Determination of 

Discharge During 

Pulsating Flow

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1869-D

Prepared in cooperation 

with the California 

Department of Water 

Resources
Determination of 

Discharge During 

Pulsating Flow

By T. H. THOMPSON

RIVER HYDRAULICS

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1869-D

Prepared in cooperation 

with the California 

Department of Water 

Resources

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1968
UNITED STATES DEPARTMENT OF THE INTFRIOR 

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY 

William T. Pecora, Director

For sale by the Superintendent of Documents, U.S. Government Printing Office 

Washington, D.C. 20402 - Price 15 cents (paper cover)
CONTENTS

Page

Abstract.._ _______________________________________________________ Dl

Introduction. _____________________________________________________ 2

Purpose and scope.__________.-__-__-_____..__-_____-_____-__-_- 2

Acknowledgments _____________________________________________ 5

Characteristics of pulsating flow.___-_--____-______-__________--_-__- 5

Determination of discharge_________________________________________ 6

Description of the Haines Creek instrumentation __________________ 6

Description of controlled test in Santa Anita Wash ________________ 7

Method of computing discharge-______-_-_-_--____-__--__--_____ 10

Discussion of results-__________________________________________ 15

Proposed methods of determining discharge.__________________________ 17

Photographic method __________________________________________ 17

Depth-recorder method.._______________________________________ 19

Dye-dilution method.__________________________________________ 19

Discussion of proposed methods _________________________________ 20

Summary and conclusions __________________________________________ 21

Selected references.________________________________________________ 22

ILLUSTRATIONS

Page 

FIGURE 1. Index map showing observation sites on Santa Anita Wash

in Arcadia, Calif________________-____-----_-___-__ D3

2. Photograph showing pulsating flow in Santa Anita Wash

above Sierra Madre Wash.__________________________ 4

3. Sketch showing dimensions and configuration of grid on

channel wall-______________________________________ 7

4-6. Discharge hydrographs at observation sites on Santa Anita 

Wash:

4. Above Sierra Madre Wash (O.S. 201)-____-__-__- 9

5. At Colorado Blvd. (O.S. 202)____________________ 10

6. At Longden Ave. (O.S. 203)_____________________ 11

7. Schematic sketch of longitudinal water-surface profile and

list of elements observed.__________________________ 12

8. Schematic sketch of general component layout for photographic method of determining discharge.-____________ 18
IV CONTENTS

TABLES

Page 

TABLE 1. Channel geometry at observation sites on Santa Anita Wash

flood control channel-_-_----_-----_-__-_-__-_-_-_--_ D8

2. Computation of discharge in the overriding wave (Qa) at

O.S. 201_-______-___-_---_-___-_-_-___---_-_-_-____ 12

3. Computation of discharge in the shallow-depth (overrun)

part of flow (Qi) and total discharge at O.S. 201,_______ 13

4. Computation of discharge in the overriding wave (Q.) at

O.S. 202_________________.__________-___-__________ 14

5. Computation of discharge in the shallow-depth (overrun)

part of flow (Qi) and total discharge at O.S. 202________ 15

6. Computation of discharge in the overriding wave (Qw) at

O.S. 203____-_____--_-_-___--_____________---_-____ 15

7. Computation of discharge in the shallow-depth (overrun)

part of flow (Qi) and total discharge at O.S. 203________ 16

8. Observed wave characteristics at approximate time of

maximum discharge.._______.-_________---____-___ 17

SYMBOLS

AI Cross-sectional area of channel (ft2)

b Width of channel (ft)

Dj Depth between waves (ft)

Dz Depth at wave front (ft)

g Gravitational constant

hw Height of wave crest (ft)

Lw Length of overriding wave from crest to trough (ft)

Lr Length of measuring reach (ft)

n Roughness factor (Manning)

Qi Discharge in shallow-depth part of flow (cfs)

Qw Discharge in overriding wave (cfs)

R Hydraulic radius (ft)

S Slope of channel (ft/ft)

t b Time interval between arrivals of crest and trough at observation point.

	(sec)

tw Time required for wave crest to travel length of measuring reach (sec)

Tj, Time interval between arrivals of consecutive wave fronts at observation

	point (sec) 

F Velocity (fps) 

Vi Shallow-depth velocity (fps) 

Vw Average wave velocity (fps)
RIVER HYDRAULICS

DETERMINATION OF DISCHARGE DURING 

PULSATING FLOW

By T. H. THOMPSON

ABSTRACT

Pulsating flow in an open channel is a manifestation of unstable-flow conditions in which a series of translatory waves of perceptible magnitude develops 

and moves rapidly downstream. Pulsating flow is a matter of concern, in the 

design and operation of steep-gradient channels. If it should occur at high stages 

in a channel designed for stable flow, the capacity of the channel may be inadequate at a discharge that is much smaller than that for which the channel was 

designed. If the overriding translatory wave carries an appreciable part of the 

total flow, conventional stream-gaging procedures cannot be used to determine 

the discharge; neither the conventional instrumentation nor conventional 

methodology is adequate.

A method of determining the discharge during pulsating flow was tested in the 

Santa Anita Wash flood control channel in Arcadia, Calif., April 16, 1965. 

Observations of the dimensions and velocities of translatory waves were made 

during a period of controlled reservoir releases of about 100, 200, and 300 cfs 

(cubic feet per second). The method of computing discharge was based on 

(1) computation of the discharge in the overriding waves and (2) computation of the discharge in the shallow-depth, or overrun, part of the flow. Satisfactory results were obtained by this method. However, the procedure used 

separating the flow into two components and then treating the shallow-depth 

component as though it were steady has no theoretical basis. It is simply an 

expedient for use until laboratory investigation can provide a satisfactory 

analytical solution to the problem of computing discharge during pulsating flow.

iSixteen months prior to the test in Santa Anita Wash, a robot camera had 

been designed and programed to obtain the data needed to compute discharge 

by the method described above. The photographic equipment had been installed 

in Haines Creek flood control channel in Los Angeles, Calif., but it had not been 

completely tested because of the infrequency of flow in that channel. Because 

the Santa Anita Wash tests afforded excellent data for analysis, further development of the photographic technique at Haines Creek was discontinued.

Three methods for obtaining the data needed to compute discharge during 

pulsating flow are proposed. In two of the methods the photographic method and 

the depth-recorder method the dimensions and velocities of translatcry waves

Dl
D2 RIVER HYDRAULICS

are recorded, and discharge is then computed by the procedure developed in this 

report. The third method the.constant-rate-dye-dilution method yields the discharge more directly. The discharge is computed from the dye-injection rate and 

the ratio of the concentration of dye in the injected solution to tl a concentration 

of dye in the water sampled at a site downstream. The three methods should be 

developed and tested in the Santa Anita Wash flood control channel under controlled conditions similar to those in the test of April 1965.

INTRODUCTION

Pulsating flow in an open channel, as defined by Chow (1959, p. 

580), is a manifestation of unstable-flow conditions in which a series 

of translatory waves of perceptible magnitude develops and moves 

rapidly downstream. These translatory waves, commonly called roll 

waves or slug flow, are a matter of concern to the designer of steepgradient channels. Channels are invariably designed for stable flow. 

Should pulsating flow occur at high stages in a channel so designed, 

the channel capacity may be inadequate at a discharge much smaller 

than the design flow.

If the overriding translatory wave carries an appreciable part of 

the total flow, conventional stream-gaging methods cannot be used to 

determine the discharge. Conventional water-stage recorders of either 

the float or pressure-sensing type do not react quickly enough to record the rapidly fluctuating stage; depths and velocities change too 

rapidly to permit discharge measurement by current meter; no stagedischarge relation exists for pulsating flow; and the commonly used 

formulas for computing stable open-channel dischp.rge are not 

applicable.

PUBPOSE AND SCOPE

The purpose of the study was twofold: to develop and test a method 

for computing discharge during pulsating flow and to suggest instrumentation for automatic monitoring of the hydraulic elements needed 

to compute flow under such conditions of instability.

Pulsating-flow data were obtained by Los Angeles County Flood 

Control District personnel during observations April 16,1965, at three 

observation sites (O.S. 201, O.S. 202, and O.S. 203) on the Santa Anita 

Wash flood control channel in Arcadia, Calif, (fig. 1). These data 

were used to test a method of computing discharge that is based on the 

dimensions of the translatory waves and on the velocity of the wave 

fronts. The rate of flow was controlled during the test period by releases from the reservoir upstream at Big Santa Anita D-,m. Observations were made during rates of release of about 100, 200, and 300 cfs 

(cubic feet per second). Figure 2 shows pulsating flow at the upstream 

observation site (O.S. 201) at a discharge of about 200 cfs. The depth
DETERMINATION OF DISCHARGE DURING PULSATING FLOW

118°02'

D3

34°10'

EXPLANATION

O.S.202

FIGUKE 1. Observation sites on Santa Anita Wash in Arcadia, Calif.
D4 RIVER HYDRAULICS

of the water ahead of the wave front at this discharge was 0.25 foot 

and the total depth of water in the wave front was 0.9 foot.

Prior to this test, efforts had been made to determine discharge during preiods of pulsating flow in Haines Creek flood control channel in 

Los Angeles, by photographically documenting wave dimensions and 

wave velocities. A prolonged drought, however, afforded few runoff 

events for observing and testing the photographic equipment. After 

16 months of sporadic operation the opportunity for the controlled 

test on Santa Anita Wash presented itself, and shortly afterwards the 

camera was removed from the installation at Haines Creek. The instrumentation suggested later in this report was based in part on the 

experience gained at Haines Creek, but primarily on the observations 

made during the Santa Anita Wash test and on the subsequent analysis of that data.

FIGURE 2. Pulsating flow in Santa Anita Wash above Sierra Madre Wash (O.g. 

201) in Arcadia, Calif., April 16, 1965. Discharge was about 200 cfs.
DETERMINATION OF DISCHARGE DURING PULSATING FLOW D5 

ACKNOWLEDGMENTS

This study was made under a cooperative agreement between the 

U.S. Geological Survey and the State of California, Department of 

Water Resources. The report was prepared by the Water Resources 

Division of the Geological Survey, under the general supervision of 

Walter Hofmann, former district chief for California, and hif successor, R. S. Lord. Technical supervision was provided by S. E. Rantz, 

research hydrologist, who gave helpful advice and guidance durin g the 

data analysis and report preparation. The Haines Creek instrumentation was developed and supervised by Winchell Smith, hydrologist.

The valuable assistance and data furnished by the Los Angeles 

County Flood Control District are gratefully acknowledged.

CHARACTERISTICS OF PULSATING FLOW

Little is known concerning unstable or pulsating flow. From laboratory experiments and field observations many investigators have 

definitely established that unstable flow will develop only in steep 

channels. Craya (1952), Vedernikov (Powell, 1948), and others agree 

that some perturbation or disturbance to stable flow, no matter how 

slight, is required to trigger instability, and a length of channel downstream is required for pulsating flow to develop. The investigators 

also believe that the minimum length of channel required for this 

development depends on the channel geometry and on the degree of 

instability that the flow has attained on entering the reach of channel. 

For example, engineers of the Los Angeles County Flood Control District (oral commun., 1965) have observed that large flow disturbances, 

caused by stream-borne debris lodged in a steep channel, can cause 

pulsating flow to develop in a very short reach. The length of channel 

required for the development of pulsating flow also increases with the 

depth of the flow. Field observations of spectacular examples of pulsating flow have been documented. Holmes (1936) described waves, in 

steep stormflow channels, whose heights ranged from 5 to 8 feet. The 

wedge-shaped waves were several hundred feet long, and the channel 

between waves was dry or nearly so.

Many investigators have derived an index of instability based en laboratory data; that is, a numerical index that describes the conditions 

required for incipient instability. An obvious index would be based 

on Froude number; steep slopes are required for instability. Cl annel 

geometry has an effect, however, and because flumes in the various laboratories vary in size, almost every investigator has his own index 

of instability. Koloseus and Davidian (1966) used Froude number and 

relative roughness; Thomas (1940) and Mayer (1959) used Froude
D6 RIVER HYDRAULICS

number and slope; Vedernikov (Powell, 1948) used Froude number 

and a shape factor; and Escoffier and Boyd (1962) usei slope, bottom 

width, roughness, and discharge.

Observation of the waves in Santa Anita Wash during the field test 

bears out the statements about the development of pulsating flow. Small 

perturbations of the water surface originated on the steep spillway of 

the debris-basin dam at the upstream end of the flood control channel 

(fig. 1). Perturbations did not grow significantly until the flow became disturbed in a superelevated bend in the channel, 2,000 feet downstream from the dam. In this bend the flow showed the pattern of reflected waves that is typical of all supercritical flows in bends. As the 

flow passed the bend, pulsating flow became fully developed, and a 

continuous series of relatively large translatory waves moved down 

the channel. The wave fronts were perpendicular to the channel walls. 

The waves were not evenly spaced and occasionally one wave would 

overtake another, but in general the waves maintained their spacing 

and underwent little attenuation in the upper 2-mile re^ch of channel. 

As the channel became wider and the slope decreased in the lower reach, 

there was slight attenuation of the waves and the period between waves 

increased.

DETERMINATION OF DISCHARGE

DESCRIPTION OF THE HAINES GREEK INSTRUMENTATION

The first efforts by the Geological Survey to document and determine discharge during pulsating flow were made from January 1964 

to August 1965 in the concrete-lined Haines Creek flood control channel in the Tujunga area of Los Angeles, Calif. Instrumentation at the 

test site consisted of a 24-mm robot camera mounted so that vertical 

and horizontal changes in camera position occurred during a programed cycle. A 200-foot film magazine was used, and the camera 

operated at a rate of 6 frames per second. The control and power units 

were housed in a 48-inch corrugated-metal pipe shelter on the left 

bank, with the camera set in an appended box which projected over the 

channel. Photographs were taken through windows in the box that 

faced upstream and across the channel.

A grid, as shown in figure 3, was painted on the channel wall opposite the camera so that water depths could be read. Upstream from 

the grid, vertical stripes were painted on the wall at 20-foot intervals 

for a distance of 400 feet so that wave velocities could be computed.

The camera unit was turned on and off manually, but the filming 

sequence was automatic. It was programed as follows:

1. Three frames were exposed while the camera was oriented in an 

upstream direction.

2. The camera was rotated to face at right angles to the channel.
DETERMINATION OF DISCHARGE DURING PULSATING FLOW D7

3. Thirty-six exposures were made at a rate of six per second at halfsecond intervals a total of 12 seconds.

4. After 4 minutes the camera was reoriented upstream and the sequence was repeated.

The Haines Creek project was discontinued for several reasons after 

the Santa Anita Wash observations were made. As already mentioned, 

a continuing drought during the testing phase afforded few runoff 

events to observe, and therefore the equipment could not be thoronghly 

tested. Furthermore, the Santa Anita Wash test provided an c^portunity to study excellent data which might be years in coming at the 

Haines Creek site. After observing the flow at Santa Anita Wash and 

analyzing the data, the author concluded that the programed secuence 

from the camera would not provide adequate documentation for determining discharge. Also, the manual on-off switch was a serious 

shortcoming. An automatic triggering mechanism in the channel was 

needed to activate the camera when flow reached a significant magnitude. The absence of such a device resulted in the loss of data whQ,n an 

observer was unable to reach the site during a period of flow.

DESCRIPTION OF CONTROLLED TEST IN SANTA ANITA WASH

The Santa Anita Wash flood control channel, in which tests were 

made April 16, 1965, is rectangular in cross section and is concrete 

lined. The geometry of the channel at the three observation sites is 

given in table 1. The heading "Length of measuring reach" refers to 

the short reach at each observation site where the travel times of the

I T ^^

 30'-

FIGUEE 3. Dimensions and configuration of grid on channel wall.
D8 RIVER HYDRAULICS

traiislatory waves were measured. At each observation site a grid, 

similar to that shown in figure 3, was painted on the channel wall opposite the observer so that water depths could be read.

TABLE 1. Channel geometry at observation sites on Santa Anita Wash flood control

channel

Length of Channel properties 

Location of observation site measuring

reach, L, Width, 6 Depth Slope, S 

(ft) (ft) (ft) (ft/ft)

Santa Anita Wash above Sierra Madre

Wash (O.S. 201) __________________ 102.5 28.0 11.0 0.0251

Santa Anita Wash at Colorado Blvd. (O.S.

202), 0.78 mile downstream from O.S.

201____________________ _____________ 118.5 30.0 13.0 .0174

Santa Anita Wash at Longden Ave. (O.S.

203), 2.42 miles downstream from O.S. 

202_________________________________ 115.5 35.0 14.5 .0092

Water was released into the channel from the reservoir at Big Santa 

Anita Dam (fig. 1), 2.2 miles upstream from O.S. 201, at controlled 

rates of about 100, 200, and 300 cfs. These discharges were modified by 

storage in the Santa Anita debris basin, 1.0 mile upstream from O.S. 

201, with the result that the discharge hydrograph for the flood control channel did not show abrupt changes in discharge, but instead 

took the shape shown in figure 4. The discharge hydrographs in figures 4, 5, and 6 were constructed from current-meter discharge measurements made continuously during the test on the crept of the debrisbasin dam. Adjustments for travel time from the dam to each observation site were made on the basis of staff-gage observations made at 

all sites throughout the test. Because of storage in the concrete channel 

there may be some error in time and discharge in figures 4-6 during 

the three steep rises and the steep fall of the hydrographs. Storage, 

however, should have only negligible effect on the three plateaus of 

nearly steady flow. There was no inflow into the channel from Sierra 

Madre Wash, which joins Santa Anita Wash a short distance downstream from O.S. 201 (fig. 1).

Figure 7 is a schematic, sketch of the water-surface profile in the 

flood-control channel and lists the elements observed at each observation site. Water depths, as mentioned, were read on the painted grids, 

and travel times were obtained by stopwatch. Observations were made 

at intervals of approximately 15 minutes, at which time depths and 

travel time characteristics were recorded for each of five consecutive 

waves. In addition, floats were thrown into the channel at varying distances behind the wave crests and their passage through the reach was 

timed. The floats traveled at approximtely the same velocity as the
DETERMINATION OF DISCHARGE DURING PULSATING FLOW D9

wave crests at O.S. 201, a result that was not surprising because the 

waves tended to maintain their spacing and shape in the reach of channel between the upstream bend and Sierra Madre Wash. At O.S. 202 

and O.S. 203 observed float velocities were slightly slower than wavecrest velocities. No observations of velocity were made at depths corresponding to Z>i in figure 7. Further documentation of flow character320

280

240

200

O 160 

oo

D 

O

120

80

40

O

Computed discharge 

O

Measured discharge

1300 1400 1500 1600 1700 

CLOCK TIME, IN HOURS

1800 1900

FIGURE 4. Discharge hydrograph at Santa Anita Wash above Sierra Madre 

Wash. (O.S. 201) April 16, 1965, and plot of discharges computed from observations of flow.

285-953
D10 RIVER HYDRAULICS

istics was obtained in motion pictures and still photograpl ^ of the flow. 

Motion pictures, as well as graphs of observed water depths plotted 

against time, showed that the longitudinal profile of the waves was 

slightly concave upward. To simplify the computation of discharge, 

however, the waves were assumed to have a simple wedge shape, as 

illustrated in figure 7. For each wave, the dimensions of th°> wedge were 

chosen so as to match closely the shape and volume of wate^ in the wave.

METHOD or COMPUTING DISCHARGE

The method for computing discharge comprised the following steps: 

1. Computation of discharge in the overriding wave (Qw }.

The volume of water in the overriding wave was computed, and 

this volume was divided by the average period of the wave (Tp ) to 

obtain the discharge.

280

240 -

200 -

!:; 160 -

120 -

1300 1400 1500 1600 1700

CLOCK TIME, IN HOURS

1800 1900 2000

FIGURE 5. Discharge hydrograph at Santa Anita Wash at Colorado Blvd. (O.S. 

202) April 16, 1965, and plot of discharges computed from observations of 

flow.
DETERMINATION OF DISCHARGE DURING PULSATING FLOW Dll

2. Computation of discharge in the shallow-depth, or overrun, part 

of the flow (&).

The product of the depth (Z>i) and the width of channel (&) was 

multiplied by a derived velocity to obtain the discharge.

3. Computation of total discharge.

Total discharge was computed by adding the discharges determined in steps 1 and 2.

The method is explained by reference to the computations for O.S. 

201 in tables 2 and 3. No computations were attempted where values 

of Z>! were less than 0.2 foot, because of the difficulty of getting reliable 

depth observations at such shallow flows.

280

240

200

160

120

80

40

0

Computed discharge

1300 1400 1500 1600 1700 1800 

CLOCK TIME, IN HOURS

1900 2000

FIGURE 6. Discharge hydrograph at Santa Anita Wash at Longden Ave. (O.S. 

203) April 16, 1965, and plot of discharges computed from observations of 

flow.
D12 RIVER HYDRAULICS

The first six columns of table 2 represent observed data. These observations, other than clock time, were subject to random error because the 

height of a fluctuating water surface is difficult to read against a coarse 

grid. Consequently, average values of depth and time for each sequence of five waves were plotted against time of observation, and 

the plotted points were averaged with a smooth curve. The values for 

each element in columns 2-6, corresponding to clock time in the first 

column, were picked from the curves and recorded in table 2. Values of

Measuring reach (L r)

FIGURE 7. Schematic sketch of longitudinal water-surface profile and list of elements observed.

Elements observed: 

Di=depth between waves (ft). 

D2=depth at wave front (ft). 

Tp=time interval between arrivals of consecutive wave fronts at observation

point (sec). 

ti, time interval between arrivals of erest "a" and trough "I" at observation

point (sec).

* w =time required for wave crest "a" to travel length of measuring reach 

(sec). For example, length of measuring reach at O.?, 201 is 102.5 ft.

TABLE 2. Computation of discharge in the overriding wave (Qv) at O.S. 201

Column 7=£, -4- column 6 

Column 8= Column 5 X column 7 

Column 9=Column 3 column 2

Column 10=6/2 X column 8 X column 9 

Column 11=Column 10 -*- column 4

(1)

Clock 

time

(hr)

1400

1424

1440

1450

1510

1530

1543

1600

1614

1627

1638

(2)

Di

(ft)

0.20

.23

.24

.25

.25

.26

.30

.42

.45

.28

.20

(3)

D*

(ft)

0.76

.85

.90

.92

.97

1.01

1.04

1.08

1.07

.78

.71

(4)

p (sec)

15.0

13.8

13.0

12.3

11.4

10.2

9.7

9.6

9.9

12.6

15.0

(5)

*b 

(sec)

6.0

6.6

7.0

7.2

7.5

7.5

7.3

6.8

6.2

5.6

4.2

(6)

t v 

(sec)

6.1

5.4

5.2

5.0

4.8

4.6

4.5

4.4

4.7

5.4

6.2

(7)

vw

(fps)

16.8

19.0

19.7

20.5

21.4

22.3

22.8

23.3

21.8

19.0

16.5

(8)

Lw

(ft)

101

125

138

148

160

167

166

158

135

106

69

(9)

h w

(ft)

0.56

.62

.66

.67

.72

.75

.74

.66

.62

.50

.51

(10) (11)

Overriding wave

Volume

(ft')

792

1,085

1,275

1,388

1,613

1,754

1,720

1,460

1,172

742

493

Qw

(cfs)

53

79

98

113

141

172

177

152

118

59

33
DETERMINATION OF DISCHARGE DURING PULSATING FLOW D13

tw in column 6 are an average of the observed travel times of several 

floats through the measuring reach.

Columns Y-ll of table 2, together with the headnotes, illustrate the 

computation of discharge in the overriding wave (Qw)- Vie, in column 

7, is the velocity of the wave through the 102.5-foot measuring reach. 

The dimensions of the wedge-shaped wave are Lw (column 8), hw 

(column 9), and the width of the channel (b) at O.S. 201, which is 

28.0 feet. In column 10 these three dimensions were combined to give 

the volume of water in the wave, and in column 11 the computed volumes were combined with the wave period (Tp ) to give the discharge 

in the overriding wave.

TABLE 3. Computation of discharge in the shallow-depth (overrun) part of flow 

(Qi) and total discharge at O.S. 201

Column 2=Column 2, table 2 

Column 3=6 X column 2 

Column 4 = + Di)

Column 5 = Column 7, table 2 column 4 

Column 6=Column 3 X column 5

Column 7= Column. 6 + column 11, table 2 

Column 8=(1.486/«)S^,R2$ 

Column 9=Column 3 X column 8 

Column 10=Column 9 + column 11, table 2

(1)

Clock

time

(hr)

1400

1424

1440

1450

1510

1530

1543

1600

1614

1627

1638

(2)

-Di

(ft)

0.20

.23

.24

.25

.25

.26

.30

.42

.45

.28

.20

(3)

A ,

 f*-\

(ft»)

5.6

6.4

6.7

7.0

7.0

7.3

8.4

11.8

12.6

7.8

5.6

(4)

celerity

(fps)

7.7

8.0

8.3

8.3

8.7

8.9

8.7

7.9

7.6

6.9

7.2

(5)

V,

(fps)

9.1

11.0

11.4

12.2

12.7

13.4

14.1

15.4

14.2

12.1

9.3

(6)

First trial

<?i

(cfs)

51

70

76

85

89

98

118

182

179

94

52

(7) (8) (9) (10)

Second trial

Total

discharge

(cfs)

104

149

174

198

230

270

295

334

297

153

85

Vi

(fps)

8.0

8.7

9.0

9.2

9.2

9.4

10.4

12.9

13.5

10.0

8.0

Qi

(cfs)

45

56

60

64

64

69

87

152

170

78

45

Total

discharge

(cfs)

98

135

158

177

205

241

264

304

288

137

78

Table 3 illustrates the computation of discharge in that part of the 

flow that is overrun by the translatory waves. The accuracy of the 

computed values is open to question because water is not an ideal 

(nonviscous) fluid and the transfer of momentum across streamlines 

causes the shallow-depth velocities (Vi) to be affected by the velocity 

of the translatory waves. Column 2 of table 3 lists depths (D^, and in 

column 3 these depths have been multiplied by the width of the 

channel (6) to give the cross-sectional area (Ai). The velocity, Vi, was 

determined by two methods. First, the theoretical celerities of the 

wave fronts, or velocities of the waves relative to Vi, were determined 

from the equation for a rectangular channel:

Wave celerity- V(fif/2) (A/A) (A+A)

where g is the acceleration of gravity. These wave celerities, shown 

in column 4 of table 3, were subtracted from observed wave velocities,
D14 RIVER HYDRAULICS

Vu, obtained from table 2. The differences, representing tl e theoretical 

velocities of the shallow-depth part of the flow (Fi), are listed in 

column 5. Column 6 lists the shallow-depth discharges (Qi) corresponding to values of FI, and in column 7 these discharges were added to 

the previously computed wave discharges (Qw) to give total discharge. 

The computed total discharges in column 7 are much larger than 

the discharges shown by the hydrograph in figure 4. Because there was 

little likelihood of serious error in the determination of discharge in 

the overriding wave (Qw), another method of computing the shallowdepth part of the flow (Qi) was tried. In this second trial, V\ was 

computed from the Manning equation:

where n is the roughness coefficient, S is the slope, and R is the 

hydraulic radius. Column 8 of table 3 gives the values of Fi computed 

from the Manning equation, and columns 9 and 10 show the corresponding values of Qi and total discharge. The computed values of 

total discharge from column 10 have been plotted in figure 4; they 

fit the hydrograph satisfactorily.

Similar computations of discharge were made for O.S. 202 (tables 4 

and 5) and O.S. 203 (tables 6 and 7). At these sites the values of FI 

determined from the celerity equation were much too low to give total 

discharges that matched the hydrographs in figures 5 r,nd 6. When 

values of Fi computed by the Manning equation were used, satisfactory 

results were obtained. The total discharges obtained in the second 

trial computations are plotted in figures 5 and 6.

TABLE 4. Computation of discharge in the overriding wave (Qw"> at O.S. 202

Column 7=£r -5- column 6 Column 10=6/2 X column 8 X column 9 

Column 8=Column 5 X column 7 Column 11=Column 10 * column 4 

Column 9= Column 3 column 2

(1)

Clock 

time

(hr)

1333

1340

1405

1415

1435

1446

1456

1505

1512

1530

1539

1550

1557

1609

1617

1625

(2)

(ft)

0.25

.27

.30

.31

.35

.39

.41

.42

.44

.50

.51

.52

.52

.51

.48

.40

(3)

(ft)

0.38

.52

.60

.62

.75

.85

.96

.99

1.00

1.05

1.08

1.10

1.11

1.10

1.03

.95

(4)

T * P

(sec)

38.0

32.0

26.2

24.8

22.8

22.0

21.2

20.5

20.0

19.0

18.8

18.8

19.0

21.0

23.0

30.0

(5)

tt

(sec)

8.4

7.8

7.4

7.6

8.1

8.4

8.7

8.8

9.0

9.3

9.4

9.6

9.8

10.0

10.8

11.8

(6)

t w

(sec)

12.5

11.0

9.7

8.8

8.1

7.7

7.6

7.5

7.4

7.1

7.0

6.7

6.7

6.6

6.8

8.0

(7)

(fps)

9.5

10.8

12.2

13.5

14.6

15.4

15.6

15.8

16.0

16.7

16.9

17.7

17.7

18.0

17.4

14.8

(8)

Lw

(ft)

80

84

90

102

118

129

135

139

144

155

159

170

173

180

188

175

(9)

h w

(ft)

0.13

.25

.30

.31

.40

.46

.55

.57

.56

.55

.57

.58

.59

.59

.55

.55

(10) (11)

Overriding wave

Volume

(ft')

156

315

405

474

708

890

1,110

1,190

1,210

1,280

1,360

1,480

1,530

1,590

1,550

1,440

Qw

(cfs)

4

10

15

19

31

40

52

58

60

67

72

79

81

76

67

48
DETERMINATION OF DISCHARGE DURING PULSATING FLOV D15 

DISCUSSION OF RESULTS

The method of computing total discharge values from the wave 

discharge and Manning's equation gave results that fit the measureddischarge hydrograph fairly well. A value of ?i=0.010 was used in the 

Manning equation. This value was calculated by the Los Angeles 

County Flood Control District (written commun., 1966) using float 

velocities determined from motion pictures taken during the Santa

TABLE 5. Computation of discharge in the shallow-depth (overrun) part of flow 

(Qi) and total discharge at O.S. 202

Column 2=Column 2, table 4 

Column 3=6 X column 2

Column 4 = V (0/2) (DtID i) (Dt + Di) 

Column 5=Column 7, table 4 column 4 

Column 6=Column3 X column 5

Column 7=Column 6 + column 11, table 4 

Column 8=(1.486M)Si/2jR2/3 

Column 9=Column3 X columns 

Column 10=Column 9 + column 11, table 4

(1)

Clock

time

(hr)

1333

1340

1405

1415

1435

1446

1456

1505

1512

1530

1539

1550

1557

1609

1617

1625

(2)

Di

(ft)

0.25

.27

.30

.31

.35

.39

.41

.42

.44

.50

.51

.52

.52

.51

.48

.40

(3)

A ,

^T-l

(ft")

7.5

8.1

9.0

9.3

10.5

11.7

12.3

12.6

13.2

15.0

15.3

15.6

15.6

15.3

14.4

12.0

(4)

VV&V6

celerity

(fps)

3.9

5.0

5.4

5.5

6.2

6.6

7.2

7.3

7.3

7.2

7.4

7.4

7.5

7.5

7.2

7.2

(5)

Vl

(fps)

5.6

5.8

6.8

8.0

8.4

8.8

8.4

8.5

8.7

9.5

9.5

10.3

10.2

10.5

10.2

7.6

(6)

First trial

Qi

(cfs)

42

47

61

74

88

103

103

107

115

142

145

161

159

161

147

91

(7) (8) (9) (10)

Second trial

Total

discharge

(cfs)

46

57

76

93

119

143

155

165

175

209

217

240

240

237

214

139

Vl

(fps)

7.7

8.1

8.7

8.8

9.6

9.7

10.6

10.8

11.1

12.1

12 2

12.4

12.4

12.2

11.8

10.5

Qi

(cfs)

58

66

78

82

101

114

130

136

146

182

187

194

194

187

170

126

Total

discharge

(cfs)

62

76

93

101

132

154

182

194

206

249

259

273

275

263

237

174

TABLE 6. Computation of discharge in the overriding wave (Qw) at O.S. 208

Column 7=Lr * column 6 

Column 8=Column 5 X column 7 

Column 9=Column 3 column 2

Column 10=6/2 X column 8 X colunn 9 

Column 11= Column 10 -r column 4

(1)

Clock 

time

(hr)

1415

1430

1445

1500

1515

1530

1545

1600

1615

1630

1645

1700

(2)

Vi

(ft)

0.30

.33

.35

.43

.47

.50

.55

.60

.58

.50

.40

.30

(3)

D2

(ft)

0.5G

.62

.72

.85

.90

.92

1.01

1.15

1.15

.95

.74

.57

(4)

P 

(sec)

43

45

48

50

51

54

55

55

52

49

44

40

(5)

(sec)

17.0

21.0

23.5

25.0

26.5

27.5

28.0

28.0

27.5

25.0

21.0

19.0

(6)

t w

(sec)

11.5

11.0

10.5

10.0

9.6

9.2

8.8

8.3

8.3

9.2

10.6

12.2

(7)

vw

(fps)

10.0

10.5

11.0

11.6

12.0

12.6

13.1

13.9

13.9

12.6

10.9

9.5

(8)

Lw

(ft)

170

220

258

290

318

346

367

389

382

315

229

180

(9) (10) (ID

Overriding wave

(ft)

0.26

.29

.37

.42

.43

.42

.46

.55

.57

.45

.34

.27

Volurre

(ft3)

77o

1, 120

1,670

2,131

2,390

2,550

2,950

3,750

3,810

2, 760

1,600 94°5

Q

(cfs)

18

25

35

43

47

47

54

68

73

56

36

24
D16 RIVER HYDRAULICS

Anita Wash, tests. The float velocities were corrected to mean velocity 

by assuming the velocity varied directly with the logarithm of the 

distance from the channel bottom. A higher value of n would give a 

slightly better fit of computed values to the measured hydrograph 

during the highest discharges at O.S. 201; however, the results of 

computations at O.S. 202 and O.S. 203 seem to support the use of

As stated earlier, the observed values of flow used to compute discharge wTere selected from curves determined by plotting average 

dimension and velocity values of each five-wave sequence against time 

of observation. The author recognized that a sum of discharge computations for each wave of the sequence would probably give more 

accurate results, but the data collected were not of sufficient detail or 

accuracy to permit this approach.

The inconsistent results obtained when the equation for wave celerity 

was used to compute the velocity at minimum depth (V-C) is probably 

due to the fact that the celerity equation is valid only when the flowT 

is uniform on each side of the wave front. Uniformity does not exist 

upstream from the wave front during pulsating flow.

The observed wave characteristics at the three observation sites at 

the approximate time of maximum discharge are shown in table 8. 

The most striking difference is in the wave velocity.

TABLE 7. Computation of discharge in the shallow-depth (overrun) part of flow 

(Qi) and total discharge at O.S. 203

Column 2=Column 2, table 6 

Column 3=6 X column 2 

Column 4 = A

Column 5=Column 7, table 6 column 4 

Column 6=Column 3 X column 5

Column 7= Column 6 + colurrn 11, table 6 

Column 8=(1.486/71)5^.???$ 

Column 9=Column 3 X colurrn 8 

Column 10=Column 9 + coluirn 11, table 6

(1)

Clock

time

(hr)

1415

1430

1445

1500

1515

1530

1545

1600

1615

1630

1645

1700

(2)

Di

(ft)

0.30

.33

.35

.43

.47

.50

.55

.60

.58

.50

.40

.30

(3)

A, ^ 1 

(ft 2)

10.5

11.6

12.2

15.0

16.4

17.5

19.2

21.0

20.3

17.5

14.0

10.5

(4)

WQTTO Wave

celerity

(fps)

5.1

5.4

6.0

6.4

6.5

6.5

6.8

7.4

7.4

6.7

5.8

5.2

(5)

V,

(fps)

4.9

5.1

5.0

5.2

5.5

6.1

6.3

6.5

6.5

5.9

5.1

4.3

(6)

First trial

<?i

(cfs)

51

59

61

78

90

107

121

136

132

103

71

45

(7) (8) (9) (10)

Second trial

Total

discharge

(cfs)

69

84

96

121

137

154

175

204

205

159

107

69

Vi

(fps)

6.3

6.7

7.0

8.0

8.5

8.8

9.4

9.9

9.7

8.8

7.6

6.3

<?i

(cfs)

66

78

85

120

139

154

180

208

197

154

107

66

Total

discharge

(cfs)

84

103

120

163

186

201

234

276

270

210

143

90
DETERMINATION OF DISCHARGE DURING PULSATING FLOV D17 

TABLE 8. Observed wave characteristics at approximate time of maximum discharge

Time of Depth Depth at Wave

Observation site observa- between wave velocity,

tion waves, front, DI Vw (fps)

(hr) Dl (ft) (ft)

Santa Anita Wash above Sierra Madre 

Wash (O.S. 201)-.__--_____. ______ -_

Santa Anita Wash at Colorado Blvd. 

202)_._____ _ __

Santa Anita Wash at Longden Ave. 

203) ____________________________

(O.S.

(O.S.

1600 

1609 

1615

0. 5 

. 5 

. 6

1. 1 

1. 1 

1. 1

23. 3 

18. 5 

13. 9

In summary, the methodology used to determine the discharge of 

pulsating flow is adequate, but more accurate observations would improve the discharge computation. In the following section tvo methods are described for recording the elements necessary to determine 

discharge during pulsating flow by the method developed in this 

study. A third method that has some promise for determining the 

discharge is the dye-dilution method. It is also briefly discussed in the 

section that follows.

PROPOSED METHODS OF DETERMINING DISCHARGE 

PHOTOGRAPHIC METHOD

Discharges may be determined from measurements of channel geometry and observations of wave velocity and shape, maximum depth, 

and minimum depth, as shown in the Santa Anita Wash test. Tests 

of the robot camera at Haines Creek have shown that the programed 

sequence used at that site was inadequate to define all the elements 

needed. Following is a brief description of the filming sequence and 

equipment necessary to determine discharges from motion pictures. 

A schematic sketch, shown in figure 8, gives the general component 

layout. The operating sequence is as follows:

1. Unit is turned on automatically when minimum water level is 

maintained at some preselected depth. The time the unit is turned 

on is recorded on a clock-driven chart.

2. A power-driven 16-inm camera, with automatic electric-eye diaphragm, operates for 180 seconds at a rate of 10 framer per second. Floodlights and a timing light operate simultaneously 

with the camera

3. After 15 minutes, the sequence is repeated.

4. When minimum water level recedes below the preselected depth, 

the unit is shut off automatically and the time is recorded on the 

clock-driven chart.
D18 RIVER HYDRAULICS

Grid (as shown in fig. 3)

FIGURE 8. Schematic sketch of general component layout for photographic 

method of determining discharge.

The automatic on-off switch consists of a pair of contacts, set at a 

predetermined depth and connected to a relay switch which closes 

when the contacts are submerged. A delay unit between the contacts 

and the relay switch prevents a fluttering of the power to the sequence 

unit when the water level is fluctuating near the depth of the contacts. 

The time on and off is marked on a clock-driven chart with the aid of 

a second relay switch attached to a marking pen. When the relay switch 

is closed, the sequence unit supplies power to the floodlights, timing 

light, and camera for 180 seconds every 15 minutes. This allows filming 

of at least five waves with an average period of 36 second? or less. The 

timing light, set in the field of view of the camera, will flash for 0.2 

second every 10 seconds to aid in determining the exact number of 

frames per second exposed. There are 40 frames per foot of 16-mm film; 

the film required for 4 hours of operation (3 minutes every 15 minutes) 

is 720 feet. Magazines up to 900 feet are available for some cameras.
DETERMINATION OF DISCHARGE DURING PULSATING FLOW D19

The floodlights, which are necessary for filming at night, require a high 

power source for operation. Therefore, a large rechargeable battery 

pack or, preferably, line powT er is required.

A frame-by-frame analysis of each wave will yield wave-frent velocity and wave dimensions, including maximum and minimum depth. 

If debris is present, surface velocities at various points behind the 

wave front may also be determined. Discharge for each wave is computed seperately and then averaged for the five-wave sequence.

DEPTH-RECORDER, METHOD

The depth recorder is an analog device which transforms electrical 

current received from a depth-sensing unit to a measure of the corresponding depth of water. The depth-sensing unit consists of several 

electrical contacts set at known intervals above the channel bottom. 

When the lowest contact is submerged, a connecting relay switch is 

closed, the unit is turned on, and a known amount of current is supplied to an analog recorder. As more contacts are submerged, more 

current is supplied. The recorder is calibrated so that incoming current is proportional to depth of water. The resultant depth of water 

is plotted against time on moving graph paper. A magnetic-tape recorder in parallel with the graphic recorder would facilitate data 

analysis with a digital computer. The time on and off are each recorded on the chart.

Analysis of data from a single depth sensor would yield a graph 

of stage only. To determine wave velocity and compute wave shape, 

a second sensor is installed at a known distance downstream frcm the 

first. Data from both sensors are recorded simultaneously. The velocity 

of the wave is equal to the distance between sensors divided by the wave 

travel time.

An a-c power supply would be preferable to battery power to prevent 

polarization of the contacts on the depth sensor.

DYE-DILUTION METHOD

The Geological Survey and others have successfully measured the 

discharge of stable flow by injecting dye of a known concentration 

into a stream at a constant rate.1 Downstream from the injection site, 

at a distance sufficiently long to allow complete mixing of the dye and 

water, samples of the water are taken. The dye concentrations of the 

samples are determined by fluorometer, and a simple computation 

gives the discharge of the stream. Studies by F. A. Kilpatrick of the

1 Cobb, B. D., and Bailey, J. F., 1965. Measurement of discharge by dye-dilution irethods : 

unpub. data to be published as Techniques of Water Resources Investigations of the U.S. 

Geological Survey.
D20 RIVER HYDRAULICS

Geological Survey (written commun., 1966) have shown that the 

process can be automated to obtain continuous measurement of discharge, but no automatic dye-injection equipment is operational at 

this time.

Three types of automatic samplers may be used. Two types are operational; they are the individual-sample bottling system (U.S. InterAgency Committee 011 Water Resources, 1963, p. 124) and a chartrecorder asembly (such as that made by G. K. Turner Associates, Palo 

Alto, Calif.) connected to an open-door flow-through fluorometer. Both 

require line or battery power. The third type, being developed at Colorado State University by T. M. Zorich (written commun., 1966), shows 

considerable promise for use in isolated sites. Dye is injected to the 

stream at a constant rate using a Mariotte vessel. After the dye and 

water are well mixed, water is diverted from the channel by gravity 

flow past a capillary-feed stylus. The stylus applies a sample of the 

water to a clock-driven roll of filter paper, and the filter paper is later 

processed through a fluorometer. The last two methods provide a continuous recording of dye concentration.

There is some question as to how successful the dye-dilution 

method would be for measuring pulsating flow. In thQ, Santa Anita 

Wash flood control channel, the method would undoubtedly be successful if the dye were injected and sampled upstream from the bend 

where the translatory waves become fully developed. However, its 

success there would not prove the general applicability of the method 

in a channel carrying pulsating flow. The method should be tested 

under controlled-flow conditions, similar to those of April 16, 1965, 

by the injection of dye downstream from the above-mentioned bend 

in the Santa Anita Wash channel.

DISCUSSION OF PROPOSED METHODS

Successful analysis of motion picture data from the Santa Anita 

Wash test has shown that discharge during pulsating flow can be 

computed by the photographic method. The proposed instrumentation 

has some limitations which should be considered when selecting a 

method. The system is complex, and when it is operating under field 

conditions, regular maintenance and testing would be required to prevent failure of any component. The small film magazine capacity 

precludes a continuous monitoring of the flow and during prolonged 

flow, film must be replaced frequently. Hence, the method is not 

adaptable for use in isolated areas. Frame-by-frame rnalysis of the 

film to determine discharge is tedious and time consuming.

As for the depth-recorder method, the depth-sensing part of the 

analog recorder has not been developed and tested. Providing that this 

part of the system can be made operational, this method has certain
DETERMINATION OF DISCHARGE DURING PULSATING FLOW D21

obvious advantages over the photographic method. Stage is continuously recorded in a form that can be easily analyzed not in 15- 

minute samples, as in the photographic method. Since the contacts are 

exposed in the channel, regular testing and maintenance would be required. The system requires line power which may prevent its use in 

isolated areas.

The dye-dilution method has not been tested during pulsating flow. 

Should the method prove successful, computations of discharge from 

the measurement of dye concentration are simple and direct. The 

dye-dilution method is the only method of the three discussed which 

does not require line power for operation except when the sampling 

is done with a continuous-recording flow-through fluorometer; it is 

therefore adaptable to field use at isolated sites. The dye-dilution 

method may not be applicable in smooth channels during high flows 

when pulsating flow is not fully developed and the turbulence is low; 

the channel may not be long enough for adequate mixing of the dye. 

The length of channel necessary for adequate mixing of the dye would 

increase if the turbulence decreased for any reason. This might occur, 

for example, if pulsating flow became less pronounced with an increase 

in discharge. If the distance for adequate mixing is such that tributary 

inflow between the points of dye injection and sampling is significantly 

large, the method cannot be used.

SUMMARY AND CONCLUSIONS

A method of determining the discharge during pulsating flow was 

tested in the Santa Anita Wash flood control channel in Arcadia, 

Calif., April 16, 1965. Observations of the dimensions of the translatory waves and of the velocity of the wave fronts were made at three 

sites during a period of known discharges ranging from 0 to 270 cfs. 

The method of computing discharge was based on (1) computation 

of the average discharge of the overriding wave, as determined from 

its observed volume and period, and (2) computation of the discharge 

in the shallow-depth, or overrun, part of the flow, the velocity of 

which was determined from the Manning equation. The method gave 

satisfactory results. The procedure used separating the flow into two 

components and then treating the shallow-depth component as though 

it were steady has no theoretical basis. It is, however, an expedient 

for use until laboratory investigation can provide a satisfactory analytical solution to the problem of computing discharge during pulsating flow.

Earlier, a robot camera had been designed and programed to obtain 

the data needed to compute discharge by the method described above. 

The photographic equipment had been installed in Haines Creek flood 

control channel, but it had not been completely tested because of £he
D22 RIVER HYDRAULICS

infrequency of flow in that channel. Because of a prolonged drought 

at Haines Creek and because the Santa Anita Wash teste demonstrated 

the inadequacy of the operating sequence of the camera, development 

of the camera technique was discontinued at the Haines Creek site.

Three methods are proposed for obtaining the data needed to compute discharge during pulsating flow. Two of them the photographic 

method and the depth-recorder method are based on observations of 

wave size, shape, and velocity. From these observations the discharge is 

computed by the procedure described above. The third method proposed discharge determination by dye dilution is based on the relation between rate of flow and measured dye concentration at a site 

downstream from a point of constant-rate injection. Each method has 

limitations which must be considered when selecting instrumentation 

for determining discharge during pulsating flow.

Each of the three methods proposed for determining discharge during pulsating flow should be developed and tested under conditions 

similar to those in the Santa Anita Wash test of April 16, 1965.

SELECTED REFERENCES

Chow, Ven Te, 1959, Open-channel hydraulics: New York, McGraw-Hill, 680 p.

Craya, A., 1952, The criterion for the possibility of roll-wave formation in Gravity 

waves : Natl. Bur. Standards Circ. 521, p. 141-151.

Dressier, B. F., 1949, Mathematical solution of the problem of roll waves in inclined open channels: Commun. on Pure and Appl. Matt ematics, v. 2, p. 

149-194.

 1952, Stability of uniform flow and roll-wave formation, in Gravity waves : 

Natl. Bur. Standards Circ. 521, p. 237-241.

Escoffier, F. F., and Boyd, M. B., 1962, Stability aspects of flow in open channels: 

Am. Soc. Civil Engineers Proc., Hydraulics Div. Jour., v. 88, no. HY6, p. 

145-166.

Holmes, W. H. ( 1936, Traveling waves in steep channels: Civil Eng., v. 6, no. 7, 

p. 467-168.

U.S. Inter-Agency Committee on Water Resources, Subcommittee on Sedimentation, 1963, A study of methods used in measurement and analysis of sediment 

loads in streams; Report 14, Determination of fluvial sed'ment discharge: 

151 p.

Iwasa, Y., 1954, The criterion for instability of steady uniform flows in open 

channels: Kyoto Univ., Mem. Fac. Eng., v. 16, no. 4, p. 264-275.

Koloseus, H. J., and Davidian, Jacob, 1966, Free-surface instability correlations: 

U.S. Geol. Survey Water-Supply Paper 1592-C, 72 p.

 1966, Roughness-concentration effects on flow over 1 ydrodynamically

rough surfaces: U.S. Geol. Survey Water-Supply Paper 1592-D, 21 p. 

Mayer, P. G., 1959, Roll waves and slug flow in inclined open channels: Am. Soc.

Civil Engineers Proc., Hydraulics Div. Jour., v. 89, no. HY4, p. 99-141. 

Powell, R. W., 1948, Vedernikov's criterion for ultra-rapid flovT : Am. Geophys.

Union Trans., v. 29, no. 6, p. 882-886. 

Thomas, H. A., 1940, The propagation of waves in steep prismatic conduits:

Iowa Univ., Hydraulics Conf. Proc., p. 214-229.

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