEIRS OF IRREGULAR SECTION. 109

Discharge per foot of crest, Francis formula for Merrimac dam, compared icith Cornell 

experiments on similar cross section.

Depth on

crest, H.

0.15

.20

.25

.30

.35

.40

.45 

.50

.55 

.60

.65 

.70

.75

.80

Q per foot of 

crest, in cusecond, 

Francis.

0. 1653

.2567

.3611

.4774

.6043

.7431

.8877 

1. 0430

1.206 

1 37Q

1. 5581 

1. 7452

1. 9395

2. 1408

Coefficient C

Q=Ci

Francis's 

formula.

2.845

2.871

2.889

2.905

2.913

2.937

2.940 

2.940

2.956 

9 q««

2.973 

2.980

2.986

2.992

i in formula

Cornell experiments.

3.05

3.06

3.07

3.08

3.09

3.11

3.12

3.13

3.135 

3.14

3.14 

3.14

3.14

3.15

Depth on

crest, //.

0/85

.90

.95

1.00

1.15

1.25

1.50 

1.75

' 2.00

2.50

3.00 

3.50

4.00

Q per foot of 

crest, in cusecond. 

Francis.

'2. 3490

2. 5636

2. 7846

3.0121

3. 7500

4. 2378

5. 6012

8. 6975

16. 1750

25. 1200

Coefficient 

Q=C

Francis's 

formula.

2.997

3.002

3.007

3. 012

3.041

3. 033

3. 048

3.075

3.113

3.140

7i in formula

Cornell experiments.

3.15

3.15

3.15

3.13

3.12

3.10 

3.12

3.14

Q 9f)

3.26 

3.31

3. 36

Aside from Blackwell's experiments the Francis formula for the 

Merrimac dain was until recently the only one available for a large 

dam of irregular section, and for want of more appropriate data it has 

been used for the calculation of discharge over many forms of weirs 

of irregular section, and in spite of Francis's explicit caution, it has 

been applied where the heads differed widely from those used in the 

original experiments.

Considering the limited experiments on which it is based, Francis's 

Merrimac dam formula gives good agreement with the much more 

'extended experiments on a similar section made at Cornell hydraulic 

laboratory.
110 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

FLOW OVER WEIRS WITH BROAD CRESTS. 

THEORETICAL FORMULA OF UN WIN AND FRIZELL.<*

Consider a weir of such breadth that the nappe becomes of sensibly 

uniform depth in the portion J3C, iig. 8, the upstream corner of the 

weir being rounded to prevent vertical contraction and the surface 

slightly inclined downstream so that it becomes parallel with the 

surface of the nappe 13C.

FIG. 8. Broad-crested weir.

The fall causing the velocity Fin the section BC is If d. 

lows that if v is the mean velocity in SO

It folIn this equation Q is 0 when d Q or d=H. There must, therefore, be an intermediate value of d for which Q will be a maximum. 

Differentiating we find for the condition of a maximum,

dQ=Q=iL K 

dd

d ^ Giving H d - and d=^ \ or, for maximum discharge, one-third

the head would be expended in producing the velocity of flow, With

this value of d the expression for discharge becomes

Q-- ZLH

3V3

or if

(58)

In this formula frictional resistance has been neglected. The discharge given is the maximum for the conditions, and would result only 

if the stream discharges itself in accordance with the "principle of 

least energy.'1

Black well's experiments, given elsewhere, show a considerably 

larger coefficient for weirs 3 feet broad, slightly inclined downward, 

than for those with horizontal crests.

« Given by W. C. Unwin, in article 1 on Hydrodynamics in Ency. Brit. Independently derived by 

J. P. Frizell. See his Water Power, pp. 198-200.
WEIE8 WITH JBKOAD CEESTS.

Let d=KH, then from the formula first given

Ill

(59)

The theoretical coefficient Cl can be computed from this equation if 

A''has been determined experimentally.

From profiles taken in connection with United States Deep Waterways experiments at Cornell University hydraulic laboratory in 1899 

the following values of D and d for broad-crested weirs have been 

scaled and the ratio d D computed. D was taken 4 feet upstream 

from the upper face of the weir, and does not include velocity of 

approach correction; values of d: and da were taken at the lower-crest 

lip and center of crest, respectively. The value of da at center of 

crest has been used in the computations.

Values of D and dfor broad-crested weirs.

D ^!^^^^S=~3? j ^-^^

\

\

, B ----. 

-*» '

Broad-crested weir.

^^^____

?

j '

8

1

K-0.il'

^%$N

Broad-crested weir.

1

2

o

4

5

6

1

2

3

4

5

6

D

0.90

1.15

1. oU

2.60

3.55

5.15

1.00

1.32

1.98

2.85

3.90

4.65

di

0.35

.45

. /o

1.20

1.72

2.20

.35

.53

.75

1.08

1.50

2.10

d>

0.52

.68

1.14

1.75

2.52

3.15

.50

.70

.98

1.70

2.50

3.10

'-1

0.58

.59

£0

.67

.71

.61

.50

.50

.50

.60

.64

.61

For low heads a sudden drop begins near the upstream crest corner 

and terminates at a distance 1.5 to 2 I) below the upstream corner. 

From this point to within a distance about equal to D from the downstream crest corner the surface is nearly parallel with the crest. 

If the width of crest is not greater than 2.5 to 3 D the nappe passes 

over the broad crest in a continuous surface curve, becoming more 

nearly convex as the ratio D\ B increases.

For low heads Cornell experiment 13, crest 6.56 feet wide, with 

rounded upstream corner, complies very well with the theory of dis-
112 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

charge in accordance of the principle of least energy. The coefficient 

computed as above is

C{ = 8.02 X 0.585^/1^07585 

= 8.02X0.585X0.6442 

= 3.02

The experimental coefficient with head corrected for velocity of 

approach is 2.82.

The following additional data may be cited:

Trautwine" quotes data of Elwood Morris, C. E., for Clegg's dam, 

Cape Fear River, North Carolina. Horizontal crest 8.42 feet wide, 

vertical faces. 77=1.25 feet, d (throughout central portion of crest) 

= 0.50 foot, ^7 #=0.40.

Thos. T. Johnston b gives data of elaborate profiles of the nappe for 

Desplaines River dam, Illinois. Horizontal planed stone coping, vertical downstream face; upstream face batter, 1 2:1. 7T= 0.587 foot. 

(7=0.315 to 0.307 foot in central, nearly level portion at distances 1.5 

to 4 feet from upstream edge of crest. Johnston and Cooley deduce 

the coefficient (7=1.69 for this case.

BLACKWELL'S EXPERIMENTS ON DISCHARGE OF WATER OVER BROAD

CRESTED WEIRS.

Experiments made by Thomas E. Blackwell/ M. Inst. C. E., are of 

interest as being probably the first recorded for weirs with broad crests. 

The discharge was volumetrically measured, and the conditions were 

generally favorable to accuracy. The experiments were made on a 

side pond of the Keunet and Avon Canal, 106,200 square feet surface 

area, closed by a lock at each end, the water being admitted from time 

to time as required, the relation between area of reservoir and volume 

of discharge being such that there was no sensible variation in water 

level during an experiment.

The weir was constructed in a dock to which the water had access 

through an irregularly shaped channel 40 feet in width, cut off from 

the main pond by a submerged masonry wall 9 feet wide, situated 25 

feet upstream from the weir, having its top 18 inches to 20 inches 

below water surface.

The water level in the pond being constant when outflow took place, 

the weir, which had a crest adjustable in a vertical plane, was set with 

its crest level at the depth below water surface desired for an experiment, by means of adjusting screws at the ends of the weir; the water

« Engineers' Pocket Book.

& Johnston, T. T., and Cooley, E. L., New experimental data for flow over a broad-crest dam: Jour. 

Western Soc. Engrs., vol. 1, Jan., 1896, pp. 30-51.

cOriginal paper before Institution of Civil Engineers of London, reprinted in the Journal of the 

Franklin Institute, Philadelphia, March and April, 1852.
WEIRS WITH BROAD CRESTS. 113

was then allowed to waste through the weir until a uniform regimen 

of flow was established.

A gaging tank having a floor of brick laid in cement, with plank 

sides, and M9.39 cubic feet capacity, was erected at the foot of the 

weir. At a given signal the lid of this tank was raised, the time 

noted, and the rate of filling of the tank recorded by several observers. Such leakage from the tank as occurred was separately measured 

and allowed for. There was no correction for velocity of approach or 

for end contractions.

The wind was so slight as to be negligible, except during one series 

when there was a brisk wind blowing downstream. The experimenter 

states that parallel experiments on a quiet day indicated an increase of 

about 5 per cent in discharge due to this wind.

The crest of the thin-edged weir consisted of an iron plate barely 

one-sixteenth inch thick. A square-top plank 2 inches thick was 

attached to the weir, and an apron of deal boards, roughly planed so 

as to form an uninterrupted continuation downstream, constituted the 

wide-crested weir used in the experiments.

The coefficient C^ from Blackwell's experiments has been worked 

out and is given in the following table. The measured depths taken 

in inches have also been reduced to feet. 

IRK 150 06 12
114 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

Blackwell's experiments on broad-crested weirs, Kennet and Avon Canal, England, 1850.

Weir.

II.'

Thin plate, 10 feet 

long.

VII.

L=S feet. r*^

Series VII, average.

VIII.

L=B feet.

rr--- 5l°f>e '"a ---"Tj

1 -"I

IX.

£=10 feet.

Series VIII-IX, avhead, Measured 

inches. in. oMcn^oJtoM o eniix <w M «sINS 

1

2 

3

4 

5 

7 

8 

9

itof 

2 

4 

6

8

Head, feet. in 

0.083 

.167 

.25 

.333 

.417 

.667 

.750

.083 

.167 

.250 

.333 

.500 

.583 

.750

.083 

.167 

.250 

.333 

.417 

.583 

.667 

.750

.073 

.167 

.333 

.500 

.667

II

0.104 

.292 

.429 

.675 

.935 

1.691 

1. 842

.060 

.194 

.360 

.468 

1.005 

1.254 

1.729

.070 

.199 

.359 

.443 

.743 

1.222 

1.426 

1.709

.060 

.181 

.530 

1.028 

1.476

M

0.539 

.535 

.428 

.437 

.433 

.387 

.353

.311 

.355 

.359 

.303 

.354 

.351 

.332

.363 

.364 

.358 

.287 

.344 

.342 

.327 

.328

.311 

.330 

.343 

.362 

.338

11^

4.32 

4.29 

3.43 

3.50 

3.47 

3.10 

2.83

2.49 

2.85 

2.88 

2.43 

2.84 

2.82 

2.66

2.71

2.91 

2.92 

2.87 

2.30 

2.76 

2.74 

2.62 

2.63

2.49 

2.65 

2.75 

2.90 

2.71

2.71

Weir.

X.

L=% feet.

.,.,,,, , .....,,,

! i

XI. 

L=6 feet.

Se

XII.

l-i J.U Ic^Jl.

Hori-LOntal

ries X, XI, XII, average ..............

Measured head, 

inches. in 

1 

2 

3 

4 

5 

6 

7 

8 

9

ItoU 

3

4

7 

9 

10 

12

1

5 

6 

8 

9 

10

ti 

 c"

g 

a

0.083 

.167 

.250 

.333 

.417 

.500 

.583 

.667 

.750

.093 

.250 

.333 

.500 

.583 

.750 

.883 

1.0

.083 

.167 

.417 

.500 

.667 

.750 

.833

II

Qj ^ 

O^ £*

0.058 

.175 

.295 

.431 

.689 

.947 

1.162 

1.369 

1.642

.071 

.329 

.511 

.963 

1.191 

1.670 

1.895 

2.495

.049 

.174 

.745 

.972 

1. 362 

1.688 

1.847

M

0.301 

.321 

.294 

.279 

.319 

.334 

.325 

.313 

.317

.328 

.331

.331

.310 

.311

.254 

.319 

-.345 

.342 

.312 

.324 

.303

?

tes 

i^

2.57 

2.36 

2.24 

2.56 

2.68 

2.61 

2.51 

2.54

2.63 

2.65

2.65

2.49 

2.49

2.04 

2.56 

2.77 

2.74 

2.50 

2.60 

2.43

52.03 

2.48

EAST INDIAN ENGINEERS' FORMULA FOR BROAD-CRESTED WEIRS. a

This formula is

Q=- M'LH V 2 gH, or if V 2 </=8.02, Q=6.35M'LJI^= C^LH^ (60)

o

Where M'=coefficient for thin-edged weir 6 =0.654 0.01ZT,

a+i (61)

«Mullins, Gen. Joseph, Irrigation Manual, Madras, 1890.

&See table giving values of M and equivalent values of C, p. 22.
WEIRS WITH BROAD ORES 18. 115

Experimental data not given. This formula gives values of M' or 

O1 decreasing as breadth of crest B increases, and for low heads 

increasing to a maximum for a head of about 1 foot, then slowly 

decreasing.

The formula reduces to

. (62)

For £=0

 C\=G

which differs by the ratio B, from the equivalent value of C for a 

thin-edged weir.

Values of coefficient Cl of discharge over a broad-crested weir and of R, the ratio of the 

former to Cthe coefficient of discharge over a thin-edged iveir, by Mullins's formula.

\ B i Ifoot.

TT \

feet.\

1

2

3

4

5

6

7

8

9

10

\ B

rj- \^

feet/\

1

2

3

4

5

6

7

8

9

10

-R

0.975

.983

.988

.990

.992

.993

.994

.994

.995

.995

Ci

3.359

3. 335

3.296

3. 253

3.204

3. 155

3.104

3.054

3.003

2.950

5 feet.

R

0.925

.95

.962

.97

.975

.978

.981

.983

.985

.986

C'i

3.182

3.22

' 3. 213

3.193

3.149

3.11

3.06

3.017

2.97

2.92

2 feet.

R

0.962

.975

.981

.985

.988

.989

.991

.992

.992

.993

Ci

3.316

3.307

3.275

3.237

3.191

3.144

3. 095

3. 045

2. 995

2.944

3 feet.

R

0. 95

.967

.975

'.98

.983

.986

.988

.989

.99

.991

Ci

3.319

3.279

3. 255

3.220

3.177

3.132

3.085

3.037

2.988

2.937

6 feet. 1 7 feet.

R

0. 912

.942

.956

.965

.971

.975

.978

.980

.982

.984

Ci

3.144

3.194

3.192

3.171

3.137

3. 098

3. 057

3. 012

2.966

2.916

R

0.9

.933

.95

.96

.967

.971

.975

.977

.98

.982

C'i

3.100

3.166

3.171

3. 154

3. 123

3.080

3.046

2. 999

3.019

2.910

4 feet.

R

0. 938

.958

.969

.975

.979

.982

.984

.986

.988

.989

Ci

3.230

3. 251

3. 234

3.204

3.164

3.121

3.075

3.028

3. 042

2.930

8 feet.

R

0.888

.925

.944

. 955

.962

.968

.972

.975

.978

.979

f\

3.057

3.138

3.150

3.138

3.110

3.076

3.036'

2. 994

2.950

2.093

The values of C^ given in the above table have been deduced from 

the corresponding values of C for a thin-edged weir by Mullins's 

formula. The ratio R may, if desired, be applied approximately to 

correct values of C derived from other standard weir formulas.
116 WEIK EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

FTELEY AND STEARNS EXPERIMENTS ON BROAD-CRESTED WEIRS. a

The formula of Fteley and Stearns is based on five series of experiments made in the Sudbury River conduit, Boston, 1877, on weirs 2, 3, 

4, 6, and 10 inches wide, respectively. Suppressed weirs 5 feet long 

were used, the depths being as follows:

Fteley and Steams

Width of 

crest, in 

inches.

2

3

4

6

10

Number 

of experiments.

7

21

25

22

17

Range of depth observed 

on broad crests, in feet.

From 

0. 1158

.1307

.1318

.1320

.1352

To 

0. 2926

.4619

.6484

. 8075

.8941

The results are given by the authors in the form of a table of corrections to be added algebraically to the measured head for the broadcrested weir to obtain the head on a thin -edged weir that would give 

the same discharge.

Fteley and Stearns's correction c may be found approximately from 

the formula

c = 0.2016V[(0.807 ^-//)2+ 8] 0.1876 B . . (63) 

or if &=0.2016, m =0.1 876, n= 0.2146, O= 0.807, then

Q= CL [H-mB+lf.j(OB-B¥+nl?'¥ . . . (64) 

If the head on a broad-crested weir is IT, the discharge will be

Q=CL(H+c$ ....... (65)

C being the coefficient of discharge for thin-edged weirs. 

If C-i is the coefficient for the broad weir, then we may also write

Hence

H

(66)

From formula (66) have been calculated Fteley and Stearns's coefficients for weirs with nappe adhering to crest for use in the formula

a Fteley and Stearns, Experiments on the flow of water, etc.: Trans. Am. Soc. C. E., vol. 12, pp. 86-96.
WEIRS WITH BROAD CRESTS. 117

correction for velocity of approach being made by adding 1.5 to the A<7 

measured head to obtain H. a

Values of the ratio 4f of the coefficient of discharge for a broad-crested tveir, by Fteley 

and, Stearns's experiments, to that for a, thin-edged weir.

H

0.0 

.1 

2 

.3 

A 

.5 

.6 

.7 

.8 

.9 

1.0 

1.1 

1.2 

1.3 

1.4 

1.5

Width of crest, in inches.

3

0. 7466 

.8234 

.9172 

.9963

..........

4 6

0. 74798 

.7878 

.8524 

.9201 

.9806 

1. 0030

0. 7562 

.7740 

.8003 

.8353 

.8781 

. 9230 

.9391 

.9997 

1. 0317

8

0.7589 

.7679 

.7809 

.8003 

. 8255 

. 8567 

.8911 

.9245 

.9553 

.9835 

1.0090 

1. 0317 

1.0499

10

0. 7576 

.7644 

.7727 

. 7850. 

.8003 

.8199 

.8424 

.8695 

.8983 

.9406 

.9508 

.9732 

.9948 

1. 0148 

1.0317

12

0. 7576 

.7624 

' .7685 

.7768 

.7865 

.7989 

.8150 

.8339 

. 8552 

.8824 

.9027 

.9252 

.9465 

. 9657 

.9850

BAZIN's FORMULA AND EXPERIMENTS ON BROAD-CRESTED WEIRS.

These included series of about 20 periods each for depths not 

exceeding 1.4 feet on weirs of 0.164, 0.328, 0.656, 1.315, 2.62, and

3

6.56 feet breadth of crest. The coefficient Ol in the formula Q C^LH* , 

deduced from a recomputation of the experiments on weirs 2.46 feet 

high, using the Francis velocity of approach correction, is given 

on PL IV.

Other experiments were made for the four narrower weirs with 

heights 1.148 and 1.64 feet, to determine the comparative velocity of 

approach effect.

Bazin shows that if the nappe is free from the downstream face of 

the weir it may assume two forms: (1) It may adhere to the horizontal 

crest surface; (2) it may become detached at the upstream edge in 

such a manner as to flow over the crest without touching the 

downstream edge. In the second case the influence of the flat crest 

evidently disappears and the discharge is like that over a thin-edged 

weir. The nappe usually assumes this fqrm when the depth I) exceeds 

twice the breadth of crest B, but it may occur whenever the depth 

exceeds \R. Between these limits the nappe is in a state of instability; it tends to detach itself from the crest, and may do so under the

« Fteley and Stearns's formula for a thin-edged weir has been used to calculate Q in deriving these 

coefficients, the experiments having been made under conditions similar to those under which their 

formula was derived.
118 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

influence of any external disturbance, as, for example, the entrance of 

air or the passage of a floating object over the weir.

When the nappe adheres to the crest, the coefficient C\ depends 

chiefly on the ratio DIB and may be represented by the formula

C; = O (0.70+0.185 DlB] ..... (67) 

in which C is the coefficient for a thin-edged weir.

When 2)iJl=I.50 to 2, Gj (7=0.98 to 1.07 if the nappe adheres to 

crest, or Cj C'=1.00 if nappe is detached, and for Z>/jff>2, GJ (7=1.00. 

Between the limits D=^1.5I> and 7>=2Z? the value which the coefficient G^ will assume in a particular case is uncertain. Bazin considers 

that his formula gives accurate results for adhering nappes with 

breadth of crests up to 2 or 3 feet. For a crest 6.56 feet wide and 

D 1.476 feet he finds the result by formula (67) 93.4 per cent of 

that given directly by the experiment.

Values of the ratio C\/f!, for a broad-crested weir, wilh, <t.dli,ermg nappe, by Basin's

formula,. 0

DjB

0.0

.1

.2

.3

.4

.5

.6

.7

.8

.9

1.0

1.1

1.2

1.3

1.4

1.5

Ci/C'=0.700+0.185 DIB.

0

0.700

. 7185

.7370

.7555

.7740

.7925

.8110

.8295

.8480

.8665

.8850

. 9035

.9220

.9405

.9590

.9775

0.01

0. 7018

.7204

.7388

.7574

.7758

.7944

.8128

.8314

.8498

.8684

.8868

.9054

. 9238

.9424

.9608

.9794

0.02

0. 7037

.7222

.7407

.7592

.7777

.7962

.8147

.8332

.8517

.8702

.8887

.9072

.9257

.9442

. 9627

.9812

0.03

0. 7056

. 7240

.7426

.7610

.7796

.7980

.8166

.8350.

.8536

.8720

.8906

.9090

.9276

.9460

.9646

.9830

0.04

0. 7074

.7259

.7444

.7629

.7814

.7999

.8184

.8369

. 8554

.8739

.8924

.9109

.9294

.9479

.9664

.9849

0.05

0. 7092

.7278

. 7462

.7648

.7832

.8018

.8202

.8388

.8572

.8758

.8942

.9128

.9312

.9498

.9682

.9868

0.06

0. 7111

.7296

.7481

.7666

.7851

.8036

.8221

.8406

.8591

.8776

. 8961

.9146

.9331

.9516

.9701

.9886

0.07

0. 7130

.7314

.7500

.7684

.7870

.8054

.8240

.8424

.8610

.8794

.8980

.9164

.9350

. 9534

.9719

.9904

0.08

0. 7148

.7333

.7518

.7703

.7888

.8073

.8258

.8443

.8628

.8813

.8998

.9183

. 9368

. 9553

.9738

.9923

0.09

0. 7166

. 7352

.7536

.7722

.7906

.8092

.8276

.8462

.8646

.8832

.9016

. 9202

.9386

. 9572

.9756

. 9942

«If there is velocity of approach, the value of DIB, not H/B, should be used as an argument. The 

ratio rye may be applied in a formula which includes the velocity of approach correction, either in 

the head H or in the coefficient.

BaznTs formula gives ratios which continually increase as H 

increases, B remaining constant, and which continually decrease as 

B increases, II remaining constant. It gives, however, a constant 

ratio for all widths or heads where the ratio III B is unchanged.

Compared with their respective standard weir formulas, Mullins's 

formula gives for a broad-crested weir a continuously decreasing ratio 

of discharge as B increases from zero, H remaining constant, and a 

continuously increasing discharge as II increases from zero, B remaining constant; Fteley and Stearns's experiments give a discharge ratio 

which is less than unity, but which varies in an irregular manner, 

depending on the head and breadth of weir.
WEIRS WITH BROAD CRESTS. 119

On referring to PI. IV, in which the Bazin coefficients are given in a 

form comparable with the experiments of the United States Geological 

Survey, it will be noticed that, except for the lowest heads, the coefficient curves are simple linear functions of the head. The rate of 

increase of the coefficients as the head increases grows rapidly less 

as the breadth of the weir increases, indicating that for a very broad 

weir the coefficient would be sensibly constant throughout the range 

of stability of the nappe.

For the narrower weirs the coefficients tend to increase rapidly 

almost from the start toward the value for a thin-edged weir or 

detached nappe. For the weirs 2.62 and 6.56 feet breadth of crest 

the total variation in the coefficient for the range of heads covered by 

the experiments is comparatively small. The average coefficients are 

as follows:

Average Bazin coefficients, broad-crested weirs.

Bazin series 

No.

113

114

115

Crest width, 

in feet.

1.312

2.624

6.56

Range of head, in feet.

From 

Lowest.

0.35

.55

To 

0.60

( .85 

1 1.32

Highest.

Average constant coefficient, C\.

2.64

2.59 

2.62

« 2. 58

a Coefficient increases slowly throughout.

The average coefficients show a fair agreement with the constant 

coefficient for broad-crested weirs with stable nappe deduced from the 

experiments of the United States Geological Survey (page 120).

EXPERIMENTS OF THE UNITED STATES GEOLOGICAL SURVEY ON BROADCRESTED WEIRS.

The method of conducting these experiments and the detailed results 

are given on pages 95-107. The coefficient curves are presented on Pis. 

XXVIII to XXXII. It may be remarked here that the models were 

larger and the range of breadth of crest and depth of flow experimented upon was greater than in the earlier experiments described. 

In general, the laws of behavior of the nappe pointed out by Fteley 

and Stearns and Bazin were confirmed.
120 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

The following table presents a resume of the results:

Resume of United States Geological Survey experiments on broad-crested weirs.

Series.

40

Breadth 

of crest, 

in feet.

0.479

47 . 927

41

46

45

44

42

43

1.646

3.174

5.875

8.980

12. 239

16. 302

43a 16.302

Average.

Nappe 

unfor 

heads

less 

than

values 

below,

in feet.

0.3

.3

.7

.5

.5

.5

.4

Nappe

toched 

from

crest 

at

head, 

in feet.

0.8

1.8

2.8

Coefficient C\ varies between 

the limits 

Head, in feet. | Coefficient.

From 

0.3

.3

.7

.5

.5

.5

o

.3

.4

To 

0.8

1.8

2.8

1.3

.9

2.0

2.0

1.1

1.0

From 

2.64

2.57

2.56

2.70

2.72

2.73

j 2.62

1 2.73

2.72

To 

3.32

3.31

3.32

2.64

2.64

2.62

J 2.72

1 2.63

2.64

Coefficient 

constant.

Above 

head,

in feet.

0.8

1.8

2.8

1.3

.9

2.0

2.0

I "

1.0

a

3.32

3.31

3.32

Increases

2.64

2.62

o 2. 64

2.64

2.634

a Coefficient shows tendency to increase slowly with head.

b Edges of planed and matched boards not flush. Crest smoothed in series 43o.

The deductions that follow have been based on a consideration of 

earlier experiments as well as those here given for the first time.

1. For depths below 0.3 to 0.5 foot the nappe is very unstable, 

owing probably to magnified effect of crest friction and to the varying 

aeration or adhesion of the nappe to the downstream weir face.

2. For heads from 0.5 foot to 1 or 2 feet for very broad weirs, or 

from 0.5 foot to the point of detachment for narrower weirs, the 

coefficient is somewhat variable and changes in an uncertain manner. 

For the broader weirs, the range of variation of Cl between the depths 

indicated is narrow, from 2.73 to 2.62.

3. When the nappe becomes detached the coefficient remains nearly 

identical with that for a thin-edged weir. For the narrower weirs the 

coefficient increases rapidly within the range of tendency to detachment indicated by Bazin, i. e., for heads between D and 2.ZX

4. On the broader weirs for depths exceeding 1 to 2 feet up to the 

limit of the experiments (about 5 feet), the experiments indicate a sensibly constant coefficient for all depths. Where there is any tendency 

to variation within the range indicated there is a gradual increase 

in Cr

For weirs of 5 to 16 feet breadth the experiments show no conspicuous tendency for the coefficient C\ to change with variation in either 

If or B, the range of value of C\ being from 2.62 to 2.64.

The line of detachment of the nappe % for a weir of 5 feet breadth 

would be 7.5 to 10 feet head or perhaps more, and a higher head for
WEIRS WITH BROAD CRESTS. 121

broader crests. If this depth were ever reached it may be surmised 

that the coefficient C\ would increase to about 3.33 at the point of 

detachment. It would also appear, as is in fact indicated in Bazin's 

formula, that the coefficient should very slowly increase with II and 

decrease as B increases, independent of the tendency to detachment 

of the nappe, and owing to the decreased relative effect of crest friction and contraction.

The United States Geological Survey experiments indicate that this 

effect is of relatively little significance for large heads and broad weirs, 

and hence a constant coefficient covering a wide range may be safoiy 

adopted.

The average coefficient, 2.64, which we have tentatively chosen for 

weirs exceeding 3 feet in breadth under heads exceeding- 2 feet, may 

apparently be applied for considerably lower heads for weirs of 5 feet 

or more crest breadth with but small error.

TABLE OF DISCHARGE OVER BROAD-CRESTED WEIRS WITH STABLE NAPPE.

A table has been calculated, using 6^=2.64 and covering heads varying by 0.1 foot increment from zero to 10 feet (p. 177). It is considered applicable'for weirs of 3 feet or more crest breadth when HiB 

lies between the general limits 0.25 to 1.5. The coefficient 2.64 gives 

a discharge 79.2 per cent of that for a thin-edged weir by the Francis 

formula. The relative discharge obtained by other formulas and 

experimenters is shown in the following table:

Comparison of broad-crested weir formulas and experiments giving percentage of 

discharge over a thin-edged tveir."

Formula or experiments.

U. S. Deep Waterways exU. S. Geological Survey ex1 foot width. ' 2.62 feet width.

HIB=0. 5

1T=0, 5

96.7

79.2

81.0

1.0

1.0

97.5

88.2

88.5

90.3

1.5 0.5

1.5

98.0

97.8

97.5

1.0

1.31 2.62

79.2 

82.8 

(-79.5

88.5 

93.3 

c81.3

1.5

3.93

97.8 

114.1 

C86.7

6.56 feet width.

0.25

1.64

93.2

74.6 

72.0 

79.2

0.5

3.28

95.5

79.2 

71.1 

d 79. 2

1.0

6.56

97.5

88.5 

72.3 

d79.2

1.5

9.84

98.2

97.8 

73.2 

rf79.2

o No velocity of approach.

ft East Indian engineers' formula, given in Mnllins's Irrigation Manual, Madras Presidency. 

<-Weir 2.17 feet broad. 

feet broad.

Considering the low heads used, it may be noted that before Bazin's 

experiments only those of Blackwell included a weir breadth sufficient 

to eliminate the early tendency to detachment and permit the existence 

of the stable period for which a constant coefficient applies.
122 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

Blackwell's experiments on weirs 3 feet broad indicate a maximum 

coefficient C\ of 2.65 to 2.77 for a head of about 0.5 foot, decreasing as 

the head increased.

The experiments of the United States Deep Waterways Board on 

models with 2. 62 and 6. 56 foot crest width are shown on PL XV. For 

the narrower weir the coefficient increased uniformly with the head. 

The nappe did not leave the crest, although the experiments were 

continued to the limit H\B 2, at which stage the coefficient exceeded 

that for a thin-edged weir. For the broader weir the coefficients are 

much less variable and the curves indicate that the coefficients approach 

a constant as the breadth of crest is increased.

It will be noted that considerable care must be exercised in determining the condition of the nappe for broad-crested weirs of inconsiderable width, while for those of greater breadth the wind may exert 

considerable influence on the nappe on the broad crest under lower 

heads. The constant coefficient 2.64 has been deduced from experiments on weirs with smooth, planed crests and sharp upstream crest 

angles. The effect of crest roughness on weir discharge is discussed 

on page 133.

EFFECT OF ROUNDING UPSTREAM CREST EDGE.

Experiments by Fteley and Stearns a indicate that the effect of 

rounding the upstream crest corner is to virtually lower the weir, by 

allowing the water to pass over with less vertical contraction. To 

determine the discharge over a thin-edged weir, with upstream crest 

corner rounded to a radius J?, add to the measured head the quantity

K=0.70fi ........ (68)

The above formula was deduced by Fteley and Stearns from experiments on weirs with crest radii of one-fourth, one-half, and 1 inch. 

For heads not exceeding 0.17. 0.26, and 0.45 foot, respectively, the 

nappe adhered to the crest, and the formula does not apply.

The correction formula (68) is equivalent to increasing the discharge coefficient in the ratio

or nearly in the ratio

H+R

H '

A second series of experiments was made with rounded upstream 

edges of similar radii applied to a crest 4 inches wide, giving the correction formula for this case,

........ (69)

« Experiments on the flow of water, etc.: Trans. Am. Soc. C. E., vol. 12, pp. 97-101.
EFFECT OF BOUNDING UPSTREAM CREST EDGE. 123

where JT is a correction to be added to the measured head before 

applying the formula for discharge over the broad-crested weir. This 

formula is applicable for depths of not less than 0.17 and 0.26 foot, 

respectively, on weirs with radii of one-fourth and one-half inch. 

Fteley and Stearns's formulas show the effect to decrease with the 

breadth of crest. It also decreases, when expressed as a percentage, 

with the head. These formulas are probably applicable to weirs with 

smaller, though not to those with greatly larger, radii than those of the 

experimental weirs.

Bazin experimented upon two weirs, duplicated in the United States 

Deep Waterways experiments, having crest widths of 2.624 and 6.56 

feet, respectively, with an upstream crest radius of 0.328 foot (PL 

IV).

Broad-crested weirs with rounded upstream com,e.r.

Head, in 

feet.

0.25

.50

1.00

1.50

1.50

2.00

3.00

4.00

5.00

6.00

Coefficient C\, Bazin's experiments.

Crest width, 2.62 feet.

With angle 

crest.

2.52

2.59

2.64

2.69

With rounded crest.

2.85

2.95

3.00

3.04

Crest width, 6.56 feet.

With angle 

crest.

2.40

2.515

2.575

2.635

With rounded crest.

2.58

2.76

2.89

2.92

Coefficient C\, United States Deep Waterways experiments.

2.67

2.75

2.93

3.11

3.30

Nappe free.

2.92

3.00

3.17

3.34

3.51

3.00

2.39

2.41

2.44

2.47

2.50

2.53

2.81

2.81

2.81

2.81

2.81

2.81

United States Deep Waterways series 14 and 15, PI. XV, show the 

effect of rounding the upstream crest corner, radius 0.33 foot, on a 

model of the Rexford flats, New York, dam. In this case, with a 

weir 22 feet broad with 6:1 slope on each face, the effect of rounding 

becomes comparatively slight, the average increase being about 2 

per cent.

United States Geological Survey experiments, series Nos. XXXV 

and XXXVI, PI. XXVI, show the effect of the addition of a 4-inch 

radius (0.33 foot), quarter-round extension to the upstream face of the 

model of an ogee-section dam, having 4.5 feet crest width, 4.5 :1 slope.
124 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

Effect of rounded upstream crest corner on an ogee dam.

Head, in

feet.

0.50

1.00

2.00

3.00

Chambly 

model, 

series 35.

3.18

3.30

3.42

3.49

Same, with 

rounded upstream crest 

corner.

3.22

3.34

3.51

3.64

Difference 

per cent of 

Francis's 

coefficient.

+1.5

+1.2

+2.7

+4.5

EXPERIMENTS ON WEIRS WITH DOWNSTREAM SLOPE, OR APRON, 

OF VARYING INCLINATION.

Aside from the experiments of Blackwell on weirs with very slight 

inclination and a few series by other experimenters on weirs of irregular section involving aprons, the data on this subject are limited to 

those of Bazin's experiments.

Bazin selected a number of weir types, each having a constant top 

width, height, and upstream inclination and applied to each a number 

of different downstream slopes. a

TRIANGULAR WEIRS WITH VERTICAL UPSTREAM FACE AND SLOPING

APRONS.

Such weirs are occasionally used, as where the apron slopes to the 

stream bed in log slides. A similar form in which the downstream 

slope terminates at a greater or less distance from the vertical upstream 

face is not uncommon, and to this form the Bazin experiments may 

probably be applied, provided the breadth of the sloping apron is considerable. The experiments are of special interest, however, as showing the effect of attaching a sloping apron to the downstream face of 

a thin-edged weir, and by inference affording an indication of the effect 

of a similar apron attached to any form of cross section. The results 

of Bazin's experiments recomputed on the basis of the Francis formula are shown on PI. V.

Four series of experiments on weirs 2.46 feet high are included. 

If or all these series the coefficient C tends to remain nearly constant 

for the range of heads covered, 0.2 foot to 1.5 feet, there being a slight 

increase in C with the lower heads only.

Two series on weirs 1.64 feet high are also given. In series 145, 

slope of apron 3 :1, there is a general increase in coefficient with head 

below 0.9 foot. Series 138, for a weir 1.64 feet high, is duplicated on 

a weir 2.46 feet high, and the latter series is given preference in the 

general curve. The lower weirs indicate in both cases slightly higher 

coefficients, possibly owing to the incomplete elimination of the effect 

of excessive velocity of approach.

« Bazin did not attempt to collate the results extensively. His general resume has been translated 

by the writer, and may be found in Kept. U. S. Board of Engineers on Deep Waterways, pt. 2,1900, 

pp. 646-658.
WEIRS WITH VARYING DOWNSTREAM SLOPE. 125

The average constant coefficients for the several series are shown in 

the following table:

Mean coefficients, triangular weirs with varying apron slope.

Series.

136

137

138

145

141

,142

Height.

2.46

2.46

1.64

1.64

2.46

2.46

Slope.

1 :1

2:1

2:1

3 :1

5:1

10:1

Range of head.

From 

0.3

.3

.7

.9

.6

.75

To 

1.40

1.6

1.5

1.5

1.5

1.5

Range of C.

i

From 

3.84

3.48

3.56

3.39

3.08

2.90

To 

3.88

3.52

3.58

3.41

3.14

2.93

Average (J.

3.85 '

3.50

3.57

3.40

3.13

2.91

3.9

3.8 

3.7 

3.6 

3.5 

3.4 

3.3 

3.2 

3.1 

3.0 

2.9 

2.8

-i

\

\

\

i

\

\

\ ,

\

\

\

f

\

A Weir

O J>

v ^

N

/'

k,

1T

vo £^ *. i <\i ^

i

s 1.64 

Z.4-6

\

ft. high.

H 

^

\

» 

^

\ \~~^

\

Form of experimental weirs

\

-? - 3,85 

- So.,z

^ ^,

\ ^^

^**.-- ^. ^1

- ^

---.

 ^

12 34667 8 9 10 11 1

Slope or batter of downstream face of weirs. 

FIG. 9. Coefficient curve for triangular weirs.

The mean coefficients have also been plotted on fig. 9 and a general 

curve drawn. This curve becomes approximately a straight line when 

plotted on logarithmic cross-section paper. Its equation expressed in 

logarithmic form is

^ 3.85 //,QS 

<- $*.* ........ w

where $ is the batter or slope of apron.
126 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

If $=6, then, solving by logarithms,

log 6=0.7781513 

log 6°-12 =0.0933782

log 00^=9.9066218 

log 3.85=0.5854607 

log O= 0.4920825 

(7=3.105

Fig. 9 gives C=3.07; the difference is 1 per cent. 

The following conclusions deduced from the recomputed data conform in general with those of Bazin:

1. For steep apron slopes where the nappe tends to break free, the 

apron materially increases the discharge by permitting a partial 

vacuum to be formed underneath the nappe.

2. For flat apron slopes the conditions approach those for a horizontal crest.

3. For an apron slope of about 3 :1, the discharge is nearly the same 

as for a thin-edged weir.

4. For slopes greater than 3 :1 the apron diminishes the discharge 

the amount of diminution increasing as the slope becomes flatter.

TRIANGULAR WEIRS WITH UPSTREAM BATTER 1:1 AND VARYING SLOPE

OF APRON.

Three series of experiments by Bazin are included (PL IX), all made 

from weirs 1.64 feet high. The results are comparable among themselves, but owing to the high velocity of approach their general applicability is less certain.

Series No. 161, downstream slope 1:1, shows a generally decreasing 

coefficient with an apparent tendency to become constant through a 

narrow range of heads, from 0.5 to 0.9 foot, with C about 4.11.

Series No. 163 and 165, with apron slopes of 2:1 and 5:1, give coefficient lines, which may be fairly represented by the constants 3.82 and 

3.47, respectively. These coefficients compare with those for vertical 

weirs with the same apron slopes as follows:

Comparative coefficients.

Batter of 

apron.

1:1

2:1

5:1

Vertical 

face.

C

3.85

3.53

3.13

Face inclined 1:1. 

C

4.11

3.82

3.47

Difference, 

per cent, 

Francis's 

coefficient.

+ 7.8

+ 8.7

+10.2
WEIRS WITH VARYING DOWNSTREAM SLOPE. 127

EXPERIMENTS ON WEIRS OF TRAPEZOIDAL SECTION WITH UPSTREAM

SLOPE OF 

SLOPES.

HORIZONTAL CREST, AND VARYING DOWNSTREAM

Five series of Bazin's experiments on weirs 2.46 feet high, with 

crest width of 0.66 foot, are shown on Pis. VI and VII. The curves 

indicate coefficients increasing with the head, the rate of increase being 

more rapid for the steeper apron slopes. There is a tendency to 

depression at about 0.4 foot head, representing, possibly, the point at 

which the nappe changes from adhering to depressed condition on the 

downstream face. The curves are all convex, and apparently approach 

a constant, which was not, however, reached within the limit of experiments, except, perhaps, for the flattest slope of 5:1. The coefficients 

increase in value as the steepness of the apron slope increases.

Three series of experiments 011 weirs similar to those above described, 

but with flat crests 1.317 feet wide, are shown on PL VIII. The 

coefficient curves are of uncertain form for heads below 0.6 foot. 

For greater heads they may be represented by inclined straight 

lines. The coefficients increase uniformly with the head, the initial 

values for 0.6 foot head being nearly the same for the several slopes, 

the increase being more rapid for the steeper downstream slopes.

It may be seen from the following table that increased width of the 

flat crest, as compared with that of the preceding weir, causes a 

decrease in the discharge,

Comparative coefficients at 1-foot head, weirs with flat crests and %:1 upstream slope.

Slope of 

apron.

1:1 

2:1 

3:1 

4:1 

5:1 

6:1

Crest width, in feet.

0.66

3.52 

3.38 

3.265 

3.205 

3. 195

1.317

2.985

2.94

2.93

COMBINATION OF COEFFICIENTS FOR WEIRS WITH COMPOUND

SLOPES.

Series 163 for an apron slope 2:1 represents a weir form which would 

be produced by placing, vertical face to vertical face, a weir with back 

slope 1:1 and a weir with apron slope 2:1. For the former, Bazin's 

experiments indicate 10 per cent excess discharge over that for a thinedged weir, and for the latter (from PL V) C= 3.50, equivalent to 5
128 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

per cent excess over a thin-edged weir. If the discharge over the 1:1 

upstream slope was similarly increased by the addition of an apron, 

C would be 3.66X1.05=3.84. PL IX indicates <7=3.82.

The above method of determining the coefficient for weir of irregular cross section by combining the coefficients for two principal elements of which it is composed, as separate weirs, is restricted in its 

application and may lead to inconsistencies.

WEIRS WITH VARYING SLOPE OF UPSTREAM FACE.

Experiments were made by Bazin on thin-edged weirs inclined at 

various angles. Bazin found the ratio of the coefficient of discharge 

to that for a vertical thin-edged weir to be sensibly constant for all 

heads within the limits of his experiments, 0.0 to 1.5 feet. Bazin's 

results were expressed in the form of a modulus by which to multiply 

the coefficient for a vertical weir to obtain that for an inclined weir. 

Assuming the Francis coefficient 3.33 to apply to a vertical weir, the 

coefficients for weirs of various inclinations would be as follows:

Coefficients for inclined weirs, Bazin's experiments.

Upstream inclination of the weir

Vertical weir. ......................

Downstream inclination of the weir. .-

"1 horizontal to 1 vertical... 

2 horizontal to 8 vertical. ..

1 horizontal to 3 vertical. . .

1 horizontal to 3 vertical. . .

2 horizontal to 3 vertical . . .

1 horizontal to 1 vertical . . .

2 horizontal to 1 vertical. . .

4 horizontal to 1 vertical...

Bazin' s 

modulus.

0. 93 

.94

.96 

1.00

1.04

1.07

1.10

1.12

1.09

C

3.097 

3.130

3.197 

3.330

3.463

3.563

3.663

3.996

3.630

On PI. XVI are shown the results of United States Deep Waterways 

experiments on weirs 4.9 feet high, having horizontal crests 0.67 foot 

broad, and with various inclinations of the upstream slope. .The 

experiments cover heads from 1.75 to 5.2 feet, but only 3 or 4 points 

are given on each coefficient curve. The results indicate in a gerieral 

way, however, nearly constant coefficients for each inclination of the 

upstream face. The values of the coefficients are considerably smaller 

than those obtained by Bazin, whose experiments were on weirs 2.46 

feet high with sharp crests.

Pis. X, XI, and XII show the results of experiments of Bazin on 

weirs of irregular section, with various upstream slopes. PI. X 

includes 5 series of experiments on weirs 1.64 feet high, with sharp
WEIRS WITH VARYING UPSTREAM SLOPES. 129

crest angles, and 2 : 1 downstream slopes. The coefficient curves 

show a depression period at from 0.3 to 0.7 foot head, beyond which 

the coefficients may be fairly represented by constants up to 1.5 foot 

head (the limit of the experiment). A general curve showing the 

constant coefficient in terms of a downstream slope or batter has been 

added. This indicates a maximum coefficient of discharge for an 

upstream slope of about 2.6 : 1. Bazin found, for thin-edged weirs, 

with inclined downstream slopes, a maximum coefficient for an inclination of 30°, or If : 1.

Pis. XI and XII show coefficient curves for weirs having the same 

upstream slopes as in PL X, but 2.46 feet high, and with flat crests 

0.67 foot wide. The coefficient curves are convex outward, indicating 

that they may approach constant values at some point beyond the 

limits of the experiments. The marked difference in character of 

these coefficient curves, as compared with those in the preceding 

group, is notable. For weirs with flat crests 0.67 foot wide the 

coefficients for a given head uniformly increase as the slope becomes 

flatter up to a batter of about If : 1. They are also greater for all 

heads within the limit of the experiments than the coefficients for 

weirs with sharp crest angles. The comparative values are indicated 

in the following table:

Comparative coefficients, weirs with varying upstream slope.

Upstream 

slope.

Vert.

*:1

i:l

1 :1

2:1

Pl.X,

sharp crest, 

2 : 1 downstream 

slope; average confident.

C

3.58

3.68

3.72

3.83

3.87

Pis. XI and XII, 0.67 feet

crest width, 2 : 1 downstream slope.

Head, in feet.

0.5

C

2.78

2.87

2.92

3. 03

3.13

1.0

C

3.26

3.34

3.38

3.42

3.43

1.5

C

3.51

3.56

3.62

3.65

3.61

It will be seen that the addition of the flat crest has an effect in this 

case similar to that observed in Pis. VI and VIII, showing the results 

of experiments by Bazin on weirs with various downstream slopes.

United States Deep Water ways series No. 7, PL XVII, may be 

compared with Bazin's series No. 178, shown on PL XI. The former 

gives a coefficient of 3.55 for a head of 2 feet on a weir 4.895 feet 

high, the coefficient slowly increasing with the head. The latter gives 

a coefficient of 3.6 for a head of 1.5 feet, decreasing rapidly as the 

head decreases.

JBB 150 06 13
130 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

United States Geological Survey series No. XXXIX, PL XXVIII, 

and United States Deep Waterways series No. 18, PL XVIII, represent weirs with vertical downstream faces and inclined crests. The 

upstream slope does not, however, extend back to the bottom of the 

channel of approach, but is cut oft' abruptly by a vertical .upstream 

face. The average coefficients deduced from these series have been 

plotted on a general curve on PL XVI, the coefficients agreeing 

closely with those of the United States Deep Waterways experiments 

on weirs of similar upstream slope, extending to the channel bottom. 

United States Geological Survey series No. XXX represents the Dolgeville dam, with rounded crest removed, leaving a trapezoid with crest 

6 feet broad and 1 foot lower at upstream than at downstream edge. 

The coefficient is not constant, but apparently approaches a constant 

value of about 3.25 for heads exceeding 3 feet. United States Deep 

Waterways series No. 18 represents a model of the spillwa}^ of the 

Indian Lake dam, having a crest 7 feet wide, 1.5 feet lower at 

upstream than at downstream edge, which gives an average constant 

coefficient of 3.42.

It is suggested that if the upstream slope of an inclined weir is continued back 6 feet or more and terminates in a vertical upstream face, 

the discharge coefficient will not differ materially from that for an 

upstream slope extending to the channel bottom.

DAMS OF OGEE CROSS SECTION, PLATTSBURG-CHAMBLY TYPE.

The United States Geological Survey experiments on dams of this 

type are shown on Pis. XXIII to XXVII. Cross sections of the 

various dams, with lines indicating the comparative size of the models 

used in the United States Geological Survey experiments, are shown 

on PL XXXIV. Cross sections of other ogee dams used as weirs are 

shown on PL XXXV.

This class includes dams with downstream crest radius sufficiently 

large to retain the nappe always in contact, yet not so large as to simulate a broad flat crest. We thus exclude the Dolgeville section on 

the one hand, in which the nappe as observed in the existing dam partially or completely breaks free near the crest for other than very low 

stages, and on the other hand, the Austin dam, with a crest radius of 

20 feet, which appears, from the meager data available, to lie outside 

this class.

We have arranged the available data in order, advancing with 

decreased breadth and increased inclination of sloping upstream face.

The coefficients for various depths are as follows:
U. 8. GEOLOGICAL SURVEY WATER-SUPPLY PAPER NO. 150 PL. XXXIV

(£)

Smooth stone

COMPARATIVE SIZE OF MODELS AND SECTIONS OF OGEE DAMS.
U. 8. GEOLOGICAL SURVEY

23.'a*

BEAVER RIVER;N. Y.

TRENTON FALLS, N. Y.

HANN AW A FALLS, N.Y.

I.O''-.

Crest, Trenton FalJs

tfonk Falls, N.Y. 

Crest section

CROSS SECTIONS OF OGEE DAMS.
U. S. GEOLOGICAL SURVEY 

Experimental data

Series Symbol ' Crests/ope Crest wi

.U. S. G. S. 3O-3I A P/attsburg (RegJ 47 3 

32-33 e " (Spec) 2-7 3 

" 34 a " '/ /.04:/ 3 

" 35 o Chambfy 4.5-7 4.5

Head- feet

COEFFICIENT DIAGRAM FOR OGEE D
DAMS OF OGEE CROSS SECTION. 131

Comparative coefficients, dams of ogee cross section.

Approx. constant coeffiBreadth of slope, in feet .

Experiment ..............

Head 0. 5 foot ....... 

1.0 foot.......

9 0 fppt

3. 0 feet .......

4. 0 feet .......

5. 0 feet .......

6. 0 feet -.-.-..

/ChainI bly.

3.435 

4.5 

4J:1 

2.0

3.30 

3.42 

3.49 

3.53

--.-,.-

Platteburg.

3.48 

3 

4:1 

3.0 

U.S.G.8. 

«30

3.22 

3.43

3.42 

3.47 

3. 52

Modified Plattsburg.

3.48 

3 

4:1 

3.0 

U.S.G.S. 

631

3.29 

3.35 

3.43 

3.54 

3.62

3.48

0

4:1 

3.0 

Mean of 

30-31

3.36 

3.385 

3.45 

3.53

3.70 

3 

2:1 

3.0

U.S.G.S.

632

3.29 

3.37 

3.51 

3.57 

3.67 

3.73

3.70 

3 

2:1 

3.0

U.S.G.S.

«33

3.22 

3.44

3.67 

3.72 

3.74

3.70 

3 

2:1 

3.0 

Mean of 

32-33

3.255 

3.405 

3.59 

3. 645 

3. 705

3 

1.04 : 1 

3.0 

U.S.G.S. 

34

3.46 

3.75

3.87 

3.88

«15.969 feet crest length, without end contraction. 

6 7.979 feet crest length, one end contraction.

It appears that the rounded crest changes the character of the law 

of coefficient from a value tending toward a constant for each back 

slope to a slowly increasing function of the head. Compared with the 

constant coefficients for weirs with similar upstream slopes extending 

back to canal bottom, and with vertical faces, we find that the constant values deduced for these cases correspond with the values of the 

varying coefficients for ogee sections at a medium head of 2 to 4. feet.

By plotting the data for weirs of ogee section on logarithmic crosssection paper the following convenient approximate formula has been 

deduced, applicable for weirs with 2 or 3 feet crest radius and upstream slopes 3 to 4.5 feet broad. S indicates the batter ratio of the

si one horizontal run hiupe, = ; . 

vertical rise

C= [3.62-0.16(^-1; ..... (71)

If £=2:1 .#=4.0 6^=3.46X4^

log 4^=0.030103 (7=3.46x1.0716=3.70. 

The experiments give £7= 3.74.

EXPERIMENTS ON DISCHARGE OVER ACTUAL DAMS.

On PI. XXXVII are shown the results of a number of experiments 

made by measuring the discharge over existing dams by means of floats 

or current meters. Aside from those for the Austin, Tex., dam, the 

data have been collected by Mr. George T. Nelles. a

"Discussion of paper by G. W. Rafter on the flow of water over dams: Trans. Am. Soc. C. E., vol. 44, 

pp. 359-362.
132 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS. 

BLACKSTONE RIVER AT ALBION, MASS.

This is a timber dam 217 feet long, with horizontal crest 1 foot wide, 

vertical downstream face, and upstream slope covered with riprap. 

Discharge was measured by current meter 500 feet below dam, and the 

depth was measured by hook gage 20 feet upstream from crest. Coefficients have not been corrected to eliminate velocity of approach. 

They illustrate the uncertainty of discharge for broad-crested weirs of 

small width under low heads.

MIJSKINGUM RIVER, OHIO.

Discharge was measured by rod floats in a cross section 500 feet 

above the dams, which are constructed of timber cribs filled with stone.' 

Data by Maj. W. H. Bixby, U. S. Army.

Discharge data for Muskingum River dams.

Number of 

dam.

3 

4

j

8

Length 

on crest, 

in feet.

848 

535

472

515

Mean 

height, in 

feet.

12.6 

15.9

14.2

16.0

Area of 

discharge 

section, in 

square 

feet.

7, 765 

8,360

8,230

7, 330

Discharge, 

in cubic 

feet per 

second.

18, 118 

25, 559

21, 015

22, 310

Mean 

velocity, 

in feet per 

second.

2.333 

3.045

2. 553

3.044

Pall over 

dam, in 

feet.

8.00 

6.70

7.00

5.16

Observed 

depth on 

crest, in 

feet.

2.86 

4.66

4.40

5. 90

Coefficient C.

4.419 

4.723

4.812

3. 015

The depth on crest has not been corrected to eliminate velocity of 

approach.

OTTAWA RIVER DAM, CANADA.

Data by T. C. Clark, C. E. Dam 30 feet high, with upstream and 

downstream faces planked and sloping 3:1, forming sharp crest angle 

at junction.

Discharge data for Ottawa River dam.

Length of 

dam, in feet.

1, 600

1,760

Depth on 

crest, in feet.

2.5

10.0

Discharge, 

in cubic feet 

per second.

26, 000

190, 000

Discharge 

coefficient f.

4.106

3. 408

These data are notable as giving the only authentic value of discharge over a dam under so great a head as 10 feet. The high coefficient found for a head of 2.5 feet renders the results somewhat doubtful.
U. S. GEOLOGICAL SURVEY WATER-SUPPLY PAPER NO. 150 PL. XXXVII

0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 

Corrected head H in feet.

TAYLOR-HOWARD EXPERIMENTS ON DAM AT AUSTIN, TEX

XD=

.2 .4 .6 .8 1.0 

Corrected head H in feet.

EXPERIMENTS OF DWIGHT PORTER, BLACKSTONE RIVER, ALBION, MASS.

r<rr>t}er cribs . stone

~"<&,

(35--

4.0 5.0

Corrected head H in feet.

MUSKINGUM RIVER DAM. DATA BY MAJ. W. H. BIXBY, U. S. A.

EXPERIMENTS TO DETERMINE COEFFICIENT C FROM ACTUAL DAMS.
ROUGHNESS OF CREST. 133

AUSTIN, TEX., T>AM. a

A series of current-meter measurements of the discharge over this 

dam were made in January and March, 1900. Several observations at 

each depth have been combined. The resulting mean coefficients are 

given in the following table:

Discharge coefficients for the Austin, Tex., dam.

Date.

1900.

Jan. 15

Jan. 18

Jan. 26

Mar. 28

Mar. 28 

Average

Number.

1

2

3

4

5

H

1.09

.72

.42

1.44

1.32

H= depth 

at crest 

of dam.

0.838

.625

.33

1.04

.96

Range of variation of C.

From 

3.09

3.00

3.06

3.32

3.26

To 

3.14

3.11

3.13

3.36

3.33

Number of determinations.

4

11

_^

3

5

Average 

value 

of C.

3.132

3. 053

3.112

3.333

3.302

3.186

ROUGHNESS OF CREST.

The models used in weir experiments have usually been constructed 

of planed and matched timber. In actual dams a wide variety of conditions exist, including, in the order of roughness, sheet-steel crests, 

boards smoothed by wear and rendered slippery by water soaking and 

fungus growths, unplaned boards, dressed masonry, formed concrete, 

rubble and undressed ashlar, with earth, cobble, or broken-stone 

approaches. For the determination of the extent, if any, to which 

the coefficient applying .for a smooth-crested dam must be modified to 

apply to any of these conditions, the following data are available.

UNITED STATES DEEP WATERWAYS, SERIES 7 AND 8 (PL. XVIl).

Model dams, 4.9 feet high, 2:1 slope on both faces. The mean coefficients are about 1 per cent greater for crest of planed boards than 

for crest covered with one-fourth-inch mesh wire cloth.

a Taylor, T. U., the Austin dam: Water-Sup, and Irr. Paper No. 40, U. S. Geol. Survey, 1900, p. 33.
134 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

CROTON DAM, ROUND-CREST SECTION, MODEL A (PL. XIX).

Crest rounded, radius 10 feet. Upstream slope about 6:1.

Comparative coefficients with varying roughness, Croton round crest.

Series .....

Head, in 

feet.

0.25

.50

1.00

1.50

2.00

2.50

3.00

l

Smooth-pine 

crest.

3.34

3.24

3.21

3.21

3.21

3.21

3.21

la

Unplanedplank crest 

and slope. o

2.84

2.91

3.04

3.12

3.15

3. 15

2

Brokenstone slope, 

unplaned 

crest.

3.18

3.18

3.19

3.20

3.21

3.22

3.22

3

Brokenstone slope, 

wire cloth 

on crest.

3.16

3.09

3.15

3.15

3.15

3.15

3.15

a This series appears doubtful. R. E. H. 

CROTON DAM, ANGULAR SECTION, MODEL B (PL. XX).

Apron slope 1.25:1, upstream slope 6.2-4:1 for 13 feet, then rough, 

and slope about 4:1 to bottom.

Comparative coefficients, varying roughness, Oroton annular crest.

\ 

x > Series.

Head. ^

\

0.25

.50

1.00

1.50

2.00

2.50

Unplaned 

plank.

3.61

3.63

3.67

3.68

3.70

3.70

Unplaned 

plank, 

rough -stone 

approach.

3.66

3.66

3.66

3.66

3.66

The data given above are somewhat discordant, but indicate that in 

general the decrease in discharge resulting from the roughness of the 

various materials forming the crests and approaches of dams will not 

exceed from 1 to 2 per cent for low heads, and usually decreases as 

the depth of overflow increases.
WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS. 135

FALLS.

Bellasis a presents the following analysis for a fall in which there is 

neither a raised weir nor a lateral reduction in section. If v is the 

mean velocity at CD, near to AB, then <v is both the velocity of 

approach and the velocity in the weir formula

v=-<

FIG. 10. Fall.

where G is a coefficient of velocity.

ac

Making or=1.00 and c=O.T9.

(72)

The depth Dis to be measured so near ^..5 that the water shall have 

acquired its velocity of efflux. The depth will, of course, be affected 

by the surface curve, the upstream extension of which will be longer 

according as the slope of the leading channel is flatter, being very 

great for a horizontal channel. The formula needs experimental verification, but affords a convenient basis of approximation of the flow 

through troughs and sluices and over aprons and falls.

Experimental data for c are needed.

a Hydraulics, p.
136 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

WEIR CURVED IN PLAN.

Milldams of both wood and masonry are often constructed to bow 

upstream, sometimes to secure the added strength of arched form, or to 

secure additional spillway length, or to follow the crest of a favorable 

rock ledge, or to throw the ice-bearing current away from intake 

gates. The dam may follow the arc of a circle, or, as is common with 

timber dams, there may be an abrupt angle in the plan of the dam. 

Fig. 11 shows a graphical comparison of curved and angle dams with a 

straight dam across the same channel, the former being each 13.5 per 

cent longer than the straight created dam.

If such an arched spillway opens out of a broad, deep pond, the discharge over it would be greater than for a straight overfall very nearly 

in proportion to the excess in length of the arc as compared with the 

length of its chord.

When the stream is confined in a restricted channel, the increased 

velocity of approach above the longer spillway will become a factor. 

Thus if two dams one straight, the other arched were placed in the 

same straight, uniform channel, and the depth on crest measured at 

the same distance upstream from each, then, with the same measured 

head on both, the velocity of approach to the arched dam would be

FIG. 11. Weir curved or angular in plan.

greater nearly in the same proportion that its length of crest and discharge are greater than for the straight crest. Properly corrected for 

velocity of approach, the arched dam will give a correct measurement 

of the discharge, the length of the arc being used as the crest length. 

When the length of the arc greatly exceeds the channel width, the 

velocity of approach may become excessive, introducing uncertainty 

as to the proper correction coefficient, difficulty in measuring the head, 

and an uplifting of the central swifter-flowing portion of the stream 

surface.

The circular overflow lip of a vertical artesian-well casing is sometimes used to approximate the flow, the measured depth of water 

above the lip of the pipe, together with its circumference, being used 

in the weir formula. a

a Experiments showing the discharge over a circular weir to be proportional to the length of the 

arc were made by Simpson at Chew Magna, Somersetshire, England, 1850, not recorded in detail.
WEIR EXPERIMENTS, COEFFICIENTS, AND FOBMULAS. 137

SUBMERGED WEIRS.

THEORETICAL FORMULA.

In a "submerged," ''drowned," "in complete, "or " partial" weir the 

water on the downstream side stands above the crest level.

The submerged weir is not extensively used as a device for stream 

gaging. A knowledge of the relations of head, rise, and discharge 

of such weirs is, however, of great importance in works of river 

improvement, canals, etc., and the leading formulas are here presented. 

It may be added that for situations where head can not be sacrificed, 

precluding the use of an ordinary weir, and where the velocity is not 

a continuous function of the depth, as in race ways, making a channelrating curve inapplicable, the use of submerged weirs to measure or 

control the discharge merits consideration. Their use for such purposes as the equable division and distribution of water in power canals 

has hitherto been very restricted, owing to the lack of experimental 

coefficients.

Let Zf=Head on upstream side, corrected for velocity of approach. 

D=Measured head, upstream side of weir. 

d= Measured head, downstream side of weir, or the depth of

drowning, taken below the ressault. 

Z = Difference of elevation, upstream and downstream sides

=H-d.

P Height of weir above channel bottom. 

Z = Length of weir crest, feet. 

'V =Mean velocity of approach.

A = Head on a thin-edged weir that would give the same discharge. 

M' and C' coefficients of discharge for a submerged weir.

FIG. 12. Submerged weir.

The theoretical formula of Dubuat for discharge is obtained by 

regarding the overflow as composed of two portions, one through the 

upper part D d, treated as free discharge, the other through the 

lower part d, treated as flow through a submerged orifice.

Combining the two discharges,

Q=

IBS 150 06 14
138 WEIK EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

By reducing, including- a coefficient, and using the head //corrected 

for velocity of approach, we have the general formula for a submerged 

weir.

Z . . . (73)

The head due to the velocity of retreat should in strictness be subtracted from the depth of submergence d. This is not commonly done, 

however, in the experiments, where the u^ual method of producing the 

submergence is by damming and retarding the water below. In practice, if the velocity of retreat is large, the correction should be made.

The theory of formula (73) makes 6" = 1.5 times the value of the 

coefficient C in the free portion of the discharge." This value is 

adopted by Dubuat and Weisbach.

D'Aubuissou gives <?' = 1.43(7.

Francis's early experiments make 6" = 1.38t Y.

From gage records of large rock-tilled crib dams on Kentucky 

River, having planked upstream slope 3:1 and vertical steps below 

crest height of dams about 20 feet, heads 4 to 7.5 feet, mean 5.3 

feet Nelles found results as follows:

Dam No. 3, water falling slowly 4 days, C' \.5C.

Dam No. 2, water falling slowly 3 days, £' = 1.53 £'.

Dam No. 1, water rising and falling slowly 5 days, C'=- 1.466'.

FTELEY AND STEARNS SUBMERGED-WEIR FORMULAE

Fteley and Stearns use the base formula

Z ...... (74)

Coefficients for the above formula were derived from experiments on 

thin-edged weirs, by Fteley and Stearns and by J. B. Francis, and 

give correct results for weirs for which the free discharge would be 

correctly calculated by the Francis formula.

The head on upstream side varied from 0.3251 to 0.9704 foot, and

TT varied from 0.063 to 0.081 with air under nappe, and from 0.077

to 0.975 with no air under nappe, and in applying the formula the 

same conditions should be complied with. The authors comment that 

where sufficient head can not be obtained for a weir of the usual freedischarge type, a submerged weir may be used, provided that the head 

does not vary greatly.

« See valuable discussion of submerged weirs by Geo. T. Nelles in Trans. Am. Soc. C. E., vol. 44, 

pp. 359-383. *

b Fteley and Stearns, Experiments on the flow of water, etc.: Trans. Am. Soc. C. E., vol. 12, pp. 

101-108.
SUBMERGED WEIKS. 139

Froni a large-scale curve Fteley and Stearns derive the following 

table of coefficient C\ for formula (74):

Fteley and Stearns's coefficients for submerged weirs.

d 

H

0.0

.1

.2

.3

.4

.5

.6

.7

.8

.9

0.00

3.365

3.286

3.214

3.155

3. 113

3.092

3.092

3.122

3.190

0.01

3. 330

3.359

3.278

3.2C7

3.150

3.110

3.091

3.093

3.127

3.200

0.02

3.331

3.352

3.271

3.201

3. 145

3.107

3.090

3.095

3.131

3.209

0.03

3.335

3. 343

3.264

3.194

3.140

3.104

3.090

3. 097

3.137

3.221

0.04

3. 343

3. 335

3. 256

3.188

3.135

3.102

3. 089

3.099

3.143

3.233

0.05

3.360

3.327

3.249

3.182

3.131

3. 100

3.089

3.102

3.150

3.247

0.06

3.368

3.318

3.241

3.176

3.127

3.098

3.089

3.105

3.156

3.262

0.07

3.371

3.310

3.234

3.170

3.123

3.096

3.090

3.109

3.164

3.280

0.08

3.372

3.302

3. 227

3. 165

3.119

3.095

3.090

3.113

3.172

3.300

0.09

3.370

3.294

3. 220

3. 159

3.116

3.093

3.091

3.117

3.181

3.325

Where -^ is less than 0.15 O is not sensibly affected by submergence. Jz

Where -j^ is from 0.5 to 0.8 C may be taken at 3.10.

Correction for velocity of approach was made by the formula

v2 

I£=2)-{-~. No correction was made for velocity of retreat.

. g ' 

The formula is probably applicable to larger darns and greater depths

by selecting proper values of C\ -=, being a relative quantity. Jd

A number of empirical formulae for submerged-weir discharge are 

also used.

CLEMENS HERSCHEL'S FORMULA. a

HerschePs formula, based on experiments of J. B. Francis, 1848, 

Fteley and Stearns, 1877, and J. B. Francis, 1883, is

0=3.33 (75)

In this formula the measured head 6 is reduced to an equivalent head 

that would give the same discharge over a free overflow. The value of

the coefficient N=-fr depends on the proportional submergence-TTaHershel, Clemen, The problem of the submerged weir: Trans. Am. Soc. C. E., vol. 14, May, 1885, 

pp. 190-196.

b Corrected for velocity of approach by method for Francis's formula before applying in above 

formula.
140 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

The values of this ratio, together with their probable error, are 

given below.

Coefficient N, Herschel's submerged-weir formula. «

d

D

0.0

.1

.2

.3

.4

.5

.6

. 7

.8

.9

0.0

1.000

1.005

.985

.959

.929

.892

.846

.787

.703

.574

[0.01

1.004

1.003

.982

.956

.926

.888

.841

.780

.692

.557

0.02

1.006

1.002

.980

.953

.922

.884

.836

.773

.681

.539

0.03

1.006

1.000

.977

.950

.919

.880

.830

.766

.669

.520

0.04

- 

1.007

.998

.975

.947

.915

.875

.824

.758

.656

.498

0.05

1.007

.996

.972

.944

.912

.871

.818

.750

.644

.471

0.06

1.007

.994

.970

.941

.908

.866

.813

.742

.631

.441

0.07

1.006

.992

.967

.938

.904

.861

.806

0.08

1.006

.989

.964

.935

.900

.856

.800

. 732 . 723

. 618 . 604

. -402 . 352

0.09

1.005

.987

.961

.932

.896

.851

.794

.714

.590

.275

a Values for exceeding 0.80 less accurately determined.

^=0.02 to 0.14, variation of jy= ±0.005

=0.15 to 0.22, variation of ^T= ±0.008 

=0.24 to 0,32, variation of ^=±0.012 

=0.33 to 0.41, variation of #"=±0.015 

= 0.42 to 0.59, variation of ^=±0.018 

= 0.60 to 0.65, variation of ^=±0.017 

=0.66 to .071, variation of ^=±0.014 

=0.72 to .084, variation of N= ±0.011

to 0.007.

to 0.010. 

to 0.014. 

to 0.017.

to 0.015. 

to 0.012. 

to 0.009.

This table indicates that for depths of submergence not exceeding 

20 per cent, the head will not ordinarily be increased more than 2 

per cent.

The discharge over a submerged weir, according to Herschel's for3

mula, bears the ratio JV1 to that over an unsubmerged weir under the 

same head.

THE CHANOINE ANJD MARY FORMULA.

(76)

This expression has a form similar to that for the ordinary formula 

for submerged orifices. It is applicable only under conditions identical with those for which M' has been determined. a

« Van Nostrand's Eng. Mag., vol. 34, p. 176.
SUBMERGED WEIRS. 141

R. H. RHIND'S FORMULA."

. . (77)

This may be reduced to the theoretical formula (73) by omitting 

the correction for velocity of approach.

BAZIN'S FORMULAS. 6

By duplicating, with various depths of submergence, his experiments on thin-edged weirs Bazin deduced the following expressions 

for the coefficients for submerged weirs to be applied in the discharge 

formula

Q=m'LD

Let P represent, as heretofore, the height of weir crest above channel bottom, the coefficient in being that which would apply to the 

same weir with free discharge.

(1) Accurate formula with small values of d:

...... (78)

(2) Accurate formula with large values of d:

m'=m [(1.08+0.18 ^ ^f^~\ ..... (79)

(3) Approximate formula for all cases:

05 +0.21 ~\ -\T (80)

The above formulas are for weirs without end contractions.

The coefficient m contains the correction for velocity of approach of 

the free-discharge weir, and m! contains the necessary factor (if any) 

for the resulting modification of the velocity of approach effect, when 

the weir becomes drowned. The}T are only strictly accurate, therefore, when m' is substituted for m in Bazin's formula.

In Bazin's formulas the height P of the weir enters as a controlling 

factor in (1), and is present less prominently in (2) and (3).

The modification by drowning is made to depend on -p- in (2), and

7~i sJ

on this ratio and that of the cube root of ^ jointly in formula (3).

It is often difficult to determine P or to apply these formulas to a 

weir fed by a large pond and having end contractions.

a Proc. Inst. Civil Engineers, 1886.

6 Bazin, H., Experiences nouvelles sur l'6coulement en deversoir, 6.me art., Ann. Fonts et Chauss6es, 

M£moires et Documents, 1898.
142 WEIR EXPERIMENTS, COEFFICIENTS AND FORMULAS.

Assume P-=<x> 

Then (2) becomes

3 /Z

.... (81)

and differs from (3) when similarly reduced only in the substitution 

of 1.05 for 1.08 as a coefficient.

If

Ex. 7>=4' 

m = 0.425

m' = 0.425 XI. 08 I- =0.364,

the discharge being 89.4 per cent of that over an unsubmerged weir 

under the same head.

Comparison of submerged-iceir foi^mulas. f>

d, feet. .......... 

H, feet ..........

din.............

Fteley-Stearns . .

.25 

2.0

i

.50 

2.0

j

.75 

2.0

.25 

1.0

.50 

1.0

.75 

1.0

?

Percentage of nnsubmerged-weir discharge.

99.91 

100. 15 

100.43

95.06 

95. 83 

95. 40

89.29 

90.56 

89.78

95. 01 

95. 83 

95. 40

82.61 

84.24 

83.34

64.02 

64.95 

66. 15

« Weir assumed to be very high so that there is no velocity of approach or of retreat. The coefficient of discharge for a thin-edged weir with free discharge has been taken at 3.33 for the FteleyStearns and Hersohel formulas.

INCREASE OF HEAD BY SUBMERGED WEIRS.

Any of the submerged-weir formulas may be transformed into 

expressions giving the rise in water level caused by the construction 

of a submerged weir in a channel or canal; in this form they are most 

useful in the design of slack-water navigation works.

RANKINE'S FORMULAS/' 

Weir not drowned, with flat or slightly rounded crest:

3 , approximate

Weir drowned:

First approximation 

Second approximation 

(82)

(83)

« Civil Engineering, p. 689.
SUBMERGED WEIRS. 143 

COLONEL DYAS'S FORMULA. 3

This is intended to determine the height of a weir on the crest of a 

fall in an irrigation or other canal to maintain a desired uniform depth 

and slope.

J9=Depth on weir, feet.

^L= Depth of uniform channel, feet.

P X D Height of weir necessary.

^4 = Area uniform channel section, feet.

R= Hydraulic radius, feet.

/$'= Slope or fall in feet, per foot.

Z= Length of weir crest, feet.

-. .... (84) V L J

If .1=1000 X=10' ^=8.33 #=0.001 Z=100, 

D = |-900X

= 9.0856-1.0441 = 8.04 

P= 10 8.04=1.96 feet.

In this case length of weir equals width of channel, and the velocity 

of approach would be the mean velocity, which by Kutter's formula 

will vary, sa\T , from 8 to 10 feet per second under the conditions, 

depending on the value of the coefficient of roughness n. This would 

make the flow in the channel 8,000 to 10,000 cubic feet per second.

As a check on the calculated depth D, it will be found that the flow 

over a weir 100 feet long under a head 8.04 feet (corrected for the 

iarge velocity of approach) will also be from, say, 8,000 to 10,000 

cubic feet per second, depending upon the coefficient used in "the weir 

formula.

SUBMERGED WEIRS OF IRREGULAR SECTION.

For certain forms of irregular weirs having vertical downstream 

faces, the discharge when subject to submergence may probably be 

approximated by applying the ratio of drowned to free discharge for 

a thin-edged weir similarly submerged as a correction to the coefficient for free discharge over the weir in question. For broad-crested 

weirs or weirs with aprons this method probably will not be applicable.

BAZIN'S EXPERIMENTS.

For many of the model weirs of irregular section for which free - 

discharge coefficients were obtained by Bazin, duplicate series of coefficients with various degrees of submergence were also obtained.

a. Wilson, H. M., Irrigation in India: Twelfth Ann. Kept. U. 8. Ueol. Survey, 1890-91, pt. 2, p. 482.
144 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

Many of these data have been reduced to English units by Nelles. a 

Evidently each form of weir section will require a special formula or 

table of coefficients, and little more can be done than to refer to the 

original data for each specific case.

By way of general illustration of the character of submergence effect 

on weirs of irregular section, the writer has deduced the following 

roughly approximate formulas from Bazin's experiments on triangular 

weirs with vertical upstream faces and sloping aprons. The weirs 

were 2.46 feet high and the end contractions were suppressed. Coefficient curves for free discharge are given on PL V. 

Three series are included:

Series 195, batter of face 1 :1.

Series 196, batter of face 2:1.

Series 19T, batter of face 5 :1.

Experiments in which the proportional submergence -^ was nearly 

the same were grouped, and the average values of A, J9, and d were 

determined. From these the mean values of -=- and -= were computed 

and platted and a straight-line formula deduced.

(85)

1=0.08+0.17,5. 

The initial effect occurs when

d 0.17,5-0.20 D~ 17,5+0.08 

In the above formulas A is the measured head on a weir with free 

overflow, having the same form of cross section, that would give the 

same discharge. D is the depth on the submerged weir, d is the depth 

of submergence, and B is the batter or slope of the apron.

DATA CONCERNING EAST INDIAN WEIRS.

The following data compiled by Nelles 6 are derived from observations on actual dams under heads unusually great. The calculated 

coefficients in the ordinary weir formula (a)

in the theoretical submerged-weir formula

*Z)

and in the Rhind formula (77) are given in columns 14, 13, and 12, 

respectively (p. 145), the observed head being corrected for velocity of 

approach.

aTrans. Am. Soc. C. E., vol. 44, pp. 359-383. 6 Loc. cit.
Data of submerged flow at certain large masonry da

Name of river.

1

Brahmini ..... 

Byturnee ......

Do......... 

Brahmini ..... 

Manhanuddy.. 

Do.........

Do.........

Name of weir.

2

T>Q * f I Q

Byturnee.....

Burrah .......

Brahmini .... 

Kajooree ..... 

Manhanuddy.

Make-up of weir.

Description of 

parts.

3

[Total overfall .... 

fSeines ............

iMain bodv ....... 

(Total overfall ....

[Total overfall .....

(Shore sluices ..... 

I Center sluices ....

[Total overfall ....

iPiers .............

IMain body ....... 

[Total overfall ....

Length 

of 

parts, 

in feet.

L

4

665 

60 

9 

100 

14 

912 

1,026 

100 

14 

412 

526 

389 

140 

3,471 

4,000 

90 

324 

3,433 

 3,847

190 

460 

5,696 

6,346 

255 

118 

1,607 

1,980

Depth of crest 

below water 

surface in 

Upper 

pool, 

feet.

1)

5

19.21 

8.21 

2.21 

18.10 

7.68 

10.84

18.60 

8.18 

11.34

18.02 

8.02 

10.52

30. 675 

25. 615 

20. 115

17.45 

17.20 

11.45

13.30 

4.30 

8.30

Lower 

pool, 

feet.

d

6

18.35 

7.35 

1.35 

15.29

4.87 

8.03

15.20

4.78 

7.94

17.55 

7.55 

10.05

30.15 

25.09 

19.59

15.45 

15.20 

9.45

12.10 

3.10 

7.10

Observed 

fall over 

weir, feet.

D-d=z

1

1 0.86 

I 2.81

I 3.40

i 0.47

1 0. 525

i 2.00

1 1.20

Mean 

surface 

velocity of 

approach, 

in feet 

per 

second.

V

8

7.84

1.13

6.99

11.62

7.74

6.65

Correspond - 

ingveloeitv of" 

head, 

feet.

i»

tg

9

0.96

8.51

0.76

2.10

0.93

0.68

Calc

disch

cubic 

se

For 

bod

f... ..

1 

V"" "

(""""
146 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS. 

UNITED STATES DEEP WATERWAYS EXPERIMENTS. a

These experiments were made in 1899 at Cornell University hydraulic laboratory on a model having completely rounded profile, being a 

design for a submerged dam for regulation of Lake Erie.

The coefficient curve for free discharge is given on PI. XVI. The 

absolute coefficients and the relative discharge with various degrees of 

submergence are shown below. The Francis formula is used.

Q= C'LH*.

Absolute coefficients.

D

Feet.

0.0

.1

. 2

.3

.4

 .5

.6

.7

.8

.9

Submergence from 

backwater.

Feet:

0.00

.66

1.32

1.98

2.64

3. 30

3.99

4.62

5.28

5.94

c"

3.70

3.67

3.64

3.60

3.54

3.47

3. 36

3.17

2.88

2.30

Relative coefficients, United Stales Deep Waterways submerged-weir model.

d 

H

0.0

.1

.2

.3

.4

C" 

G

1.000

.991

.983

.972

.956

/; 

H

0.5

.6

.7

.8

.9

1.0

C'

T!

0.937

.907

.856

.778

.621

C is the coefficient for free discharge over a similar weir under the 

same head.

WEIR DISCHARGE UNDER VARYING HEAD.

Problems of weir discharge under varying head occur in the design 

of storage reservoirs for river regulation, and in determining the maximum discharge of streams.

a Kept. U. S. Board of Engineers on Deep Waterways, pt. 1, p. 291.
WEIR DISCHABGE TINDER VAKYTNO HEAD. 147

An effort has been made in the present chapter to record the various 

working formulas resulting from the solution of this mathematically 

difficult portion of the theory of the weir, and to give numerical data 

to facilitate calculations.

It is assumed that there is no velocity of approach, or, if any, that the 

head has been corrected therefor. The weir coefficient is also assumed 

to continue constant through the range of variation of the head. 

Notation :

T =Time in seconds required for the head to change between

two assigned values. 

H0 Initial depth on weir, feet. 

Ht ~ Depth on weir at the time t. 

S = Reservoir surface area, square feet. 

L = Length of overflow weir, feet. 

I = Rate of inflow to reservoir, cubic feet per second. 

Q =Rate of outflow at time t.

PRISMATIC RESERVOIR, NO INFLOW, TIME REQUIRED TO LOWER 

WATER SURFACE FROM H0 TO Ht.»

fjQ= CLH**dt= - Stiff

<?t=--^< 

OLH*

Where

T= co, when jETf=0. 

If ^=1,000,000, If0 --=± ^=0.1, ^=3.33, and Z=100,

To lower the reservoir from Jf=± to 77=1 would require 3,000 

seconds.

APPROXIMATE TIME OF LOWERING PRISMATIC OR NONPRISMATIC RESERVOIR.

Choosing small successive values of H0 Ht, we may solve this 

problem approximately, as shown in the following table:

/ TT __ 77" \ O

Time required to lower reservoir from H0 to jEft= |^ i-Q~ (88)

«Des Ingenieurs Tasehenbuch, I, 1902, p. 230.
148 WEIE EXPERIMENTS, COEB'FICIENTS, AND FORMULAS.

We may take the mean discharge between the narrow limits H0 and

(89)

or, using the average head,

In the following example we have used the latter value, and have 

made Jf0 ITt=Q.5 foot. A similar solution may be made for a noncv u O1 prismatic reservoir, using successive values of ^~L^ as the reservoir

Zi

area, and determining the increments of Tby formula (88). 

Example of varying discharge.

Ho

4.0

3.5

3.0

2.5

2.0

1.5

Ht

3.5

3.0

2.5

2.0

1.5

1.0

Average j-f

3. 75

3. 25

2.75

2.25

1.75

1.25

Q per second.

2,417.0

1, 951. 0

1, 519. 0

1,124.0

771.0

465.4

1000

^

0. 4137

. 5126

.6580

.8970

1. 2970

2. 1500

Tfor increment 

H0 -Ht.

207

256

330

448

650

1,070

Total T, in 

seconds.

207

463

793

1,241

1,891

2,961

The total time required in seconds is 2,961, as compared with 3,000 

by formula (87).

The time required, using the average Q instead of the average H\\\ 

the calculation, that is, using formula (89) instead of (90), is 2,933.5 

seconds.

The time T is directly proportional to the area of storage surface 

and inversely proportional to the length of spillway. It is also usually proportional to the value of C in the weir formula.

RESERVOIR PRISMATIC, WITH UNIFORM INFLOW.a 

GENERAL FORMULAS.

Starting with reservoir full to crest level, 7^=0, to find the time 

required for the depth of overflow to reach a given sta'ge, fl/.

aMullins, Lieut. Gen. J., Irrigation Manual, Madras Govt., 1890, App. V, pp. 214-223.
WE1E DISCHARGE UNDER VARYING HEAD. 149

When individual values of the increment H^ H^ are small, not over 

0.5 foot each, if successive values are taken, we have approximately:

t time required to rise through the increment HZ H^ 

A summation of the successive values of t required for the water to

rise each increment will give the total time of rise from H0 to Ht.

Formula (92) will give the maximum run-off from a catchment area

tributary to a reservoir if two successive values of JjTand the corresponding value of t are known. 

Formula (92) may also be used to determine T for a nonprismatic

reservoir with a variable rate of inflow by choosing such increments,

HZ H^ that the average values of $, /, and Q will be nearly correct.

Variations in the weir coefficient G may also be considered.

FORMULAS FOR TIME OF RISE TO ANY HEAD H, PRISMATIC RESERVOIR 

WITH UNIFORM INFLOW.

Several analytical solutions of this problem have been made. Starting at spillway level, let Hu equal the depth of overflow corresponding to the quantity of inflow I. The problem is stated by the following differential equation whose primitive is required:

(Rate inflow rate outflow) dt=d (increase in storage), or

...... (93)

In the solution, mathematical substitutions are necessary in order to 

render the time-outflow equation integrable in known forms. A very 

clear demonstration for a special value of Ch&s been given by Frizell. 05 

By modifying FrizelFs formula to adapt it to the use of any value of 

the weir formula, the following equation is obtained:

ZCLb T fl -. _! -. _t

-- ^=nat. log V £Zr-+V3 tan y__V3 tan -=- (94)

where l 

When H H^ the second member becomes the sum of an infinite 

and two finite quantities, Tis then infinite, and the outflow can never

"Water Power, pp. 200-203.
150 WEIE EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

become equal to the inflow, or 11 can never equal /7a , which quantity 

it approaches as a limit as T increases. Frizell places H rH^ r 

having any value less than unity, and, being very nearly unity, Jr 

will be more nearly so, and is taken as equal to unity, without great 

error, enabling the two inverse trigonometric constants to be evaluated 

in terms of arc, giving finally:

nat. log

J\r

0.88625 } 

J

(95)

Nat. log 2V= 2.302585 Iog10

E. Ludlow Gould" gives the following formula, identical with the 

above except in the form of the constant of integration:

 log

-V3

H K *

' -77^ as before.

(i+^H[l . . . . \z y b i

(96)

Gould does not consider ^jr constant, but derives

the values of the function in brackets for various values of /', from 

which the following table has been derived:

>=J nat. log ^ . (97)

Values of (£>, Gould's formula.

x

Ha

0.0 

 1

.2

g

.4

.5

.6

.7

.8

.9

0

0. 0000 

.1532

.3137

.4865

.6747

.8876

1. 1489

1. 4792

1.9141

2. 6129

l

0. 0153 

. 16854

. 3301

.5047

.6960

. 9137'

1. 1750

1. 5145

1.9658

2. 7681

2

0. 0306 

.1838

.3464

.5229

3

0. 0459 

.1992

.3628

.5411

. 7173 . 7386

.9399

1. 2012

1. 5498

2. 0176

2. 9233

.9660

1. 2322

1. 5851

2. 0715

3. 0785

4

0. 0613 

.2155

.3791

.5593

.7598

.9921

1. 2674

1. 6203

2. 1488

3. 2347

5

0. 0766 

.2319

.3955

.5775

.7811

1. 0183

1. 3027

1. 6556

2. 2262

3. 3889

6 , 7

0. 0919 

.2483

.4137

.5957

.8024

1.0444

1. 3380

1. 7073

2. 3035

3. 5441

0. 1072 

.2646

.4319

.6139

.8237

1. 0705

1. 3733

1. 7590

2. 3808

4. 0096

8

0. 1226 

.2810

.4501

.6321

.8450

1.0966

1. 4086

1.8107

2. 4582

4. 4750

Q

0. 13788 

.2973

.4683

.6534

.8663

1. 1128

1. 4439

1. 8624

2. 5355

4. 9405

« Engineering News, Dec. 5,1901, pp. 430-431.
clent

a

1 H \

\Ha)

<t

y

Y

y

'

/

,

/

/

y

/

/

/

^

A

\

/Zf\

^ \H 1 , uy

/

3

/

C

/

<U\

/

^X"

0^

IT

Ha

.1 

.2 

.3 

.4 

.5 

.6 

. 7 

.8 

.9

e(|4

s**'

**~

^r"l

" i p.f e= 1 (sin ;c) 

Jo

(H\*

WJ

0.1777 

.299 

.4053

.594

.6817 

.7653 

.8459 

.9240

^^^^

1 

" sec x dx .r

x

Arc.

0. 1786 

. 3037 

.4174

.750 

.871 

1.008 

1.178

Angle.

o i n 

10 14 00 

17 24 00 

23 25 00

42 58 30 

49 56 00 

57 46 00 

67 31 00

NOTE. 

-

1 1 1

,*, 

0.562 

.669 

.740

.880 

.914 

.946 

.974

Values plotted on 

1 1 1 1 

Corrected head H in feet. 

'DIAGRAM OF VARIABLE DISCHARGE.
WEIR DISCHARGE UNDER VARYING HEAD.

We may write formula (9)

dward suggests

T ±$

3^6Y2Z2 / 

t>

x^sm-1 y CL

\BO,J 3%j c'

the formula a

CX -L I (sin X)3 

lo

^l =sin-1 ^

v^

sec JTrfX

F/A*

T )

*aJ

151

(98)

where

This, like the preceding expressions, becomes infinity when the

integral is carried over the entire range JL = O to X-=-, conforming

2i

with the physical conditions.

The writer has evaluated this functioii for finite values of -^

Ma 

by mechanical quadrature, as shown in thd diagram, PI. XXXVIII.

The diagram illustrates the rapid rise until 

Ha is attained, occupying a comparatively

a, head closely approaching 

short time interval, while

for further increments of head the time pterval is relatively very 

great.

E. Sherman Gould b gives the same integral developed as an infinite 

series

 (ioo)

where

 (101)

a Engineering News, December 5,1901, | p. 431. 

&Engineering News, November 14, 1901 pp. 362-363.

IBS 150 06 15
152 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

If we write

then Frizell's formula may be written T=Fxp ( -jr )

E. L. Gould's formula may be written T=Fx <j> ( ~rr )

Yg£< (102)

Woodward's formula may be written T='2l<ty ( TT )

\-HaS

///\ E. Sherman Gould's formula may be written T fx ^ (~fT )

The formulas are therefore identical, the transcendental factors 

bearing the relation,

The E. L. Gould, Woodward, and E. 8. Gould formulas are applicable for any value of the ratio ^ . That of Frizell can be strictly -"«

applied only when 77 is nearly unity. In the E. S. Gould formula

^a

(

77"\ 

- converges very slowly as the argument approaches unity.

For rough calculations E. S. Gould gives the rule

where /* is the coefficient in the weir formula for reducing final 

head to mean head.

T= ....... (103)

The ratio /* of the constant mean head which would give the total 

discharge SHin the time The finds by trial. 

E. S. Gould gives the values

>u=0.67 for small values of If 

to >u=0.75 for large values of If.

Comparing the formulas,

Let 8= 1,000,000 square feet

(7=3.33=^ 

o

7=10,000 cubic feet per second 

/=-V=30*=:9.655 feet.
WEIE DISCHARGE UNDER VARYING HEAD. 153

Required the time to rise to a height H~0.9II 8.6895 feet.

Frizell (95) F=1677.6 seconds. 

E. L. Gould (96) f=1681.5 seconds. 

Woodward (99) ^=1660.2 seconds. 

E. S. Gould (approximate) (103) ^=1488.3 seconds.

The difference in the value of T by the first three formulas represents the difference in the values of the transcendental portions of the 

equations as evaluated by different methods.

The time required to rise from Ha to Ifb will be the difference of 

the times T\ and T2 by the above formulas!

NONPRISMATIC RESERVOIR, UNIFORM INFLOW.

P. P. L. oCGNNELIi.

Representing the reservoir by a cone having its apex at distance A, 

below plane of the overflow,

Area at overflow level =/S0 = 7r( 

Area at any other level =S=7t\c

I

(104)

where a is the slope of the sides, or where there is OL foot horizontal 

run to 1 foot vertical rise. From (104) wi^h £0 and a given, A may 

be determined.

Where the factor

. . (105)
154 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

E. L. GOTJLD.*

Calling I the angle of inclination of the banks, P0 the perimeter, at 

spillway level, exclusive of overflow,

* where B= P cot i

. = ?r 'cot i'2 r=-rf

(so(CL)*+B(ICLf\ nat. log Vl±VH± 

IV / 1 yr

tan"

V3

nat. log (106)

For /=90° and B=Q, the above formula reduces to (96), the equation for a prismatic reservoir.

VARIABLE INFLOW, NONPRISMATIC RESERVOIR.

This problem may be solved by dividing the reservoir into successive 

levels, and solving by the formulas previously given, as if each layer 

represented a portion of a* reservoir with a constant inflow equal to 

the average rate, or if the formulas for prismatic reservoir .are used, 

then each layer will be supposed to represent a portion of a prismatic 

reservoir of area equal to the average area of the laj^er.

Mullius's formula may often be more conveniently used and a better 

solution be obtained than by attempting to average the area and inflow, 

as would be necessary to apply the analytical formulas given.

The general differential equation for rise in time Twith a variable 

inflow and reservoir area is

(I-Q)dT=SdH ....... (107)

If we can express /as a function of T, and S and Q as functions of 

H, and integrate between the limits ZT=0, H H^ we may obtain an 

equation between H and T similar to those given for prismatic reservoirs with constant inflow.

We may write the ordinary weir formula,

Q= CLH*.
WEIR DISCHARGE UNDER VARYING HEAD. 155

The area /S can usually be readily expressed in terms of the area at 

crest level and slope of the reservoir sides (assumed constant within 

the narrow limits 0, If)', the inflow I of ten increases nearly as a linear 

function of T7 while a stream is rising rapidly; we have, then,

Qt= C

st 

rt

Substituting in (107)

(I0+fT-CLH^dT=(S0+ZajS0H+a?H^dH . (108)

The complete primitive of this differential equation can be determined only as an infinite series."

Rivers during flood usually rise rapidly and fall slowly. The timeinflow function can sometimes be approximated by a modified sinusoid.

m(l>t ...... (109)

where n=or>l

T Total duration of flood. 

^= Maximum rate of inflow.

:7^=Time elapsed from beginning of rise to maximum. 

The constants are so chosen that the arc value of the duration of 

the flood from stage I0 through to the same stage is TT, or,

(U^=7t 5=^ ...... (110)

For the maximum we will have, differentiating (109),

orprj^f . . . (Ill)

T

or = w

log . . . (112)

common logarithms being used. 

If 7=1000 and if Tm=20Q, then rc=3.322 log 5=2.322,

7T2-322 1

5= -=0.0143, and -=0.43066 1UUU fl

aSeddon, James A., C. E. (Proc. Am. Soc. C. E., vol. 24, June, 1S98, pp. 559-598), has solved equation 

(108) for the Great Lakes reservoir system, assuming an annual cycle following the law I=Im+A 

sin T, Im being the mean inflow and Tthe time arc on a circle whose circumference represents one 

year.' He also assumes Q= Q^+bH, or a linear function of the height H.
156 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

Example of variable food discharge computed by formula, (109}.

t, in seconds.

100

200

300

400

500

600

700

800

900

(W)

1.43

2.86

4.29

'5.72

7.15

8.58

10.01

11.44

12.87

log (hi)

,

0. 155336

. 456366

. 632457

. 757396

. 854306

. 933487

1. 000434

1. 058426

1. 109578

i

log (bt) "

0. 06695

. 196694

. 27259

. 32644

. 36820

. 40233

. 43100

. 45618

. 47790

1 

(W)"

1. 1667

1. 573 '

1. 8732

2. 1206

2. 3346

2. 5254

2. 6978

2. 8588

3. 0054

Angle.

0 /

66 55

90 00

107 21

121 08

133 49

144 42

155 00

163 52

172 12

i 

sin (bt) n

0. 9199

1.0000

.9938

.8560

.7216

.5779

.3907

.2779

.1320

The form of the graph of the flood may be determined by plotting 

the quantities in the last column of this table in terms of t. The 

resulting curve rises rapidly to a maximum when £=200, after which 

it descends slowly.

TABLES FOB CALCULATIONS OF WEIR DISCHARGE.

The investigations at Cornell University have greatly extended the 

limit for which weir coefficients are definitely known. The experiments of jBazin did not reach beyond 1.8 feet head maximum. The 

tables of Francis for thin-edged weirs extended to a head of 3 feet.

The experiments at Cornell have furnished the coefficients for a 

variety of weir forms for heads up to 4, 5, and 6 feet. At such heads 

the nappe form has become stable for nearty all forms of weirs. We 

may now predict the probable extension of the coefficient curves for 

higher heads with more confidence than could be done by starting 

from a lower datum.

Owing to their usefulness in the approximate determination of flood 

discharges, the weir tables have been carried up to a head of 10 feet.

In the tables here given the head is uniformly expressed in feet. 

For computing the flow over irrigation modules and other small weirs 

where the head is measured in inches, weir tables expressed with the 

inch as the argument of head are convenient. Numerous tables of 

this character are available. The following may be referred to:

The Emerson weir tables, computed by Charla A. Adams, pages 251-285 of Emerson's Hydrodynamics, published by J. and W. Jolly, Holyoke, Mass. These give 

discharge in cubic feet per minute for weirs with two end contractions having lengths 

of 2, 3, 4, 5, 6, 7, 8,10,12,16, and 20 feet. The discharge is computed by the Francis 

formula for heads from 0.001 foot to 2 feet, advancing by thousandths of a foot, with 

auxiliary table of decimal equivalents of fractional parts of inches.
TABLES FOB CALCULATING WEJE DISCHABGE. 157

The Measurement and Division of Water, Bulletin No. 27, Agricultural Experiment Station, Fort Coiling, Colo. This publication gives tables of discharge in cubic 

feet per second, computed by the Francis formula, for a weir 1 foot long, for heads 

in inches and sixteenths, from TV inch to 30 inches, with auxiliary table for end 

contractions, and for velocity of approach correction by the Fteley and Stearns 

rule (H=D-\-^h). A similar weir table for a weir 1 inch long is given. Also a table 

of discharge for Cippoletti weirs (C=3.36f), for lengths of crest sill of 1, 1.5, 2, 3, 4, 

5, and 10 feet. Head in inches and decimals with feet equivalents.

Special Instructions to Watermasters as to Measurements of Water, State Engineer's 

Office, Salt Lake City, Utah, 1896. Table of discharge, in cubic feet per second, for 

1-foot crest, based on the Francis formula, with auxiliary table for end contractions 

and velocity of approach. The head is expressed in inches and thirty-seconds (with 

equivalents in feet) for g^ inch to 36 inches. A similar table for heads in inches and 

sixteenths, from -^ to 36 inches, gives the discharge in cubic feet per second by the 

Francis formula for weirs with two end contractions and for the crest lengths of 1, 1J, 

2, 2£, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12 feet. A table for trapezoidal weirs (C=3.367) 

of various crest lengths is also given.

California Hydrography, by J. B. Lippincott, Water-Supply Paper No. 81, United 

States Geological Survey. This publication contains a table of weir discharge in 

cubic feet per second for heads, advancing by sixteenths, from ^ inch to 10 inches 

(with equivalent decimals of a foot), for weirs with two end contractions having 

crest lengths as follows: 4, 6, 9, 12, 15, and 18 inches, 2, 2.5, 3, 3.5, 4, 4.5, 5, 6, 7, 8, 

9, 10, 12, 14, 16, 18, and 20 feet. Based on the Francis formula. Also published as 

a circular.

The tables that follow are all original computations, with exception 

of the " Francis weir tables," page 162, and the table of head due to 

various velocities, page 158.

TABLE i. HEAD DUE TO VARIOUS VELOCITIES.«

This table gives values of the expression

based on the constant of gravity for the latitude and altitude of Lowell,

Mass.,

0=32.1618, i= 0.01554639. 2l7

a Francis, Lowell Hydraulic Experiments, extended.
158 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

j2

TABLE 1. Values of h= , or heads due to velocities from 0 to 4-99 feet per second.

V

0.0

.1

.2

.3

.4

.5

.6

.7

.8

.9

1.0

.1

.2

.3

.4

.5

.6

.7

.8

.9

2.0

.1

.2

.3

.4

.5

.6

.7

.8

.9

3.0

.1

.2

.3

.4'

.5

.6

.7

.8

.9

4.0

.1

.2

.3

.4

.5

.6

.7

.8

.9

0.00

0.0000

.0002

.0006

.0014

.0025

.0039

.0056

.0076

.0099

.0126

0.0155

.0188

.0224

.0263

.0305

.0350

.0398

.0449

.0504

.0561

0. 0622

.0686

.0752

.0822

.0895

.0972

.1051

.1133

.1219

.1307

0. 1399

.1494

.1592

.1693

.1797

.1904

.2015

.2128

.2245

.2365

0. 2487

.2613

.2742

.2875

.3010

.3148

.3290

.3434

.3582

.3733

0.01

0. 0000

.0002

.0007

.0015

.0026

.0040

.0058

.0078

.0102

.0129

0. 0159

.0192

.0228

.0267

.0309

.0354

.0403

.0455

.0509

.0567

0. 0628

.0692

.0759

.0830

.0903

.0979

.1059

.1142

.1228

.1316

0. 1409

.1504

.1602

.1703

.1808

.1915

.2026

.2140

.2257

.2377

0.2500

.2626

.2755

.2888

.3023

.3162

.3304

.3449

.3597

.3748

0.02

0.0000

.0002

.0008

.0016

.0027

.0042

.0060

.0081

.0105

.0132

0. 0162

.0195

.0231

.0271

.0313

.0359

.0408

.0460

.0515

.0573

0. 0634

.0699

.0766

.0837

.0910

.0987

.1067

.1150

.1236

.1326

0. 1418

.1513

.1612

.1714

.1818

.1926

.2037

. 2151

.2269

.2389

0. 2512

. 2639

.2769

.2901

.3037

.3176

0.03

0.0000

.0003

.0008

.0017

.0029

.0044

.0062

.0083

.0107

.0134

0. 0165

.0199

.0235

.0275

.0318

.0364

.0413

.0465

.0521

.0579

0.0641

.0705

.0773

.0844

.0918

.0995

.1075

.1159

.1245

.1335

0. 1427

.1523

.1622

.1724

.1829

.1937

.2049

.2163

.2280

.2401

0. 2525

.2652

.2782

.2915

.3051

.3190

.3318 .3333

.3463

.3612

.3763

.3478

.3627

.3779

0.04

0.0000

.0003

.0009

.0018

.0030

.0045

.0064

.0086

.0110

.0137

0. 0168

.0202

.0239

.0279

.0322

.0369

.0418

.0471

.0526

.0585

0.0647

.0712

.0780

.0851

.0926

.1003

.1084

.1167

.1254

.1344

0. 1437

.1533

.1632

.1734

.1840

.1948

.2060

.2175

.2292

.2413

0. 2537

.2665

.2795

.2928

.3065

.3204

.3347

.3493

.3642

.3794

0.05

0.0000

.0003

.0010

.0019

.0031

.0047

.0066

.0087

.0112

.0140

0. 0171

.0206

.0243

.0283

.0327

.0374

.0423

.0476

.0532

.0591

0.0653

.0719

.0787

.0859

.0933

.1011

.1092

.1176

.1263

.1353

0.1446

.1543

.1642

.1745

.1850

.1959

.2071

.2186

.2304

.2426

0. 2550

.2677

.2808

. 2942

.3079

.3218

.3362

.3508

.3657

.3809

0.06

0.0001

.0004

.0011

.0020

.0033

.0049

.0068

.0090

.0115

.0143

0. 0175

.0209

.0247

.0288

.0331

.0378

.0428

.0482

.0538

.0597

0. 0660

.0725

,0794

,0866

.0941

.1019

.1100

.1184

.1272

.1362

0.1456

.1E52

.1652

.1755

.1861

.1970

.2083

.2198

.2316

.2438

0.2563

.2690

.2821

.2955

.3092

.3233

.3376

.3522

.3672

.3825

0.07

0.0001

.0004

.0011

.0021

.0034

.0051

.0070

.0092

.0118

.0146

0.08

0.0001

 0005

"0012

 0022

 0036

 0052

 0072

.0095

.0120

.0149

0. 0178 0. 0181

.02J3

.0251

.0292

.0336

.0383

.0434

.0487

.0544

.0603

0.0666

.0732

.0801

.0873

.0948

.1027

.1108

.1193

.1281

.1373

0. 1465

.1562

.1662

.1766

.1872

.1981

.2094

.2210

.2328

.2450

- 0. 2575

.2703

.2835

.2969

.3106

.3247

.3390

.3537

.3687

.3840

.0216

.0255

.0296

.0341

.0388

.0439

.0493

.0549

.0609

0. 0673

.0739

.0808

.0881

.0956

.1035

.1117

.1201

.1289

.1381

0. 1475

. 1572

.1673

.1776

.1883

.1992

.2105

.2221

.2340

.2463

0.2588

.2716

.2848

.2982

.3120

.3261

.3405

.3552

.3702

.3856

0.09

0.0001

.0006

.0013

.0024

.0037

.0054

.0074

.0097

.0123

.0152

0.0185

.0220

.0259

.0300

.0345

.0393

.0444

.0498

.0555

.0616

0,0679

.0746

.0815

.0888

.0964

.1043

.1125

.1210

.1298

.1390

0.1484

.1582

.1683

.1787

.1894

.2004

.2117

.2233

.2352

.2475

0. 2601

.2729

.2861

.2996

.3134

.3275

.3420

.3567

.3717

.3871
TABLES FOE CALCULATING WEIE DISCHAEGE. 159

This value will suffice in ordinal'}' corrections for velocity of 

approach for localities in the United States.

Velocity of approach correction.

Francis, and as used in portions of this paper (approximate) .......... H D-\-h

Fteley and Stearns, contracted weir ................................. H=D^-l. 5h

Hamilton Smith, suppressed weir ................................... ff D^-l^h

Hamilton Smith, contracted weir. ................................... H=D-\-~LAh

TABLE 2. PERCENTAGE INCREASE IN DISCHARGE BY VARIOUS 

RATES OF VELOCITY OF APPROACH.

This table has been calculated from the Francis correction formula,

The percentage increase in discharge over that at the same measured head with no velocity of approach is

jyt _ 2$ 

Percentage =1UO 3 =K .... (113)
TABLE 2. Percentage increase in discharge over weirs for various rat

D=measured head in feet; .H=head corrected for velocity of approach; t>=mean velocity of app

V

h

A*

D

0.2

.4

.6

.8

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

7.0

7.5

8.0

8.5

9.0

9.5

10.0

0.4

0.0025

0.0002

0.5

0.0039

0.0003

0.6

0. 0056

0. 0005

0.7

0.0076

0.0007

0.8

0. 0099

0. 0010

0.9

0. 0126

0.0014

1.0

0.0155

0.0019

1.1

0.0188

0.0025

1.2

0.0224

0.0033

1.3

0. 0263

0. 0041

1.4

0.0305

0.0051

1.5

0. 03

0. 00

l~i H

Percentage increase in discharge=100 x - 

L ^

1.36

.68

.42

.40

.36

.24

.19

.18

.12

.10

.09

.08

.07

.06

.06

.06

.05

.04

.04

.04

.04

.04

.04

2.65

1.34

.90

.62

.56

.37

.29

.22

.19

.16

.14

.12

.11

.10

.09

.08

.08

.07

.07

.06

.06

.06

.05

3.72

1.90

1.29

.92

.79

.53

.41

.31

.27

.23

.20

.18

.16

.14

.13

.12

.11

.10

.10

.09

.09

.08

.08

5.02

2.58

1.74

1.26

1.07

.72

.59

.43

.36

.31

.27

.24

.22

.20

.17

.17

.15

.14

.14

.13

.12

.11

.10

6.44

3.33

2.22

1.62

1.40

.94

.72

.56

.47

.40

.35

.31

.27

.27

.23

.22

.20

.18

.18

.16

.16

.15

.14

8.16

4.20

2.86

2.11

1.76

1.19

.91

.71

.60

.47

.45

.40

.35

.33

.30

.28

.26

.24

.23

.21

.20

.19

.19

9.77

5.11

3.44

2.66

2.15

1.45

1.11

.88

.74

.63

.55

.49

.44

.40

.37

.34

.32

.29

.28

.26

.25

.23

.23

12.19

6.16

4.14

3.12

2.59

1.75

1.34

1.06

.89

.76

.67

.60

.54

.49

.45

.41

.39

.36

.34

.31

.30

.28

.27

14.13

7.20

4.88

3.70

3.05

2.07

1.58

1.25

1.06

.90

.80

.71

.65

.58

.53

.49

.46

.43

.40

.38

.36

.34

.31

16.34

8.39

5.69

4.33

3.56

2.43

1.84

1.47

1.24

1.06

.93

.83

.75

.68

.63

.58

.54

.50

.47

.44

.42

.40

.37

18.62

9.62

6.56

4.98

4.10

2.79

2.13

1.70

1.43

1,23

1.08

.96

.83

.78

.72

.67

.63

.58

.55

.51

.48

.46

.44

20.

10.

7.4

5.6

4.6

3.2

2.4

1.9

1.6

1.4

1.2

1.

.9

.9

.8

.7

.7

.6

.6

.5

.5

.5

.5
V

h

A«

D

0.2

.4

.6

.8

1.0

1.6

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

7.0

7.5

8.0

8.5

9.0

9.5

10.0

2.3 

0. 0822 

0.0235

2.4 

0. 0895 

0.0268

2.5 

0.0972 

0. 0303

2.6 

0. 1051 

0. 0340

2.7 

0.1133 

0. 0381

2.8 

0. 1219 

0. 0426

2.9 

0. 1307 

0. 0472

3.0 

0. 1399 

0.0524

3.1 

0. 1494 

0. 0576

3.2 

0.1592 

0.0634

3.3 

0. 1

0.06

Percentage increase in discharge= 100 1 1 .

42.03

23.06

16.12

12.44

10.23

7.06

5.44

4.36

3.68

3.18

2.80

2.50

2.26

2.10

1.90

1.75

1.63

1.53

1.44

1.35

1.28

1.22

1.15

44.91

24.77

17.64

13.42

11.04

7.65

5.85

4.73

3.99

3.44

3.04

2.71

2.46

2.24

2.06

1.90

1.78

1.66

1.56

1.47

1.39

1.32

1.25

48.00

26.59

18.68

14.46

11.90

8.28

6.32

5.11

4.31

3.73

3.28

2.93

2.66

2.42

2.22

2.06

1.92

1.80

1.69

1.59

1.51

1.43

1.36

51.14

28.45

20.02

15.51

12.77

8.84

6.79

5.50

4.64

4.01

3.54

3.16

2.87

2.61

2.40

2,22

2.07

1.94

1.82

1.72

1.62

1.54

1.47

54.22

30.29

37.16

16.58

13.66

9.47

7.28

5.90

4.98

4.31

3.80

3.40

3.09

2.80

 2.58

2.39

2.23

2.13

1.96

1.85

1.78

1.66

1.57

57.34

32.19

22.80

17.77

14.57

10.03

7.78

6.32

5.33

4.61

4.07

3.64

3.30

3.01

2.77

2.56

2.39

2.22

2.10

1.98

1.87

1.78

1.68

60.64

34.15

24.17

18.89

15.51

10.78

8.30

6.74

5.70

4.93

4.35

3.89

3.53

3.21

2.96

2.74

2.55

2.39

2.25

2.12

2.0

1.90

1.80

63.68

36.09

25.60

19.95

16.46

11.46

 8.83

7.17

6.07

5.25

4.63

4.14

3.76

3.43

3.09

2.94

2.67

2.55

2.40

2.26

2.14

2.03

1.94

67.34

38.15

27.12

21.16

17.47

12.12

9.39

7.63

6.45

6.04

4.93

4.41

4.00

3.63

3.36

3.17

2.96

2.73

2.56

2.41

2.28

2.17

2.05

70.65

40.22

28.62

22.26

18.47

12.88

9.93

8.09

6.84

5.93

5.23

4.68

4.25

3.88

3.57

3.31

3.09

2.88

2.72

2.56

2.43

2.30

2.19

73

42

30

23

19

13

10

8

7

6

5

4'

4

4

3

3

3

3

.2

2

2

2

2
162 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

To use this table the discharge corresponding to the measured head 

D may be taken directly from Table 3 or 4 and the quantity so obtained 

increased by the percentage indicated in Table 2. This table is especially useful where the velocity of approach is measured directly. If 

the velocity of approach is determined from the approximate discharge

by the formula V= A-> successive approximate corrections may lie 

required.

Table 2 shows directly the relative error introduced by various velocities of approach. The large error introduced by moderate velocities 

with low heads and the comparatively small error resulting from 

higher velocities under great heads are conspicuous.

TABLES 3 AND 4. DISCHARGE OVER A THIN-EDGED WEIR BY 

THE FRANCIS FORMULA.

These tables give the discharge in cubic feet per second, for a crest 

length of 1 foot, without contractions, computed by the formula

TABLE 3. Discharge over a thin-edged weir per foot of crest.

Head H, feet.

0.00

.01

.02 

.03

.04

.05

.06

.07

.08

.09

0.10

.11

.12

.13

.14

.15

.16

.17

.18

.19

0.20

.21

.22

.23

.24

.25

.26

.27

.28

.29

.000

0. 0000

.0033

.0094

.0173

.0266

.0372

.0489

.0617

.0753

.0899

0.1053

.1215

.1384

.1561

.1744

.1935

.2131

.2334

.2543

.2758

0.2978

.3205

.3436

.3673

.3915

.4162

.4415

.4672

.4934

.5200

.001

0.0001

.0038

.0101

.0182

.0276

.0384

.0502

.0630

.0768

.0914

0. 1069

.1231

.1402

.1579

.1763

.1954

.2151

.2355

.2564

.2780

0.3001

.3228

.3460

.3697

.3940

.4187

.4440

.4698

.4960

.5227

.002

0.0003

.0044

.0109

.0191

.0287

.0395

.0514

.0643

.0782

.0929

0. 1085

.1248

.1419

.1597

.1782

.1973

.2171

. 2375

.2586

.2802

0. 3023

.3250

.3483

.3721

.3964

.4213

.4466

.4724

.4987

.5254

.003

0.0005

.0049

.0116

.0200

.0297

.0406

.0527

.0657

.0796

.0944

0. 1101

.1265

.1436

.1615

.1801

.1993

.2191

.2396

.2607

.2823

0. 3046

. 3274

.3507

.3745

.3989

.4238

.4491

.4750

.5013

.5281

.004

0.0008

.0055

.0121

.0209

.0307

.0418

.0539

.0670

.0811

.0960

0. 1117

.1282

.1454

.1633

.1820

.2012

.2212

.2417

.2628

.2845

0. 3068

.3297

.3530

.3769

.4014

.4263

.4517

.4776

.5040

.5308

.005

0. 0012

.0061

.0132

.0218

.0318

.0430

.0552

.0684

.0825

.0975

0.1133

.1299

.1472

.1652

.1839

.2032

.2232

.2438

.2650

.2867

0. 3091

.3320

.3554

.3794

.4038

.4288

.4543

.4802

.5067

.5336

.006

0. 0015

.0067

.0140

. 0227

.0329

.0441

.0565

.0698

.0840

.0990

0. 1149

.1316

.1489

.1670

.1858

.2052

.2252

.2459

.2671

.2890

0. 3113

.3343

.3578

.3818

.4063

.4313

.4568

.4828

.5093

.5363

.007

0. 0020

.0074

.0148

.0237

.0339

.0453

.0578

.0712

.0855

.1006

0. 1166

.1333

.1507

.1689

.1877

.2072

.2273

.2480

.2693

.2912

0. 3136

.3366

.3601

.3842

.4088

.4339

.4594

.4855

.5120

.5390

.008

0.0024

.0080

.0156

.0247

.0350

.0465

.0590

.0725

.0869

.1022

0. 1182

.1350

.1525

.1707

.1896

.2091

.2293

.2501

.2714

.2934

0. 3159

.3389

.3625

.3866

.4113

.4364

.4620

.4881

.5147

.5417

.009

0.0028

.0087

.0164

.0256

.0361

.0477

.0604

'. 0739

.0884

.1037

0. 1198

.1367

.1543

.1726

.1915

.2111

.2314

.2522

.2736

.2956

0. 3182

.3413

.3649

.3891

.4138

.4389

.4646

.4907

.5174

.5444
TABLES FOE CALCULATING WEIE DISCHARGE. 163

TABLE 3. Discharge over a thin-edged weir per foot of crest Continued.

Head If, feet.

0.30

.31

.32

.33

.34

.35

.36

.37

.38

.39

0.40

.41

.42

.43

.44

.45

.46

.47

.48

.49

0.50

.51

.52

.53

.000

0.5472

.5748

.6028

.6313

.6602

.6895

.7193

.7495

.7800

.8110

0.8424

.8742

.9064

.9390

.9719

1. 0052

1. 0389

1.0730

1. 1074

1.1422

1. 1773

1.2128

1. 2487

1. 2849

.54 1.3214

.55

.56

.57

.58

.59

0.60

.61

.62

.63

.64

.65

.66

.67

.68

.69

0.70

.71

.72

.73

.74

.75

.76

. 77

.78

.79

1. 3583

1.3955

1. 4330

1. 4709

1.5091

1.5476

1.5865

1.6257

1.6652

1. 7050

1. 7451

1. 7855

1. 8262

1. 8673

1.9086

1.9503

1. 9922

2.0344

2. 0770

2. 1198

2. 1629

2.2063

2.2500

2. 2940

2.3382

.001

0. 5499

.5775

.6056

.6341

.6631

.6925

.7223

.7525

.7831

.8142

0. 8456

.8774

.9096

.9422

.9752

1. 0086

1.0423

1. 0761

1. 1109

1.1457

1. 1809

1.2164

1.2523

1.2885

1. 3251

1. 3620

1.3992

1. 4368

1. 4747

1. 5130

1. 5515

1.5904

1.62%

1. 6691

1.7090

1. 7491

1.78%

1. 8303

1. 8714

1.9128

1. 9544

1.9964

2. 0387

2, 0812

2. 1241

2. 1672

2. 2107

2.2544

2.2984

2.3427

.002 .003

0.5527 0.5554

. 5803 . 5831

.6085 .6113

. 6370 . 6399

.6660

.6954

.7253

.7555

.7862

,8173

0.8488

.8806

.9129

.9455

.9785

1.0119

1.0457

1. 0798

1.1143

1. 1492

1.1844

1. 2200

1.2559

1.2921

1. 3287

1. 3657

1.4030

1.4406

1. 4785

1. 5168

1.5554

1.5943

1. 6335

1. 6731

1. 7130

1.7531

1. 7936

1.8344

1.8755

1. 9169

1. 9586

2.0006

2. 0429

2.0855

2.1284

2. 1716

2.2150

2.2588

2.3028

2. 3471

.6689

.6984

.7283

.7586

.7893

.8204

0. 8519

.8838

.9161

.9488

.9819

1. 0153

1.0491

1. 0833

1.1178

1. 1527

1.1879

1. 2235

1. 2595

1.2958

1. 3324

1. 3694

1. 4067

1.4444

1. 4823

1. 5206

1. 5593

1. 5982

1.6375

1. 6771

1.7170

1.7572

1. 7977

1. 8385

1. 8796

1. 9211

1. 9628

2.0048

2. G472

2. 0898

2. 1327

2. 1759

2.2194

2.2632

2.3072

2.3515

.004

0. 5582

.5859

.6141

.6428

.6719

.7014

.7313

.7616

. 7924

.8235

0. 8551

.8870

.9194

.9521

.9852

1. 0187

1. 0525

1. 0867

1. 1213

1.1562

1. 1915

1. 2271

1. 2631

1. 2994

1. 3361

1. 3731

1. 4105

1. 4481

1. 4862

1. 5245

1. 5631

1.6021

1. 6414

1.6810

1. 7210

1. 7612

1.8018

1. 8426

1.8838

1. 9252

1. 9670

2. 0091

2. 0514

2.0941

2. 1370

2. 1802

2.2237

2. 2675

2. 3116

2.3560

.005

0. 5609

.5887

.6170

.6457

.6748

' .7043

.7343

.7647

.7955

.8267

0. 8583

.8903

.9226

.9554

.9885

1. 0220

1. 0559

1.0901

1. 1248

1. 1597

1. 1950

1.2307

1. 2667

1. 3031

1. 3398

1. 3768

1.4142

1. 4519

1.4900

1. 5283

1. 5670

1. 6060

1.6454

1.6850

1.7250

1. 7652

1.8058

1. 8467

1.8879

1. 9294

1. 9712

2. 0133

2.0557

2. 0983

2. 1413

2. 1846

2. 2281

2. 2719

2. 3161

2.3604

.006

0. 5637

.5915

.6198

.6486

.6777

.7073

.7373

.7678

.7986

.8298

0. 8615

.8935

.9259

.9587

.9919

1.0254

1. 0593

1.0936

1. 1282

1. 1632

1.1986

1.2343

1. 2703

1.3067

1. 3435

1. 3806

1.4180

1.4557

1.4938

1.5322

1. 5709

1. 6100

1. 6493

1. 6890

1. 7290

1. 7693

1. 8099

1.8508

1.8920

1.9336

1. 9754

2. 0175

2. 0599

2.1026

2. 1456

2.1889

2. 2325

2. 2763

2.3205

2.3649

.007

0. 5664

.5943

.6227

. .6515

.6807

.7103

.7404

.7708

.8017

.8330

0. 8646

.8967

.9292

.9620

.9952

1.0288

1.0627

1. 0970

1. 1317

1.1668

1. 2021

1. 2379

1. 2740

1. 3104

1. 3472

1. 3843

1.4217

1. 4595

1. 4976

1. 5361

1. 5748

1. 6139

1. 6533

1. 6930

1. 7330

1.7733

1. 8140

1.8549

1. 8962

1. 9377

1. 9796

2.0217

2. 0642

2. 1069

2. 1499

2. 1932

2. 2369

2.2807

2. 3249

2.3694

.008

0. 5692

. 5972

.6255

.6544

.6836

. 7133

.7434

.7739

.8048

.8361

0. 8678

.8999

.9324

.9653

.9985

1.0321

1. 0661

1. 1005

1. 1352

1. 1703

1.2057

1. 2415

1.2776

1. 3141

1.3509

1.3880

1. 4255

1. 4633

1. 5014

1. 5399

1. 5787

1. 6178

1. 6572

1. 6970

1. 7370

1. 7774

1. 8181

1. 8590

1.9003

1. 9419

1.9838

2.0260

2.0684

2. 1112

2. 1543

2. 1976

2. 2412

2.2851

2. 3293

2. 3738

.009

0.5720

.6000

.6284

.6573

.6866

.7163

.7464

.7770

.8079

.8393

0. 8710

.9032

.9357

.%86

1.0019

1.0355

1.0696

1. 1039

1. 1387

1. 1738

1.2093

1.2451

1. 2812

1. 3177

1. 3546

1. 3918

1.4293

1. 4671

1.5053

1. 5438

1. 5826

1. 6217

1. 6612

1. 7010

1. 7410

1. 7814

1.8221

1. 8632

1.9045

1. 9461

1.9880

2. 0302

2.0727

2. 1155

2. 1586

2. 2019

2.2456

2. 2896

2.3338

2. 3783
164 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

TABLE 3. Discharge over a thin-edged weir per foot of crest Continued.

Head H, feet.

0.80

.81

.82

.83

.84

.85

.86

.87

.88

.89

0.90

.91

.92

.93

.94

.95

.96

.97

.98

.99

1.00

.01

.02

.03

.04

 .05

.06

.07

.08

.09

1.10

.11

.12

.13

.14

.15

.16

.17

.18

.19

1.20

.21

.22

.23

.24

.25

.26

.27

.28

.29

.000

2.3828

2. 4276

2.4727

2.5180

2. 5637

2.6096

2.6558

2.7022

2.7490

2. 7959

2.8432

2. 8907

2.9385

2. 9865

3.0348

3.0834

3. 1322

3. 1813

3. 2306

3.2802

3.3300

3.3801

3.4304

3. 4810

3.5318

3.5828

3.6342

3.6857

3. 7375

3.7895

3. 8418

3.8943

3. 9470

4.0000

4.0532

4.1067

4. 1604

4. 2143

4.2684

4. 3228

4. 3774

4. 4322

4. 4873

4. 5426

4.5981

4.6538

4.7098

4. 7660

4. 8224

4.8790

.001

2. 3872

2.4321

2. 4772

2.5226

2.5683

2. 6142

2.6604

2.7069

2.7536

2.8007

2.8479

2.8955

2. 9433

2.9914

3.0397

3.0883

3. 1371

3. 1862

3.2355

3.2851

3.3350

3. 3851

3.4354

3.4860

3. 5369

3.5880

3.6393

3. 6909

3. 7427

3.7947

3.8470

3. 8996

3. 9523

4.0053

4.0586

4. 1120

4. 1657

4. 2197

4.2738

4. 3282

4.3829

4. 4377

4. 4928

4.5481

4. 6036

4. 6594

4.7154

4. 7716

4.8280

4.8847

.002

2. 3917

2.4366

2.4817

2.5271

2.5728

2.6188

2.6650

2. 7116

2. 7583

2.8054

2. 8527

2.9003

2.9481

2.9962

3.0445

3.0931

3. 1420

3. 1911

3.2405

3.2901

3.3400

3. 3901

3.4405

3.4911

3.5420

3. 5931

3.6444

3. 6960

3. 7479

3. 8000

3. 8523

3.9048

3. 9576

4. 0106

4.0639

4. 1174

4.1711

4. 2251

4. 2793

4.3337

4.3883

4. 4432

4.4983

4.5537

4.6092

4.6650

4.7210

4. 7772

4.8337

4.8903

.003

2.3962

2.4411

2.4862

2. 5317

2. 5774

2. 6234

2.6697

2. 7162

2. 7630

2. 8101

2. 8574

2.9050

2. 9529

3. 0010

3.0494

3.0980

3.1469

3.1960

3.2454

3.2951

3.3450

3. 3951

3.4455

3. 4962

3.5471

3.5982

3. 6496

3. 7012

3. 7531

3. 8052

3. 8575

3. 9101

3.9629

4.0160

4.0692

4. 1228

4. 1765

4. 2305

4.2847

4.3392

4.3938

4.4487

4.6038

4. 5592

4.6148

4.6706

4. 7266

4. 7829

4.8393

4.8960

.004

2.4006

2.4456

2.4908

2.5363

2.5820

2.6280

2.6743

2.7209

2. 7677

2.8148

2. 8622

2.9098

2. 9577

3.0058

3.0542

3.1029

3. 1518

3.2010

3.2504

3.3001

3. 3500

3.4002

3.4506

3. 5013

3. 5522

3.6033

3.6547-

3. 7064

3. 7583

3.8104

3. 8628

3.9154

3.9682

4. 0213

4. 0746

4.1281

4. 1819

4. 2359

4.2901

4.3446

4.3993

4.4542

4.5094

4.5647

4. 6203

4.6762

4. 7322

4.7885

4.8450

4.9017

.005

2.4051

2. 4501

2.4953

2.5408

2. 5866

2.6327

2.6790

2. 7256

2. 7724

2. 8195

2. 8669

2. 9146

2. 9625

3. 0107

3. 0591

3.1078

3. 1567

3. 2059

3.2554

3.3051

3.3550

3.4052

3.4557

3.5063

3.5573

3. 6085

3.6599

3. 7116

3. 7635

3. 8156

3.8680

3.9206

3.9735

4.0266

4.0799

4.1335

4. 1873

4. 2413

4.2956

4.3501

4.4048

4.4597

4. 5149

4. 5703

4. 6259

4.6818

4. 7378

4. 7941

4.8506

4.9074

.006

2.4096

2.4546

2.4999

2.5454

2. 5912

2. 6373

2.6836

2.7302

2. 7771

2.8243

2. 8717

2.9194

2. 9673

3. 0155

3.0639

3. 1127

3. 1616

3. 2108

3.2603

3.3100

3.3600

3.4102

3. 4607

3.5114

3.5624

3. 6136

3. 6651

8, 7167

3. 7687

3. 8209

3.8733

3.9259

3.9788

4.0319

4.0853

4. 1389

4. 1927

4. 2467

4,3010

4. 3555

4.4103

4.4652

4.5204

4. 5759

4. 6315

4. 6874

4. 7435

4. 7998

4.8563

4.9131

.007

2.4141

2. 4591

2. 5044

2.5500

2.5958

2.6419

2.6883

2.7349

2. 7818

2.8290

2. 8764

2.9241

2. 9721

3.0203

3.0688

3. 1175

3. 1665

3.2158

3. 2653

3.3150

3.3650

3. 4153

3.4658

3. 5165

3.5675

3. 6187

3.67Q2

3. 7219

3. 7739

3. 8261

3. 8785

3. 9312

3. 9841

4. 0372

4.0906

4.1442

4. 1981

4. 2522

4.3065

4.3610

4.4158

4.4707

4.5260

4.5814

4.6371

4.6930

4. 7491

4.8054

4. 8620

4. 9187

.008

2. 4186

2.4636

2.5089

2.5545

2.6004

2.6465

2. 6929

2.7396

2. 7865

2.8337

2. 8812

2.9289

2. 9769

3.0252

3. 0737

3.1224

3. 1714

3. 2207

3.2702

3. 3200

3.3700

3.4203

3.4708

3. 5216

3. 5726

3. 6239

3.6754

3. 7271

3. 7791

3. 8313

3.8838

3. 9365

3.9894

4.0426

4.0960

4. 1496

4. 2035

4.2576

4. 3119

4.3665

4.4212

4.4763

4.5315

4.5870

4.6427

4. 6986

4.7547

4.8111

4. 8676

4.9244

.009

2.4231

2.4681

2. 5135

2. 5591

2.6050

2. 6511

2. 6976

2.7443

2. 7912

2.8385

2.8860

2. 9337

2. 9817

3.0300

3. 0785

3. 1273

3. 1764

3. 2257

3. 2752

3. 3250

3. 3751

3. 4254

3.4759

3.5267

3. 5777

3.6290

3. 6805

3. 7323

3.7843

3.8365

3. 8890

3.9418

3. 9947

4.0479

4. 1013

4.1550

 4. 2089

4.2630

4.3173

4. 3719

4. 4267

4.4818

4.5370

4. 5925

4. 6482

4.7042

4. 7603

4.8167

4.8733

4.9301
TABLES FOR CALCULATING WEIR DISCHARGE. 165 

TABLE 3. Discharge over a thin-edged weir per foot of crest Continued.

Head If, feet. .000

1.30

.31

4.9358

4. 9929

. 32 5. 0502

.33 5.1077

. 34 5. 1664

. 35 5. 2233

.36 5.2814

.37 ' 5.3398

.38 i 5.39S4

.39

1.40

.41

.42

.43

.44

.45

.46

.47

.48

.49

1.50

.51

.52

.53

.54

.55

.56

.57

.58

.59

1.60

.61

.62

.63

.64

.65

.66

.67 .

.68

.69

1.70

.71

.72

.73

.74

.75

.76

.77

.78

.79

5.4572

5. 5162

5. 5754

5. 6348

5. 6944

'5. 7542

5. 8143

5. 8745

5. 9350

5.9957

6. 0565

6. 1176

6. 1789

6. 2404

6. 3020

6.3639

6, 4260

6,4883

6.5508

6. 6135

6. 6764

6.7394

6. 8027

6.8662

6.9299

6. 9937

7. 0578

7. 1221

7.1865

7.2512

7. 3160

7.3810

7. 4463

7.5117

7. 5773

7. 6431

7. 7091

7. 7752

7.8416

7.9081

7. 9749

.001

4. 9415

4.9986

5.0559

5. 1134

5.1712

5. 2291

5.2873

5. 34.56

5.4142

5.4630

5. 5221

5.5813

5. 6407

5.7004

5. 7602

5. 8203

5. 8806

5. 9410

6. 0017

6. 0626

6. 1237

6. 1850

6.2465

6.3082

6. 3701

6. 4322

6. 4945

6.5570

6. 6198

6.6827

6.7458

6. 8091

6. 8726

6. 9363

7.0001

7. 0642

7.1285

7.1930

7. 2576

7. 3225

7. 3876

7. 4528

7. 5182

7.5839

7. 6497

7. 7157

7. 7819

7.8482

7. 9148

7. 9816

.002

4. 9472

5.0043

5. 0616

5.1192

5. 1769

5.2349

5.2931

5. 3515

5.4101

5. 4689

5.5280

5. 5872

.003

4.9529

5.0100

5. 0674

5. 1249

5. 1827

5. 2407

5. 2989

5. 3573

5. 4160

5.4748

5. 5339

5. 5932

5.6467 1 5.6526

5. 7064

5.7662'

5. 8263

5. 8866

5. 9471

6.0078

6.0687

6. 1298

6. 1912

6. 2527

6.3144

6.3763

6.4385

6.5008

6. 5633

6. 6260

6. 6890

6. 7521

6. 8154

6. 8789

6. 9426

7.0065

7.0706

7. 1349

7. 1994

7. 2641

7.3290

7. 3941

7.4593

7.5248

7.5904

7.6563

7. 7223

7.7885

7.8549

7. 9215

7. 9882

5. 7123

5. 7722

5. 8323

5. 8926

5. 9532

6. 0139

6. 0748

6. 1360

6. 1973

6. 2588

6.3206

6. 3825

6.4447

6. 5070

6.56%

6. 6323

6. 6953

6.7584

6.8217

6.8853

6. 9490

7. 0129

7. 0771

7. 1414

7. 2059

7.2706

7.3355

7.4006

7. 4659

7.5313

7.5970

7. 6628

7.7289

7. 7951

7.8615

7. 9281

7.9949

.004

4.9586

5. 0158

5. 0731

5. 1307

5. 1885

5. 2465

5. 3048

5. 3632

5. 4219

5.4807

5. 5398

5. 5991

5.6586

5. 7183

5. 7782

5. 8384

5. 8987

5. 9592

6. 0200

6.0809

6.1421

6.2034

6. 2650

6.3268

6.3887

6.4509

6. 5133

.005

4.9643

5.0215

5. 0789

5.1365

5.1943

5. 2523

5. 3106

5. 3691

5.4277

5.4866

5. 5457

5.6050

5. 6646

5. 7243

5.7842

5. 8444

5. 9047

5. 9653

6. 0261

6. 0870

6.1482

6.20%

6. 2712

6.3330

6. 3949

6. 4571

6. 5195

6. 5758 6. 5821

6. 6386 6. 6449

6. 7016 6. 7079

6. 7647

6. 8281

6.8916

6.9554

7.0193

7. 0835

7. 1478

7. 2124

7. 2771

7. 3420

7.4071

7. 4724

7.5379

7. 6036

7.6694

7. 7355

7. 8018

7. 8682

7.9348

8. 0016

6. 7711

6.8344

6. 8980

6. 9618

7.0258

7.0899

7. 1543

7. 2188

7. 2836

7.3485

7. 4136

7. 4789

7.5445

7. 6102

7.6760

7. 7421

7.8084

7. 8748

7. 9415

8.0083

.006

4.9700

5.0272

5.0846

5.1423

5. 2001

5. 2582

5. 3164

5. 3749

5.4336

5.4925

5. 5516

5. 6110

5. 6705

5. 7303

5. 7902

5. 8504

5. 9108

5. 9714

6.0322

6. 0931

6. 1543

6. 2157

6. 2773

6. 3391

6. 4012

6. 4634

6.5258

6.5884

6. 6512

6. 7142

6. 7774

6.8408

6.9044

6. 9682

7. 0322

7.0963

7. 1607

7. 2253

7.2901

7.3550

7. 4201

7. 4855

7.5510

7. 6167

7.6826

7. 7487

7.8150

7.8815

7. 9482

8.0160

.007

4. 9757

5. 0330

5. 0904

5.1480

5.0259

5. 2640

5. 3223

5. 3808

5. 4395

5.4984

5. 5576

5. 6169

5. 6765

5. 7363

5. 7962

5.8564

5. 9168

5. 9774

6. 0382

6.0993

6. 1605

6. 2219

6. 2835

6.3453

6.4074

6. 4696

6. 5320

6. 5946

6. 6575

6. 7205

6.7837

6. 8471

6. 9108

6. 9746

7. 0386

7. 1028

7. 1672

7. 2318

7. 2965

7. 3615

7.4267

7. 4920

7. 5576

7. 6233

7. 6892

7.7554

7. 8217

7. 8882

7.9648

8. 0217

.008

4.9814

5. 0387

5.0961

5. 1638

5. 2117

5. 2698

5.3281

5.3866

5.4454

5.5043

5. 5635

5. 6229

5. 6825

5.7423

5. 8023

5. 8625

5. 9229

5.9835

6.0443

6.1054

6.1666

6. 2280

6.2897

6. 3515

6.4136

6. 4758

6.5383

6.6009

6. 6638

6. 7268

6. 7901

6.8535

6. 9171

6. 9810

7.0450"

7.1092

7. 1736

7.2382

7.3030

7.3680

7. 4332

7. 4986

7. 5641

7. 6299

7. 6958

7. 7620

7.8283

7. 8948

7.9615

8.0284

.009

4. 9872

5.0444

5. 1019

5. 1596

5. 2175

5.2756

5.3340

5. 3925

5. 4513

5. 5102

5. 5694

5.6288

5.6884

5.7482

5.8083

5. 86a5

5.9289

5.98%

6.0504

'6. 1115

6. 1727

6. 2342

6.2959

6. 3577

6. 4198

6. 4821

6.5445

6. 6072

6. 6701

6.7331

6.7964

6. 8598

6.9235

6. 9874

7.0514

7. 1156

7.1801

7. 2447

7. 3095

7.3745

7. 4397

7.5051

7. 5707

7. 6365

7. 7024

7. 7686

7. 8349

7. 9015

7. 9682

8. 0351
166 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

TABLE 3. Discharge over a thin-edged weir per foot of crest Continued.

Head H, feet.

1.80

.81

.82

.83

.84

.85

.86

.87

.88

.89

1.90

.91

.92

.93

.94

.95

.96

.97

.98

.99

2.00

.01

.02

.03

.04

.05

.06

.07

.08

.09

2.10

.11

lli

.13 .14"

.15

.16

.17

.18

.19

2.20

.21

.22

.23

.24

.25

.26

.27

.28

.29

.000

8.0418

8. 1089

8. 1762

8. 2437

8. 3113

8. 3792

8. 4472

8. 5154

8.5838

8. 6524

8. 7212

8. 7901

8. 8592

8. 9285

8. 9980

9. 0677

9. 1375

9. 2075

9. 2777

9.3481

9. 4187

9. 4894

9. 5603

9. 6314

9. 7026

9. 7741

9. 8457

9. 9174

9. 9894

10. 062

10. 134

10. 206

10. 279

10. 352

10. 425

10. 498

10. 571

10. 645

10. 718

10. 792

10. 866

10.940

11.015

11.089

11. 164

11.239

11.314

11.389

11.464

11. 540

.001

8. 0485

8. 1156

8. 1829

8. 2504

8. 3181

8. 3860

8.4540

8.5223

8. 5907

8. 6593

8. 7281

8. 7970

8. 8662

8. 9355

9. 0050

9. 0747

9.1445

9. 2145

9. 2848

9.3552

9. 4257

9. 4965

9. 5674

9. 6385

9. 7098

9. 7812

9.8528

9. 9246

9. 9966

10. 069

10.141

10. 214

10. 286

10. 359

10. 432

10. 505

10. 579

10.652

10.726

10.800

10. 874

10.948

11.022

11. 097

11. 171

11.246

11.321

11. 396

11. 472

11. 547

.002

8. 0552

8. 1223

8. 1897

8. 2572

8. 3249

8.3928

8. 4608

8.5291

8. 5975

8. 6661

8. 7349

8. 8039

8. 8731

8.9424

9.0119

9. 0816

9.1515

9.2216

9. 2918

9. 3622

9. 4328

9. 5036

9. 5745

9.6456

9. 7169

9. 7884

9.8600

9. 9318

10.004

10. 076

10.148

10. 221

10. 293

10. 366

10. 439

10. 513

10. 586

10. 659

10. 733

10.807

10. 881

10,955

11. 030

11.104

11. 179

11.254

11. 329

11.404

.008

8. 0619

8. 1291

8. 1964

8. 2640

8. 3317

8. 3996

8. 4677

8. 5359

8. 6044

8. 6730

8. 7418

8. 8108

8. 8800

8.9494

9.0189

9. 0886

9. 1585

9. 2286

9.2988

9. 3693

9.4399

9. 5106

9. 5816

9. 6527

9. 7240

9. 7955

9. 8672

9.9390

10.011

10.083

10. 156

10. 228

10. 301

10.374

10.447

10. 520

10. 593

10. 667

10. 741

10. 814

10. 888

10. 963

11.037

11.112

11. 186

11.261

11. 336

11.412

11.479 11.487

11. 555 11. 562

.004

8. 0686

8. 1358

8. 2032

8. 2707

8. 3385

8.4064

8.4745

8.5428

8.6112

8. 6799

8. 7487

8. 8177

8.% 8869

8. 9563

9. 0259

9. 0956

« 9. 1655

9. 2356

9. 3059

9.3763

9. 4469

9. 5177

9. 5887

9. 6599

9. 7312

9. 8027

9. 8744

9. 9462

10. 018

10.090

10. 163

10. 235

10. 308

10. 381

10. 454

10. 527

10. 601

10. 674

10. 748

10. 822

10. 896

10. 970

11.045

11.119

11.194

11. 269

11. 344

11.419

11. 494

11. 570

.005

8. 0753

8. 1425

8. 2099

8. 2775

8. 3452

8.4132

8. 4813

8.5496

8. 6181'

8. 6868

8. 7556

8. 8246

8. 8939

8. 9633

9. 0328

9. 1026

9.1725

9.2426

9. 3129

9.3834

9. 4540

9. 5248

9. 5958

9. 6670

9. 7383

9.8098

9.8815

9. 9534

10. 025

10. 098

10.170

10. 243

10. 315

10. 388

10. 461

10. 535

10. 608

10.682

10. 755

10. 829

10. 903

10. 978

11. 052

11. 127

11.201

11. 276

11. 351

11. 427

11.502

11. 578

.006

8.0820

8. 1493

8. 2167

8. 2842

8. 3520

8. 4200

8. 4881

8.5564

8.6250

8. 6936

8. 7625

8.8316

8. 9008

8.9702

9. 0398

9. 1096

9. 1795

9. 2496

9. 3199

9.3904

9. 4611

9.5319

9. 6029

9. 6741

9.7455

9. 8170

9.8887

9. 9606

10. 033

10. 105

10. 177

10.250

10. 323

10. 396

10. 469

10. 542

10. 615

10. 689

10. 763

10. 837

10. 911

10. 985

11. 059

11. 134

11.209

11. 284

11. 359

11.434

11. 510

11.585

.007

8.0888

8. 1560

8.2234

8.2910

8.3588

8. 4268

8. 4949

8.5633

8. 6318

8. 7005

8. 7694

8. 8385

8. 9077

8. 9772

9. 0468

9.1165

9. 1865

9.2567

.008

8.0955

8. 1627

8.2302

8.2978

8. 3656

8. 4336

8. 5018

8. 5701

8. 6387

8. 7074

8. 7763

8.8454

8. 9147

8. 9841

9. 0537

9. 1235

9. 1935

9. 2637

9.3270 9.3340

9. 3975

9. 4682

9. 5390

9. 6100

9. 6812

9. 7526

9. 8242

9. 8959

9. 9678

10.040

10. 112

10. 185

10.257

10. 330

10. 403

10. 476

10. 549

10.623

10. 696

10. 770

10. 844

10. 918

10. 992

11. 067

11. 141

11.216

11.291

11. 366

11. 442

11.517

11. 593

9.4045

9. 4752

9. 5461

9. 6171

9. 6884

9. 7598

9. 8313

9. 9031

9. 9750

10.047

10.119

10. 192

10. 264

10. 337

10.410

10. 483

10. 557

10. 630

10. 704

10. 777

10. 851

10. 926

11.000

11.074

11. 149

11.224

11.299

11. 374

11. 449

11. 525

11.600

.009

8. 1022

8. 1695

8. 2369

8. 3046

8. 3724

8.4404

8. 5086

8. 5770

8. 6455

8. 7143

8. 7832

8. 8523

8. 9216

8. 9911

9.0607

9.1305

9.2005

9. 2707

9.3411

9. 4116

9. 4823

9.5532

9. 6243

9. 6955

9. 7669

9. 8385

9. 9103

9. 9822

10. 054

10. 127

10.189

10. 272

10.344

10. 417

10. 491

10. 564

10. 637

10. 711

10. 785

10. 859

10. 933

11.007

11. 082

11. 156

11. 231

11.306

11.381

11.457

11. 532

11. 608
TABLES FOE CALCULATING WEIE DISCHAEGE. 167

TABLE 3. Discharge over a thin-edged iveir per foot of crest Continued.

Head H, feet.

2.30

.31

.32

.33

.34

.35

.36

.37

.38

.39

2.40

.41

.42

.43

.44

.45

.46

.47

.48

.49

2.50

.51

.52

.53

.54

.55

.56

.57

.58

.59

2.60

.61

.62

.63

.64

.65

.66

.67

.68

.69

2.70

.71

.72

.73

.74

.75

.7ti

. 77

.78

.79

.000

11. 615

11. 691

11. 767

11. 843

11. 920

11. 996

12. 073

12. 150

12. 227

12. 304

12. 381

12. 459

12. 536

12.614

12. 692

12. 770

12. 848

12.927

13.005

13.084

13. 163

13.242

13. 321

13.401

13.480

13. 560

13.640

13. 720

13.800

13.880

13. 961

14.041

14. 122

14. 203

14. 284

14.356

14. 447

14.528

14. 610

14. 692

14. 774

14.856

14. 938

15. 021

15. 103

15. 186

15. 269

15. 352

15. 435

15. 519

.001

11. 623

11.699

11. 775

11. 851

11. 927

12. 004

12. 081

12. 157

12.234

12. 312

12. 389

12. 466

12.544

12. 622

12.700

12. 778

12. 856

12. 935

13. 013

13. 092

13. 171

13.250

13. 329

13. 409

13.488

13. 568

13. 648

13. 728

13. 808

13. 888

13. 969

14.049

14. 130

14. 211

14. 292

14. 373

14. 455

14.536

14. 618

14. 700

14. 782

14. 864

14. 946

15.029

15. 112

15. 194

15.277

15. 360

15.443

15. 527

.002

11. 631

11. 706

11.783

11. 859

11. 935

12. 012

12.088

12.165

12. 242

12. 319

12. 397

12. 474

12. 552

12. 630

12. 708

12. 786

12.864

12. 942

13. 021

13. 100

13. 179

13.258

13.337

13. 417

13. 496

13. 576

13.656

13. 736

13. 816

13.896

13. 977

14. 057

14. 138

14. 219

14.300

14.382

.14.463

14.545

14. 626

14. 708

14. 790

14. 872

14. 955

15. 037

15. 120

15. 203

15. 285

15. 369

15. 452

15. 535

.003

11. 638

11. 714

11.790

11. 866

11.943

12. 019

12. 096

12. 173

12. 250

12. 327

12. 404

12. 482

12. 560

12. 637

12. 715

12. 794

12. 872

12.950

13. 029

13. 108

13. 187

13. 266

13.345

13. 424

13. 504

13.584

13.664

13.744

13.824

13.904

13.985

14. 065

14. 146

14. 227

14.308

14. 390

14. 471

14.553

14. 634

14. 716

14. 798

14.881

14. 963

15. 045

15. 128

15. 211

15. 294

15. 377

15. 460

15.544

.004

11. 646

11. 722

11.798

11. 874

11.950

12. 027

12. 104

12. 181

12.258

12. 335

12. 412

12.490

12.667

12. 645

12. 723

12. 801

12. 880

12. 958

13. 037

13. 116

13. 195

13. 274

13.353

13. 432

13. 512

13. 592

13. 672

13. 752

13. 832

13. 912

13. 993

14. 074

14.154

14. 235

14. 316

14. 398

14. 479

14.561

14. 643

14. 725

14. 807

14.889

14. 971

15.054

15. 136

15. 219

15. 302

15. 385

15. 468

15. 552

.005

11. 653

11.729

11. 805

11. 882

11. 958

12. 035

12. Ill

12. 188

12.265

12. 342

12. 420

12. 497

12. 575

12. 653

12. 731

12. 809

12. 888

12. 966

13. 045

13. 124

13. 202

13. 282

13. 361

13.440

13. 520

13.600

13. 680

13. 760

13.840

13.920

14. 001

14.082

14. 162

14.243

14. 325

14. 406

14. 487

14. 569

14. 651

14. 733

14. 815

14. 897

14. 979

15. 062

15. 145

15.227

15. 310

15. 394

15. 477

15. 560

.006

11. 661

11. 737

11. 813

11. 889

11. 966

12.042

12. 119

12. 196

12. 273

12.350

12. 428

12. 505

12.583

12. 661

12. 739

12. 817

12. 895

12. 974

13. 053

13. 131

13. 210

13.290

13. 369

13. 448

13.528

13. 608

13.688

13. 768

13.848

13. 928

14. 009

14.090

14. 171

14. 252

14. 333

14.414

14. 496

14. 577

14.659

14. 741

14. 823

14. 905

14. 988

15. 070

15. 153

15. 236

15. 319

15. 402

15. 485

15. 569

.007

11. 669

11. 744

11. 821

11.897

11. 973

12.050

12. 127

12. 204

12. 281

12. 358

12. 435

12. 513

12. 591

12. 669

12. 747

12. 825

12. 903

12. 982

13. 060

13. 139

13. 218

13.297

13. 377

13. 456

13. 536

13. 616

13. 696

13. 776

13.856

13. 936

14. 017

14. 098

14. 179

14.260

14.341

14. 422

14. 504

14.585

14. 667

14. 749

14.831

14. 913

14. 996

15. 078

15. 161

15. 244

15. 327

15. 410

15. 494

15. 577

.008

11. 676

11. 752

11. 828

11. 904

11. 981

12. 058

12. 134

12. 211

12. 288

12. 366

12.443

12. 521

12. 598

12. 676

12. 754

12. 833

12. 911

12. 990

13. 068

13. 147

13. 226

13. 305

13. 385

13. 464

13. 544

13. 624

13. 704

13. 784

13.864

13. 944

14. 025

14. 106

14. 187

14. 268

14. 349

14. 430

14. 512

14. 594

14. 675

14. 757

14.839

14. 922

15.004

15. 087

15. 169

15. 252

15.335

15. 419

15.502

15. 585

.009

11. 684

11. 760

11.836

11. 912

11. 989

12. 065

12. 142

12. 219

12. 296

12. 373

12. 451

12. 528

12. 606

12.684

12. 672

12.840

12. 919

12. 997

13. 076

13.155

13. 234

13. 313

13. 393

13. 472

13. 552

13. 632

13.712

13. 792

13. 872

13. 953

14. 033

14.114

14. 195

14.276

14.357

14.438

14. 520

14. 602

14. 684

14. 766

14.848

14. 930

15. 012

15. 095

15. 178

15. 261

15.344

15. 427

15. 510

15. 594

IEB 150 06 16
168 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

TABLE 3. Discharge over a thin-edged weir per foot of crest Continued.

Head H, feet.

2.80

.81

.82

.83

.84

.85

.86

.87

.88

.89

2.90

.91

.92

.93

.94

.95

.96

.97

.98

99

3.00

.000

15. 602

15. 686

15. 769

15. 853

15. 938

16. 022

16. 106

16.191

16. 275

16. 360

16. 445

16. 530

16. 616

16. 701

16.787

16. 872

16. 958

17.044

17. 130

17. 217

17. 3033

.001

15. 610

15. 694

15. 778

15. 862

15. 946

16. 030

16. 115

16. 199

16. 284

16. 369

16.454

16. 539

16. 624

1C. 710

16. 795

16.881

16.967

17. 053

17. 139

17. 225

.002

15. 619

15. 702

15. 786

15. 870

15. 954

16. 039

16. 123

16. 208

16. 292

16. 377

16. 462

16. 547

16. 633

16.718

16.804

16. 890

16. 975

17. 062

17.148

17. 234

.003

15. 627

15. 711

15. 795

15. 879

15. 963

16. 047

16. 132

16. 216

16. 301

16.386

16. 471

16. 556

16. 641

16. 727

16.812

16. 898

16. 984

17. 070

17. 156

17. 243

.004

15. 635

15. 719

15. 803

15. 887

15. 971

16. 056

16. 140

16. 225

16. 309

16. 394

16. 479

16. 565

16. 650

16. 735

16. 821

16. 907

16.993

17. 079

17.165

17. 251

.005

15. 644

15. 728

15. 811

15. 895

15. 980

16. 064

16. 148

16. 233

16. 318

16. 403

16. 488

16. 573

16. 658

16. 744

16. 830

16. 915

17.001

17. 087

17. 174

17. 260

.006

15.652

15. 736

15. 820

15,904

15. 988

16. 072

16. 157

16. 242

16. 326

16. 411

16. 496

16. 582

16. 667

16. 752

16. 838

16. 924

17. 010

17. 096

17. 182

17. 269

.007

15. 661

15. 744

15. 828

15. 912

15.997

16. 081

16. 165

16. 250

16. 335

16. 420

16. 505

16. 590

16. 675

16. 761

16. 847

16. 932

17. 018

17. 105

17. 191

17.277

.008

15. 660

15. 753

15. 837

15. 921

16. 005

16. 089

16. 174

16. 258

16. 343

16. 428

16. 513

16. 599

16.684

16. 770

16. 855

16. 941

17. 027

17. 113

17. 199

17. 286

.009

15. 677

15. 761

15. 845

15. 929

16. 013

16. 098

16. 182

16. 267

16. 352

16. 437

16. 522

16. 607

16. 693

16. 778

16. 864

16. 950

17. 036

17. 122

17. 208

17. 295

TABLE 4. Discharge over a thin-edged weir per foot of crest.

Head 

H, feet.

0.0

.1

_ 2

.3

.4

.5

.6

7

.8

.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

.00

0.0000

.1053

. 2978

. 5472

.8424

1.1773

1. 5476

1. 9503

2. 3828

2. 8432

3. 3300

3. 8418

4. 3774

4. 9358

5. 5162

6. 1176

6. 7394

7. 3810

8. 0418

8. 7212

.01

0.0033

.1215

.3205

.5748

.8742

1. 2128

1.5865

1.9922

2. 4276

2. 8907

3. 3801

3. 8943

4. 4322

4. 9929

5. 5754

6. 1789

6. 8027

7. 4463

8.1689

8. 7901

.02

0. 0094

.1384

.3436

.6028

. 9084

1. 2487

1. 6257

2. 0344

2. 4727

2.9385

3.4304

3. 9470

4. 4873

5. 0502

5. 6348

6. 2404

6. 8662

7. 5117

8. 1762

8. 8592

.03

0. 0173

.1561

.3673

.6313

.9390

1. 2849

1. 6652

2. 0770

2. 5180

2. 9865

3. 4810

4.0000

4. 5426

5. 1077

5.6944

6. 3020

6. 9299

7. 5773

8.2437

8. 9285

.04

0. 0266

.1744

. 3915

.6602

.9719

1. 3214

1.7050

2. 1198

2. 5637

3. 0348

3. 5318

4. 0532

4. 5981

5. 1654

5. 7542

6. 3633

6. 9937

7. 6431

8.3113

8.9980

.05

0. 0372

.1935

. 4162

.6895

1. 0052

1. 3583

1.7451

2. 1629

2.6096

3. 0834

3. 5828

4. 1067

4. 6538

5. 2233

5. 8143

6. 4260

7. 0578

7. 7091

8. 3792

9. 0677

.06

 

0.0489

.2131

.4415

.7193

1. 0389

1. 3955

1. 7855

2. 2063

2. 6558

3. 1322

3. 6342

4. 1604

4. 7098

5. 2814

5. 8745

6. 4883

7.1221

7. 7752

8. 4472

9. 1375

.07

0. 0617

.2334

.4672

.7495

1. 0730

1.4330

1. 8262

2.2500

2. 7022

3. 1813

3. 6857

4. 2143

4. 7660

5. 3398

5. 9350

6.5508

7. 1865

7. 8416

8. 5154

9. 2075

.08

0. 0753

.2543

.4934

.7800

1. 1074

1.4709

1. 8673

2.2940

2. 7490

3.2306

3. 7375

4. 2384

4. 8224

5. 3984

5. 9957

6. 6135

7.2512

7.9081

8.5838

8. 2777

.09

0.0899

.2758

.5200

.8110

1. 1422

1.5091

1.9086

2.3382

2. 7959

3.2802

3. 7895

4. 3228

4. 8790

5. 4572

6. 0565

6. 6764

7. 3160

7. 9749

8.6524

9. 3481
TABLES FOE CALCULATING WEIE DISCHARGE. 169

TABLE 4. Discharge over a thin-edged weir per foot of crest Continued.

Head 

H, feet.

2.0

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3.0

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

4.0

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

5.0

5.1

5.2

5.3

5.4

5.5

5.6

5.7

5.8

5.9

6.0

6.1

6.2

6.3

6.4

6.5

6.6

6.7

6.8

6.9

.00

9. 4187

10.1340

10. 8660

11. 6150

12. 3810

13. 1630

13. 9610

14. 7740

15. 6020

16. 4450

17. 3033

18. 1754

19. 0619

19. 9624

20. 8777

21. 8045

22. 7456

23. 6999

24. 6673

25. 6473

26. 6400

27. 6453

28. 6626

29. 6926

30. 7342

31. 7878

32. 8534

33. 9307

35.0193

36. 1182

37. 2304

38. 3529

39. 4865

40. 6310

41. 7866

42. 9523

44. 1292

45. 3166

46,5141

47. 7226

48. 9407

50. 1694

51.4082

52. 6570

53. 9157

55. 1832

56. 4625

57. 7505

59. 0482

60. 3556

.01

9. 4894

10. 2060

10. 9400

11. 6910

12. 4590

13. 2480

14. 0410

14. 8560

15. 6860

16. 5300

17. 3899

18. 2634

19. 1515

20. 0533

20. 9690

21. 8980

22. 8405

23. 7962

24. 7645

25. 7459

26. 7399

27. 7466

28. 7652

29. 7962

30. 8391

31.8941

32. 9607

34. 0373

35. 1288

36. 2297

37. 3423

38.4658

39. 6004

40. 7462

41. 9024

43. 0700

44. 2474

45. 4359

46. 6347

47. 8438

49. 0632

50. 2930

51. 5324

52. 7822

54. 0419

55. 3116

56. 5910

57. 8801

59. 1788

60. 4868

.02

9. 5603

10. 2790

11.0150

11. 7670

12. 5360

13. 3210

14. 1220

14. 9380

15. 7690

16. 6 J 60

17. 4698

18.3516

19. 2410

20. 1442

21. 0613

21. 9917

22. 9354

23. 8924

24. 8621

25. 8748

26. 8401

27. 8478

28. 8678

29. 9001

30.9440

32. 0003

33. 0679

34. 1475

35. 2354

36. 3406

37.4542

88. 5787

39. 7146

40. 5281

42. 0186

43. 1871

44. 3659

45.5554

46. 7552

47. 9653

49. 1858

50. 4162

51. 6570

52. 9077

54.1684

55. 4392

56. 7192

58.0093

59. 3090

60. 6183

.03

9. 6314

10. 3520

11. 0890

11. 8430

12. 6140

13. 4010

14. 2030

15. 0210

15. 8530

16. 7010

17. 5634

18. 4399

19. 3307

20. 2354

21. 1538

22. 0856

23. 0306

23. 9887

24. 9600

25. 9437

26. 9404

27. 9494

28. 9703

30.0040

31. 0493

32. 1065

33. 1755

34. 2560

35.3480

36. 4515

37. 5661

38. 6919

39. 8288

40. 9766

42. 1352

43. 3043

44. 4845

45. 6746

46. 8757

48. 0869

49. 3083

50. 5401

51. 7818

53. 0336

54. 2950

55. 5667

56. 8478

58. 1388

59. 4428

60. 7499

.04

9. 7026

10. 4250

11.1640

11. 9200

12. 6920

13. 4800

14. 2840

15. 1030

15. 9380

16. 7870

17. 6503

18. 5285

19.4206"

20. 3267

21. 2464

22. 1795

23. 1259

24. 0852

25. 0576

26. 0429

27. 0406

28. 0509

29. 0732

30. 1079

31. 1545

32. 2128

33. 2830

34. 3646

35. 4578

36. 5624

37. 6783

38. 8052

39. 9430

41. 0919

42. 2517

43. 4219

44. 6030

45. 7945

46. 9963

48. 2084

48. 4312

50. 6637

51. 9034

53. 1591

54. 4219

55. 6943

56. 9766

58. 2687

59. 5700

60. 8814

.06

9. 7741

10. 4980

11.2390

11. 9960

12. 7700

13. 5600.

14. 3650

15. 1860

16. 0220

16. 8720

17. 7376

18. 6170

19. 5105

20. 4179

21. 3390

22. 2734

23. 2211

24. 1818

25. 1555

26. 1422

27. 1412

28. 1525

29. 1761

30. 2118

31. 2597

32. 3193

33. 3906

34. 4735

35. 5677

36. 6736

37. 7905

38. 9184

40. 0576

41. 2074

42. 3683

42. 5394

44. 7216

45. 9140

47. 1172

48. 3303

49. 5537

50. 7875

52. 0313

53. 2850

54.5487

55. 8221

57. 1055

58.3982

59. 7009

61. 0129

.06

9.8457

10. 5710

11. 3140

12. 0730

12. 8480

12. 6400

14. 4470

15. 2690

16. 1060

16. 9580

17. 8248

18. 7056

19. 6007

20. 5C95

21. 4319

22. 3677

23. 3167

24. 2787

25. 2537

26. 2414

27. 2417

28. 2544

29. 2790

30. 3163

31. 3649

32. 4259

33. 4985

34. 5824

35. 6780

36. 7845

37. 9027

39. 0319

40. 1718

41. 3230

42. 4848

43. 6573

44. 8404

46. 0339

47. 2380

48. 4522

49. 6766

50. 9114

52. 1531

53. 4109

54. 6756

55. 9500

57. 2340

58. 5281

59. 8314

61. 1445

.07

9. 9174

10. 6450

11.3890

12. 1500

12. 9270

13. 7200

14. 5280

15. 3520

16. 1910

17. 0440

17. 9124

18. 7945

19. 6910

20. 6011

21. 5248

22. 4618

23.4122

24. 3756

25. 3520

26. 3410

27.3423

28.3563

29. 3823

30. 4205

31. 4705

32. 5324

33. 6064

34. 6913

35. 7882

36. 8961

38. 0153

39. 1455

40.2867

41. 4386

42. 6017

43. 7752

44. 9593

46. 1538

47. 3589

48. 5744

49. 7999

51. 0356

52. 2813

53. 5371

54.8025

56. 0779

57. 3233

58. 6580

59. 9623

61. 2763

.08

9. 9894

10. 7180

11.4640

12. 2270

13. 0050

13. 8000

14. 6100

15. 4350

16. 2750

17. 1300

18. 0000

18. 8838

19. 7812

20. 6930

21. 6180

22. 5564

23. 5081

24. 4728

25. 4502

26. 4405

27. 4432

28. 4582

29. 4855

30. 5251

31. 5764

32. 6393

33. 7143

34.8005

35. 8984

37. 0073

38. 1275

39. 2591

40. 4012

41.5544

42. 7186

43. 8931

45. 0782

46. 2740

47. 4798

48. 6963

49. 9230

51. 1595

52. 4062

53. 6630

54. 9297

56. 2061

57. 4921

58.7882

60. 0935

61. 4082

.09

10. 0620

10.7920

11.5400

12. 3040

13. 0840

13. 8800

14. 6920

15. 5190

16. 3600

17. 2170

18. 0876

18. 9727

19. 8718

20. 7849

21. 7113

22. 6510

23. 6040

24. 5697

25. 5488

26.5401

27.5411

28.5604

29. 5890

30. 6297

31. 6820

32. 7462

33. 8225

34. 9097

36. 0086

37. 1188

38.2404

39. 3726

40. 5161

41. 6703

42. 8355

44. 0109

45. 1974

46. 3939

47. 6010

48. 8185

50.0462

51. 2837

52. 5314

53. 7892

55. 0569

56. 3343

57. 6213

58.9180

60. 2244

61. 5404
170 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

TABLE 4. Discharge, over a thin-edged weir per foot of crest Continued.

Head 

jff, feet.

7.0

7.1

7.2

7.3

7.4

7.5

7.6

7.7

7.8

7.9

8.0

8.1

8.2

8.3

8.4

8.5

8.6

8.7

8.8

8.9

9.0

9.1

9.2

9.3

9.4

9.5

9.6

9.7

9.8

9.9

10.0

.00

61. 6736

62. 9986

64.3343

65. 6793

67. 0336

68. 3969

69.7695

71. 1508

72. 5414

73. 9410

75. 3492

76. 7665

78. 1924

79. 6273

81.0705

82.5224

83. 9833

85. 4521

86. 9297

88. 4192

89. 9100

91. 4125

92.9237

94. 4428

95. 9703

97. 5057

99. 0492

100. 6010

102. 1607

103. 7282

105. 3039

.01

61.8048

63. 1318

64. 4685

65.8145

67. 1694

68. 5337

69.9070

71. 2896

72.6809

74. 0815

75. 4908

76. 9087

78. 3356

79. 7711

81.2154

82. 6682

84. 1298

85. 5996

87. 0778

88. 5647

90. 0599

91. 5633

93. 0782

94. 5950

96.1234

97. 6596

99. 2040

100. 7565

102. 3169

103. 88.53

105. 4618

.02

61. 9370

63. 2650

64. 6027

65. 9493

67. 3053

68. 6706

70. 0449

71. 4282

72.8208

74. 2220

75. 6320

77.0509

78.4788

79. 9153

81. 3602

82. 8141

84. 2763

85. 7472

87. 2264

88. 7139

90. 2064

91. 7142

93. 2267

94. 7475

96. 2766

97. 8138

99. 3589

100. 9123

102. 4734

104. 0429

105. 6199

.03

62. 0692

63. 3992

64. 7369

66.0845

67. 4415

68. 8078

70. 1827

71. 5670

72. 9603

74. 3626

75. 7735

77. 1934

78. 6220

80. 0592

81. 5054

82. 9600

84. 4228

85. 8947

87. 3745

88.8630

90. 3599

91. 8650

93. 3785

94.9000

96. 4298

97. 9679

99. 5141

101. 0678

102. 6299

104.2000

105. 7781

.04

62. 2017

63. 5317

64. 8711

66. 2197

67. 5777

68. 9447

70. 3209

71. 7059

73. 1002

74. 5031

75. 9150

77. 3360

78. 7655

80. 2034

81.6503

83.1058

84. 5697

86. 0455

87. 5231

89. 0126

90. 5101

92.0159

93. 5304

95. 0529

96.5833

98. 1021

99. 6689

101. 2237

102. 7868

104. 3575

105. 9363

1

.05

62. 3343

63. 6653

65.0056

66. 3552

67. 7139

69. 0818

70. 4591

71. 8451

73. 2400

74. 6439

76. 0569

77. 4784

78. 9087

80. 8479

81. 7955

83. 2517

84. 7165

86. 1897

87. 6716

89.1617

90. 6602

92.1671

98. 6822

95.2054

96. 7368

98. 2763

99. 8241

101. 3799

102. 9433

104. 5121

106. 0945

.06

62. 4671

63. 7991

65.1268

66. 4908

67. 8504

69. 2794

70. 5973

71. 9843

73. 3802

74. 7848

76. 1987

77. 6210

79. 0522

80. 4921

81.9406

83. 3979

84. 8634

86. 3376

87. 8204

89. 3113

90.4778

92. 3183

93. 8341

95. 3582

96. 8903

98. 4308

99. 9793

101. 5357

103. 1001

104. 6726

106. 2530

.07

62. 6000

63. 9327

65. 2750

66. 6263

67. 9869

69. 3566

70. 7355

72. 1235

73.5201

74. 9260

76. 3406

77. 7638

79. 1957

80. 6366

82. 0862

83.5440

85. 0106

86. 4854

87. 9689

89. 4608

90. 9609

92. 4694

93. 9863

95. 5111

97. 0442

98.5853

100. 1344

101. 6919

103. 2570

104. 8304

106. 4115

.08

62. 7329

64. 0665

65. 4095

66. 7618

68. 1235

69.4941

70. 8737

72. 2627

73. 6603

75. 0669

76. 4824

77.9067

79. 3396

80. 7811

82. 2314

83.6902

85. 1578

86. 6336

88. 1178

89. 6103

91. 1115

92. 6206

94. 1384

95. 6639

97. 1977

98. 7398

100. 2899

101. 8481

103. 4141

104. 9882

106. 5700

.09

62. 8657

64. 2004

65.5444

66. 8977

68. 2600

69. 6316

71.0123

72. 4743

73.8005

75. 2081

76. 6243

78. 0496

79. 4834

80. 9260

82. 3769

83. 8367

85.3049

86. 7815

88. 2666

89. 7602

91. 2620

92. 7721

94. 2906

95. 8171

97.3516

98. 8943

100. 4455

102.0042

103. 5710

105. 1461

106. 7285

When applied to a weir with N end contractions, the measured 

crest length U should be reduced by the formula

When applied to a weir having appreciable velocity of approach, 

the measured head should be corrected by the correction formula of 

Francis (see p. 15), or by one of the simpler approximate equivalents; 

or the correction may be applied as a percentage to the discharge, by 

the use of Table 2.

Table 3, taken from Lowell Hydraulic Experiments, by James B. 

Francis, gives the discharge for heads from zero to 3 feet, advancing 

by thousandths.

Table 4 is original and gives the discharge for heads from zero to 

10.09 feet, advancing by himdredths.
TABLES FOR CALCULATING WEIR DISCHARGE. 171

By increasing the quantities from either table 1 per cent, the discharge by the Cippoletti formula will be obtained,

Q= 3.36f Lfft.

In calculating discharge by this formula, the head should be corrected for velocity of approach by the formula

TABLES 5 AND 6. THREE-HALVES POWERS.

These tables of three-halves powers (cubes of the square roots) were 

prepared by the writer to facilitate the calculation of discharge over 

weirs of various forms, by the use of coefficients taken from the 

diagrams that accompany this paper and the base formula

Q= CLIF.

TABLE 5. Three-halves powers for numbers 0 to 1.49.

Numbers.

0.00

.01

.02

.03

.04

.05

.06

.07

.08

.09

0.10

.11

.12

.13

.14

.15

.16

.17

.18

.19

0.20

.21

.22

.23

.24

.25

.26

.27

.28

.000

0. 0000

.0010

.0028

.0052

.0080

.0112

.0147

.0185

.0226

.0270

0. 0316

.0365

.0416

.0469

.0524

.0581

.0640

.0701

.0764

.0828

0. 0894

.0962

.1032

.1103

.1176

.1250

.1326

.1403

.1482

. 29 . 1562

.001

0. 0001

. 00118

.00304

.00548

.00832

. 01155

. 01508

. 01891

. 02304

. 02746

0. 03209

. 03701

.04213

. 04745

.05297

.05869

. 06451

. 07073

. 07704

.08346

0.09008

. 09690

. 10391

. 11103

.11834

. 12576

.13337

.14109

.1490

. 15701

.002

0.0002

. 00136

. 00328

.00576

.00864

. 01190

. 01546

. 01932

.02348

. 02792

0. 03258

. 03752

.04266

.04800

.05354

. 05928

.06522

. 07136

.07768

.08412

0. 09076

. 09760

. 10462

. 11176

.11908

. 12652

. 13414

.14188

.1498

. 15782

.003

0.0003

.00154

.00352

.00604

.00896

. 01225

.01584

. 01973

.02392

.02838

0.03307

.03803

. 04319

.04855

.05411

. 05987

.06583

. 07199

. 07832

. 08478

0.09144

.09830

. 10533

. 11249

. 11982

. 12728

. 13491

.14267

.1506

.15863

.004

0. 0004

. 00172

. 00376

. 00632

.00928

. 01260

. 01622

. 02014

. 02436

.02884

0. 03356

.03854

. 04372

. 04910

.05468

.06046

. 06644

. 07262

. 07896

.08544

0. 09212

. 09900

.10604

. 11322

. 12506

. 12804

.13568

. 14346

.1514

. 15944

.005

0.0005

. 00190

.00400

.00660

.00960

. 01295

.01660

.02055

,02480

.02930

0. 03405

.03905

.04425

.04965

. 05525

.06105

. 06705

. 07325

. 07960

. 08610

0. 09280

.09970

. 10675

. 11395

. 12130

.12880

. 13645

.14425

.1522

. 16025

.006

0.0006

.00208

.00424

.00688

. 00992

. 01330

. 01698

. 02096

.02524

. 02976

0.03454

. 03956

.04478

. 05020

. 05582

. 06164

. 06766

. 07388

.08024

. 08676

0. 09348

.10040

. 10746

. 11468

. 12204

. 12956

. 13722

.14504

1530

. 16106

.007

0.0007

.00236

. 00448

.00716

. 01024

.01365

. 01736

.02137

.02568

.03022

0. 03503

.04007

.04531

.05075

. 05639

.06223

.06827

. 07451

. 08088

. 08742

0. 09416

. 10110

. 10817

. 11541

. 12278

. 13032

. 13799

. 14583

.1538

. 16187

.008

0.0008

. 00244

. 00472

. 00744

. 01056

.01400

. 01774

. 02178

.02612

.03068

00355.2

.04058

.04584

.05130

. 05696

. 06282

.06888

. 07514

. 08152

.08808

0.09484

.1018

.10888

. 11614

. 12352

. 13108

.13876

. 14662

.1546

.16268

.009

0. 0009

. 00262

.00496

. 00772

.01088

.01435

. 01812

. 02219

. 02656

. 03114

0. 03601

.04109

.04637

.05185

.05753

. 06341

.06949

.07577

. 08216

.08874

009552

.1025

. 10959

. 11687

.12426

. 13184

. 13953

. 14741

.1554

. 16349

.010

0. 0010

.0028

.0052

.0080

.0112

.0147

.0185

.0226

.0270

.0316

0.0365

.0416

.0469

.0524

.0581

.0640

.0701

.0764

.0828

.0894

0. 0962

.1032

.1103

.1176

.1250

.1326

.1403

.1482

.1562

.1643
172 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS. 

TABLE 5. Three-halves powers for numbers 0 to 1.49 Continued.

Numbers.

0.30

.31

.32

.33

.34

.35

.36

.37

.38

.39

0.40

.41

.42

.43

.44

.45

.46

.47

.48

.49

0.50

.51

.52

.53

.54

.55

.56

.57

.58

.59

0.60

.61

.62

.63

.64

.65

.66

.67

.68

.69

0.70

.71

.72

.73

.74

.75

.76

.77

.78

.79

.000

0. 1643

.1726

.1810

.1896

.1983

.2071

.2160

. 2251

. 2342

. 2436

0. 2530

. 2625

.2722

. 2820

.2919

.3019

.3120

. 3222

.3325

.3430

0.3536

.3642

.3750

.3858

.3968

.4079

.4191

.4303

.4417

.4532

0. 4648

.4764

.4882

.5000

.5120

.5240

.5362

.5484

.5607

.5732

0. 5857

.5983

.6109

.6237

.6366

.6495

.6626

.6757

.6889

.7022

.001

0. 16513

. 17344

.18186

.19047

.19918

. 20799

. 21691

. 22601

. 23514

. 24454

0. 25395

. 26347

. 27318

. 28299

. 2929

. 30291

. 31302

. 32323

. 33355

.34406

0. 35466

.36528

. 37608

.38690

. 39791

.40902

.42022

.43144

.44285

.45436

0. 46596

. 47758

. 48938

. 50120

.5132

. 52522

.53742

. 54963

. 56195

.57445

0. 58696

. 59956

. 61218

. 62499

. 63789

. 65081

. 66391

. 67702

.69023

.70353

.002

0. 16596

.17428

. 18272

. 19134

.20006

.20888

. 21782

. 22692

. 23608

. 24548

0. 25490

.26444

. 27416

. 28397

.2939

. 30392

.31404

. 32426

. 33460

. 34512

0. 35572

. 36636

. 37716

.38800

. 39902

. 41014

. 42134

. 43258

. 44400

.45552

0. 46712

. 47876

.49056

.50240

.5144

. 52644

. 53864

.55086

. 56320]

.5757

0. 58822

.60082

.61346

. 62628

. 63918

. 65212

. 66522

. 67834

. 69156

. 70486

.003

0. 16679

. 17512

. 18358

. 19221

. 20094

. 20977

. 21873

. 22783

. 23702

. 24642

0. 25585

.26541

.27514

. 28497

.2949

.30493

. 31506

. 32529

. 33565

.34618

0. 35678

.36744

. 37824

.38910

. 40013

. 41126

. 42246

. 43372

.44515

. 45668

0. 46828

. 47994

. 49174

.5036

.5156

. 52766

. 53986

. 55209

. 56445

. 57695

0. 58948

. 60208

. 61474

. 62757

.64047

. 65343

. 66653

. 67966

. 69289

. 70619

.004

0. 16762

.17596

.18444

.19308

. 20182

. 21066

. 21964

. 22874

. 23796

. 24736

0. 25680

.26638

. 27612

.28596

.2959

. 30594

. 31608

. 32632

. 33670

.34724

0. 35784

. 36852

.37932

.39020

. 40124

. 41238

.42358

.43486

.44630

.45784

0. 46944

.48112

.49292

.5048

.5168

.52888

.54108

.55332

. ,56570

. 57820

0. 59074

. 60334

. 61602

. 62886

. 64176

. 65474

. 66784

. 68098

.69422

. 70752

.005

0. 16845

. 17680

.18530

. 19395

.20270

. 21155

. 22055

. 22965

. 23890

.24830

0. 25775

.26735

. 27710

. 28695

.2969

. 30695

. 31710

. 32735

. 33775

.3483

0. 35890

. 36960

.38040

. 39130

. 40235

. 41350

. 42470

.43600

.44745

.45900

0. 47060

.48230

.49410

.5060

.5180

.53010

.54230

.55455

. 56695

. 57945

0. 59200

.60460

. 61730

.63015

.64305

. 65605

. 66915

.68230

. 69555

.70885

.006

0. 16928

. 17764

. 18616

. 19482

.20358

. 21244

. 22146

. 23056

.23984

. 24924

0. 25870

. 26832

. 27808

.28794

.2979

.30796

. 31812

.32838

. 33880

.34936

0. 35996

. 37068

. 38148

.39240

.40346

. 41462

. 42582

. 43714

.44860

. 46016

0. 47176

.48348

. 49528

.5072

.5192

. 53132

.54352

. 55578

. 56820

. 58070

0. 59326

.60586

. 61858

. 63144

. 64434

. 65736

.67046

. 68362

. 69688

. 71018

.007

0. 17011

. 17848

. 18702

. 19569

. 20446

. 21333

. 22237

.2317

.24078

. 25018

0. 25965

. 26929

.27906

.28893

.2989

. 30897

. 31914

.32941

. 33985

.35042

0. 36102

. 37176

.38256

. 39350

.40457

. 41574

.42694

. 43828

.44975

. 46132

0. 47292

.48466

.49646

.5084

.5204

.53254

. 54474

. 55701

.56945

. 58195

0. 59452

. 60712

. 61986

. 63273

.64563

.65867

. 67177

.68494

. 69821

. 71151

.008

0. 17094

. 17932

. 18788

. 19656

. 20534

. 21422

.22328

.23238

.24172

. 25112

0. 26060

. 27026

.28004

. 28992

.2999

. 30998

.32016

. 33044

.34090

. 35148

0. 36208

. 37284

. 38364

.39460

. 40568

. 41686

. 42806

. 43942

.45090

.46248

0. 47408

.48584

. 49764

.5096

.5216

. 53376

.54596

.55824

. 57070

. 58320

0. 59578

.60838

. 62114

. 63402

.64692

. 65998

. 67308

. 68626

.69954

.71284

.009

0. 17177

.18016

. 18874

. 19743

. 20622

. 21511

. 22419

. 23329

. 24266

.25206

0. 26155

. 27123

. 28102

.29091

.3009

.31099

. 32118

. 33147

.34195

. 35254

0.36314

. 37392

.38472

.3957

.40679

.41798

. 42918

.44056

. 45205

. 46364

0. 47524

. 48702

.49882

.5108

.5228

.53498

.54718

.55947

. 57195

.58445

0. 59704

.60964

. 62242

.63531

. 64821

. 66129

. 67439

. 68758

.70087

. 71417

.010

0. 1726

.1810

.1896

.1983

.2071

.2160

. 2251

.2342

.2436

.2530

0. 2625

. 2722

.2820

. 2919

.3019

.3120

.3222

.3325

.3430

.3536

0. 3642

.3750

.3858

.3968

.4079

.4191

.4303

.4417

.4532

.4648

0. 4764

.4882

.5000

.5120

.5240

.5362

.5484

.5607

.5732

.5857

0.5983

.6109

.6237

.6366

.6495

.6626

.6757

.6889

.7022

. 7155
TABLES FOR CALCULATING WEIR DISCHARGE.

TABLE 5. Ihree-Tialves powers for numbers 0 to 1.49 Continued.

173

Numbers.

0.80

.81

.82

.83

.84

.86

.86

.87

.88

.89

0.90

.91

.92

.93

.94

.95

.96

.97

.98

.99

1.00

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08

1.09

1.10

1.11

1.12

1.13

1.14

1.15

1.16

1.17

1.18

1.19

1.20

1.21

1.22

1.23

1.24

1.25

1.26

1.27

1.28

1.29

.000

0. 7155

.7290

.7426

.7562

.7699

.7837

.7975

.8115

.8255

.8396

0. 8538

.8681

.8824

8969

.9114

.9259

.9406

.9553

.9702

.9850

1.0000

1. 0150

1. 0302

1. 0453

1. 0606

1. 0759

1. 0913

1. 1068

1. 1224

1. 1380

1. 1537

1. 1695

1.1853

1. 2012

1. 2172

1. 2332

1. 2494

1. 2656

1. 2818

1. 2981

1. 3145

1. 3310

1.3475

1.3641

1.3808

1. 3975

1.4144

1. 4312

1. 4482

1. 4652

.001

0. 71685

. 73035

. 74387

. 75757

. 77128

. 78508

.79890

.8129

. 82691

.84102

0. 85523

. 86953

. 88385

. 89835

. 91285

. 92737

.94207

. 95679

. 97168

.9865

1. 0015

1. 01652

1. 03171

1. 04683

1. 06213

1.07744

1. 09285

1. 10836

1. 12396

1. 13957

1. 15528

1. 17108

1. 18689

1. 20280

1. 21880

1. 23482

1. 25102

1. 26722

1. 28343

1. 29974

1. 31615

1. 33265

1. 34916

1.36577

1. 38247

1.39919

1. 41608

1. 4329

1. 4499

1. 4669

.002

0. 71820

.7317

. 74524

. 75894

. 77266

. 78646

. 80030

.8143

.82832

. 84244

0. 85666

. 87096

.8853

89980

.9143

. 92884

. 94354

.95828

. 97316

.9880

1. 0030

1. 01804

1. 03322

1. 04836

1.06366

1.07898

1.09440

1. 10992

1. 12552

1.14114

1. 15686

1.17266

1. 18848

1. 20440

1. 22040

1. 23644

1. 25264

1. 26884

1. 28.506

1. 30138

1.31780

1. 33430

1. 35082

1.36744

1. 38414

1. 40088

1. 41776

1. 4346

1.4516

1. 4686

.003

0. 71955

. 73305

. 74661

. 76031

. 774C4

. 78784

.80170

.8157

. 82973

.84386

0. 85809

. 87239

. 88675

Qfll QC . yuizo

. 91575

. 93081

. 94501

. 95977

.97464

. 9895

1. 0045

1. 01956

1. 03473

1. 04989

1. 06519

1. 08052

1. 09595

1. 11148

1. 12708

1. 14271

1. 15844

1. 17424

1. 19007

1. 20600

1. 22200

1. 23806

1. 25426

1. 27046

1.28669

1. 30302

1. 31945

1. 33595

1. 35248

1. 36911

1. 38581

1. 40257

1. 41944

1.4363

1.4533

1. 4703

.004

0. 72090

. 73440

. 74798

. 76168

. 77542

. 78922

.8031

.8171

. 83114

. 84528

0. 85952

. 87382

. 88S20

9027(.

. 91720

. 93178

. 94648

. 96126

. 97612

.9910

1.0060

1. 02108

1. 03624

1. 05142

1. 06672

1. 08206

1.09750

1. 11304

1. 12864

1. 14428

1.16002

1. 17582

1. 19166

1. 20760

1. 22360

1. 23968

1. 25588

1. 27208

1. 28832

1. 30466

1. 32110

1. 33760

1.35414

1. 37678

1. 38748

1. 40426

1. 42112

1. 4380

1. 4550

1. 4720

.005

0. 72225

. 73575

. 74935

. 76305

. 7768Q

. 79060

.8045

. 8185

. 83255

. 84670

0. 86095

. 87525

. S89C.5

90415

. 91865

. 93325

. 94795

. 96275

. 97760

.9925

1. 0075

1. 02260

1. 03775

1. 05295

1. 06825

1. 08360

1. 09905

1. 11460

1. 13020

1. 14585

1. 16160

1.17740

1. 19325

1. 20920

1. 22520

1.24130

1. 25750

1. 27370

1. 28995

1. 30630

1. 32275

1. 33925

1. 35580

1. 37245

1. 38915

1. 40595

1. 42280

1. 4397

1. 4567

1. 4737

.006

0. 72360

. 73710

.75072

.76443

. 77818

. 99198

.8059

. 8199

.83396

. 84812

0. 86238

. 87668

. 89110

QASfi . yuoo

.92010

. 93472

. 94942

.96424

. 97908

.9940

1. 0090

1.02412

1.03926

1. 05448

1. 06978

1. 08514

1.10060

1.11616

1.13176

1. 14742

1. 16318

1. 17898

1.19484

1. 21080

1.22680

1. 24292

1. 25912

1. 27532

1.29158

1. 30794

1. 32440

1. 34090

1. 35746

1. 37412

1. 39082

1. 40764

1. 42448

1. 4414

1. 4584

1.4754

.007

0. 72495

. 73845

.75209

. 76579

. 77956

. 79336

.8073

.8213

. 83537

. 84954

0. 86381

. 87811

. 89255

. 90705

. 921.55

. 93619

. 95089

. 96573

.98056

.9955

1. 0105

1. 02564

1. 04077

1. 05601

1. 07131

1.08668

1.10215

1. 11772

1. 13332

1. 14899

1. 16476

1. 18056

1. 19643

1. 21240

1.22840

1.24454

1. 26074

1. 27694

1. 29321

1. 30958

1. 32605

1. 34255

1. 35912

1. 37579

1. 39249

1. 40933

1.42616

1. 4431

1. 4601

1. 4771

.008

0. 72630

. 73980

. 75346

. 76716

.78094

. 79474

.8087

.8227

. 83678

.85096

0. 86524

.87954

.8940

QAQFi 

. VUoO

.9230

. 93766

. 95236

. 96722

. 9S204

.9970

1. 0120

1. 02716

1. 04282

1. 05754

1. 07284

1. 08822

1. 10370

1. 11928

1. 13488

1. 15056

1. 16634

1. 18214

1. 19802

1. 21400

1. 23000

1. 24616

1. 26236

1. 27856

1. 29484

1. 31122

1. 32770

1. 34420

1. 36078

1. 37746

1. 39416

1. 41102

1. 42784

1. 4448

1.4618

1.4788

.009

0. 72765

. 74115

. 75483

.76853

. 78232

. 79612

.8101

.8241

. 83819

.85238

0. 86667

.88097

. 89545

90995

. 92445

. 93913

.95383

. 96871

. 98352

.9985

1.0135

1.02868

1. 04379

1.05907

1. 07437

1. 08976

1. 10525

1. 12084

1. 13644

1. 15213

1. 16792

1. 18372

1. 19961

1. 21560

1. 23160

1. 24778

1. 26398

1. 28018

1. 29647

1.31286

1. 32935

1. 34585

1. 36244

1. 37913

1.39583

1. 41271

1.42952

1. 4465

1. 4635

1. 4805

.010

0.7290

.7425

. 7562

.7699

.7837

.7975

. 8115

. 8255

.8396

. 8538

0. 8681

.8824

.8969

.9114

.9259

.9406

.9553

.9702

.9850

1.0000

1. 0150

1. 0302

1. 0453

1.0606

1. 0759

1. 0913

1. 1068

1. 1224

1. 1380

1. 1537

1. 1695

1. 1853

1.2012

1. 2172

1. 2332

1. 2494

1. 2656

1. 2818

^. 2981

1. 3145

1. 3310

1.3475

1. 3641

1.3808

1. 3975

1. 4144

1. 4312

1.4482

1. 4652

1.4822
174 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

TABLE 5. Time-halves poicersfor numbers 0 to 1.49 Continued.

Numbers.

1.30

1.31

1.32

1.33

1.34

1.35

1.36

1.37

1.38

1.39

1.40

1.41

1.42

1.43

1.44

1.45

1.46

1.47

1.48

1.49

.000

1. 4822

1. 4994

1. 5166

1. 5338

1.5512

1. 5686

1. 5860

1. 6035

1. 6211

1. 6388

1.6565

3.6743

1.6921

1. 7100

1. 7280

1. 74(10

1.7611

1.7823

1. 8005

1. 8188

.001

1.48392

1.50112

1. 51832

1. 53554

1. 55294

1. 57034

1. 58775

1. 60526

1. 62287

1. 64057

1. 65828

1. 67608

1. 69389

1. 7118

1. 7298

1. 74781

1.76592

1. 78412

1.80233

.002

1. 48564

1. 50284

1. 52004

1. 53728

1. 55468

1. 57208

1. 58950

1. 60702

1. 62464

1. 64234

1. 66006

1.67786

1. 69568

1. 7136

1. 7316

1. 74962

1. 76774

1. 78594

1. 80416

.003

1. 48736

1. 50456

1.52176

1. 53902

1. 55642

1. 57382

1. 59125

1. 60878

1. 62641

1. 64411

1. 66184

1. 67964

1. 69747

1. 7154

1. 7334

1. 75143

1.76956

1. 78776

1. 80599

.004

1.48908

1.50628

1. 52348

1. 54076

1. 55816

1. 57556

1. 59300

1. 61054

1. 62818

1. 64588

1. 66362

1. 68142

1. 69926

1. 7172

1. 7352

1. 75324

1. 77138

1. 78958

1. 80782

.005

1. 49080

1.50800

1. 52520

1. 54250

1. 55990

1. 57730

1. 59475

1. 61230

1. 62995

1.64765

.006

1. 49252

1. 50972

1. 52692

1.54424

1.56164

1. 57904

1.59650

1. 61406

1. 63172

1. 64942

1.66540] 1.66718

1. 68320

1. 70105

1. 7190

1. 7370

1.75505

1. 77320

1. 79140

1. 80965

1. 68498

1.70284

1.7208

1.7388

1.75686

1. 77502

1. 79322

1.81148

.007

1. 49424

1. 51144

1. 52864

1.54598

1. 56338

1. 58078

1. 59825

1. 61582

1. 63349

1. 65119

1. 66896

1.68676

1. 70463

1.7226

1.7406

1. 75867

1. 77684

1. 79504

1. 81331

.008

1. 49596

1. 51316

1. 53036

1. 54772

1. 56512

1. 58252

1.60000

1. 61758

1. 63526

1. 65296

1. 67075

1. 68854

1. 70642

1. 7244

1. 7424

1. 76048

1. 77866

1. 79686

1. 81514

.009

1. 49768

1. 51488

1. 53208

1. 54946

1. 56686

1.58426

1. 60175

1. 61934

1. 63703

1. 65473

1. 67252

1. 69032

1. 70821

1. 7262

1. 7442

1. 76229

1. 78048

1. 79868

1. 81697

.010

1.4994

1. 5166

1. 5338

1. 5512

1. 5686

1.5860

1. 6035

1. 6211

1. 6388

1.6565

1. 6743

1. 6921

1. 7100

1. 7280

1. 7460

1. 7641

1. 7823

1. 8005

1. 8188

TABLE 6. Three-halves powers for numbers from 0 to

%.

0.00 

.01 

.02

.03

.04

.05

.06

.07

.08

.09

0.10

.11

.12

.13

.14

.15

.16

.17

.18

0

0.0000 

.0010 

.0028

.0052

.0080

.0112

.0147

.0185

.0226

.0270

0. 0316

.0365

.0416

.0469

.0524

.0581

.0640

.0701

.0764

1

1.0000 

1.0150 

1. 0302

1.0453

2

2. 8284 

2. 8497 

2. 8710

2. 8923

1. 0606 2. 9137

1. 0759

1. 0913

1. 1068

1. 1224

1.1380

1. 1537

1. 1695

1. 1853

1. 2012

1. 2172

1. 2332

1. 2494

1.2656

1. 2818

. 19 . 0828'l. 2981

2. 9352

2. 9567

2. 9782

2.9998

3. 0215

3.0432

3. 0650

3.0868

3. 1086

3.1306

3. 1525

3. 1745

3. 1966

3. 2187

3. 2409

3

5. 1962 

5.2222 

5.2482

5. 2743

5.3004

5. 3266

5. 3528

5. 3791

5.4054

5. 4317

5. 4581

5.4845

5, 5110

5. 5375

5.5641

5. 5907

5. 6173

5. 6440

5. 6708

5. 6975

4

8.0000 

8.0300 

8.0601

8. 0902

8. 1203

8.1505

8.1807

8. 2109

8. 2412

8. 2715

8.3019

8. 3323

8. 3627

8. 3932

8. 4237

8. 4542

8.4848

8. 5154

8.5460

8. 5767

5

11. 1803 

11. 2139 

11. 2475

11. 2811

11. 3148

11. 3485

11. 3822

11. 4160

11. 4497

11. 4836

11. 5174

11. 5513

11. 5852

11. 6192

11. 6532

11. 6872

11. 7213

11. 7554

11. 7895

11. 8236

6

14. 6969 

14. 7337 

14. 7705

14. 8073

14. 8442

14. 8810

14. 9179

14. 9549

14. 9919

15. 0289

15. 0659

15. 1030

15. 1400

15. 1772

15. 2143

15. 2515

15. 2887

15. 3260

15. 3632

15. 4005

7

18. 5203 

18.5600 

18. 5997

18. 6394

18. 6792

18. 7190

18. 7589

18. 7988

18. 8387

18. 8786

18. 9185

18. 9585

18. 9985

19. 0386

19. 0786

19.1187

19. 1589

19. 1990

19. 2392

19. 2794

8

22. 6274 

22. 6699 

22. 7123

22. 7548

22. 7973

22. 8399

22. 8825

22. 9251

22. 9677

23.0103

23. 0530

23. 0957

23. 1384

23. 1812

23. 2240

23. 2668

23. 3096

23. 3525

23. 3954

23. 4383

9

27.0000 

27. 0450 

27.0890

27. 1351

27. 1802

27. 2253

27. 2705

27. 3156

27. 3608

27. 4060

27. 4512

27. 4965

27. 5418

27. 5871

27. 6324

27. 6778

27. 7232

27. 7686

27. 8140

27. 8595

10

31. 6228 

31. 6702 

31. 7177

31. 7652

31. 8127

31. 8602

31. 9078

31. 9554

32. 0030

32.0506

32. 0983

32. 1460

32. 1937

32. 2414

32. 2892

11

36. 4829 

36. 5326 

36.5824

36. 6322

36. 6820

36. 7319

36. 7818

36. 8317

36. 8816

36. 9315

36. 9815

37. 0315

37. 0815

37. 1315

37. 1816

32. 3370, 37. 2317

32. 3848

32. 4326

32. 4804

32. 5283

37. 2817

37. 3319

37. 3820

37. 4322
TABLES FOR CALCULATING WEIR DISCHARGE. 175 

TABLE 6. Three-halves powers for numbers from Oto 12 Continued.

\0.20

.21 

.22

.23

.24

0

0. 0894 

.0962 

.1032

1103

.1176

.25 | .1250

.26 139fi

.27 1403

.28 1489

.29

0.30

.31

. 1562

0. 1643

1796

1

1. 3145 

1. 3310 

1. 3475

1. 3641

1.3808

1. 3975

1.4144

1. 4312

1.4482

1. 4652

1. 4822

1 . 4994

. 32 . 1810 1. 5166

.33

.34

.35

. 1896 1. 5338

. 1983 1. 5512

.20711.5686

. 36 . 2160 1. 5860

.37

.38

.39

0.40

.41

.42

.43

.44 '

.45

*.46

.47

.48

.49

0.50

.51

.52

.53

.54

.55

.56

.57

.58

.59

0.60

.61

.62

.63

.64

.65

.66

.67

.68

.69

. 2251

.2342

.2436

0. 2530

. 2625

.2722

.2820

.2919

.3019

.3120

.3222

.3325

.3430

0.3536

.3642

.3750

.3858

.3968

.4079

.4191

.4303

.4417

.4532

0.4648

.4764

.4882

.5000

.5120

.5240

.5362

.5484

.5607

.5732

1

1. 6035

1. 6211

1.6388

1. 6565

1. 6743

1.6921

1.7100

1.7280

1. 7460

1. 7641

1. 7823

1.8005

1. 8188

1. 8371

1. 8555

1.8740

1.8925

1. 9111

1.9297

1.9484

1. 9672

1.9860

2. 0049

2.0238

2.0429

2. 0619

2. 0810

2.1002

2. 1195

2. 1388

2.1581

2.1775

2. 1970

-

3. 2631 

3. 2854 

3. 3077

3. 3301

3. 3525

3. 3750

3. 3975

3. 4201

3.4427

3.4654

3. 4881

3. 5109

3.5337

3. 5566

3. 5795

3. 6025

3.6255

3. 6486

3. 6717

3. 6949

3. 7181

3. 7413

3.7646

3.7880

3. 8114

3.8349

3.8584

3. 8819

3.9055

3. 9292

3. 9529

3. 9766

4.0004

4. 0242

4.0481

4. 0720

4.0%0

4.1200

4.1441

4. 1682

4.1924

4. 2166

4. 2408

4. 2651

4. 2895

4. 3139

4.3383

4. 3628

4.3874

4. 4119

8

5. 7243 

5. 7512 

5. 7781

5.8050

5.8320

5.8590

5. 8861

5. 9132

5. 9403

5. 9675

5. 9947

6. 0220

6.0493

6. 0767

6. 1041

6. 1315

6. 1590

6. 1865

6. 2141

6. 2417

6. 2693

6. 2970

6. 3247

6. 3525

6.3803

6. 4081

6.4360

6.4639

6. 4919

6. 5199

6.5479

6. 5760

6.6041

6. 6323

6.6605

6.6887

6. 7170

6. 7453

6. 7737

6.8021

6.8305

6.8590

6. 8875

6. 9161

6.9447

6. 9733

7.0020

7. 0307

7. 0595

7.0883

1

4

8. 6074 

8. 6382 

8. 6690

8. 6998

8. 7307

8. 7616

8. 7925

8. 8235

8.8545

8. 8856

8. 9167

8.9478

8. 9790

9. 0102

9. 0414

9. 0726

9. 1040

9.1353

9. 1667

9. 1981

9. 2295

9.2610

9.2925

9.3241

9.3557

9. 3873

9. 4189

9.4506

9. 4824

9. 5141

9. 5459

9. 5778

9. 6097

9. 6416

9. 6735

9.7055

9. 7375

9. 7695

9. 8016

9. 8337

9.8659

9. 8981

9. 9303

9.%26

9.9949

10. 0272

10. 0596

10. 0920

10. 1244

10. 1569

5

11. 8578 

11. 8920 

11. 9263

11. 9606

11. 9949

12. 0293

12.0636

12. 0981

12. 1325

12. 1670

12. 2015

12. 2361

12. 2706

12. 3053

12. 3399

12. 3746

12. 4093

12. 4440

12. 4788

12. 5136

12.5485

12. 5833

12. 6182

12. 6532

12. 6882

12. 7232

12. 7582

12. 7933

12.8284

12. 8635

12. 8986

12. 9338

12. 9691

13. 0043

13. 03%

13. 0749

13. 1103

13. 1457

13. 1811

13. 2165

13. 2520

13. 2875

13. 3231

13. 3587

13. 3943

13. 4299

13.4656

13.5013

13. 5370

13. 5728

6

15. 4379 

15. 4752 

15. 5126

15.5501

15. 5866

15. 6250

15. 6616

15. 7001

15. 7376

15. 7752

15. 8129

15. 8505

15. 8882

15. 9260

15. 9637

16. 0015

16. 0393

16. 0772

16. 1150

16. 1529

16. 1909

16. 2288

16. 2668

16. 3048

16. 3429

16. 3810

16. 4191

16. 4572

16. 4954

16. 5336

16. 5718

16. 6101

16. 6484

16. 6867

16. 7250

16. 7634

16. 8018

16. 8402

16. 8787

16. 9172

16. 9557

16. 9943

17. 0328

17. 0714

17. 1101

17. 1488

17. 1874

17. 2172

17. 2649

17. 3037

7

19. 3196 

19. 3599 

19. 4002

19. 4405

19. 4808

19. 5212

19. 5576

19. 6021

19. 6425

19. 6830

19. 72a5

19. 7641

19.8046

19. 8452

19. 8858

19. 9265

19. 9672

20. 0079

20. 0486

20. 0894

20. 1302

20. 1710

20. 2118

20. 2527

20. 2936

20. 3345

20. 3755

20. 4165

20. 4575

20. 4985

20. 5396

20.5807

20. 6218

20. 6630

20. 7041

20. 7453

20. 7866

20. 8278

20. 8691

20. 9104

20. 9518

20. 9931

21.0345

21.0759

21. 1174

21. 1589

21. 2004

21. 2419

21.2834

21. 3250

8

23. 4812 

23. 5242 

23. 5672

23. 6102

23. 6533

23. 6963

23. 7394

23. 7825

23. 8257

23. 8689

23. 9121

23. 9553

23. 9986

24. 0418

24. 0851

24. 1285

24. 1718

24. 2152

24. 2586

24. 3021

24. 3455

24. 3890

24. 4325

24. 4761

24. 5196

24.5632

24.6068

24. 6505

24. 6941

24. 7378

24. 7815

24. 8253

24.8691

24. 9129

24. 9567

25. 0005

25. 0444

25. 0883

25. 1322

25. 1762

25.2202

25.2642

25.3082

25. 3522

25. 3963

25.4404

25.4845

25. 5287

25. 5729

25. 6171

9

27. 9050 

27. 9514 

27. 9960

28. 0416

28. 0872

28. 1328

28. 1784

28. 2241

28. 2698

28. 3155

28.3612

28. 4069

28. 4527

28. 4985

28. 5444

28. 5902

28. 6361

28. 6820

28. 7279

28. 7739

28. 8199

28. 8659

28. 9119

28. 9579

29. 0040

29. 0501

29. 0962

29. 1424

29.1885

29. 2347

29. 2810

29. 3272

29. 3735

29. 4198

29.4661

29. 5124

29. 5588

29. 6052

29. 6516

29. 6980

29. 7445

29. 7910

29. 8375

29.8841

29. 9306

29. 9772

30.0238

30.0704

30. 1171

30.1638

10

32. 5762 

32. 6241 

32. 6720

39 7200

3? 7680

32. 8160

R9 8640

3?, 9121

3? 9600

33.0083

33. 0564

33 1046

33. 1527

33. 2009

33. 2492

33 2974

33 3457

33. 3940

33.4423

33. 4906

33. 5390

33. 5874

33. 6358

33.6842

33. 7327

33. 7811

33. 8297

33. 8782

33. 9267

33. 9753

34.0239

34. 0725

34. 1211

34. 1698

34. 2185

34. 2672

34. 3159

34. 3647

34. 4135

34. 4623

34. 5111

34. 5599

34.6088

34. 6577

34. 7066

34. 7557

34.8045

34.8535

34.9025

34. 9516

11

37. 4824 

37. 5326 

37. 5828

37. 6331

37. 6833

37. 7336

37. 7840

37. 8343

37. 8847

37. 9351

37. 9855

38. 0359

38.0864

38.1369

38. 1874

38. 2379

38.2884

38. 3390

38. 38%

38.4402

38. 4908

38. 5415

38. 5922

38. 6429

38. 6936

38.7443

38. 7951

38. 8459

38. 8967

38.9475

38.9984

39. 0493

39. 1002

39. 1511

39. 2020

39. 2530

39. 3040

39. 3550

39. 4060

39. 4571

39. 5082

39. 5593

39. 6104

39. 6615

39. 7127

39. 7639

39. 8151

39. 8663

39. 9176

39. 9689
176 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS. 

TABLE 6. Three-halves powers for numbers from 0 to 12 Continued.

\%v>v?

0.70 

.71 

.72

.73

. 74

.75

.76

.77

.78

.79

0.80

.81

.82

. 83

.84

.85

.86

.87

.88

.89

0.90

.91

.92

.93

.94

.95

.96

.97

.98

.99

1.00

U

0. 5857 

.5983 

.6109

.6237

.6366

.6495

.6626

.6757

.6889

.7022

0. 7155

.7290

.7425

. 7562

.7699

.7837

.7975

.8115

.8255

.8396

0. 8538

.8681

.8824

.8969

.9114

.9259

.9406

.9553

.9702

.9850

1.0000

1

2. 2165 

2. 2361 

2. 2558

2. 2755

2. 2952

2. 3150

2. 3349

2.3548

2.3748

2. 3949

2.4150

2. 4351

2. 4553

2. 4756

2. 4959

2. 5163

2. 5367

2. 5572

2. 5777

2. 5983

2.6190

2.6397

2. 6604

2. 6812

2. 7021

2. 7230

2.7440

2. 7650

2. 7861

2.8072

2.8284

2

4. 4366 

4. 4612 

4.4859

4.5107

4. 5355

4.5604

4. 5853

4. 6102

4. 6352

4.6602

4. 6853

4. 7104

4.7356

4. 7608

4. 7861

4. 8114

4. 8367

4. 8621

4. 8875

4. 9130

4. 9385

4.9641

4. 9897

5.0154

5. 0411

5.0668

5. 0926

5. 1184

5.1443

5. 1702

5. 1962

3

7. 1171 

7. 1460 

7. 1749

7. 2038

7. 2328

7. 2618

7.2909

7. 3200

7. 3492

7. 3783

7. 4076

7. 4368

7. 4661

7. 4955

7.5248

7. 5542

7.5837

7. 6132

7. 6427

7. 6723

7. 7019

7. 7315

7.7702

7.7909

7.8207

7. 8505

7.8803

7. 9102

7.9401

7. 9700

8.0000

*

10. 1894 

10. 2214 

10. 2545

10. 2871

10. 3197

10.3524

10. 3851

10. 4178

10. 4506

10. 4834

10. 5163

10. 5492

10. 5812

10.6150

10. 6480

10. 6810

10. 7141

10. 7472

10. 7803

10. 8134

10. 8466

10. 8798

10. 9131

10. 9464

10. 9797

11. 0131

11. 0464

11.0799

11. 1133

11. 1468

11. 1803

5

13. 6086 

13. 6444 

13. 6803

13. 7161

13. 7521

13. 7880

13. 8240

13. 8600

13. 8961

13. 9321

13. 9682

6

17. 3425 

17. 3814 

17. 4202

17. 4591

17. 4981

17. 5370

17. 5760

17. 6150

17. 6541

17. 6931

17. 7322

14.0044'l7.7714

14. 0406^7. 8105

14. 0768

14. 1130

14. 1493

14. 1856

14. 2219

14. 2582

14. 2946

14. 3311

14. 3675

14. 4040

14. 4405

14.4770

14. 5136

14.5502

14. 5869

14. 6235

14. 6602

14. 6969

17. 8507

17. 8889

17. 92S2

17. 9674

18.0067

18. 0461

18.0854

18. 1248

18. 1642

18.2037

18. 2432

18. 2827

18. 3222

18, 3617

18. 4013

18.4409

18, 4806

18. 5203

7

21. 3666 

21. 4083 

21. 4499

21.4916

21. 5333

21. 5751

21. 6169

21. 6587

21. 7005

21. 7423

21. 7842

21. 8261

21. 8681

21. 9100

21. 9520

21. 9940

22. 0361

22. 0781

22. 1202

22. 1623

22. 2045

22.2467

22.2889

22. 3311

22. 3733

22. 4156

22. 4579

22. 5003

22. 5426

22. 5850

22. 6274

8

25. 6613 

25. 7056 

25.7499

25. 7942

25. 8395

25. 8828

25. 9272

25. 9716

26. 0161

26. 0605

26. 1050

26. 1495

26. 1941

26. 2386

26. 2832

26. 3278

26. 3725

26. 4171

26. 4618

26. 5065

26.5523

26. 5960

26, 6408

26. 6856

26. 7305

26. 7753

26. 8202

26. 8651

26.9100

26. 9550

27.0000

9

30. 2105 

30. 2572 

30. 3040

30. 3507

30. 3975

30.4444

30. 4912

30. 5381

30.5850

30. 6319

10

35. 0006 

35. 0497 

35. 0988

35. 1479

35. 1971

35. 2462

35.2954

35.3446

35. 3939

35.4431

30.678935.4924

30. 725835. 5417

30. 7728 35. 5911

30. 8198 35. 6404

30. 8669 35. 6898

30.913935.7392

30. 9610 35. 7886

31. 0081 35. 8380

31. 0553 35. 8875

31. 1024

31. 1496

35. 9370

35. 9865

31. 1968 36. 0360

31. 2441 36. 0856

31. 2913 36. 1352

31. 3386 36. 1848

31. 3850 36. 2344

31. 4332 36. 2841

31. 4806 36. 3337

31. 5280 36. 3834

31.575436.4331

31. 6228 36. 4829

11

40. 0202 

40. 0715 

40. 1228

40. 1742

40. 2256

40. 2770

40. 3284

40. 3798

40.4313

40. 4828

40. 5343

40. 5859

40. 6374

40.6890

40. 7406

40. 7922

40. 8439

40. 8955

40.9472

40.9989

41. 0507

41. 1024

41. 1542

41. 2060

41. 2578

41. 3097

41, 3615

41. 4134

41. 4653

41. 5173

41. 5692

The tables of three-halves powers may conveniently be used 

in conjunction with Oelle's Rechentafeln, or similar tables of the 

products of pairs of factors. G will usually be constant, or nearly so. 

Entering- Crelle's tables with G or CL as an argument, the discharge

3

corresponding to values of H^ read from the tables here given may be 

taken out directly, and usually with sufficient precision at least for 

1 foot length of crest, without any arithmetical computation. Table 5

gives H 

sandths.

for values of ZTfrom zero to 1.5 feet, advancing by thouIn Table 6 the increment is 0.1 foot, and the range zero to

12 feet. Should IT* be required for larger values of H, it ma}^ be 

found from the three-halves power of \H, by the formula

(114)
TABLES FOR CALCULATING- WEIR DISCHARGE. 177

TABLE 7. FLOW OVER BROAD-CRESTED WEIRS, WITH STABLE

NAPPE.

This table gives values of 

where

The derivation of this coefficient is given in connection with discussion of broad-crested weirs (pp. 119-121). It may be applied to broadcrested weirs of any width of cross section exceeding 2 feet within such 

limiting heads that the nappe does not adhere to the downstream face 

of the weir for low heads nor tend to become detached with increased 

head. Under the latter condition the coefficient increases to a limit 

near the value which applies for a thin-edged weir, a point being 

finally reached where the nappe breaks entirely free from the broad 

crest and discharges in the same manner as for a thin-edged weir. 

The coefficient, 2.64, may often be applied for weirs exceeding 2-feet 

crest width and for heads from 0.5 foot up to 1.5 or 2 times the breadth 

of weir crest. If corrections for the velocity of approach are required 

the Francis correction formula, or its equivalent, should be used.

TABLE 7. Weir discharge per foot of crest length. 

[Coefficient Ci=2.64.]

Head H, feet.

\

0.00 

.01 

.02 

.03

.04

.05

,06

.07

.08

.09

0.10

.11

.12

.13

.14

.15

.16

.17

.18

.19

I

0

0.000 

.003 

.007 

.014

.021

.030

.039

.049

.060

.071

0.083

.096

.110

.124

.138

.153

.169

.185

.202

.218

'

2.64 

2.68 

2.72 

2.76

2.80

2.84

2.88

2.92

2.96

3.00

3.04

3.09

3.13

3.17

3.21

3.26

3.30

3.34

3.38

3.43

2

7.47 

7.52 

7.58 

7.64

7.69

7.75

7.81

7.86

7.92

7.98

8.03

8.09

8.15

8.21

8.26

8.32

8.38

8.44

8.50

8.56

3

13.7 

13.8 

13.8 

13.9

14.0

14.1

14.1

14.2

14.3

14.3

14.4

14.5

14.5

14.6

14.7

14.8

14.8

14.9

15.0

15.0

4

21.1 

21.2 

21.3 

21.4

21.4

21.5

21.6

21.7

21.8

21.8

21.9

22.0

22. 1

22.2

22.2

22.3

22.4

22.5

22.6

22.6

5

29.5 

29.6 

29.7 

29.8

29.9

30.0

30.0

30.1

30.2

30.3

30.4

30.5

30.6

30.7

30.8

30.8

30.9

31.0

31.1

31.2

6

38.8 

38.9 

39.0 

39.1

39.2

39.3

39.4

39.5

39.6

39.7

39.8

39.9

40.0

40.1

40.2

40.3

40.4

40.5

40.6

40.7

7

48.9 

49.0 

49.1 

49.2

49.3

49.4

49.5

49.6

49.7

49.8

49.9

50.0

50.2

50.3

50.4

50.5

50.6

50.7

50.8

50.9

8

59.7 

59.8 

59.9 

60.1

60.2

60.3

60.4

60.5

60.6

60.7

60.8

61.0

61.1

61.2

61.3

61.4

61.5

61.6

61.8

61.9

9

71.3 

71.4 

71.5 

71.6

71.7

71.9

72.0

72.1

72.2

72.3

72.5

72.6

72.7

72.8

72.9

73.1

73.2

73.3

73.4

73.5

10

83.5 

83.6 

83.7 

83.9

84.0

84.1

84.2

84.4

84.5

84.6

84.7

84.9

85.0

85.1

85.2

85.4

85.5

85.6

85.7

85.9
178 WEEK EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

TABLE 7. Weir discharge per foot of crest length Continued.

Head H, feet.

x

\

0.20 

.21 

. 22 

.23

.24

.25

.26

.27

.28

.29

0.30

.31

.32

.33

.34

.35

.36

.37

.38

.39

0.40

.41

.42

.43

.44

.45

.46

.47

.48

.49

0.50

.51

.52

.53

.54

.55

.56

.57

.58

.59

0.60

.61

.62

.63

.64

.65

.66

.67

.68

.69

0

0.236 

.254 

.272 

.291

.310

.330

.350

.370

.391

.412

0.434

.456

.478

.500

.524

.547

.570

.594

.618

.643

0.668

.693

.719

.744

.771

.797

.824

.851

.878

.905

0.934

.961

.990

1.02

1.05

1.08

1.11

1.14

1.17

1.20

1.23

1.26

1.29

1.32

1.35

1.38

1.42

1.45

1.48

1.51

!

i

3.47 

3.51 

3.56 

3.60

3.64

3.69

3.73

3.78

3.82

3.87

3.91

3.96

4.06

4.05

4.10

4.14

4.19

4.23

4.28

4.33

4.37

4.42

4.47

4.51

4.56

4.61

4.66

4.70

4.75

4.80

4.85

4.90

4.95

5.00

5.04

5.09

5.14

5.19

5.24

5.29

5.34

5.39

5.44

5.49

5.54

5.60

5.65

5.70

5.75

5.80

8.61 

8.67 

8.73 

8.79

8.85

8.91

8.97

9.03

9.09

9.15

9.21

9.27

9.33

9.39

9.45

9.51

9.57

9.63

9.69

9.75

9.82

9.88

9.94

10.0

10.1

10.1

10.2

10.2

10.3

10.4

10.4

10.5

10.6

10.6

10.7

10.8

10.8

10.9

10.9

11.0

11.1

11.1

11.2

11.2

11.3

11.4

11.4

11.5

11.6

11.6

3

15.1 

15.2 

15.2 

15.3

15.4

16.5

15.5

15.6

15.7

15.8

15.8

15.9

16.0

16.0

16.2

16.2

16.3

16.3

16.4

16.5

16.6

16.6

16.7

16.8

16.8

16.9

17.0

17.1

17.1

17.2

17.3

17.4

17.4

17.5

17.6

17.7

17.7

17.8

17.9

18.0

18.0

18.1

18.2

IS. 2

18.3

18.4

18.5

18.6

18.6

18.7

4

22.7 

22.8 

22.9 

23.0

23.0

23.1

23.2

23.3

23.4

23.4

23.5

23.6

23.7

33.8

23.9

24.0

24.1

24.1

24.2

24.3

24.4

24.4

24.5

24.6

24.7

24.8

24.9

24.9

25.0

25.1

25.2

25.3

25.4

25.4

25.5

25.6

25.7

25.8

25.9

26.0

26.0

26.1

26.2

26.3

26.4

26.5

26.6

26.6

26.7

26.8

5

31.3 

31.4 

31.5 

31.6

31.7

31.8

31.8

31.9

32.0

32.1

32.2

32.3

32.4

32.5

32.6

32.7

32.8

32.8

32.9

33.0

33.1

33.2

33.3

33.4

33.5

33.6

33.7

33.8

33.9

34.0

34.0

34.1

34.2

34.3

34.4

34.5

34.6

34.7

34.8

34.9

35.0

35.1

35.2

35.3

35.4

35.4

35.5

35.6

35.7

35.8

6

40.8 

40.9 

41.0 

41.0

41.1

41.2

41.3

41.4

41.5

41.6

41.7

41.8

41.9

42.0

42.1

42.2

42.3

42.4

42.5

42.6

42.7

42.8

42.9

43.0

43.1

43.2

43.3

43.4

43.6

43.6

43.7

43.8

44.0

44.1

44.2

44.3

44.4

44.5

44.6

44.7

44.8

44.9

45.0

45.1

45.2

45.3

45.4

45.5

45.6

45.7

1

51.0 

51.1 

51.2 

51. 3

51.4

51.5

51.6

51.7

51.9

52.0

52.1

52.2

52.3

52.4

52.5

52.6

52.7

52.8

52.9

53.0

53.1

53.2

53.4

53.5

53.6

53.7

53.8

53.9

54.0

54.1

54.2

54.3

54.4

54.6

54.7

54.8

54.9

55.0

55.1

55.2

55.3

55.4

55.5

55.6

55.7

55.9

56.0

56.1

56.2

56. 3 (

8

62.0 

62.1 

62.2 

62 3

62.4

62.6

62.7

62.8

62.9

63.0

63.1

63.2

63.4

63.5

63.6

63.7

63.8

63.9

64.0

64.2

64.3

64.4

64.5

64.6

64.7

64.8

65.0

65.1

65.2

65.3

65.4

65.5

65.6

65.8

65.9

66.0

66.1

66.2

66.3

66.5

66.6

66.7

66.8

66.9

67.0

67.2

67.3

67.4

67.5

67.6

9

73.7 

73.8 

73.9 

74.0

74.1

74.3

74.4

74.5

74.6

74.8

74.9

75.0

75.1

75.2

75.4

75.5

75.6

75.7

75.8

76.0

76.1

76.2

76.3

76.4

76.6

76.7

76.8

76.9

77.0

77.2

77.3

77.4

77.5

77.7

77.8

77.9

78.0

78.2

78.3

73.4

78.5

78.6

78.8

78.9

79.0

79.1

79.3

79.4

79.5

79.6

10

86.0 

86.1 

86.2 

86.4

86.5

86.6

86.8

86.9

87.0

87.1

87.3

87.4

87.5

87.6

87.8

87.9

88.0

88.2

88.3

88.4

88.5

88.7

88.8

88.9

89.0

89.2

89.3

89.4

89.6

89.7

89.8

90.0

90.1

90.2

90.3

90.5

90.6

90.7

90.8

91.0

91.1

91.2

91.4

91.5

91.6

91.8

91.9

92.0

92.1

92.3
TABLES FOB CALCULAT1NU WEIR DISCHARGE. 

TABLE 7. Weir discharge per foot of crest length Continued.

179

HeadB", feet.

\

0.70

.71 

.72 

.73

.74

.75

.76

.77

.78

.79

0.80

.81

.82

.83

.84

.85

.86

.87

.88

.89

0.90

.91

.92

.93

.94

.95

.96

.97

.98

.99

1.00

0

1.55 

1.58 

1.61 

1.65

1.68

1.71

1.75

1.78

1.82

1.85

1.89

1.92

1.96

2.00

2.03

2.07

2.10

2.14

2.18

2.22

2.25

2.29

2.33

2.37

2.41

2.44

2.48

2.52

2.56

2.60

2.64

1

5.85 

5.90 

5.96 

6.01

6.06

6.11

6.16

6.22

6.27

6.32

6.38

6.43

6.48

6.54

6.59

6.64

6.70

6.75

6.80

6.86

6.91

6.97

7.02

7.08

7.13

7.19

7.24

7.30

7.36

7.41

7.47

2

11.7 

11.8 

11.8 

' 11.9

12.0

12.0

12.1

12.2

12.2

12.3

12.4

12.4

12.5

12.6

12.6

12.7

12.8

12.8

12.9

13.0

13.0

13.1

13.2

13.2

13.3

13.4

13.4

13.5

13.6

13.6

13.7

3

18.8 

18.9 

" 18.9 

19.0

19.1

19.2

19.2

19.3

19.4

19.5

19.6

19.6

19.7

19.8

19.9

19.9

20.0

20.1

20.2

20.2

20.3

20.4

20.5

20.6

20.6

20.7

20.8

20.9

21.0

21.0

21.1

4

26.9 

27.0 

27.1 

27.2

27.2

27.3

27.4

27.5

27.6

27.7

27.8

27.8

27.9

28.0

28.1

28.2

28.3

28.4

28.5

28.5

28.6

28.7

28.8

28.9

29.0

29.1

29.2

29.3

29.3

29.4

29.5

5

35.9 

36.0 

36.1 

36.2

36.3

36.4

36.5

36.6

36.7

36.8

36.9

37.0

37.1

37.2

37.3

37.4

37.4

37.5

37.6

37.7

37.8

37.9

38.0

38.1

38.2

38.3

38.4

38.5

38.6

38.7

38.8

6

45.8 

45.9 

46.0 

46.1

46.2

46.3

46.4

46.5

46.6

46.7

46.8

46.9

47.0

47.1

47.2

47.3

47.4

47.5

47.6

47.7

47.8

48.0

48.1

48.2

48.3

48.4

48.5

48.6

48 7

48.8

48.9

7

56.4 

56.5 

56.6 

56.7

56.8

57.0

57.1

57.2

57.3

57.4

57.5

57.6

57.7

57.8

58.0

58.1

58.2

58.3

58.4

58.5

58.6

58.7

58.8

59.0

59.1

59.2

59.3

59.4

59.5

59.6

59.7

8

67.7 

67.9 

68.0 

68.1

68.2

68.3

68.4

68.6

68.7

68.8

68.9

69.0

69.2

69.3

69.4

69.5

69.6

69.7

69.9

70.0

70.1

70.2

70.3

70.4

70.6

70.7

70.8

70.9

71.0

71.2

71.3

9

79.8 

79.9 

80.0 

80.1

80.2

80.4

80.5

80.6

80.7

80.9

81.0

81.1

81.2

81.4

81.5

81.6

81.7

81.9

82.0

82.1

82.2

82.4

82.5

82.6

82.7

82.9

83.0

83.1

83.2

83.4

83.5

10

92.4 

92.5 

92.7 

92.8

92.9

93.0

93.2

93.3

93.4

93.6

93.7

93.8

94.6

94.1

94.2

94.4

94.5

94.6

94.7

94.9

95.0

95.1

95.3

95.4

95.5

95.6

95.8

95.9

96.0

96.2

96.3
180 WEIE EXPEEIMENTS, COEFFICIENTS, AND FORMULAS.

TABLE 8. BACKWATER CAUSED BY A DAM OR WEIR.

In a channel of uniform depth, width, and slope, let

D Original uniform depth.

d%= Depth at the dam or obstruction.

d1 Depth at a point upstream. 

/= Distance upstream to the point dr 

w = Width of channel.

^= Distance upstream to the "hydrostatic limit." 

S= Natural uniform slope or inclination of water surface and

stream bed, assumed parallel.

g= Acceleration of gravity. __ 

G Coefficient in the Chezy or slope formula v= C^RS,

where B is the ' hydraulic radius=-^ea of se.ctio,n 

wetted perimeter

The value of G varies for rivers from about 50 to 140. 

The distance upstream from the obstruction at which the depth will 

be dj_ may be found by the formula

I is a function of -=, whose value c#n be expressed mathematically

only as a transcendental equation. The numerical values of this function are given in Table 8. F± will be found opposite the argument

^, and

FIG. 15. Concave backwater surface.

The inverse problem of tinding the depth at any given distance / 

upstream can be solved only by successive trials.

Using the above equation, a series of values of d1 may be determined 

giving in tabular form the corresponding values of I. From this data 

the form of the surface curve may be graphically shown or the depth 

of back piling at any point may be interpolated.

If #=5, <Zg =10, (7=75, £=0.0001, ^=0.5, and ^8 =0.1318,

1 i-
TABLES FOE CALCULATING WEIE DISCHABGE. 181

Column (2) in the following- table gives the values of I for various 

values of d^ computed by means of formula (115).

Form of backwater curve above a dam.

d

Depth, 

feet 

d).

9

8

7

6.5

6.4

6.3

6.2

6.1

6

5.5

5.3

5.1

I

Distance 

from dam 

todepthdi, 

feet 

(2).

11,687

24, 277

38, 526

47, 024

48, 798

50, 796

52, 813

55, 010

57,240

67, 252

82,940

101,255

 S

Hydrostatic 

depth at 

distance /, 

feet 

(3).

8.83

7 57

6.15

5.30

5.13

a 5. 00

a 5. 00

a 5. 00

a 5. 00

a 5. 00

a 5. 00

«5.00

dj-«

Depth of 

"back 

piling," 

feet 

W-

.17

.43

.85

1.20

1.27

1.30

1.20

1.10

1.00

.50

.30

.10

a Above hydrostatic limit.

If the pond formed by the dam were level, the hydrostatic depth 

at any distance upstream would be

-^- sn (116)

Column (3) in the above table shows this factor for the several values 

of I. The true "back piling" or rise due to the surface curvature is 

expressed by the difference d1 S^ as given in colump (4).

This quantity has a maximum value at the hydrostatic limit, or terminus of the level pond, where d D.

Its location is such that if l^ is the distance upstream from the dam

sin 8

(117)

FIG. 16. Convex backwater surface.

In the example given the hydrostatic limit occurs at a distance 

^=50,000 feet above the dam, at which point the maximum back piling 

of about 1.31 feet occurs.

Above the hydrostatic limit the depth of back piling is d^ D.
182 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

When £> JTs ........ (118)

the pond surface will not be concave, but a remou or hydraulic 

jump will occur, having #, height

(119)

where vl is the mean velocity corresponding to dr To find the distance upstream to the point where the jump occurs, solve equation 

(115) for the value of <#2 , found by formula (119).

If £=0.004 67=100 -^1=0.003216.

if,/ -1 r-- 7? ^22 j . w o ~110~~

= 13.49 feet per second.

Let the depth at the dam be 10 feet; using </8 as found above as the 

terminal depth in formula (115), we obtain

10-Y.48

0.004

P= 0.1318

Z=630+(250-311)X 0.126 = 622.3 feet.

The hydrostatic limit in this case is

10 _ 5

If the channel above an obstruction consists of successive reaches 

having different slopes or cross sections, the depth at the head of the 

first reach or level may be found by the method outlined, and using 

this as the initial depth d%, a similar solution may be made for the 

second and succeeding levels. a

a Table 8 has been extended from Bresse's original table by interpolation. Demonstrations of tlie 

formulas here given may be found in Merriman's or Bovey's Hydraulics. In case of a fall a different 

function must be employed. Its values will be found in the works mentioned.
TABLES FOB CALCULATING WEIR DISCHARGE. 183

TABLE 8. Backivater function (F) for a dam or ohstmction.

[The column headings are hundredths for values of -^ from zero to 0.-9, thousandths for values of -g 

from 0.30 to 0.899, and ten-thousandths for values of from 0.900 to 0.999.]..

I)

d

0.0

.1

.2

0.30

.31 '

.32

.33

.34

.35

.36

.37

.38

.39

0.40

.41

.42

.43

.44

.45

.46

.47

.48

.49

0.50

.51

.52

.53

.54

.55

.56

.57

.58

.59

0.60

.61

.62

.63

.64

.65

.66

.67

.68

.69

0

F

0. 0000

.0050

. 0201

0. 0455

.0486

.0519

.0553

.0587

.0623

.0663

.0699

.0738

.0779

0. 0821

.0865

.0909

.0995

.1003

. 1052

.1102

.1154

.1207

.1262

0.1318

.1376

. 1435

.1497

.1560

.1625

.1692

.1761

.1832

.1905

0. 1980

.2058

.2138

.2221

.2306

.2395

.2486

.2580

.2667

.2778

1

F

0.0001

.0061

.0221

0. 0458

.0489

.0522

.0556

.0591

. 0627

.0664

.0703

.0742

.0783

0. 0825

.0869

.0918

.0996

.1008

.1057

.1107

.1159

.1212

.1268

0. 1324

.1382

2

F

0.0003

.0072

.0243

0. 0461

.0493

.0526

.0560

.0594

.0630

.0668

.0707

.0746

.0787

0.0830

.0874

.0926

.0997

.1013

.1062

.1112

.1165

.1218

.1273

0. 1330

.1388

.1441 .1447

. 1503 . 1510

. 1566 . 1573

. 1632 . 1638

.1699

.1768

.1839

.1912

0. 1988

.2066.

.2146

.2230

.2315

.2404

.2495

.2589

.2678

.2788

.1706

.1775

.1847

.1920

1

3

F

0. 0005

.0085

.0266

0. 0464

.0496

.0529

.0563

.0698

. 0634

.0672

.0711

.0750

.0792

0.0834

.0878

.0935

.0997

.1018

.1067

.1118

.1170

.1224

.1279

0.1335

.1394

.1454

.1516

.1580

.1645

.1713

.1782

.1854

.1928

0. 1996 0. 2003

.2074

.2155

.2238

.2324

.2413

.2505

.2597

.2689

.2799

.2082

.2163

.2246

.2333

2422

.2514

.2606

.2700

.2810

4

F

0. 0009

.0098

.0290

0. 0467

.0499

.0533

5

F

0. 0013

.0113

.0314

0. 0471

.0503

.0536

. 0567 . 0570

.0601

.0638

.0676

.0715

.0754

.0796

0. 0839

.0883

.0943

.0998

.1023

.1072

.1123

.1175

.1229

.1284

0. 1341

.1400

.1460

.1522

.1586

.1652

.1720

.1789

.1861

.1935

0. 2011

. 2090

.2171

.2255

.2342

.2431

.2524

.2615

.2711

.2820

.0605

.0642

.0680

.0718

.0758

.0800

0.0843

.0887

.0952

.0999

.1030

.1077

.1128

.1180

.1234

.1290

0.1347

.1406

.1466

.1528

.1592

.1658

. 1726

.1796

.1868

.1942

0. 2019

.2098

.2180

.2264

.2350

.2440

.2533

.2624

.2722

.2830

6

F

0. 0018

.0128

.0340

0. 0474

.0506

. 0539

.0573

.0609

.0645

.0683

.0722

.0763

.0804

0. 0847

.0891

.0961

.1000

.1032

.1082

.1133

.1186

.1240

.1296

0. 1353

.1411

.1472

.1535

.1599

. 1665

.1733

.1804

.1876

.1950

0. 2027

.2106

.2188

.2272

.2359

.2450

.2542

.2632

.2734

.2841

1

7

F

0. 0026

.0045

.0367

0. 0477

.0509

. 0543

.0577

.0612

.0649

.0687

.0726

.0767

.0808

0.0852

.0896

.0969

.1001

.1037

.1087

.1138

.1191

.1246

.1301

0. 1359

.1417

.1478

.1541

.1606

.1672

.1740

.1811

.1883

.1958

0. 2035

.2114

.2196

.2280

.2368

.2459

.2552

.2641

.2745

.2852

8 9

F

0. 0034

.0162

.0395

0. 0480

.0512

.0546

. 0580

.0616

.0653

.0691

.0730

.0771

.0813

0.0856

.0900

.0978

.1001

.1042

.1092

.1144

.1196

.1215

.1307

0. 1364

.1423

. 1485

.1547

.1612

.1679

.1747

.1818

.1890

.1965

0.2042

. 2122

.2204

.2289

.2377

.2468

.2561

.2650

.2756

.2862

F

0.0042

.0181

.0425

0.0483

.0516

.0550

.0584

.0619

.0656

.0695

.0734

.0775

.0817

0. 0861

.0905

.0986

.1002

.1047

.1097

.1149

.1202

. 1256

.1312

0. 1370

.1429

.1491

.1554

.1618

.1685

.1754

.1825

.1898

.1972

0.2050

.2130

.2213

.2298

.2386

.2477

.2571

.2658

.2767

.2872

IBB 150 06 17
184 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

TABLE 8. Backwater function (F) for a dam or obstruction Continued.

[The column headings are hundredths for values of from zero to 0.29, thousandths for values of _

a a

from 0.30 to 0.899, and ten-thousandths for values of from 0.900 to 0.999.]

J o

0.70

.71

.72

.73

.74

. 75

.76

.77

.78

.79

0.80

.81

.82

. 83

.84

.85

.86

.87

.88

.89

0.900

.901

.902

.903

.904

.905

.'906

.907

.908

.909

0.910

.911

.912

.913

.914

.915

,916

.917

.918

.919

0. 920

.921

.922

.923

.924

. 925

. 926

.927

.928

. 929

F

0. 2883

1

F

0.2894

.2991 t .3002

.3104 .3116

. 3221

. 3343

.3470

. 3603

.3741

.3886

.4039

0. 4198

. 4367

. 4544

.4733

. 4932

. 5146

. 5374

. 5619

, 5884

.6173

0. 6489

. 6522

.6556

.6590

. 6625

.6660

.6695

.6730

.6766

.6802

0. 6839

.6876

.6914

.6952

.6990

.7029

.7069

.7109

.7149

.7190

0. 7231

.7273

»>

F

0. 2905

.3013

.3127

.3233 .3245

. 3356 . 3368

.3483 ; .34%

.3617 .3630

. 3755 . 3770

. 3901 . 3916

. 4055 . 4070

0. 4215

.4384

0. 4231

.4402

. 4563 . 45x1

. 4753 . 4772

. 4953 . 4974

.5168 .5191

.5398

.5645

.5912

. 6204

0. 6492

.6525

.6559

.6594

.6629

.6664

.6698

.6734

.6770

.6806

0.6843

.6880

.6918

.6956

.6994

.7033

.7073

.7113

,7153

.7194

0. 7235

.7277

.7315 ' .7319

. 7358 . 7362

. 7401 j . 7405

. 7445 . 7450

, .7490 .7494

. 7535

. 7581

.7540

. 5422

.5671

.5940

. 6235

0. 6496

. 6529

.6563

.6597

. 6632

.6667

.6702

. 6737

.6773

.6809

0. 6846

.6884

. 6922

.6960

.6998

.7037

.7077

.7117

.7157

.7198

0. 7239

.7281

. 7324

.7367

.7410

.7454

.7499

. 7544

1 .7586 .7590

8 4

F F

0.2915 0.2926

. 3025 . 3036

. 3139 . 3150

.3258

.3381

.3510

.3644

. 3784

. 3932

.4086

0. 4248

.4419

.4600

. 3270

.3393

. 3523

. 3657

.3799

.3947

.4101

0. 4264

.4437

. 401S

.4792 .4811

. 4995

. 5213

.5446

.5697

.5969

. 6265

0.6499

. 6532

.6566

.6600

.6636

.6670

.6706

.6741

. 6777

.6813

0.6850

.6887

. 6925

.6963

.7002

.7041

.7081

. 7121

.7161

.7202

0. 7244

. 7286

.7328

.7371

.7414

.7458

. 5016

. 5236

.5470

. 5723

. 5997

.62%

.6502

. 6536

.6570

.6604

.6639

.6674

.6709

.6744

,6780

.6817

0.6854

. 6891

.6929

. 6967

.7006

.7045

.7085

. 7125

. 7165

. 7206

0. 7248

.7290

.7332

.7375

.7419

.7463

.7504 ! .7508

. 7549

. 7595

.7628 .7633 .7637 .7642

. 7553

.7600

.7647

5

F

0. 2937

.3047

.3162

. 3282

.3406

.3536

.3671

.3813

. 3962

.4117

0. 4281

.4454

.4637

(i

F

0. 2948

.3058

.3174

.3294

.3419

.3549

.3685

. 3828

. 3977

.4133

0. 4298

. 4472

. 4656

. 4S31 . 4851

. 5037 . 5059

.5258 ' .5281

.5494

. 5749

.6025

. 6327

0.6506

.6539

.6573

.6608

. 6642

.6678

. 6712

.6748

.6784

. 6820

0.6858

.6895

.6933

.6971

.7010

.7049

.7089

.7129

.7170

.7210

0. 7252

, 7294

.7336

.7380

.7423

.7468

. 7512

.7558

.7604

. 7652

. 5519

. 5776

. 6055

. 6359

0. 6509

. 6542

. 6576

. 6611

.6646

.6681

.6716

. 6752

.6788

.6824

0. 6861

.6899

.6937

. 6975

.7013

. 7053

.7093

. 7133

.7174

. 7215

0. 7256

.7298

.7341

.7384

. 7427

. 7472

.7517

. 7563

.7609

.7656

'

F

0. 2959

.3070

.3186

.3306

.3432

. 3563

. 3699

.3842

.3993

T4149

0. 4315

.4490

.4675

.4871

. 5081

.5304

.5544

.5803

.6084

.6392

0. 6512

. 6546

. 6580

.6614

. 6650

. 6684

. 6720

.6755

.6791

. 6828

0. 6865

.6903

.6941

.6979

.7017

.7057

.7097

.7137

.7178

. 7219

0. 7260

.7302

.7345

.7388

. 7432

. 7476

. 7522

. 7567

.7614

.7661

8 9

F ' F

0. 2969 0. 2980

.3081 . .3093

.3197 | .3209

. 3319 . 3331

. 3444 . 3457

. 3576 . 3590

. 3713 . 3727

. 3857 . 3871

.4008

.4166

. 4024

. 4182

0. 4333 0. 4350 '

. 4508 . 4526

. 4695 . 4714

. 4892 . 4912

.5102 .5124

. 5328 . 5351

. 5569

.5830

.6114

. 6424

0. 6515

. 6549

.5594

.5857

.6143

. 6457

0. 6519

.6553

. 6583 . 6587

.6618 .6622

.6653

.6688

. 6723

. 6759

.6795

. 6832

0. 6869

.6906

.6944

. 6982

.7021

.7061

.7101

.7141

, 7182

. 7223

0. 7265

.7307

.7349

.7392

.7436

.7481

. 7526

. 7572

.7619

-.7666

. 6656

. 6692

. 0726

. 6762

.6798

.6835

0. 6872

.6910

.6948

.6986

. 7025

.7065

.7105

.7145

.7186

. 7227

0. 7269

. 7311

. 7354

.7397

.7441

.7486

.7530

. 7576

. 7623

.7670
TABLES FOE CALCULATING WEIR DISCHARGE. 185

TABLE 8. Backwater function (F) for a dam or obstruction Continued. 

[The column headings are hundredths for values of from zero to 0.29, thousandths for values of 

from 0.30 to 0.899, and ten-thousandths for values of ^ from 0.900 to 0.999.]

D

d

0.980

.931

.932

.933

.934

.935

.936

.937

.938

.939

0.940

.941

.942

.943

.944

.945

.946

.947

.948

.949

0. 950

.951

.952

.953

.954

.955

.956

.957

.9,58

.959

0.960

.961

.962

.963

.964

.965

.966

.967

.968

.969

0.970

.971

.972

.973

.974

.975

.976

.977

.978

.979

0

F

0. 7675

.7723

.7772

.7821

.7871

.7922

.7973

.8026

.8079

.8133

0. 8188

.8244

.8301

.8359

.8418

.8478

.8539

.8602

.8665

.8730

0. 8795

.8863

.8931

.9002

.9073

.9147

.9221

.9298

.9376

. 9457

0. 9539

.9624

.9709

.9799

.9890

. 9985

1. 0080

1.0181

1. 0282

1.0389

1.0497

. 1

F

0. 7680

.7728

.7777

.7826

.7876

.7927

.7978

.8031

.8084

.8138

0.8194

.8250

.8307

.8365

.8424

.8484

.8545

.8608

.8672

.8736

0.8802

.8870

.8938

.9009

.9080

.9154

.9229

.9306

.9384

.9465

0.9548

.9632

.9718

.9808

.9899

.9994

1.0090

1.0191

1.0292

1. 0399

1.0508

1.0610 1.0622

1. 0727

1.0848

1.0974

1. 1105

1. 1241

1. 1383

1. 1531

1. 1686

1. 0739

1. 0861

1. 0987

1. 1119

1.1255

1. 1398

1.1546

1.1702

2 3

F

0. 7685

.7724

.7782

.7831

.7881

.7932

.7984

.8037

.8090

.8144

0. 8199

.8256

.8313

.8371

.8430

.8490

.8552

.8615

.8678

.8743

0. S809

.8877

.8945

.9016

.9088

.9162

. 9236

.9314

.9392

.9473

0. 9556

.9641

.9727

.9H17

.9909

1.0004

1.0100

1. 0201

1.0303

1. 0410

1.0519

1.0633

1. 0751

1.0873

1.1000

1. 1132

1. 1269

1. 1413

1. 1562

1.1718

F

0. 7689

.7738

.7787

.7836

.7886

.7937

.7989

.8042

.8095

.8150

0. 8205

. 8262

.8318

.8377

.8436

.8496

.8558

. 8621

.8684

.8750

. 8815

.8883

. 8952

.9023

 9095

.9169

.9244

.9321

.9400

.9482

0. 9564

.9650

.9736

. 9826

.9918

1. 0013

1.0110

1.0211

1.0314

1. 0421

1. 0530

1. 0645

1. 0763

1.0880

1. 1013

1. 1146

1. 1284

1. 1427

1. 1578

1.1735

4

F

0. 7694

.7743

.7792

.7841

.7891

. 7942

.7994

.8047

.8101

.8155

0. 8210

.8268

.8324

. 8383

.8442

.8502

.8564

.8627

.8691

. 8756

0. 8822

5

F

0. 7699

.7748

.7796

.7846

.7896

.7948

. 8000

.8052

.8106

.8160

0. 8216

.8274

.8330

.8388

.8448

.8508

.8570

.8634

.8698

.8762

0. 8829

.8890 | .8897

.8959

.9030

.9103

.9177

. 9252

. 9329

.9408

.9490

0. 9573

. 9658

. 9745

.98&5

.9928

1. 0023

1.0120

1. 0221

.8966

.9038

.9110

.9184

.9260

.9337

.9416

.9498

0. 9582

.9666

.9754

.9844

.9937

1. 0032

1. 0130

1. 0231

1.0324 1.0335

1.0432

1.0542

1.0657

1. 0775

1.0898

1. 1026

1. 1159

1. 1298

1.1442

1. 1593

1. 1751

1. 0443

1.0553

1. 0668

1. 0788

1. 0911

1. 1040

1. 1173

1. 1312

1. 1457

1. 1608

1. 1767

6

F

0. 7704

. 7752

.7801

.7851

.7902

.7953

.8005

.8058

.8111

.8166

0. 8222

.8280

.8336

.8394

.8454

.8515

.8577

.8640

.8704

.8769

0. 8836

.8904

.8974

.9045

.9117

.9191

.9267

.9345

.9425

.9506

0. 9590

.9675

.9763

.9854

.9947

1. 0042

1. 0140

1. 0241

1.0346

1. 0453

1.05G5

1.0«80

1.0800

1. 0924

1.1053

1. 1187

1. 1326

1. 1472

1. 1624

1.1783

7

F

0. 7709

.7757

.7806

.7856

.7907

. 7958

.8010

.8063

.8117

.8172

0. 8227

.8286

.8342

.8400

.8460

.8521

.8583

.8646

.8710

.8776

0.8843

.8911

.8981

.9052

.9125

.9199

.9275

. 9353

.9433

.9514

0. 9598

.9684

.9772

.9863

.9956

1. 0051

1. 0150

1. 0251

1. 0356

1.0464

1. 0576

1.0692

1.0812

1.0936

1. 1066

1.1200

1.1340

1. 1487

1.1640

1. 1799

8

F

0. 7713

.7762

.7811

.7861

.7912

.7963

.8015

.8068

.8122

.8177

0.8233

.8292

.8347'

.8406

.8466

.8527

.8589

.8652

.8717

.8782

0.8849

.8917

.8988

.9059

.9132

.9206

.9283

.9360

.9441

. 9523

0. 9607

.9692

.9781

.9872

.9966

1.0061

1.0160

1. 0261

1.0367

1. 0475

1. 0587

. 1. 0704

1. 0824

1. 0949

1. 1079

1. 1214

1.1355

1. 1501

1. 1655

9

F

0. 7718

.7767

.7816

.7866

.7917

.7968

.8021

.8074

.8128

.8182

0- 8238

.8298

.8353

.8412

.8472

.a°S33

. 8596

. 8659

.8724

.8788

0. 8856

.8924

.8995

.9066

.9140

.9214

.9290

.9368

.9449

. 9531

0.9616

.9700

.9790

.9881

.9975

1.0070

1. 0170

1.0271

1. 0378

1.0486

1. 0598

1.0715

1.0836

1. 0961

1. 1092

1. 1227

1. 1369

1. 1516

1. 1670

1. 1816 1. 1832

i
186 WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

TABLE 8. Backwater function (F) for a dam or obstruction Continued. 

[The column headings are hundredths for values of ~ from zero to 0.29, thousandths for values

of from 0.30 to 0.899, and ten-thousandths for values of £. from 0.900 to 0.999.1 d d

d

0.980

.981

.982

.983

.984

.985

.986

.987

.988

.989

0.990

.991

.992

.993

.994

.995

.996

.997

.998

.999

1 ftftft J.. \J\J\J

0

F

1. 1848

1. 2019

1. 2199

1.2390

1. 2592

1. 2807

1. 3037

1. 3284

1. 3551

1.3841

1. 4159

1. 4510

1. 4902

1. 5348

1.5861

1.6469

1. 7213

1. 8172

1.9523

2. 1834

1

 

F

1. 1865

1. 2037

1.2218

1. 2410

1. 2614

1.2830

1. 3062

1. 3311

1.3580

1.3873,

1. 4194

1.4549

1. 4947

1.5399

1.5922

1.6543

1. 7309

1. 8307

1. 9754

"

F

1. 1882

1. 2055

1. 2237

1.2430

1.2635

1.2853

1. 3086

1.3337

1. 3609

1. 3905

1.4229

1. 4588

1.4991

1.5451

1. 5983

1. 6618

1. 7405

1. 8442

1.9985

3

F

1. 1899

1.2073

1. 2256

1. 2451

1.2656

1. 2876

1. 3111

1. 3364

1.3638

1. 3936

1. 4264

1. 4628

1. 5036

1. 5502

1. 6043

1. 6692

1. 7501

1. 8577

2. 0216

4

F

1.1916

1.2091

1. 2275

1. 2471

1. 2678

1.2899

1. 3136

1.3391

1. 3667

1. 3968

1. 4299

1. 4667

1.5080

1.5553

1. 6104

1. 6767

1. 7597

1. 8712

2. 0447

.

 >

F

I. 1934

1. 2109

1.2294

1.2491

1.2700

1. 2922

1.3160

1. 3418

1. 3696

1.4000

1.4334

1. 4706

1. 5125

1. 5604

1. 6165

1. 6841

1. 7692

1.8848

2. 0678

F

1. 1951

1.2127

1. 2314

1.2511

1.2721

- 1. 2945

1. 3185

1. 3444

1. 3725

1.4032

1. 4370

1. 4745

1. 5170

1. 5656

1. 6226

1. 6915

1. 7788

1.8982

2. 0910

'

F

1.1968

1. 2145

1. 2333

1. 2531

1. 2742

1.2968

1.3210

1. 3471

1.3754

1. 4064

1. 4405

1.4784

1.5214

1.5707

1. 6287

1. 6990

1.7884

1. 9118

2. 1141

8

F

1.1985

1. 2163

1. 2352

1. 2552

1.2764

1.2991

1. 3235

1. 3498

1. 3783

1. 4095

1.4440

1. 4824

1.5259

1.5758

1. 6347

1. 7064

1. 7980

1. 9253

2. 1378

F

1. 2002

1.2181

1. 2371

1.2572

1. 2786

1. 3014

1. 3259

1. 3524

1. 3812

1. 4127

1. 4475

1. 4863

1. 5303

1.5810

1. 6408

1. 7139

1.8076

1. 9388

2. 1603
INDEX.

Albion, Mass., dam at, flow over........... 132

Angular weirs. See Weirs, angular.

Approach, channel of. See Channel of approach, 

section of, definition of................. 7

velocity of. See Velocity of approach.

Austin, Tex., dam at, flow over ............ 133

Authorities cited, list of.................... 1»

Backwater, depth of...................... 180-182

depth of, figures showing ............ 180.181

table showing .................... 183-186

Bazin, H., base formula of.................. 9

coefficients of, for thin-edged weirs.... 61

plate showing...................... 32

correction of, for velocity of approach . 63-65 

experiments of, on effect of rounding

upstream crest edge ............ 123

on submerged weirs of irregular

section ....................... 143-144

on thin-edged weirs................ 29-31

on trapezoidal weirs................ 127

on triangular weirs............... 124-126

on weirs of irregular cross section.. 63-85 

on weirs with compound slopes.. 127-128 

on weirs with varying upstream

slopes......................... 128-129

plate showing ...................... 66

formula of, comparison of, with other

formulas........................ 40-42

for submerged weirs.............. 141-142

for thin-edged weirs................ 31-34

for weirs with end contraction, use

of............................... 45

formulas and experiments of, on broadcrested weirs ................. 117-119

Bellasis, , on falls......................... 135

Blaokstone River, dam on, flow over....... 132

Blackwell, T. E., experiments of, on broadcrested weirs ............. 112-114,122

Boileau, P., experiments of, on thin-crested

weirs ........................... 21-22

formula of, compared with other formulas .......................... 40-41

Broad-crested weirs. See Weirs, broadcrested.

Castel, M., experiments of, on thin-crested

weirs ........................... 20-21

formula of, compared with other formulas .......................... 40-41

Chambly dam,- model of, experiments on.. 101

Channel, leading, definition of............. 7

Channel of approach, definition of......... 7

depth of, for weir gaging............... 50

energy in, distribution of .............. 17-20

velocity in, distribution of............. 16-17

t^hanoine and Mary, formula of, for submerged weirs................... 140

Cippoletti, Cesare, formula of. for trapezoidal weiis .................... 48-19

weir of, definition of ................... 47-48

Clegg's dam, flow over ..................... 112

Coefficients, relations of.................... 9

Compound weirs, flow over ................ 46

Contracted weirs, definition of............. 7

Contraction, crest, definition of............ 8

Contraction, end, formula for.............. 44-45

Contraction, vertical, definition of......... 8

effect of................................. 13-14

Cornell University, hydraulic laboratory of,

description of................... 86-87

hydraulic laboratory of, experiments

at........................... 39,85-107

experiments at, plate showing ..... 86

Crest, character of.......................... 52

roughness of, corrections for......... 133-134

Crest, contraction, definition of..... ...... 8

Croton dam, crest of, correction for........ 134

model of, flow over, experiments on.... 90-94

flow over, experiments on, plate

showing ..........T.............. 94

Dams, backwater caused by, depth of.... 180-182

backwater caused by, depth of, table

showing ...................... 183-186

Dams, actual, flow over, experiments on.. 131-133 

flow over, experiments on, plate showing.............................. 132

Dams, model, crest of, correction for....... 133

flow over, experiments on.............. 88-90

Dams, submerged, data concerning ...... 144-145

D'Aubuisson, J. F., formula of .............. 21

Deep Waterways, U. S. Board of Engineers 

on, experiments of, on submerged weirs................... 146

experiments of, on weirs of irregular

section ......................... 85-90

on weirs with varying upstream

slopes......................... 128-130

plate showing...................... 90

formula of, for broad-crested weirs... 121-122 

Definitions of terms ........................ 7-8

Desplaines River dam, flow over ........... 112

Dimensions, methods of expressing ........ 8

187
188 INDEX.

Page. 

Dolgeville dam, model of, experiments

on.............................. 102

Dyas, , formula of, for submerged weirs .. 143 

Dyer, C. W. D., and Flinn, A. D., experiments of........................ 45,48

East Indies, engineers of, formulas of, for

broad-crested weirs ...... 114-116,121

formulas of, for thin-edged weirs....... 22,40

submerged dams in, data concerning ........................... 144-145

Energy, distribution of, in channel of approach.......................... 17-20

Error, effect of, in determining head....... 53

Essex County,dam of, model of, experiments

on ............................ 107-109

Palls, flow over............................. 135

flow over, figure showing .............. 135

Farm Pond, Mass., experiments on thin- i 

edged weirs at.................. 28

Flinn, A. D., and Dyer, C. W. D., experiments of........................ 45,48

Flow, method of expressing................ 8

Formulas, comparison of................... 40-42

list of................................... 9

Francis, J. B., base formula of ............. 9

experiments and formulas of, for thincrested weirs ................... 23-26

formula of, compared with other formulas.............................. 40

discharge by, table showing...... 162-171

for end contractions................ 44

for weirs of irregular cross sections. 62 

weir of, diagram showing .............. 51

Francis, J. B., and Smith, H., formula of,

for thin-edged weirs............ 37

Francis, J. B., Stearns, F. P., and Fteley, 

A., formula of, for thin-edged 

weirs .................. 26,29,34,40-41

Freeman, J. R., experiments of ............ 90-94

Frizell, J. P., formula of, for broad-crested

weirs ......................... 110-112

Fteley and Stearns. See Stearns and Fteley. 

Fteley, A., Stearns, F. P., and Francis, J. 

B., formula of, for thin-edged 

weirs .................. 26,29,34,40-41

Gaging, accuracy of........................ 53-58

requirements for....................... 49-53

Geological Survey, United States, experiments of, on broad - crested 

weirs ......................... 119-121

 experiments of, on rounding upstream

edge .......................... 123,124

on weirs of irregular section....... 98-107

on weirs with varying slopes....... 130

plate showing...................... 106

Go'ild, E. L., formula of, for discharge

from nonprismatic reservoir.... 154

formula of, for discharge from prismatic reservoir................ 150-152

Gould, E. S., formula of, for discharge

from prismatic reservoir ..... 151-152

Hart and Hunking, formula of, for thinedged weirs .................... 25-26

Page.

Head, determination of, error in, effect of. 53-57 

determination of, error in, effect of,

plate showing .................. 54

effect of velocities on, table showing. 157-159 

increase in, effect of.................... 39-40

from submerged weir............. 142-143

variation in ...................... 53-54,97-98

diagram showing................... 150

effectof........................... 146-156

Herschel, C., formula of, for submerged

weirs ......................... 139-140

Horton, R. E., experiments of............. 95-107

Hunking and Hart, formula of, for thinedged weirs .................... "25-26

Inclined weirs. See Weirs, inclined.

India. See East Indies.

Inflow, effect of, on reservoir............. 148-156

Johnston, T. T., on flow over Desplaines

River dam...................... 112

Lawrence, Mass., dam at, model of, experiments on ..................... 107-109

dam at, model of, experiments on, plate

showing........................ 106

Leading channel, definition of............. 7

Lesbros, experiments of, on thin-edged

weirs ........................... 21

formulas of, compared with other formulas. .......................... 41

Lesbros and Poncelet, experiments of, on

thin-edged weirs ............... 21

Lowell, Mass., experiments at...... 23-26,107-109

Merrimac River, dam on, experiments on.. 106 

dam on, experiments on, plate showing. 106 

Metz, Germany, experiments on thin-edged

weirs at......................... 21-22

Morris, Elwood, on Clegg's dam............ 112

Muskingum River, dam on, flow over...... 132

Nappe, definition of........................ 7

form of, modifications of ............... 60-61

modifications of, plate showing .... 60

Nelles, George T., data collected by........ 131

Notation, explanation of................... 8-9

O'Connell, P. P. L., formula of, for discharge

from nonprismatic reservoir.... 153

Orifice, flow through ....................... 12-13

flow through, figure showing............. 12

Ottawa River dam, Canada, flow over...... 132

Parabolic law of velocity, application of, to

weirs ........................... 12

Parmley, W. C., formula of, compared with

other formulas ................. 40-41

formula of, for thin-edged weirs ....... 37-38

Plattsburg dam, model of, experiments on. 98-100

Poncelet and Lesbros, experiments of, on

thin-edged weirs ......... ... 21

Rafter, G. W., experiments of.............. 85-90

Rankine, W. J., formula of, for submerged

weirs ........................... 142

Reservoirs, loweriiig of, time required for. 146-156 

Rhind.R. H.,formulaof,forsubmergedweirs 141
INDEX. 189

Pa.se.

Section of approach, definition of.......... 7

Sharp-crested weirs, definition of .......... 7

Smith, Hamilton, base formula of.......... 9

formula of, for thin-edged weirs ....... 22,

34-36,40-41,44 

Smith, H., and Francis, J. B., formula of,

for thin-edged weirs...... ..... 37

Stearns, F. P.,andFteley,A.,baseformulaof 9 

experiments of, 011 broad-crested

weirs ......................... 116-117

on rounding upper crest.......... 122-124

on thin-edged weirs ............... 26-29

formula of, compared with other formulas. ............................. 40-41

for submerged weirs.............. 138-139

for thin-edged weirs................ 34

Stearns, F. P., Fteley, A., and Francis, J. B.,

formula of, for thin-edged weirs. 26, 

29,34,40-41

Submerged weirs. See Weirs, submerged. 

Suppressed weirs, definition of............. 7

Thin-edged weirs. See Weirs, thin-edged.

Thomson, James, experiments of, on coefficient of contraction for thinedged weirs .................... 46-47

Three-halves powers, table of............ 171-176

Torricelli, G., theory of..................... 10-11

theory of, application of, to weir, figure

showing........................ 11

Toulouse, France, experiments at, on thinedged weirs .................... 20-21

Trapezoidal weirs. See Weirs, trapezoidal.

Triangular weirs. See Weirs, triangular.

United States Board of Engineers on Deep 

Waterways. See Deep Waterways.

United States Geological Survey, flee Geological Survey.

Unwin, W. C., formula of, for broad-crested

weirs ......................... 110-112

Velocities, method of expressing........... 8

Velocity, parabolic law of, application of.. 12 

Velocity of approach, distribution of....... 16-17

distribution of, figure showing......... 16

effect of, on weir discharge ......... 14-20,58

correction for .......... 14-16,41-43,63-66

table showing (Hunking and 

Hart formula)................ 26

table showing (Parmly and 

Bazin formula)............... 38

table showing.................... 159-162

energy of, distribution of .............. 17-20

formulas for............................ 14-20

head due to, table'showing .......... 157-159

Vertical contraction. See Contraction, vertical.

Weir section, definition of.................. 8

Weirs, aprons of, variation in, effect of .. 124-127 

backwater caused by, depth of....... 180-182

Page. 

Weirs, backwater caused by, depth of, table

showing...................... 183-186

definition of............................ 7

discharge over, relative approximate.. 10-11 

variation in ...................... 146-154

diagram showing............... 150

flow over, calculation of, tables for .. 156-186 

measurement of, formulas for...... 9,

9-12,11-13,40-43 

theory of ........................... 10-14

gaging at. See Gaging, 

head on. See Head. 

Weirs, angular, flow over .................. 136

flow over, figure showing .............. 136

Weirs, broad-crested, edge of, rounding of,

effect of ........................ 124

flow over............................. 110-122

figure showing ..................... 110

table showing.................... 177-179

Weirs, compound, flow over................ 46

Weirs, contracted, definition of ............. 7

Weirs, curved, flow over ................... 136

flow over, figure showing .............. 136

Weirs, East Indian, flow over ............ 144-145

flow over, figures showing ............. 145

Weirs, fiat-top, mod els of, experiments on. 103-105 

Weirs, inclined, flow over................ 127-130

flow over, figure showing .............. 57

Weirs, irregular, flow over, experiments on. 61-110 

flow over, formulas for, basic .......... 62-63

use of............................ 1....... 59

Weirs, of sensible crest width, flow over... 52 

Weirs, ogee cross-sectioned, flow over.... 130-131

flow over, plate showing............... 130

Weirs, submerged, flow over ............. 137-146

flow over, figure showing .............. 137

increase of head due to .............. 142-143

Weirs, suppressed, definition of ............ 7

Weirs, thin-edged, definition of ............ 7

discharge over, table showing ....... 162-171

flow over, measurement of, experiments

on and formulas for...... 20-29,31-46

measurement of, formulas for, comparison of........................... 40-41

formulas for, extension of.......... 39-40

Weirs, trapezoidal, aprons of, variation in,

effect of......................... 127

cross-section of......................... 47

flow over, formulas for, figures showing. 47 

formulas for........................ 47

Weirs, triangular, aprons of, variation in,

effect of....................... 124-126

coefficient curve for, figure showing... 125 

flow over, experiments on, figure show- * 

ing ............................. 46

experiments on and formulas for .. 46-47 

Weirs, uneven, flow over................... 57-58

Weisbach, formula of....................... 40

"Wetted underneath," definition of ....... 7

Williams, G. S., experiments by........... 90-107

Woodman, R. S., formula of, for discharge

from prismatic reservoir ..... 151-152
CLASSIFICATION OF THE PUBLICATIONS OF THE UNITED STATES 

GEOLOGICAL SURVEY,

[Water-Supply Paper No. 150.]

The serial publications of the United States Geological Survey consist of (1) 

Annual Eeports, (2) Monographs, (3) Professional Papers, (4) Bulletins, (5) 

Mineral Resources, (6) U ater-Supply and Irrigation Papers, (7) Topographic Atlas 

of United States folios and separate sheets thereof, (8) Geologic Atlas of the United 

States folios thereof. The classes numbered 2, 7, and 8 are sold at cost of publication; the others are distributed free. A circular giving complete lists may be had 

on application.

Most of the above publications may be obtained or consulted in the following ways:

1. A limited number are delivered to the Director of the Survey, from whom they 

may be obtained, free of charge (except classes 2, 7, and 8), on application.

2. A certain number are delivered to Senators and Representatives in Congress, 

for distribution.

3. Other copies are deposited with the Superintendent of Documents, Washington, 

D. C., from whom they may be had at prices slightly above cost.

4. Copies of all Government publications are furnished to the principal public 

libraries in the large cities throughout the United States, where they may be consulted by those interested.

The Professional Papers, Bulletins, and Water-Supply Papers treat of a variety of 

subjects, and the total number issued is large. They have therefore been classified 

into the following series: A, Economic geology; B, Descriptive geology; C, Systematic geology and paleontology; D, Petrography and mineralogy; E, Chemistry and 

physics; F, Geography; G, Miscellaneous; H, Forestry; I, Irrigation; J, Water 

storage; K, Pumping water; L, Quality of water; M, General hydrographic investigations; N, Water power; O, Underground waters; P, Hydrographic progress reports. 

This paper is the sixteenth in Series M, the complete list of which follows. (PP= 

Professional Paper; B=Bulletin; WS=Water-Supply Paper):

SERIES M GENERAL HYDROGRAPHIC INVESTIGATIONS. 

WS 56. Methods of stream measurement. 1901. 51 pp., 12 pis. 

WS 64. Accuracy of stream measurements, by E. C. Murphy. 1902. 99 pp., 4 pis. 

WS 76. Observations on the flow of rivers in the vicinity of New York City, by H. A. Pressey. 1902.

108 pp., 13 pis.

WS 80. The relation of rainfall to run-off, by G. W. Rafter. 1903. 104 pp. 

WS 81. California hydrography, by J. B. Lippincott. 1903. 488 pp., 1 pi. 

WS 88. The Passaic flood of 1902, by G. B. Hollister and M. O. Leighton. 1903, 56 pp., 15 pis. 

WS 91. Natural features and economic development of the Sandusky. Maumee, Muskingum, and

Miami drainage areas in Ohio, by B. H. Flynn and M. S. Flynn. 1904. 130 pp. 

WS 92. The Passaic flood of 1903, by M. O. Leighton. 1904. 48 pp., 7 pis. 

WS 94. Hydrographic manual of the United States Geological Survey, prepared by E. C. Murphy;

J. C. Hoyt, and G. B. Hollister. 1904. 76 pp., 3 pis.

WS 95. Accuracy of stream measurements (second edition), by E. C. Murphy. 1904. 169 pp.,6 pis. 

WS 96 Destructive floods in the United States in 1903, by E. C. Murphy. 1904. 81 pp., 13 pis. 

WS 106. Water resources of the Philadelphia district, by Florence Bascoin. 1904. 75 pp., 4 pis. 

WS 109. Hydrography of the Susquehanna River drainage basin, by J. C. Hoyt and R. H. Anderson.

1904. 215pp., 28 pis.

WS 116. Water resources near Santa Barbara, California, by J. B. Lippincott. 1901. 99 pp., 8 pis. 

WS 147. Destructive floods in the United States in 1904, by E. C. Murphy and others. 1905. 206 pp.,

18 pis. 

WS 150. Weir experiments, coefficients, and formulas, by R. E. Hortou. 1906. 189 pp., 38 pis.

Correspondence should be addressed to

THE DIRECTOR,

UNITED STATES GEOLOGICAL SURVEY,

WASHINGTON, D. C. 

JANUARY, 1906,

I

O