UC-NRLF C E ENGINEERINGLIBRART Att'tid - • > U. S. GEOLOGICAL SURVEY PROFESSIONAL PAPER 86 PLATE I APPARATUS ON CAMPUS OF UNIVERSITY OF CALIFORNIA. DEPARTMENT OF THE INTERIOR UNITED STATES GEOLOGICAL SURVEY GEORGE OTIS SMITH, DIBECTOR By CHARLES QILMAN JTTDg Referred to Attended to by " " whe7 file wrier /<•.„,,. ; PROFESSIONAL PAPER 86 THE TRANSPORTATION OF DEBRIS BY RUNNING WATER BY GROVE KARL GILBERT t< BASED ON EXPERIMENTS MADE WITH THE ASSISTANCE OF EDWARD CHARLES MURPHY WASHINGTON GOVERNMENT PRINTING OFFICE 1914 1HGINEERING LIBRARt o" CONTENTS. r> v, Preface Abstract Notation CHAPTER I. The observations Introduction General classification Stream traction Outline of course of experimentation Scope of experiments Accessory studies Flume traction Apparatus and material Experiment troughs Water supply The water circuit Discharge Sand feed Sand arrester Settling tank Gage for depth measurement Level for slope measurement Pitot-Darcy gage Sand and gravel Methods of experimentation General procedure for a single experiment. Details of procedure Width of channel Discharge The feeding of sand The collection of sand Determination of load Determination of slope Contractor Measurement of depth Measurement of velocity Modes of transportation Movement of individual particles Rolling . Saltation Collective movement Units Terms Load Capacity .* Competence Discharge Slope Size of debris Form ratio Duty and efficiency Symbols Table of observations on stream traction. . Tage. 9 CHAPTER II. Adjustment of observations 55 10 Observations on capacity and slope 55 13 The observational series 55 15 Errors 56 15 Intake influences 56 15 Outfall influences 56 15 Changes in apparatus 57 17 Rhythm 58 17 Slopes of debris and water surface 59 18 The logarithmic plots 59 18 Selection of an interpolation formula 60 19 The constant a and competent slope 65 19 Interpolation 72 '19 Precision 73 19 Duty 74 20 Efficiency 75 20 Observations on depth 87 20 Mode of adjustment 87 21 Precision 94 21 Mean velocity 94 21 Form ratio 94 21 Graphic computation 94 21 CHAPTER III. Relation of capacity to slope 96 22 Introduction 96 22 In channels of fixed width 96 22 The conditions 96 22 The sigma formula 96 22 The power function and the index of rela- 23 tive variation 97 24 The synthetic index 99 24 Application to the sigma function 99 25 Variation of the index 104 25 Formulation with constant coefficient 109 25 Effect of changing the unit of slope 112 26 Precision 113 26 Evidence from experiments with mixed 26 debris 113 26 Relation of index to mode of traction 115 26 In channels of similar section 116 30 The conditions 116 34 Sigma and the index 117 35 The synthetic index 119 35 Summary 120 35 Review 120 35 Duty and efficiency 121 35 CHAPTER IV. Relation of capacity to form ratio 124 35 Introduction 124 35 Selection of a formula 124 35 Maximum 124 36 Capacity and width 125 36 Capacity and depth -. . 127 36 Capacity and form ratio 128 3 CONTENTS. Page. CHAPTER IV — Continued. Discussion of experimental data 130 Scope and method of discussion 130 Sensitiveness and the index of relative vari- ation 130 Control of constants by slope 131 Control of constants by discharge 132 Control of constants by fineness 133 Special group of observations 134 Summary as to control by conditions 134 The optimum form ratio 135 Summary 136 CHAPTER V. Relation of capacity to discharge 137 Formulation and reduction 137 Measures of precision and their interpretation . . 142 Control of relative variation by conditions 143 Duty and efficiency 144 Comparison of the controlsof discharge and slope . 145 Controls of capacity 145 Controls of duty 147 Controls of efficiency 148 Summary 149 CHAPTER VI. Relation of capacity to fineness of debris 150 Formulation 150 Precision 151 Variation of the constant 153 Index of relative variation 153 Duty and efficiency ' 154 Summary 154 CHAPTER VII. Relation of capacity to velocity 155 Preliminary considerations 155 The synthetic index when discharge is constant. 157 Mean velocity versus slope 158 The synthetic index when slope is constant — 159 The synthetic index when depth is constant. . . 160 The three indexes 160 Relative sensitiveness to controls 162 Competent velocity 162 CHAPTER VIII. Relation of capacity to depth 164 Introduction 164 When discharge is constant 164 When slope is constant 164 Depth versus discharge 165 When velocity is constant 165 The three conditions compared 166 Comparison of controls by slope, discharge, and mean velocity 167 CHAPTER IX. Experiments with mixed grades 169 Adjustment and notation 169 Mixtures of two grades 172 Control by slope and discharge 175 Mixtures of more than two grades 176 A natural grade 177 Causes of superior mobility of mixtures 178 Voids 179 Fineness 180 Relation of capacity to fineness, for natural grades 180 Definition and measurement of mean fineness.. 182 Summary 184 Page. CHAPTER X. Review of controls of capacity 186 Introduction 186 Formulation based on competence 186 The form-ratio factor 190 Duty and efficiency 192 The formula of Lechalas 193 The formula 193 Discussion 194 CHAPTER XI. Experiments with crooked chan- nels 196 Experiments 196 Slope determinations 196 Forms and slopes 197 Features caused by curvature 198 CHAPTER XII. Flume traction 199 The observations 199 Scope 199 Grades of debris 199 Apparatus and methods 199 Processes of flume traction 200 Movement of individual particles. . . . 200 Collective movement 201 Table of observations 202 Adjustment of observations 203 Formulation 203 Precision 206 Discussion 206 Capacity and channel bed 206 Capacity and slope 208 Capacity and discharge 209 Capacity and fineness 210 Mixtures 212 Capacity and form ratio 213 Trough of semicircular cross section 214 Summary 215 Competence 215 W'ork of Overstrom and Blue 216 CHAPTER XIII. Application to natural streams. ... 219 Introduction 219 Features distinguishing natural streams 219 Kinds of streams 219 Features connected with curvature of channel 220 Features connected with diversity of dis- charge 221 Sections of channel 222 The suspended load 223 The two loads 230 Availability of laboratory results 233 The slope factor 233 The discharge factor 233 The fineness factor 235 The form-ratio factor 236 The four factors collectively 236 The hypothesis of similar streams 236 Summary 240 Conclusion 240 CHAPTER XIV. Problems associated with rhythm.. 241 Rhythm in stream transportation 241 Rhythm in the flow of water 242 CONTENTS. Page. CHAPTER XIV — Continued. The vertical velocity curve 244 The moving field 249 APPENDIX A. The Pitot-Darcy gage 251 Scope of appendix 251 Form of instrument 251 Rating methods 252 Rating formula 253 Page. APPENDIX B. The discharge-measuring gate and its rating 257 The gate 257 Plan of rating 258 Calibration of measuring reservoir 258 The observations 259 INDEX.. 261 TABLES. J'age. TABLE 1. Grades of de'bris 21 2. Gate readings and corresponding dis- charges 23 3. Data connected with changes in mode of transportation 33 4. Observations on load, slope, and depth.. 38 5. Values of capacity generalized from Table 4(G) 61 6. Data for the construction of curves in figures 20 and 21 62 7. Values of a in C=6,(S-i 109 17. Values of i, for mixtures of two or more grades 114 18. Comparison of values of ^ for mixtures and their components 115 19. Values of il associated with the smooth mode of traction 115 20. Relations of il to discharge, de'bris grade, and channel width 116 21. Selected data showing the relation of ca- pacity to slope when the form ratio is constant 117 22. Values of a corresponding to data in Table 21 118 23. Values of synthetic index under condi- tion that R is constant, and under con- dition that w is constant 119 23a. Comparison of parameters in C=5, (S-, 159 50. Means based on Table 49, illustrating the control of Iva by slope, fineness, and width 159 51. Values of In 160 52. Means based on Table 51, illustrating the control of Ira by fineness and width. . 160 53. Comparison of synthetic indexes for ca- pacity and mean velocity, under the several conditions of constant discharge, constant depth, and constant slope 161 54. Values of las 164 55. Means based on Table 54, illustrating the control of Ids by slope, fineness, and width 165 56. Synthetic indexes comparing the control of capacity by depth with control by discharge 165 57. Values of I,iv 165 58. Means based on Table 57, illustrating the control of Irir by mean velocity, fine- ness, and width 166 59. Comparison of synthetic indexes for ca- pacity and depth under the several conditions of constant discharge, con- stant elope, and constant mean velocity 167 60. Adjusted values of capacity for mixtures of two or more grades of d<5bris 169 61. Capacities for traction, with varied mix- tures of two grades 172 62. Percentages of voids in certain mixed grades of debris 179 63. Fineness of mixed grades and their com- ponents 180 63a. Data on subaqueous dunes of the Loire. 194 64. Comparison of slopes required forstraight and crooked channels 197 65. Grades of debris 199 Pag,-. TABLE 66. Relative speeds of coarse and fine de- bris in flume traction 200 67. Observations on flume traction, showing the relation of load to slope and other conditions 202 68. Values of capacity for flume traction, ad- justed in relation to slope 204 69. Comparison of capacities associated with different characters of channel bed 206 70. Values of /, for flume traction 208 71. Comparison of values of 7, for flume trac- tion with corresponding values for stream traction 209 72. Data illustrating the relation of capacity for flume traction to discharge 209 73. Values of capacity for flume traction, illustrating the control of capacity by fineness 210 74. Capacities for mixed grades and their components 212 75. Depths and form ratios of unloaded streams 213 76. Capacities for flume traction in troughs of different widths 213 77. Data on flume traction in a semicylindric trough 214 78. Observations by Blackwell on velocity competent for traction 216 79. Values of n in C oc Vm" based on Blue's experiments 218 80. Velocities of streams with and without fractional loads 230 81. Ratio of the suction at one opening of the Pitot-Darcy gage to the pressure at the other 254 82. Values of K and u in 77— 7/,=A'K" 254 83. Values of A in V= A^II^II, 255 ILLUSTRATIONS. Pago. PLATE I. Apparatus on campus of the University of California Frontispiece. II. De'bris used in experiments 22 III. Rough surfaces used in experiments on flume traction 200 FIGURE 1. Diagrammatic view of shorter experi- ment trough 19 2. Diagram of water circuit 20 3. The contractor 25 4. Diagrammatic view of part of experi- ment trough with glass panels and sliding screen 27 5. Appearance of the zone of saltation 27 6. The beginning of a leap, in saltation. . 28 7. Diagram of accelerations affecting a saltatory grain 28 8. Theoretic trajectory of a saltatory par- ticle 29 9. Ideal transverse section of zone of sal- tation at side of experiment trough. . 30 FIGURE 10. Longitudinal section illustrating the dune mode of traction 11. Longitudinal section illustrating the antidune mode of traction 12. Profiles of water surface, automatically recorded, showing undulations asso- ciated with the antidune mode of traction Plot of a single series of observations of capacity and elope Logarithmic plot of a series of observa- tions on capacity and slope Diagrammatic longitudinal section of outfall end of experiment trough, illustrating influence of sand arrester on water slope Diagrammatic longitudinal section of debris bed and stream, in a long trough , Page. ;j 1 33 13. 14. 16. 56 57 57 CONTENTS. FIGURE 17. Diagrammatic longitudinal section of outfall end of trough, illustrating influence of contractor 18. Profiles of channel bed, illustrating fractional rhythms associated with dunes of greater magnitude 19. Logarithmic graph of C=f(S), for grade (G), «-=0.66 foot, Q=0.734 ft.3/aec... 20. Extrapolated curves of C=f(S) for vari- ous tentative equations of interpola- tion and for slopes greater than 2.1 per cent 21 . Extrapolated curves of C=/(S) for vari- ous tentative equations of interpola- tion and for slopes less than 0.8 per cent 22. The relation of a in C=f(S-a) to the curvature of the logarithmic graph. . 23. Diagrammatic sections of laboratory troughs, illustrating relation of cur- rent depth to trough width 24. Ideal curve of competent slope in rela- tion to width of trough 25. Logarithmic plot of competent slope in relation to fineness of debris 26. Illustration of the method used to ad- just values of capacity in relation to slope by means of a logarithmic plot of observed values of capacity in re- lation to slope minus a 27. Observations of depth of current in re- lation to slope 28. Logarithmic computation sheet, com- bining relations of capacity, mean velocity, form ratio, and slope 29. Logarithmic locus of the exponential equation 30. Locus of log .y=/(log x), illustrating the nature of the index of relative varia- tion 31. Variations of i, in relation to slope 32. Variations of t, in relation to width of channel 33. Variations of i, in relation to discharge. 34. Variations of i, in relation to fineness of debris Variations of exponents i, , j, . and k in relation to slope 35. 36. form mode of traction and uniform slope 37. Curves of C=f(S) for three trough widths 38. Illustration of the relation of capacity to width of channel and to form ratio, when slope and discharge are constant 39. Cross sections of stream channels 40. Capacity for traction in relation to width of channel, when depth and slope are constant 41. Plot of equation (55) 42. Logarithmic plots of Page. 57 58 61 62 63 65 67 67 69 72 87 95 98 98 105 106 107 108 113 116 118 124 125 127 129 130 Page. FIGURE 43. Relation of capacity, C, to form ratio, R. The variation of the function C=b2(l+nR)Rm with slope 131 44. Relation of capacity, C, to form ratio, R. The variation of the function 0=62(1-0-^)^™ with discharge 132 45. Relation of capacity, C, to form ratio, R. The variation of the function C=b,(l-aR)Rm with fineness of de- bris 133 46. Relation of capacity, C, to form ratio, R. The variation of the function C=b,(l-aR)Rm with slope and dis- charge 134 47. Logarithmic plots of the relation of capacity to discharge 139 48. Ideal cross section of a stream in the ex- periment trough, illustrating the rela- tion of competent discharge to width . 140 49. Logarithmic plot of K=f(D) 141 50. Logarithmic plots of capacity for trac- tion in relation to fineness of de'bris. . 150 51. Average departures of original values of capacity from system of values readjusted in relation to fineness of debris 152 52. Vertical velocity curve, drawn to illus- trate its theoretic character near the stream's bed 155 53. Ideal profile 'of a stream bed composed of debris grains 155 54. Ideal curves of velocity in relation to depth, illustrating their relation to the zone of saltation 161 55. Tractional capacity for mixed de'bris, in relation to proportions of compo- nent grades, with associated curves of fineness and of percentages of voids. . 173 56. Tractional capacities of components of mixed grades in relation to the percentages of the components in the mixtures 174 57. Curves of capacity in relation to slope for grade (A), grade (G), and mixtures of those grades 175 58. Curves of capacity in relation to slope for grade (B), grade (F), and mixtures of those grades 175 59. Curves of capacity in relation to slope for grade (C), grade (E), and mixtures of those grades 176 60. Curves of capacity in relation to slope for a mixture of five grades, (C, D, E, F, G). Comparison of mixture curves for three discharges and of mixture curve with curves for component grades 176 61. Capacity-slope curves for related mix- tures and capacity-discharge curves for mixture and component grades. . 177 62. Capacity-slope curves for a natural grade of debris, compared with curves for sieve-separated grades 178 CONTENTS. Page. FIGURE 63. Curves showing the relations of various quantities to the proportions of fine and coarse components in a mixture of two grades of debris, (C) and (G) 179 64. Logarithmic plots of capacity in relation to linear fineness, for related mixtures of debris 181 65. Curve illustrating the range and dis- tribution of finenesses in natural and artificial grades of debris 181 66. Typical curves illustrating the distri- bution of the sensitiveness of capacity for traction to various controlling con- ditions 191 67. Plans of troughs used in experiments to show the influence of bends on trac- tion 196 68. Contoured plat of stream bed, as shaped by a current 198 69. Curves illustrating the relation of ca- pacity for flume traction to fineness of debris 211 70. Diagram of forces 224 71. Interference by suspended particle with freedom of shearing 226 72. Suggested apparatus for automatic feed of debris 241 73. Modification of vertical velocity curve by approach to outfall 244 74. Modification of vertical velocity curve by approach to contracted outfall . . 245 75. Plan of experiment trough with local contraction 245 76. Profile of water surface in trough shown in figure 75 245 77. Modification of vertical velocity curve by local contraction of channel 245 Page. FIGURE 78. Modification of vertical velocity curve when mean velocity is increased by change of slope 245 79. Modification of vertical velocity curve when mean velocity is increased by change of discharge 246 80. Modification of vertical velocity curve by changes in the roughness of the channel bed 246 81. Modification of vertical velocity curve by addition of load to stream, with corresponding increase of slope 246 82. Modification of vertical velocity curve by addition and progressive increase of load 246 83. Ideal longitudinal section of a stream, illustrating hypothesis to account for the subsurface position of the level of maximum velocity 248 84. Diagrammatic plan of suggested mov- ing-field apparatus 250 85. Longitudinal section of lower end of receiver of Pitot-Darcy gage No. 3, with transverse sections at three points 251 86. Cross section of prism of water in trough, showing positions given to gage open- ing in various ratings 255 87. Graphic table for interpolating values of A in.V=A TJH—H!, for observations made with gage 3b in different parts of a stream 256 88. Arrangement of apparatus connected with the rating of the discharge- measuring gate 257 89. Elevation and sections of gate for the measurement of discharge 258 PREFACE. Thirty-five years ago the writer made a study of the work of streams in shaping the face of the land. The study included a qualitative and partly deductive investigation of the laws of transportation of de'bris by running water; and the limitations of such methods inspired a desire for quantitative data, such as could be obtained only by experimentation with determinate conditions. The gratification of this desire was long de- ferred, but opportunity for experimentation finally came in connection with an investigation of problems occasioned by the overloading of certain California rivers with waste from hydraulic mines. The physical factors of those problems involve the transporting capac- ity of streams as controlled by various condi- tions. The experiments described in this report were thus instigated by the common needs of physiographic geology and hydraulic engineering. A laboratory was established at Berkeley, Cal., and the investigation became the guest of the University of California, to which it is indebted not only for space, within doors and without, but for facilities of many kinds most generously contributed. Almost from the beginning Mr. E. C. Murphy has been associated with me in the investigation and has had direct charge of the experiments. Before the completion of the investigation I was compelled by ill health to withdraw from it, and Mr. Murphy not only made the remain- ing series of experiments, so far as had been definitely planned, but prepared a report. This report did not incluae a full discussion of the results but was of a preliminary nature, it being hoped that the work might be con- tinued, with enlargement of scale, in the near future. When afterward I found myself able to resume the study, there seemed no im- mediate prospect of resuming experimentation, and it was thought best to give the material comparatively full treatment. It will readily be understood from this account that I am responsible for the planning of the experimental work as well as for the discussion of results here contained, while Mr. Murphy is responsible for the experimental work. It must not be understood, however, that in assuming responsi- bility for the discussion I also claim sole credit for what is novel in the generalizations. Many conclusions were reached by us jointly during our association, and others were developed by Mr. Murphy in his report. These have been incorporated in the present report, so far as they appeared to be sustained by the more elaborate analysis, and specific credit is given only where I found it practicable to quote from Mr. Murphy's manuscript. Mr. J. A. Burgess was for a short time a scientific assistant in the laboratory, and his work is described in another connection. Credit should be given to Mr. L. E. Eshleman, carpenter, and Mr. Waldemar Arntzen, mechan- ician, for excellent work in the construction of apparatus. I recall with sincere gratitude the cordial cooperation of several members of the university faculty, and the investigation is especially indebted to the good offices and technical knowledge of Prof. S. B. Christy and Prof. J. N. Le Conte. Portions of my manuscript were read by Dr. R. S. Woodward and Dr. Lyman J. Briggs, and the entire manuscript was read by Mr. C. E. Van Orstrand and Mr. Willard D. Johnson. To these gentlemen and to members of the editorial staff of the Geological Survey I am indebted for criticisms and suggestions leading to the elimination of some of the crudities of the original draft. While the aid which my work has received from many colleagues has been so kindly and efficient that individual mention seems invid- ious, my gratitude must nevertheless be expressed for valuable assistance by Mr. Fran- cois E. Matthes in the examination of foreign literature, and for the unfailing encouragement and support of Mr. M. O. Leighton, until recently in charge of the hydrographic work of the Survey. G. K. G. ABSTRACT. Scope. — The finer debris transported by a stream is borne in suspension. The coarser is swept along the channel bed. The suspended load is readily sampled and estimated, and much is known as to its quantity. The bed load is inaccessible and we are without definite information as to its amount. The primary purpose of the investigation was to learn the laws which control the movement of bed load, and especially to determine how the quantity of load is related to the stream's slope and dis- charge and to the degree of comminution of the debris. Method. — To this end a laboratory was equipped at Berkeley, Cal., and experiments were performed in which each of the three con- ditions mentioned was separately varied and the resulting variations of load were observed and measured. Sand and gravel were sorted by sieves into grades of uniform size. Deter- minate discharges were used. In each experi- ment a specific load was fed to a stream of specific width and discharge, and measurement was made of the slope to which the stream automatically adjusted its bed so as to enable the current to transport the load. The slope factor. — For each combination of discharge, width, and grade of de'bris there is a slope, called competent slope, which limits transportation. With lower slopes there is no load, or the stream has no capacity J for load. With higher slopes capacity exists; and increase of slope gives increase of capacity. The value of capacity is approximately propor- tional to a power of the excess of slope above competent slope. If S equal the stream's slope and a equal competent slope, then the stream's capacity varies as (S — a)n. This is not a de- ductive, but an empiric law. The exponent n has not a fixed value, but an indefinite series of values depending on conditions. Its range of values in the experience of the laboratory 1 Capacity is defined for the purposes of this paper as the maximum load of a given kind of de'bris which a given stream can transport. See page 35. 10 is from 0.93 to 2.37, the values being greater as the discharges are smaller or the de'bris is coarser. The discharge factor. — For each combination of width, slope, and grade of de'bris there is a competent discharge, «. Calling the stream's discharge Q, the stream's capacity varies as (Q — K)°. The observed range of values for o is from 0.81 to 1.24, the values being greater as the slopes are smaller or the debris is coarser. Under like conditions o is less than n; or, in other words, capacity is less sensitive to change3 of discharge than to changes of slope. The fineness factor. — For each combination of width, slope, and discharge there is a limit- ing fineness of de'bris below which no transpor- tation takes place. Calling fineness (or degree of comminution) F and competent fineness , the stream's capacity varies with (F — 0)P. The observed range of values for p is from 0.50 to 0.62, the values being greater as slopes and discharges are smaller. Capacity is less sensi- tive to changes in fineness of de'bris than to changes in discharge or slope. The form factor. — Most of the experiments were with straight channels. A few with crooked channels yielded nearly the same esti- mates of capacity. The ratio of depth to width is a more important factor. For any combi- nation of slope, discharge, and fineness it is possible to reduce capacity to zero by making the stream very wide and shallow or very nar- row and deep. Between these extremes is a particular ratio of depth to width, p, corre- sponding to a maximum capacity. The values of p range, under laboratory conditions, from 0.5 to 0.04, being greater as slope, discharge, and fineness are less. Velocity. — The velocity which determines capacity for bed load is that near the stream's bed, but attempts to measure bed velocity were not successful. Mean velocity was meas- ured instead. To make a definite comparison between capacity and mean velocity it is neces- ABSTRACT. 11 sary to postulate constancy in some accessory condition. If slope be the constant, in which case velocity changes with discharge, capacity varies on the average with the 3.2 power of velocity. If discharge be the constant, in which case velocity changes with slope, capacity varies on the average with the 4.0 power of velocity. If depth be the constant, in which case velocity changes with simultaneous changes of slope and discharge, capacity varies on the average with the 3.7 power of velocity. The power expressing the sensitiveness of capacity to changes of mean velocity has in each case a wide range of values, being greater as slope, discharge, and fineness are less. Mixtures. — In general, debris composed of particles of a single size is moved less freely than debris containing particles of many sizes. If fine material be added to coarse, not only is the total load increased but a greater quantity of the coarse material is carried. Modes of transportation; movement of par- ticles.— Some particles of the bed load slide; many roll; the multitude make short skips or leaps, the process being called saltation. Sal- tation grades into suspension. When particles of many sizes are moved together the larger ones are rolled. Modes of transportation; collective move- ment.— When the conditions are such that the bed load is small, the bed is molded into hills, called dunes, which travel downstream. Their mode of advance is like that of eolian dunes, the current eroding their upstream faces and depositing the eroded material on the down- stream faces. With any progressive change of conditions tending to increase the load, the dunes eventually disappear and the de'bris sur- face becomes smooth. The smooth phase is in turn succeeded by a second rhythmic phase, in which a system of hills travel upstream. These are called antidunes, and their movement is accomplished by erosion on the downstream face and deposition on the upstream face. Both rhythms of de'bris movement are initiated by rhythms of water movement. Application of formulas. — While the prin- ciples discovered in the laboratory are neces- sarily involved in the work of rivers, the labo- ratory formulas are not immediately available for the discussion of river problems. Being both empiric and complex, they will not bear extensive extrapolation. Under some circum- stances they may be used to compare the work of one stream with that of another stream of the same type, but they do not permit an esti- mate of a river's capacity to be based on the determined capacities of laboratory streams. The investigation made an advance in the direction of its primary goal, but the goal was not reached. Load versus energy. — The energy of a stream is measured by the product of its discharge (mass per unit tune), its slope, and the accel- eration of gravity. In a stream without load the energy is expended in flow resistances, which are greater as velocity and viscosity are greater. Load, including that carried in sus- pension and that dragged along the bed, affects the energy in three ways. (1) It adds its mass to the mass of the water and increases the stock of energy pro rata. (2) Its transporta- tion involves mechanical work, and that work is at the expense of the stream's energy. (3) Its presence restricts the mobility of the water, in effect increasing its viscosity, and thus con- sumes energy. For the finest elements of load the third factor is more important than the second; for coarser elements the second is the more important. For each element the second and third together exceed the first, so that the net result is a tax on the stream's energy. Each element of load, by drawing on the supply of energy, reduces velocity and thus reduces capacity for all parts of the load. This prin- ciple affords a condition by which total capacity is limited. Subject to this condition a stream's load at any time is determined by the supply of de'bris and the fineness of the available kinds. Flume transportation. — In the experiments described above — experiments illustrating stream transportation — the load traversed a plastic bed composed of its own material. Other experiments were arranged in which the load traversed a rigid bed, the bottom of a flume. Capacities are notably larger for flume transportation than for stream transportation, and their laws of variation are different. Rolling is an important mode of progression. For rolled particles the capacity increases with coarseness, for leaping particles with fineness. Capacity increases with slope and usually with discharge also, but the rates of increase are less 12 TRANSPORTATION OF DEBRIS BY RUNNING WATER. than in stream transportation. Capacity is reduced by roughness of bed. Vertical velocity curve. — The vertical distri- bution of velocities in a current is controlled by conditions. The level of maximum velocity may have any position in the upper three- fourths of the current. In loaded streams its position is higher as the load is greater. In unloaded streams its position is higher as the slope is steeper, as the discharge is greater, and as the bed is rougher. Pilot tube. — The constant of the Pitot veloc- ity gage — the ratio between the head realized and the theoretic velocity head — is not the same in all parts of a conduit, being less near the water surface and greater near the bottom or side of the conduit. NOTATION. Certain letters are used continuously in the volume as symbols for quantities, a definition accompanying the first use. These are ar- ranged alphabetically in the following list, with brief characterization and page reference. The list does not include letters used tem- porarily as symbols and defined in immediate connection with their use; and if the same letter has both temporary and continuous uses only the continuous use is here given. Page. A A constant of the Pitot-Darcy gage. 254 (A), (B), etc Grades of debris, separated by sieves 21 (A, G4), etc Mixtures of debris grades, the let- ters and figures indicating com- ponents and proportions 169 a A pressure constant (Lechalas) ... 193 i A constant capacity in equation (10) 64 &2 A constant capacity in equation (54) 129 63 A constant in equation (64) 139 *4 A constant in equation (75) 151 &5 A constant in equation (91) 186 C Capacity of a stream for traction of debris (in gm. /sec.) 35 CT Readjusted value of capacity. . . 141, 151 c Constant coefficient in y=cx1 99 c, Constant coefficient in C=e1Sil ... 109 D Mean diameter (in feet) of particles of debris 21 d Depth of current (in feet) 33 d Differential 97 E Efficiency of stream for traction of debris; capacity per unit dis- charge per unit slope; CjQS 36 e Base of Naperian logarithms 61 F Linear fineness of debris; the re- ciprocal of /) 21, 183 FI Bulk fineness of debris; the num- ber of particles in a cubic foot. 21, 183 ft. /sec Feet per second; unit of velocity.. 34 f t.3/sec Cubic feet per second ; unit of dis- charge 34 g Acceleration of gravity 225 gm./sec Grams per second ; unit of load and of capacity for load 34 B Range of fineness 96 B, HI Readings of comparator, Pitot- Darcygage 254 I Synthetic index of relative varia- tion 99 I\ /for capacity and slope 122 la /for capacity and discharge 147 /i /for capacity and fineness 153 fas /for capacity and depth, slope being constant 164 /rf« /for capacity and depth, discharge being constant 164 far /for capacity and depth, mean velocity being constant 164 /< /for efficiency and slope 122 Is /for capacity and slope, form ratio being constant 119 Iv /for capacity and mean velocity. 157 Ivs /for capacity and mean velocity, slope being constant 157 /K« /for capacity and mean velocity, discharge being constant 157 Ivd /for capacity and mean velocity, depth being constant 157 Iw / for capacity and slope, width being constant 119 i Index of relative variation; ex- ponent in y=vx* 99 t'i i for capacity and slope 99 j'2 ifor capacity and form ratio 130 i3 ifor capacity and discharge 141 t4 i for capacity and fineness 153 j Exponent in^=er' 99 ;, Exponent in C=c1Sh 109 K, I- Constants of the Pitot-Darcy gage. . 254 K A constant discharge, correspond- ing to competent discharge 139 L Length, as a dimension of units. . 139 L Load; mass of debris transported through a cross section per unit time (in gm./sec.) 35 M. Mass, as a dimension of units. . . . 139 13 14 TRANSPORTATION OF DEBRIS BY EUNNING WATER. Page. m ............... Exponent in 0=63(1 -)p ...... 151 p. e .............. Probable error .............. . ..... 89 T. ............... Ratio of circumference of circle to diameter ..................... 21 ............... A constant linear fineness, corre- sponding to competent fineness. 151 Q ............... Discharge (in ft.3/sec.) ............ 35 R ................ Form ratio; djw .................. 36 p ............... Value of R corresponding to maxi- mum capacity .................. 129 5 ................ Slope, in per cent, of stream bed or water surface ................ 34 a ...... . .......... Slope of stream bed or water surface; = fall per unit distance . 34 Page. a A constant slope, corresponding to competent slope 64 £ Sum of 228 T Time, aa a dimension of units. . . 139 U. Duty of water (in gm./sec.) for traction of debris; capacity per unit discharge; C/Q 36 V. Velocity of current (in ft. /sec.). . . . 163 T'6 Velocity of stream at contact with bed (Lechalas) 193 Ym Mean velocity; discharge-Harea of cross section; Q/dw 33, 94, 155 | ', Velocity of stream at surface (Lechalas) 194 i> Variable coefficient in y=vxi 99 r, Variable coefficient in C=v1S'1 ... 99 r3 Variable coefficient in C=v3Q 3 . . . 141 (/• Width of stream channel (in feet) . . 67 THE TRANSPORTATION OF DEBRIS BY RUNNING WATER. By GROVE KARL GILBERT. CHAPTER I.— THE OBSERVATIONS. INTRODUCTION. GENERAL CLASSIFICATION. Streams of water carry forward debris in various ways. The simplest is that in which the particles are slidden or rolled. Sliding rarely takes place except where the bed of the channel is smooth. Pure rolling, in which the particle is continuously in contact with the bed, is also of small relative importance. If the bed is uneven, the particle usually does not retain continuous contact but makes leaps, and the process is then called saltation, an expres- sive name introduced by McGee.1 With swifter current leaps are extended, and if a particle thus freed from the bed be caught by an ascending portion of a swirling current its excursion may be indefinitely prolonged. Thus borne it is said to be suspended, and the process by which it is transported is called suspension. There is no sharp line between saltation and suspension, but the distinction is nevertheless important, for it serves to delimit two methods of hydraulic transportation which follow differ- ent laws. In suspension the efficient factor is the upward component of motion in parts of a complex current. In other transportation, including saltation, rolling, and sliding, the efficient factor is the motion parallel with the bed and close to it. This second division of current transportation is called by certain French engineers entrainement but has received no name in English. Being in need of a suc- cinct title, I translate the French designation, which indicates a sweeping or dragging along, by the word traction, thus classifying hydraulic transportation as (1) hydraulic suspension and (2) hydraulic traction. i McGee, W J, Oeol. Soc. America Bull., vol. 19, p. 199, 1908. The bed of a natural stream which carries a large load of debris is composed of loose grains identical in character with those transported. The material of the load is derived from and returned to the bed, and the surface of the bed is molded by the current. When debris is transported through artificial channels, such as flumes and pipes, the bed is usually rigid and unyielding. Trifling as this difference appears, it yet occasions a marked contrast in the quan- titative laws of transportation, and in the labo- ratory the two kinds of transportation were the subjects of separate courses of experimentation. It is necessary, therefore, for present purposes, to base a second classification of hydraulic transportation on the nature of the bottom. As the bed is typically plastic in stream chan- nels and typically rigid in flumes and other artificial channels, it is convenient to call the two classes stream transportation and flume transportation . The second classification traverses the first and their combination gives four divisions — stream suspension, stream traction, flume sus- pension, and flume traction. This report treats of stream traction and flume traction. It contains the record . and discussion of a series of experiments made hi a specially equipped laboratory at the University of Cali- fornia, Berkeley, in the years 1907-1909. STREAM TRACTION. Previous to the Berkeley work little was known of the quantitative laws of stream traction. The quantity of material trans- ported has sometimes been said to be propor- tional to the square of the slope, but I have failed to discover that the statement has a re- corded basis in theory or observation. A state- 15 16 TRANSPORTATION OF DEBRIS BY RUNNING WATER. ment more frequently encountered is to the effect that the quantity varies with the sixth power of the velocity; and the origin of this assertion is not in doubt. It is an erroneous version of a deductive law commonly attribu- ted to Hopkins (1844) or Airy and Law (1885), although announced as early as 1823 by Les- lie.1 The law, as formulated by Hopkins, is that " the moving force of a current, estimated by the volume or weight of the masses of any proposed form which it is capable of moving, varies as the sixth power of the velocity"; and this law pertains not at all to the quantity of material moved, but to the maximum size of the grain or pebble or bowlder a given cur- rent is competent to move. The subject of the competence of currents, or the relation of velocities to the size of par- ticles they can move, has also been treated experimentally by several investigators, and some account of their work will be given in later chapters. About the year 1883 Deacon made, in Man- chester, England, a notable series of experi- ments in the field of stream traction. As a result of definite measurements of quantities of sand transported and of the velocities of the transporting currents, he announced 2 that the amount transported, instead of varying with the sixth power of the velocity, as had been supposed, actually varies with the fifth power. In the field of flume traction the work has been somewhat more extensive, having as its special incentive the needs arising in ore mills for the conveyance of crushed rock; and a resumS of results will be found in the chapter on flume traction. A still greater body of investigation has been conducted by German and French engineers with the use of laboratory models of river chan- nels. The French work was done largely for the purpose of testing certain rules formulated by Fargue 3 for the improvement of navigable streams. The German experiments were and still are addressed to the broader subject of i Sir John Leslie's analysis is to be found in his Elements of natural philosophy, and in the edition of 1829 occurs at pages 426-427 of volume 1. David Stevenson mentions 1823, which probably indicates the first edition. William Hopkins gives a diflerent analysis with practically the same result, and does not mention Leslie. The passage quoted occurs in Cambridge Philcs. Sec. Trans., vol. 8, p. 233, 1844, and is probably from his earliest discussion of the subject. Wilfred Airy's later but evidently independent analysis appears in Inst. Civil Eng. Proc., vol. 82, pp. 25-26, 1885, with expansion by Henry Law on pp. 29-30. » Deacon, G. F., Inst. Civil Eng. Proc., vol. 98, pp. 93-%, 1894. * Fargue, L., Annales des ponts et chauss&s, 1894. river engineering in general and include within their scope the scientific study of the ways in which rivers shape and reshape their channels. The quantitative laws of stream traction, which constitute the chief theme of the Berke- ley work, thus fall within the province of the German investigators, but their study has not been taken up. There are three German labo- ratories, all well equipped, located severally at Dresden, Karlsruhe, and Berlin.4 The flow of a stream is a complex process, involving interactions which have thus far baffled mechanical analysis. Stream traction is not only a function of stream flow but itself adds a complication. Some realization of the complexity may be achieved by considering briefly certain of the conditions which modify the capacity of a stream to transport debris along its bed. Width is a factor; a broad channel carries more than a narrow one. Velocity is a factor; the quantity of debris carried varies greatly for small changes in the velocity along the bed. Bed velocity is affected by slope and also by depth, increasing with each factor; and depth is affected by discharge and also by slope. If there is diversity of velocity from place to place over the bed, more debris is carried than if the average velocity everywhere prevails, and the greater the diversity the greater the carrying power of the stream. Size of transported particles is a factor, a greater weight of fine debris being carried than of coarse. The density of debris is a factor, a low specific gravity being favorable. The shapes of particles affect traction, but the nature of this influence is not well understood. An important factor is found in form of chan- nel, efficiency being affected by turns and curv- ature and also by the relation of depth to width. The friction between current and banks is a factor and therefore likewise the nature of the banks. So, too, is the viscosity of the water, a property varying with tempera- ture and also with impurities, whether dis- solved or suspended. The enumeration might be extended, com- plexity might be further illustrated by pointing < The equipment and work of the laboratory of river engineering of the Technical High School of Dresden are discussed in the Zeitschrift fur Bauwesen, vol. 50, pp. 343-300, 1900, and vol. 55, pp. 664-C80, 1905; the equipment of the laboratory of the Technical High School "Frederici- ana " of Karlsruhe in the same journal, vol. 53, pp. 103-136, 1903, and vol. 60, pp. 313-328, 1910; and the equipment and work of the Laboratory for River Improvement and Naval Architecture, Berlin, in vol. 56, pp. 123- 151, 323-324, 1906. THE OBSERVATIONS. 17 out the influence of conditions on one another, and the difficulty of measuring the detrital loads of streams might be dwelt upon, but enough has been said to warrant the statement that an adequate analysis with quantitative relations can not be achieved by the mere obser- vation of streams in their natural condition. It is necessary to supplement such observation by experiments in which the conditions are definitely controlled. OUTLINE OF COURSE OF EXPERIMENTATION. In the work of the Berkeley laboratory capacity for hydraulic traction was compared with discharge, with slope, depth, and width of current, and with fineness of debris; and minor attention was given to velocity and to curvature of channel. For the principal ex- periments a straight trough was used, the sides being vertical and parallel, the ends open, the bottom plane and horizontal. Through this a stream of water was run, the discharge being controlled and measured. Xear the head of the trough sand was dropped into the water at a uniform rate, the sand grains being of approxi- mately uniform size. At the beginning of an experiment the sand accumulated in the trough, being shaped by the current into a deposit with a gentle forward slope. The deposit gradually extended to the outfall end of the trough, and eventually accumulation ceased, the rate at •which sand escaped at the outfall having be- come equal to the rate at which it was fed above. The slope was thus automatically adjusted and became just sufficient to enable the particular discharge to transport the par- ticular quantity of the particular kind of sand. The slope was then measured. Measurement was made also of the depth of the current ; and the mean velocity was computed from the dis- charge, width, and depth. In a second experiment, with the same dis- charge, the sand was fed to the current at a different rate, and the resulting slope and depth were different. By a series of such ex- periments was developed a law of relation between the quantity of sand carried, or the load, and the slope necessary to carry it, this law pertaining to the particular discharge and the particular grade of sand. The same ex- periments showed also the relations of the velocity of the current to slope and load. 20921°— Xo. 80 — 14 2 Another series of experiments, employing a greater or a less discharge, gave a parallel set of relations between slope, load, and velocity. By multiplying such series the relations be- tween discharge and slope, discharge and load, and discharge and velocity were de- veloped. Then a third condition was varied, the width of channel; and finally the remaining condition under control, the size of the sand grains. Thus data were obtained for studying the quantitative relations between load, slope, discharge, width, and fineness, as well as the relations of depth and mean velocity to all others. In all, the range of conditions in- cluded six discharges, six widths of channel, and eight grades of sand and gravel, but not all the possible combinations of these were made. The actual number of combinations was 130, and under each of these were a series of measurements of load, slope, and depth. There were also limited series of experiments involving a greater number of discharges and a greater number of widths. The separate determinations of load and slope numbered nearly 1,200, and those of depth about 900. SCOPE OF EXPERIMENTS. Before proceeding to a fuller description of apparatus and experiments, let us consider to what extent the conditions of the laboratory were representative of the conditions which exist in the natural stream. The sand used came from the beds of Ameri- can and Sacramento rivers and was assumed to be representative of river sand in general. No attention was paid to the influence on traction of the form and density of grains. Each sample used was separated from the natural mixture by means of two sieves and was composed of grains which passed through a certain mesh and were arrested by a mesh slightly smaller. In the sand carried by a river near its bed the range of size is much wider. The limit of coarseness is found in those particles which the current is barely able to roll, the limit in fineness in those par- ticles which the swirls of the current are not quite able to lift into suspension; and the limits vary from point to point of the channel bed. This difference in condition was not wholly ignored, but a short supplementary 18 TRANSPORTATION OF DEBRIS BY RUNNING WATER. series of experiments was made with definite mixtures of sand of various sizes, as well as with a natural mixture. The straight channel of the laboratory differs materially from the curved channels of nature. It gives comparatively uniform depths and velocities from side to side and from point to point in the direction of the flow, while in a curved channel the depths and velocities vary greatly both across and along the channel. This difference in condition also received some attention, a short series of experiments being made with crooked and curved channels. The vertical sides of the troughs did not well represent the sloping banks of rivers, and no attempt was made to measure the qualification due to this difference. The cross section of the laboratory current was essentially a rec- tangle and the ability of the current to trans- port was found to be definitely related to the ratio between depth and width; but satisfactory connection was not made between this relation and the forms of cross section in rivers. The thalweg of a river channel traverses an alternation of deeps and shoals, the deeps being characterized by a different system of velocities and by a different line of separation between the grades of debris carried severally by sus- pension and traction; but these contrasts were touched only in a qualitative way in the work of the laboratory. Each experiment dealt with a slope in ad- justment with a particular discharge and a par- ticular grade of sand. In a natural stream the discharge is subject to variation, and its changes cause changes in the fineness of the material carried along the bottom. Load and the local slopes are ever in process of adjust- ment to the temporary conditions of discharge and fineness, but the adjustment is never com- plete. The general or average slope is adjusted to an indeterminate discharge which is neither the smallest nor the greatest. For this phase of disparity allowance is not easily made. One of the conditions affecting velocities is friction on bed and sides of channel. Friction on the bed depends partly on the roughness of the bed and partly on the consumption of energy by traction. Its laws are the same in laboratory and in river. Friction on the sides depends on the character of the channel wall and must be materially greater on river banks than on the smooth sides of the experiment trough. The magnitude of the difference was not determined. Velocities are affected by the viscosity of the water, variations in this factor being caused by differences hi temperature and by impurities in solution and in suspension. The transportation of small particles is affected by adhesion, a property varying with the min- eral character of the particles and with the impurities of the water. These factors were, ignored but are probably negligible in com- parison with the factors tested. It may be mentioned, however, that the water of the laboratory was practically free from sus- pended material, whereas that of rivers is usu- ally highly charged at the time of most active traction. These comparisons serve to show that the investigation treats of a group of important factors of the general problem of stream traction but by no means comprehends all. Its results constitute only a contribution to the subject. ACCESSORY STUDIES. Incidental and accessory to the main in- quiry were a number of minor inquiries. One pertained to the Pitot tube, a second to other methods of measuring velocity near the bot- tom, a third to the relation between the mean velocity of a loadless stream and the rough- ness of its channel bed, and a fourth to the . mechanical process of hydraulic traction. FLUME TRACTION. In the experiments on flume traction the bed of the channel was not composed of loose d4bris but was the unyielding bottom of the trough. The same apparatus was used, with appropriate modifications. In each experi- ment slope of channel was predetermined, the trough being placed with definite inclination. The bed of the channel was given a definite quality of roughness or smoothness, and the material of the load was of a particular fine- ness or of a definite mixture of sizes. With a definite discharge flowing through the trough, debris was fed to the current at a definite rate, and the rate was gradually increased until clogging occurred. The rate of feed just be- fore clogging was then recorded as the maxi- mum load under the particular conditions. The series of experiments used two widths of THE OBSERVATIONS. 19 channel, five textures of channel bed, six dis- charges, and seven grades of sand and gravel, besides mixtures. There were nearly 300 de- terminations of load. APPARATUS AND MATERIAL. EXPERIMENT TROUGHS. The trough in which most of the experiments were made was of wood, 31.5 feet long, with an inside width of 1.96 feet. The height of the sides was 1 .8 feet at the head and 1 foot at the end, the change being made by a series of steps. Its proportions and general relations are illus- trated by figure 1. The surfaces were planed and painted. At the head, where water entered, the trough was connected with a tank by a flexible joint, a groove of the under side of the trough bottom resting on a* semi- cylindric member of the tank, so as to consti- tute a hinge, and the walls of the trough being connected with the sides of the tank by a sheet of flexible rubber. Here also was a gate by which the flow from the tank could be stopped. Close to the opposite end of the trough was a cross trough 1 1 feet long, 2.5 feet wide, and 3 feet deep, rigidly attached to the experiment trough and extending below it. A rectangular opening in the bottom of the experiment trough, an opening having the width of that trough, permitted sand in transportation by the current to sink into the cross trough, which contained boxes to receive it. The width of the experiment trough was varied by means of a longitudinal partition which was given various positions. Its width at the end was also varied by means of two oblique partitions, the "outfall contractor," which merged with the sides a few feet from the end and could be adjusted as desired. For certain experiments FIGURE 1.— Diagrammatic view o( shorter experiment trough, showing relations to stilling tank (A), cross tank (jB), and settling tank (C). false bottoms were added, with surfaces spe- cially prepared as to roughness. These will be specifically described in connection with the corresponding observations. A second trough, having the same function as the one just de- scribed was 150 feet long but similar in width and style. Its sides were higher and it was not hinged at the head but remained horizon- tal. By temporary arrangements of parti- tions curved and crooked channels 1 foot wide were constructed within this trough. The shorter trough was installed in the base- ment of the Mining Building of the University of California; the longer one on the campus near by. (See PI. I, frontispiece.) The longer trough was remodeled for the experiments on flume traction. A third trough 14 feet long and 0.67 foot wide had its wooden sides replaced for a space of 3.5 feet, at midlength, by plate glass, so that observation could be made from the side. It was provided with a sliding diaphragm, to be described in another place. A fourth trough, of iron, was used only in the experiments on flume traction and will be described hi connection with those experiments. A few experiments were made also in a trough carrying the waste water of the 150-foot trough. This had a width of 0.915 foot. WATER SUPPLY. The water was taken from the municipal mains of Berkeley. As it was not practicable to draw freely on this source, a moderate supply was made to serve for a long series of experiments, being stored in a sump and pumped up as required. By repeated use it acquired a certain amount of fine detritus in suspension, but the quantity was not sufficient to obstruct the view of the experiments — or rather, when it was found obstructive, a fresh supply of clear water was substituted. THE WATER CIRCUIT. Starting from the storage tank or sump, the water was lifted by a power pump to a 20 TRANSPORTATION OF DEBRIS BY RUNNING WATER. high trough 13 feet above. In passing through this trough it was first quieted by baffles and then regulated as to surface level by means of a spillway about 13 feet wide, the overflow returning directly to the sump. At the end it sank slowly through a vertical shaft, or leg, whence it issued in a jet through an aperture regulated by a measuring gate. After spending its force against a water cushion it passed through a stilling tank and then through the experiment trough. From that it fell a short distance to a settling tank, and thence returned to the sump. In figure 2 the circuit is shown diagram- matically but without accuracy as to the arrangement and relative sizes of the parts of the apparatus. Baffles DISCHARGE. For the control and determination of the discharges used in the experiments, a measur- ing gate was provided. Near the lower end of the vertical leg of the high trough the water issued through a rectangular opening in a brass plate. The gate, also of brass, sliding along the plate, controlled the size of the opening, its motion being given by rack and pinion and its position shown by a suitable scale. The head was about 6 feet and was determinate. The gate and its calibration are described in Appendix B (pp. 257-259). The head was regulated by means of the spillway in the high trough, and the amount of overflow on the spillway was controlled by a FIGURE 2.— Diagram of water circuit. gate valve in the supply pipe just above the pump. As a check on the control, the posi- tion of the water surface was shown in an inclined glass tube outside of the high trough. The index tube being nearly horizontal, its meniscus had a magnified motion and the condition of the head could be seen at a glance. (See fig. 88, p. 257.) SAND FEED. Above the experiment trough, and near its head, was a hopper-shaped box from which sand was delivered to the current in the trough. The box ended downward in an edge which stood transverse to the trough. Along this edge were a series of openings whose size was determined by a movable notched plate of brass. Water was supplied to the sand in the hopper, both at the top and near the bottom, the amount being regulated by valves. This water came from a small reservoir that was kept full by diverting part of the jet issuing from the measuring gate, and its use therefore added nothing to the measured discharge. For some of the experiments debris was fed by hand, the quantity being regulated by means of a measuring box and a watch. SAND ARRESTER. The cross trough attached to the experiment trough and extending below it (see figs. 1 and 2) had along its bottom a track on which moved a platform car. This car carried two iron boxes to receive the sand. The boxes were rectan- gular and a little broader than the experiment trough. Openings protected by wire gauze per- mitted water to drain from them when they were lifted out. THE OBSERVATIONS. 21 SETTLING TANK. The function of the settling tank was to catch sand which was carried past the cross trough. It was fitted with a system of parti- tions providing two alternative courses through which the stream could be turned, and with a hinged partition — the "deflector" — by which the diversion was made. GAGE FOR DEPTH MEASUREMENT. A frame resting on the experiment trough bore in vertical position a slender brass rod. This was raised and lowered by rack and pinion, and its relative height could be read on a scale. Depth of water was measured by reading the scale first with the rod end at the water surface and again with it at the d6bris ' surface. LEVEL FOR SLOPE MEASUREMENT. A surveyors' level stood a few yards from the trough, about equally distant from the ends, and was used, with a light rod, to measure relative heights for the determination of slopes of water surface and sand surface. PITOT-DARCY GAGE. A pressure-gaging apparatus of the Pitot- Darcy type, but of special pattern, was used to measure velocities of current. Its two aper- tures were directed severally upstream and downstream. Its external form was designed to give the least possible resistance to the cur- rent. The reading scale, with rubber-tube connection, had a fixed position, while the receiving member was moved from point to point. A fuller description is contained in Appendix A (pp. 251-256). SAND ANP GRAVEL. The d6bris used in the experiments was obtained from three streams — Sacramento River 7 miles below the mouth of the American, American River 8 miles above its mouth, and Strawberry Creek in Berkeley. The de"bris from the creek was relatively coarse and was used only in the experiments on flume trac- tion. The mean density of the river material was 2.69; that of the creek gravel 2.53. The forms of the grains of sand are shown in Plate II. To prepare the debris for use it was sorted into grades by a system of sieves, and in the laboratory records each grade was designated by the limiting sieve numbers. Thus the grains of the 40-50 grade passed through a sieve with 40 meshes to the inch and were caught by a sieve with 50 meshes to the inch. For the sake of brevity the grades are com- monly indicated in this report by letters in parentheses — (A), (B), etc. — and the same notation is extended to mixtures of two or more sizes. Neither of these notations, however, is suited for the mathematical discussion of the laboratory data, and three others were devised. These are, first, the mean diameter of particles, designated by D; second, the reciprocal of the mean diameter, or the number of particles, side by side, in a row 1 foot long, designated by F; third, the number of particles necessary to occupy, without voids, the space of 1 cubic foot, designated by Ft. In the sense that the notation of D distinguishes by magnitudes, the notations of F and F2 distinguish by mini- tudes. F is otherwise called linear fineness, and F2 bulk fineness. To determine the several constants for a grade of debris, a sample was weighed and its particles were counted. Then, N being the number of particles in the sample, W their weight, G their density, and Wo the weight of a cubic foot of water, „ _ NOW* W Defining mean diameter as the diameter of a sphere having the volume of the average particle — »/*, In the following table the grades of sand and gravel are characterized by the several nota- tions. TABLE 1. — Grades of debris. Grade name. Sieves used in separa- tion (meshes to linen). D, mean diame- ter of particles (root). F, num- ber of particles to linear foot. Ft, number of particles to cubic foot. Range olD or F. Range of F,. (A). 50-60 o.ooioo 1,002 1,910.000.000 .13 1.44 (B . 40-50 .00123 g!2 1,023, 000. (XX) .17 1.60 (C).. 30-40 .00166 602 417,000.000 .44 2.99 (D). 20-30 .00258 388 111.500.000 .56 3.80 (E). 10-20 .00561 178 10.770.000 .95 7.41 (F). 6- 8 .0104 95.9 1,685.000 .40 2.74 (G). 4-6 .0162 61.8 451,000 .43 2.92 (H) 3- 4 .0230 43.4 156,000 .36 2.51 (I).. 1- 2 .0547 [18.3] 11,900 [2.00] i*: <»>] 15". S.<£* • ,£*•£.• ^ipP *-,=-. ; .* c J;»T S.J3, DEBRIS USED IN EXPERIMENTS THE OBSERVATIONS. 23 The gate was set by bringing an index mark opposite a graduation mark on a scale, the two marks being on brass plates in contact. The gate was controlled by rack and pinion, and considerable force was necessary to move it. The limit of error may have been 0.002 foot. The ordinary error is believed to have been less than 0.001 foot. An error of 0.001 foot in the sotting would cause an error of 0.002 ft.3/sec., or •2-J-jr of the medium discharge. The determination of head is subject to an accidental error and a systematic error. The accidental error pertains to the adjustment of water level in the high trough, by means of the valve at the pump, with observation of the tube index of water level. It was possible to give this adjustment a refinement comparable with that of the hook gage, but in practice that refinement was not attained, because a close watch was not kept on the index. It was found by experience that the fluctuations of level (occasioned by fluctuations of the electric current supplying power to the pump) were small, and they were usually neglected, a prac- tical calibration of the valve at the pump being arranged so that it could receive the proper setting for each setting of the discharge meas- uring gate. The ordinary error of the adjust- ment of the head is estimated at 0.003 foot, which would occasion an error in the discharge of 1 in 4,000. The remaining possibility of error is con- nected with the history of the apparatus. At the time of the calibration of the measuring gate the laboratory occupied temporary quar- ters. In its removal to permanent quarters there was a measurement and readjustment of the vertical distance constituting the head. Also, for the work with the long trough the measuring gate was transferred to a replica of the high trough, which may have differed in some particular affecting the constants. As the work of calibration was at no time re- peated, there was no check on the errors which may have been thus introduced. In a gen- eral way, they are probably of the same order of magnitude as the errors of adjustment of water surface. It is believed that all other errors affecting discharge are small in compari- son with that connected with the measuring gate. The vertical width of the aperture by which discharge was regulated was 2 inches. The head, measured from the middle of the aper- ture, was 6.0 feet. The horizontal dimensions of the aperture, during experimental work, ranged from 0.1 inch to 6.0 inches, and the cor- responding discharges are given in the follow- ing table : TABLE 2. — Gate readings and corresponding discharge*. Gate opening (inches). Discharge (ft.'/sec.). Gate opening (inches). Discharge (ft.'/wc.) 0.1 0.019 .5 0.272 .2 .039 .6 .290 .3 .058 .7 .308 .4 .075 .8 .327 .5 .093 .9 .345 .6 .111 2.0 .363 .7 .128 2.5 .454 .8 .146 3.0 .545 .9 .164 3.5 .639 1.0 .182 4.0 .734 1.1 .200 4.5 .828 1.2 .218 5.0 .923 1.3 .237 5.5 1.021 1.4 .255 6.0 1.119 THE FEEDING OF SAND. The fact that the hourglass has been used to measure time suggests that the flow of dry sand through an aperture may be uniform. Such a flow was not tested in the laboratory because the plan for experimentation required that sand should be used over and over, and it was not practicable to dry it. The hopper was a device intended to produce a uniform flow of wet sand. Moist sand will not flow through a small opening; but if enough water is present to more than fill the voids, adhesion is overcome and flow takes place, as in a quick- sand. The freedom of the flow depends on the amount of water. It was found difficult to maintain a uniform condition in the hopper. Another difficulty arose from clogging of the openings, and this was occasioned by shreds of wood fiber and similar impurities in the sand. The second difficulty was largely obviated, after a time, by making the openings larger and fewer; but the hopper feeding was at best not sufficiently uniform to be used in measur- ing the load carried by the experimental stream. For all experiments in which a large quantity of ddbris was carried, the material was fed to the current by hand and was measured in the feeding. A small box of known capacity was filled with the material and emptied into the current at regular intervals timed by a watch or clock. If the interval was long, the meas- ured unit was dumped on a sloping table above 24 TRANSPORTATION OF DEBBIS BY SUNNING WATER. the trough and gradually fed to the current by means of a scraper. Hand feeding had the defect of discontinuity, as well as irregularity in detail, but it had the advantage of measure- ment, and in certain experiments its meas- urement of load gave an important check on the measurement of debris delivered at the outfall end of the trough. With a perfect and stable adjustment of conditions the two should agree, and their disagreement served to show that the slope of the channel bed had not become perfectly adjusted, or else that its adjusted condition was subject to rhythmic oscillation. In some of the later work the rate of feed was measured from time to time by inter- cepting the stream of sand falling from the hopper during a definite number of seconds and weighing the sample thus caught. THE COLLECTION OF SAND. In the original construction of the apparatus for arresting the sand the opening in the bottom of the trough was covered by a coarse wire screen, which lay flush with the trough bottom. This was intended to separate the current above from the still water below and prevent the formation of eddies, which might keep the sand from settling to the collecting box and might also check the current. It fulfilled its purpose and was altogether satis- factory for currents of moderate velocity, but with high velocities it interfered with the arrest of the sand, letting a considerable fraction pass on to the settling tank. It was accordingly removed, apparently without bad results. Eddies were formed, but the antici- pated difficulties were not realized. On the whole the apparatus for arresting sand was successful. It was only with the finer debris and at the highest velocities that the fraction of load escaping to the settling trough was too large to be neglected in the weighing. DETERMINATION OF LOAD. The sand collected, in sand box and settling tank, during the period recorded by the stop watch was weighed without drying, and the gross weight was afterward corrected by an allowance for the contained water. In order to determine the proper allowance a prelimi- nary study had been made, and as a result of that study a definite procedure was adopted for bringing the wet sand to a particular "standard" condition. After the sand-collect- ing box had been lifted from the trough all water which would drain from it by gravity alone was allowed to escape. It was then removed to smaller boxes for weighing. These boxes were jarred by tapping, which caused the sand grains to readjust their contacts and settle together, excluding a part of the inter- stitial water, which appeared at the surface and was poured off. The sand was then weighed. It is of interest to note that in the condition thus adopted as a convenient stand- ard sand occupies less space than when dry, moist, or supersaturated; its voids are at a minimum. The period recorded by the stop watch was ordinarily about 10 minutes but was made less when the current was most heavily loaded, because of the limited capacity of the sand- collecting box, and was extended for the lightest loads. Its beginning was sharply defined by the shifting of the sand boxes, which could be made to coincide within a second with the starting of the watch. Its end was somewhat less definite, but the error in time is believed to be small in comparison with the whole period. The load per second was computed by dividing the total load, namely, the corrected weight of sand, by the number of seconds in the stop-watch reading. Its error included (1) the error of timing, (2) the error of stand- ardizing the sand and correcting for contained water, and (3) the error in weighing. There are no definite data bearing on its amount, and nothing better can be recorded than a general impression that the results are reliable within 2 per cent, that the precision is lower than that of the discharge measurement, and that the error in determination of load is notably less than the error, presently to be considered, in correlating load with slope. When the rate of feed was regulated by the periodic contribution of a measureful of d6- bris, the weighings of the unit, from time to time, showed inequalities from which precis- ion could be estimated. A computation indi- cated the average probable error, for a run, as about 1 per cent. This depended chiefly on the standardization, and as that was less per- fect for the debris as fed than for the debris THE OBSERVATIONS. 25 as collected, the ordinary measurement of col- lected load is presumably affected by a smaller probable error. DETERMINATION OF SLOPE. The observations of slope were made with surveyors' level and rod. The rod, made for the purpose, without unnecessary length or weight, was graduated to hundred ths of a foot and read by eye estimate to thousandths. It was held by an assistant while the observer and recorder stood at the telescope. The po- sitions were determined by a graduation of the trough, which was marked at every foot. To measure the water slope, heights of the sur- face were taken at several points along the trough. To measure the sand slope, heights were taken at intervals of «ither 2 or 4 feet, the shorter interval being used with the shorter trough. The water slope could not be meas- ured when the surface was rough. When the debris surface was rough, it was usually graded before measurement by scraping from crests into adjacent hollows. The observed heights were plotted on section paper, with relatively large vertical scale, and a straight line was drawn through or among them. The line served the purpose of a pre- liminary determination of slope, and the plots were inspected for the detection of systematic errors. As a result of this inspection a portion of the profile was selected for the determina- tion of slope, and from the observations on this portion the slope was computed by least- squares method. CONTRACTOR. As will be explained more fully in another connection, the slope measurements were af- fected (1) by systematic errors connected with the conditions under which the water entered and escaped from the trough, and (2) by acci- dental errors arising from rhythm. One of the measures used to diminish the systematic er- rors was the contraction of the current at the outfall end of the trough. The apparatus for this purpose consisted of two boards as wide as the depth of the trough and arranged as in figure 3. Their attachment to the sides was flexible, so that the degree of convergence and the width of aperture at the outfall could be modified at will. This apparatus will be called the outfall contractor. The theory and effi- ciency of the contractor will be considered in the discussion of the slope errors. FIGURE 3.— The contractor. MEASUREMENT OF DEPTH. The depth of the current was measured ai mid width and near midlength of the trough. The determination was made by means of the gage already described (p. 21), during the period of tune for which the load was measured. As the water surface was subject to rhythmic fluctuation, a series of observations of its posi- tion were made, and their mean was used. A series of observations of the position of the de- bris surface were sometimes made also, but usually only a single observation, and the read- ing obtained was subtracted from the mean of readings on the water surface. The observa- tions of the d6bris surface were subject to an error which was regarded as more serious than that of the observations of water surface be- cause, being essentially systematic, it could not be eliminated by repetition. The prssence of the gage rod in the water modified the dis- tribution of velocities, and this modification in- cluded an increase of the current's velocity a little below the end of the rod. As the bot- tom was approached by the rod, the current scoured a hollow in the bed immediately under it; and if the rod were lowered to actual con- tact, the reading would give an excessive esti- mate of depth. What was attempted was to lower the rod to a position as nearly as possible at the level of undisturbed parts of the bod surrounding the visible hollow. This was a matter of judgment, but not of confident judg- ment, because the actual bed was concealed by a cloud of saltatory debris particles. It is therefore recognized that the measurements of depth are uncertain. Whenever the water profile as well as the debris profile was surveyed, an independent 26 TRANSPORTATION OF DEBRIS BY RUNNING WATER. estimate of depth was obtained by subtracting one profile from the other. This mode of de- termination avoided the error incident to the gage work and was on the whole satisfactory, but unfortunately the number of experiments to which it could be applied was not large. During the greater part of the experimentation the importance of the water profile was not recognized, and this particular use of it was essentially an afterthought. The values from profiles being assumed to have relatively small errors, both systematic and accidental, it is possible to measure by their aid the precision of the values from gage readings. Of 118 depths which were meas- ured by both methods, the gage gave the greater value for 36, the lesser for 78 ; and the average for gage values was 0.0045 ±0.0007 foot less than the average for profile values. Independently of this apparent systematic er- ror, the probable error of a single measure- ment with the gage was ±0.007 foot. MEASUREMENT OF VELOCITY. The mean velocity of the current is compu- ted by dividing the discharge by the area of the cross section, or the product of width and depth. Its precision depends on those of the determinations of discharge, width, and depth; and as the precision for discharge and width is relatively very high, the precision of mean velocity may be regarded as identical with that of depth. The attempts to measure velocity close to the channel bed were not successful. This is much regretted, because it is believed that bed velocity is a prime factor in traction and that slope and discharge exert their influence chiefly through bed velocity. The mode of measure- ment to which most attention was given was that by the Pitot-Darcy gage, and special forms of that instrument were constructed for the purpose. The difficulty which seemed in- superable was essentially the same as that en- countered in the measurement of depth. As the instrument approached the current-molded bed of ddbris, the bed retreated, with the for- mation of a hollow. In the presence of the instrument the normal velocity at the bed did not exist. Inseparable from this difficulty is a property of the instrument. When it is held close to the bottom or side of a channel its con- stant is not the same as in the free current. The system of flow lines and velocities with which the stream passes the obstructing object determines the instrument's constant, and when that system is modified by a neighboring ob- ject the constant changes. The nature of these difficulties is such that it was not thought worth while to experiment with other gages and meters which limit the freedom of the current. Other devices tried were of one type. Small objects, such as currants or beans, only slightly denser than water, were placed in the current and watched. The lighter ones would not remain near the bottom. The heavier ones were visibly retarded when they touched the bed and were also retarded when close to the bottom by the cloud of saltatory sand, which has a slower average velocity than the water it suffuses. MODES OF TRANSPORTATION. MOVEMENT OF INDIVIDUAL PARTICLES. ROLLING. In stream traction sliding is a negligible fac- tor. The roughness of the bed causes particles that retain contact to roll. When, as in most of the experiments, the grains are of nearly uniform size, each moving grain has to sur- mount obstacles with diameter like its own, and when it reaches the summit of an obstacle it usually possesses a velocity which causes it to leap. So rolling is chiefly the mere prelude to saltation. With mixed d4bris the same is true for the finer grains, but the coarser may roll continuously over a surface composed of the finer, and the coarsest of all, those close to the limit of competence, move solely by rolling. The large particle, as it rolls over the bed of smaller particles, indents the bed, and its con- tact involves friction. The energy thus ex- pended comes from the motion of the water, and its communication depends on differential motion between water and particle. Except under special conditions, to be mentioned later, the load travels less rapidly than the carrier, and it is also true that in a load of mixed debris the finer parts outstrip the coarser. SALTATION. In stream traction the dominant mode of particle movement is saltation. Because salta- tion grades into suspension it has often been THE OBSERVATIONS. 27 explained in the same manner, by appeal to up- ward movement of filaments of current, but the recent studies have led me to entertain a different view. Before this view is presented an account will be given of certain observations which were made with the use of the trough having glass sides. Through the trough was passed a current transporting sand of uniform grade, and the conditions were such that the sand bed and water surface were smooth. In the same water floated a few fine particles and thin flakes of mica, illustrating suspension, but there was no intergradation of the two processes. Viewed from the side, the saltation was seen to occupy at the bottom of the current a space with a definite upper limit, parallel to the sand bed. Within this space — the zone of saltation — the distribution of flying grains was systematic, the cloud being dense below and thin above, but not perceptibly varying from point to point along the bed. Viewed from above, the surface of the cloud seemed uniform and level, and it all appeared to be moving in the same direction. There was no suggestion of swirls in the current. When, in looking from the side, attention was directed to the base of the zone, it was easy to watch grains that traveled half by rolling and half by skipping, and these moved quite slowly; but higher in the zone the motions were so rapid and diverse that all was a blur. To resolve this blur a sliding dia- phragm was arranged. This consisted of a short board with a hole in it. The board hung FIGURE 4. — Diagrammatic view of part of experiment trough with glass panels (A ) and sliding screen (B). C, Hole in screen. outside the wall of the trough, being supported by a cleat above in such manner that it could be slid along the trough. (See fig. 4.) The hole, about 2 niches square, gave a restricted view of the saltation zone. By sliding the board in the direction of the current and keeping the eyes opposite, a traveling field of view was obtained. Manifestly if the field traveled at the same rate as the current, any object moving with the current would appear at rest to the observer, because there would be no relative motion of observer and object; and if objects in the water were moving (horizontally) at different rates, those coinciding hi rate with the field would be seen as if at rest, while the others would be seen as moving. When the field was moved slowly the rolling grains ceased to be distinct but were replaced in distinctness by grains that seemed to bob up and down. These vibrated through a space of two or three diameters, as if repeatedly striking the bed and rebounding. In inter- preting this appearance, allowance must be FIGURE 5. — Appearance of the zone of saltation, as viewed from the side with a moving fleld. made for the fact that the grains were dis- tinctly seen because they were moving hori- zontally about as fast as the diaphragm. Their paths were really low-arching curves, and only the vertical factor remained when the hori- zontal was abstracted. It is probable also that the appearance of rebounding was largely illusory, most of the grams either stopping at the end of the leap, or else leaping next time with a different velocity. When the field was moved somewhat faster, the bobbing grains disappeared and there came into distinct vision a set of grains quite free from the bed and occupying a belt within the sal- tation zone. All the zone above and below them was blurred. In the middle of the belt vertical motion was to be discerned but wa* less con- spicuous than in the lower zone. Where dis- tinctness graded into blurring, lines of motion could be seen which were oblique and curved, the lines above the belt curving forward and those below backward, as shown in figure 5. With progressively faster motion of the field the belt of distinct vision rose higher, until the top of the zone was reached, when all the lower part was blurred. The systematic gradation of velocity and other features from the bed upward and the 28 TRANSPORTATION OF DEBRIS BY RUNNING WATER. sensible uniformity of process over the whole width of channel are not consistent with the idea that the saltation zone is invaded by eddies of large dimensions, such as would bo competent to sustain the grains by the up- ward components of their motions. If there were large ascending and descending strands of current the visible surface of the zone would be locally raised and depressed by them. We must, indeed, assume that the flow is turbulent, in the technical sense, because parallel or laminar flow is impossible with FIGURE 6. — The beginning of a leap, in saltation. velocities competent for traction, but the ed- dies may be assumed to be of small dimen- sions in relation to the depth of the zone, and the lines of flow with which saltation is con- cerned may be assumed to be approximately parallel to the general direction of the current. The explanation I would substitute for that of the uplift of grains by rising strands is that each gram is projected from the bed with an initial velocity which gives it a trajectory an- alogous to that of a cannon ball. The follow- ing fuller statement, though given with little qualification, should be understood as largely hypothetic. In figure 6 the current is supposed to move from left to right above the grains of debris shown in outline. A grain which in A is at rest appears in B in an advanced position, having been rolled upward and forward about an undisturbed grain which lies in its way. (The moving grain is doubtless more likely to roll against two other grains than a single one, but the principle is the same.) In moving to its new position the center of gravity of the gram describes a curve convex upward. The grain continuously gains in velocity, and the acceleration also increases as the direction of motion comes to make a smaller angle with the direction of the current. At each instant the accelerative force due to the current and that of gravity are combined and have a re- sultant direction; and the combined or re- sultant accelerative force may be resolved into two parts, one of which coincides in direction with the motion of the center of gravity and the other with the line joining the center of gravity and the point of contact. The last- mentioned component presses the moving gram against the stationary grain. Opposed to it is the centrifugal force arising from the curvature of the grain's path; and the point is finally reached where the centrifugal force dominates and the grain is free. Under the ordinary conditions of saltation this point is not the crest of the obstruction, but is on the upstream side, so that the grain's direction of motion at the instant of separation is obliquely upward. Thus the free grain is initially mov- ing upward as well as forward, and it has al- most literally made a leap from the bed. If the grain were at that instant released from all influences but gravity, its path before returning to the bed would be the arc of a quadric parabola with vertical axis. The actual deviation of its trajectory from the parabolic form is analogous to that observed in gunnery, for it arises from the resistance of a fluid; but the laws of resistance are not the same for air and water, and the frictional ac- celeration in one case is negative while in the other it is mainly positive. The trajectory in gunnery is shorter than the ideal parabolic arc; in saltation it is longer. Figure 8 gives diagrammatically the trajec- tory of a saltatory grain. In figure 7 AB is a portion of the same trajectory. Let the space AC represent the instantaneous velocity of the grain, and let the line AD represent in direction and length the velocity of the water FIGURE 7. — Diagram of accelerations affecting a saltatory grain. about the grain. Then, C and D being con- nected by a line, CD represents in direction and magnitude the relative velocity of water and grain, or the velocity of the water as re- ferred to the grain. By reason of the mutual resistance of water and grain, this relative mo- tion accelerates the grain, the acceleration be- ing a function of the differential velocity, the size of the grain, and other conditions. On the line CD, showing the direction of the accelera- tion, let the space CE represent its amount. Then from E draw the vertical EF, represent- THE OBSERVATIONS. 29 ing, to the same scale, the acceleration of the grain by gravity. Connect C and F; the line CF represents in magnitude and direction the resultant acceleration of the grain. These relations are independent of the particular directions of motion of the grain and the water. Let us now introduce the assumptions, believed to be practically true for the laboratory condi- tions, that the water in the region of saltation moves parallel to the bed and that its velocity increases notably with distance from the bed. In the ascending part of its path the grain en- counters filaments of the current with higher and higher velocity. This tends to increase the relative velocity, but the grain is at the same time gaining in horizontal velocity and the gain tends to diminish the relative velocity. Unless the leap is short in relation to the size of the grain, the second of these tendencies is the greater, and at the highest point of its path the grain is moving nearly as fast as the FIGURE 8. — Theoretic trajectory of a saltatory particle, the initial point being at /. Arrows indicate acceleration. water. In the descending part of its path it en- counters slower moving filaments of current, and at some point (H, fig. 8) its horizontal mo- tion may equal that of the adjacent water. Then beyond H it passes through filaments moving still more slowly, and its acceleration from the reaction of the current becomes nega- tive. The acceleration due to gravity is of course uniform and downward, and its combina- tion with that due to the current yields a system of directions and magnitudes of the type indi- cated in figure 8 by short arrows. In the shorter and lower trajectories it is probable that the critical point //is not reached. If the position of the grain before leaping (fig. 6) is such that only a relatively short roll suffices to free it, then its initial velocity is small and the angle of ascent at which it is freed is low. It has a short, flat trajectory, and its velocity at the highest point is moderate. If the original roll is longer there is time to acquire speed before the leap; the initial ve- locity is large and the angle of ascent is rela- tively high. It has a long and high trajectory and when at the crest has been accelerated to high velocity. IF a grain at the end of a leap touches the bed at a favorable point it may leap again without coming to rest, and the impetus of the first flight will thus enhance the initial velocity of the second. In the observations with the moving field the grains seen most distinctly were those which moved horizontally with the field and at the same time had little vertical motion. So each belt of distinctness contained grains at the tops of their trajectories and was practically made up of such grains. The grains producing the curved lines in figure 3 were ascending or de- scending obliquely, and their horizontal com- ponents of motion coincided with the motion of the field for an instant only. In general the observations seem to show that the summit velocities of the leaping grains increase systematically with the height of the leap, and this generalization is in perfect accord with the hypothesis that the paths of grains are determined primarily by initial impulse. Under the hypothesis the series of velocities observed by aid of the moving field are not velocities of current, for the initial velocities of grains, being caused by the current, require that the water outspeed all the grains at the bottom of the zone of saltation. At the top of the zone there must be at least a slight advantage with the current, provided the water velocities increase upward. That the water velocities do increase upward can hardly be doubted, for in sweeping along the sand the stream expends energy, and as its energy subsists in velocity, the expenditure involves retardation. More- over, the grains of sand are at the same time most numerous and slowest near the bottom of the zone, so that their effect is there greatest. In this connection it is to be observed that the width of the belt of distinct vision in the moving field (fig. 5) is greater for the upper part of the zone of saltation than for the lower. As distinct vision is limited to a certain (unde- termined) range in horizontal velocities, this fact implies that the increase in horizontal speed of sand grains with distance from the bed is less rapid in the upper part of the zone than in the lower. The preceding discussion is subject to two qualifications, the first of which is connected with the retardation of the current at the side of the trough. By reason of that retardation the zone of saltation is shallower near the side and does not include the longer and higher 30 TRANSPORTATION OF DEBEIS BY RUNNING WATER. leaps. Figure 9 gives an ideal conception of the cross section of the zone and the distribu- tion of flying grains within it. Observation from the side penetrates but a short distance into the cloud, the distance being least where the cloud is most dense. The practical limit of visual penetration may be assumed to take some such form as the line AB. Thus the FIGURE 9. — Ideal transverse section of zone of saltation at side of experi- ment trough. tract actually studied in the work with the moving field was somewhat superficial and was not in strictness a vertical section of the zone. The second qualification is connected with turbulence. In steady flow the motion at each point of a stream is constant in velocity and direction. When the general velocity exceeds a certain minimum, which for the streams we have to consider is very small, the flow is not steady, but involves eddies or vortices, which as a rule move onward with the current. In consequence of these eddies the course of each particle of water is sinuous, and the sinuous courses interweave. The flow is then said to be turbulent. Usually there are both large and small eddies, the minute ones being multi- tudinous. As the axes of whirling movements have all attitudes, the directions of motion, as a rule, have upward or downward components, and the suspension of particles of debris is due to the upward components. Particles so small that they can not come to rest on the bottom are thereby lifted and relifted and kept in the body of the water. Under the conditions arranged for the study of saltation there appeared to be no large eddies, but the zone was unquestionably pervaded by small ones, excited by the roughness of the bed and by the differential motions of water and leaping grains. With increasing strength of current the texture of turbulence would enlarge and saltation pass into suspension. With a diver- sified debris, instead of the uniform material actually used, there would be phases of action in which the paths of small grains were made sinuous by turbulence, while those of larger grains remained simple in form. The trajec- tory of saltation, as described, may therefore be regarded as a simple type of path which combines in all proportions with the sinuous type of path characterizing suspension. Through the entire zone of saltation motion is being communicated to particles of the load by the water, and there is a corresponding loss of motion by the water. That loss reduces all the stream's velocities but makes the greatest reduction in the lower part of the zone of saltation. The loss of velocity in the lower strands reduces their power to cause particles to leap. The greater the load the greater this reduction, and thus the quantity of load is automatically regulated. COLLECTIVE MOVEMENT. In the experiment used to illustrate saltation the collective movement of the sand was uni- form, the conditions of the experiment having been adjusted to that end. But it is equallv possible so to adjust them as to make the collective movement rhythmic. Uniformity is in fact an intermediate phase between two rhythmic phases, which are of contrasted types. These phases will be described. In another experiment a bed of sand was first prepared with the surface level and smooth. Over this a deep stream of water was run with a current so gentle that the bed was not disturbed. The strength of current was gradually increased until a few grains of sand began to move and then was kept steady. Soon it was seen that the feeble traction did not affect the whole bed, but only certain tracts, and after a time a regular pattern developed and the bed exhibited a system of waves and hollows. As the waves grew the amount of transportation increased, showing that, under the given conditions, the undulating surface was better adapted to traction than the plane. With such waves and hollows are associated a special mode of transportation, which is illus- trated in figure 10. A current reaching the bed at A follows the rising slope and crest of the wave to C and then shoots free, to reach THE OBSERVATIONS. 31 the bed again at D. The space overleaped between C and D is occupied by one or more slow-moving eddies. From A to C there is traction, the material being derived from the slope between A and B. At C the debris, being abandoned by the current, is dumped, and it slides by gravity down the slope CE. So the upstream face of the wave is eroded and the downstream face built out, with the result that the wave, as a surface form, travels downstream. As this is precisely what takes place when a sand hill travels under the influence of the wind, the name of the eolian hill has been borrowed, and the waves are called dunes.1 In one of the narrower troughs of the laboratory the dunes formed a single line. In a wider trough their arrangement sometimes suggested a double line, the crests of one being opposite the hollows of the other, but their arrangement continually changed. On the bed of a broad, shallow stream they are apt to have a subregular imbricated pattern.2 In a deep stream a single dune may be nearly as broad as the channel. In the laboratory the forms were inconstant, but the type was about as broad as long, with the front edge convex downstream. In natural streams the dunes show great variety in outline, some being described as longest in the direction of the current and others as greatly extended in the transverse direction. They vary in size with the size of the stream, but especially with the depth, and are transformed and remodeled with increase and reduction of discharge. The horizontal dimensions of most laboratory examples may be conveniently described in J): FIGURE 10.— Longitudinal section illustrating the dune mode of traction. inches, but river examples may require scores or hundreds of feet.3 The maximum height in the laboratory was probably 2 inches; for those in Sacramento River 2 feet has been reported, and for those in the Mississippi 22 feet. In each series of laboratory experiments to determine the relation of load to slope the initial run was made with a small load, while for the succeeding runs the load was pro- gressively increased. Enlargement of the load caused increase of slope and velocity, with decrease of depth, and these changes were accompanied by changes in the mode of trans- portation. In the earlier runs dunes were formed, and these marched slowly down the trough. Then, somewhat abruptly, the dunes ceased to appear, and for a number of runs the i This is the name chiefly used by Swiss investigators (see De Candolle, Arch. sci. phys. et nat., vol. 9, p. 242, 1883, and Forel, idem, vol. 10, p. 43, 1884), and many observers compare the subaqueous feature to the eolian; but the specific title commonly used in the United States and Great Britain is sand wane, and some French engineers employ grbee. In the present paper, dune is preferred to sand wave because there is occasion to distinguish two species of debris waves. channel bed was without waves and approxi- mately plane, although somewhat ruffled in the run immediately following the disappearance of dunes. Finally a third stage was reached in which the bed was characterized by waves of another type. These are called antidunes, because they are contrasted with dunes in their direction of movement; they travel against the current instead of with it. Their downstream slopes are eroded and their up- stream slopes receive deposit. They travel much faster than the dunes, and their profiles are more symmetric. The water surface, which shows only slight undulation in connec- tion with dunes, follows the profiles of anti- * The imbricated pattern is frequently seen beneath tidal waters, where ripple marks due to the reaction of wind waves are transformed into dunes when the tidal current sweeps across them. It is then usually to be ascribed to a difference in direction of the two actions. An elabo- rate account of its development in rivers is given by H. Blasius, who has recently investigated the whole subject of the rhythmic features of river beds. See Zeitschr. Bauwesen, vol. 60, pp. 465-472, 1910. » Arthur Hider, who studied dunes in the lower Mississippi, reported a maximum length, crest to crest, of 750 feet, a maximum height of 22 feet, and a maximum progression of 81 feet in a day. See Mississippi River Comm. Rept., 1882, pp. 83-88 (=Chief Eng. U. S. A., Kept., 1883, pp. 2194-2199). 32 TRANSPORTATION OF DEBRIS BY RUNNING WATER, dunes closely and shares their transformations.1 (See fig. 11.) Usually each antidune occupied the full width of the experiment trough; and in natural streams, so far as I have observed, they either reach from side to side of the channel or else form well-defined rows in the direction of the current. Not only is a row of antidunes a rhythm in itself, but it goes through a rhythmic fluctuation in activity, either oscillating about a mean condition or else developing paroxys- mally on a plane stream bed and then slowly declining. Paroxysmal increase starts at the downstream end of a row and travels upstream, gaining in force for a time, and the climax is accompanied by a combing of wave crests. Where the debris is very coarse, as on the out- wash plains of glaciers, a din of clashing bowl- ders is added to the roar of the water.2 Of the phases of process in the laboratory Mr. Murphy writes : Their [the dunes'] form is continually changing as they move forward; they divide and again unite, the parts traveling at different rates, and new ones form on top of rl.O the older ones. The grains roll up the gentle slope, fall over the crest, are covered by other grains, and rest until the dune again passes over them an d they are again uncov- ered. Thus the time during which they are in motion is small compared to the time during which they rest. As the velocity of the current increases, the rate of feeding being correspondingly increased, the size of the dunes and their rate of movement increase. Thus we find that when the discharge is 0.363 ft.3/sec., load 11 gin. /sec., and slope 0.32 per cent, the dunes are 7 to 9 inches long and one-half inch high and move at the rate of 0.56 foot a minute, but when the discharge is 0.734 ft.3/see. and the load 30 gm./sec., the dunes are 13 to 15 inches long, three-fourths of an inch high and move at the rate of 1.5 feet a minute. As the velocity of the current increases some of the grains leap as well as roll, and some, instead of dropping over the crest of a dune and resting, leap to the next dune. The dune grows less distinct in form and finally at a criti- cal velocity it disappears, dune motion ceases, and the sand surface becomes comparatively even. This condi- tion of even surface flow continues as the slope increases until at another critical velocity antidune movement begins. A profile of the sand surface for this kind of motion is shown in figure 11. For this experiment the trough width is 1.32 feet, discharge 0.734 ft.3/sec., load 213 gm./sec. and the sand slope 1.23 per cent. These sand waves are from 2 to 3 feet in length from crest to crest, they extend the width of the trough, and some of them are 0.5 6 '9 FIGURE 11. — Longitudinal section illustrating the antidune mode of traction. trough. 12 15 The numbers show distance in feet from the head of the experiment foot in height from crest to trough of the wave. They travel slowly upstream, some of the sand being scoured from the downstream face in the vicinity of Y (fig. 11) and deposited on the upstream face at X. Some of these waves remain for two minutes or longer, but most of them not longer than one minute. A whitecap forms on the surface of the water when the larger waves disappear. Sometimes two or more will disappear at once and leave the surface without waves for a distance of 10 feet or more. Only a portion of the sand transported takes part in the formation of these sand waves. The velocity in a wave trough is greater than near the crest. The sand grains flow nearly parallel to the bed as they pass through the 1 Antidunes, though less common than dunes, are by no means rare under natural conditions. They are described by Vaiighan Cornish (Geog. Jour. (London), vol. 13, p. 624, 1899; Scottish Geog. Mag., vol. 17, pp. 1-2, 1901), and are mentioned by John S. Owens (Geog. Jour., vol. 31, p. 424, 1908). 2 The sequence of bed characters — dune, smooth, antidune— was observed by John S. Owens in studies with natural currents in 1907, and the characters were correlated with velocities. With depths of 3 to 6 inches and a bed of sand, he noted sand ripples [dunes) when the ve- locities, measured by floats, were from 0.85 to 2.5 ft./sec., and the appearance of antidunes at a velocity of about 3 ft./sec. (Geog. Jour. (London), vol. 31, pp. 416, 424, 1908). Sainjon and Partiot, study- ing the movement of debris in the Loire, had previously observed that whereas with low velocities the entire bottom load was transported through the progress of dunes, with higher velocities the del>ris was swept along from crest to crest and the dunes were reduced In height (Annales des ponts et chaussees, 5th ser., vol. 1, pp. 270-272, 1871). trough, but at the crest they have an up and down motion as well as a forward motion. On the crest of the larger waves their forward motion is small compared with their vertical motion. * * * There is a sand movement by rotation or whirls that aids transportation. These whirls have been observed during dune motion only for smaller sizes of sand . They are of short duration, lasting usually less than one minute, but in this time one of these may scour a hole 1 to 3 inches deep and 4 to 10 inches in length. They usually start near the side of the trough, the axis inclining downstream and toward the center, making an angle of 30° to 60° with the side and a small angle at the bottom. These whirls are 3 to 5 inches in diameter and the sand grains are thrown violently up as well as downstream by them. This move- ment aids transportation by its lifting action, some of the grains being carried in suspension for a short distance by it. The change in the appearance of a loaded stream as the load is increased, the discharge remaining constant, is very striking. For no load the water surface is even and smooth. As fine sand is fed into the water at a slow rate, small sand dunes will form on the bottom and many little waves will form on the surface. As the rate of feeding is increased, the slope and velocity increasing, these waves become larger and fewer and have the shape of an inverted canoe. These canoes are side by side, the number depending on the trough width and size of waves. When the width was 1.0 foot two sets formed side by side; when THE OBSERVATIONS. 33 the width was 1.32 feet three sets formed. As the critical velocity at which dune motion ceases is approached these waves begin todisappear, and when thisvelocity is reached the water surface is waveless. This waveless condition continues as the rate of feed increases until sand motion in antidunes begins, when large waves, the width of the trough and corresponding in length to the sand waves beneath them, are formed as illustrated in figure 11. In order to show the magnitude of these surface waves, wave traces have been drawn. Some of these are given in figure 12. A sheet of galvanized iron 4 feet long and 1 foot high was divided into inch squares by lines. This plate was moistened and covered with fine dust. It was held vertical at a given place over the experiment trough and on signal was dropped into the trough and taken out again as quickly as possible. The dust was removed from that part of the plate in the water, leaving a well-defined outline of the wave. This wave trace was quickly sketched on paper by means of the lines marking the squares. Trace A, figure 12, is for zero feed ; B is for a very small feed ; C is for a larger feed, the surface being covered with the canoe- shaped waves ; D shows one of the larger waves associated with autidunes. The slopes at which the phases of traction change are lower for large streams than for small, and lower for fine debris than for coarse. The phase with smooth bed — which may con- veniently be called the smooth phase — covers a greater range of conditions with mixed debris than, with assorted. The processes associated with dunes and antidunes were briefly studied in the glazed trough and with the moving field. Trans- portation by saltation follows the entire pro- file of the antidunes but traverses the dunes only from A to C of figure 10. The velocity of saltatory grains is greatest where erosion takes place, namely, along the upstream slopes of dunes and the downstream slopes of anti- dunes, and it may reasonably be inferred that the water velocities are greatest in those places. The eye detected no difference in water 21 2Z 23 FIGURE 12.— Profiles of water surface, automatically recorded, showing undulations associated with the antidune mode of traction. Numbers show distance, in feet, from the head of experiment trough. depth over the two slopes of the antidune, and if the depth is the same so also is the mean velocity; but the ratio of bed velocity to mean velocity is known to vary with conditions. The cause of the changes in process has not been adequately investigated, but a few sug- gestions may be made. To assist in a search for controlling conditions, the factors con- nected with the two critical points — the change from dune phase to smooth and the change from smooth to antidune — were tabulated from the experiments with sand of a single grade (C) ; the positions of the critical points being estimated by Mr. Murphy at a time when the details of the experiments were freshly in mind. In Table 3 w is the width of trough, in feet ; Q the discharge, in cubic feet per second ; 8 the per cent of slope; d the depth of water in feet; Vm the mean velocity, in feet per second; L the load, in grams per second; and Ll the load per unit width. The data in this table are taken from a preliminary reduction of the observations and are less accurate than the results of the final adjustment, which appear in Table 12 (p. 75). They suffice, however, for the present purpose. TABLK 3. — Data connected with changes in mode of transportation. First critical point. Second critical point. w S d Ym L Li S A vm L £i 0.66 0.093 1.00 0.076 1.86 13 20 1.86 0.058 2.44 44 67 .66 .182 .93 .115 2.41 28 46 1.72 .092 3.01 88 134 .66 .363 .82 .190 2.91 48 73 1.51 .165 3.37 145 221 .66 .545 .74 .272 3.05 70 106 1.34 .209 3.97 184 281 .00 .182 1.15 .082 1.76 41 41 2.02 .056 3.25 111 111 .00 .363 .89 .134 2.71 70 70 1.70 .109 3.33 201 201 .00 .545 .81 .180 3.03 95 95 1.51 .142 3.75 244 244 .00 .734 .73 .227 3.24 113 113 1.34 .194 3.78 301 301 .32 .182 1.20 .068 2.03 33 25 2.15 .051 2.70 121 92 1.32 .363 1.00 .107 2.57 76 57 1.76 .099 2.77 210 159 1.32 .545 .87 .145 2.84 106 80 1.53 .111 3.71 251 190 1.32 .734 .75 .178 3.13 140 106 1.33 .149 3.73 293 222 1.96 .363 1.04 .081 2.29 67 34 1.91 .062 2.99 219 112 1.96 .734 .84 .133 2.82 138 71 1.48 .108 3.47 354 181 1.96 1.119 .74 .182 3.14 191 98 1.22 .150 3.81 418 213 20021°— No. 80— 14- 34 TRANSPORTATION OF DEBRIS BY RUNNING WATER. The factors were then plotted in various combinations on logarithmic section paper, and certain approximate numerical relations were thus discovered. The first critical point is reached when d = 0.016 Fm2'3, or when d = 0.0045 i,°'85. The second critical point is reached when d = 0.004 Fm3'3, or when .131 Dunes Contracted. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. Transition .77 Transition do... .. .do 214 r Antidunes]. . . .do. .734 38 82 86 127 151 191 176 38 84 89 156 154 192 215 240 327 365 5 4 4 4 3 3 3 3 3 3 .35 .53 .35 .48 .51 .73 .83 .81 .97 .95 1.13 1.24 10 16 16 16 16 16 16 16 16 16 .267 .207 Dunes Contracted. Do. Do. Do. Do. Do. Do. Do. Do. Do. .205 .198 .190 Smooth. . .do Antidunes. do [Antidunes] -do... do [Antidunes] 1.% .363 8.1 20 19 33 58 81 • 128 143 128 192 215 227 8.7 20 21 32 66 92 127 128 144 179 193 218 264 283 6 6 6 5 5 5 5 5 4 4 4 4 3 3 .50 .36 .41 .55 .60 .78 .94 1.10 1.17 1.18 1.38 1.50 1.59 1.73 1.77 16 16 16 16 16 16 16 16 16 16 16 16 16 16 .179 .118 .118 .103 .090 .082 .073 .077 .075 .072 .076 .067 .069 .070 .187 Dunes.. Contracted . Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. .69 .67 .78 .97 .129 .107 .097 .084 Antidunes. . . .do ...do... do [Antidunes] do -do. . .734 7.7 32 74 70 105 148 8.3 33 88 80 105 163 167 179 231 279 10 6 4 5 5 4 4 3 3 3 .18 .18 .36 .45 .49 .56 .79 .75 .94 .98 1.01 16 12 16 16 16 16 16 16 16 16 .283 .180 .156 .156 .137 .142 .135 .129 .135 .130 .293 Contracted. Do. Do. Do. Do. Do. Do. Do. Do. Do. .58 .79 .152 .146 Smooth... do Transition . . . 214 221 [ Antidunes] 1.119 94 99 186 169 180 198 221 91 102 130 188 221 229 258 291 341 346 4 4 4 3 3 3 4 3 3 3 .30 .45 .39 .57 .59 .63 .60 .81 .79 .95 1.02 16 16 16 16 16 16 16 16 16 16 .220 .207 .194 .196 .182 .187 .165 .179 .214 ""."261" Smooth [Smooth] Smooth. Contracted. Do. Do. Do. Do. Do. Do. Do. Do. Do. .33 do do Transition ...do... .176 do THE OBSERVATIONS. 39 TABLE 4 (B). — Observations on load, slope, and depth, inth debris having 15,400 particles to the gram, or grade (B). Width. Dis- charge. Load. Slope. Depth. Character of bed . Outfall. Feed. Collec- tion. Period. Water surface. lied. Dis- tance. By gage. By pro- files. Feet. 0.23 Ft.'/iec. 0.093 Om./sec. Gm.fsec. 4.0 6.7 8.1 20 33 Minutes. 9 7 7 Per cent. Per cent. 0.85 .94 1.01 1.49 2.12 Feet. 16 16 16 16 16 Feet. 0.192 .184 .174 .146 Feet. Transition Free. Do. Do. Do. Do. do Smooth do .182 4.0 18 31 32 10 6 5 7 .73 1.12 1.55 1.55 16 16 16 16 .364 .284 .230 .232 Free. Do. Do. Do. Transition do .44 .66 .093 5.3 11 18 34 53 64 6 7 4 7 5 4 .73 .90 1.18 1.62 2.31 2.38 16 16 16 16 16 16 .136 .110 .093 .070 .072 .068 Free. Do. Do. Do. Do. Do. do .do .182 8.5 16 27 73 76 7 5 6 8 4 .50 .75 .98 1.66 1.73 16 16 16 16 16 .194 .177 .146 Transition ... . Free. Do. Do. Do. Do. Smooth Transition . .139 .do .093 5.1 9.2 17 15 22 28 34 41 53 57 93 8 6 7 9 5 5 5 6 5 5 .75 .84 .23 .32 .41 .47 .63 2.01 2.09 2.17 2.96 16 16 16 12 16 16 16 16 16 16 16 .089 .080 .060 .059 .058 .050 .056 .054 .049 .058 .037 Free. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. do Transition do... do... do do Antidunes do... do .182 4.8 3.5 14 15 23 27 24 42 47 41 51 4.2 6.0 12 16 20 21 19 42 39 44 47 55 55 55 67 72 88 105 112 119 126 139 159 169 213 236 254 268 377 8 8 7 6 8 4 8 7 6 9 8 5 8 5 7 7 4 5 6 6 4 5 5 5 4 3 3 4 3 0.32 .37 .35 .36 .69 .68 .66 .72 .81 .98 1.06 1.18 1.17 1.32 1.35 1.36 1.51 1.54 1.51 2.00 1.88 2.05 2.24 2.19 2.46 2.79 3.14 3.06 2.91 2.93 4.03 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 .175 .175 0.183 .183 Contracted. Do. Do. Do. Free. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. do. ... .137 .113 .110 .107 .113 .101 .112 .107 Transition 55 .103 .do 81 .do .091 .082 .083 .077 Antidunes .' .do 166 Antidunes .115 .0% .082 [Antidunes] ...do do... .do do .do ...do .do .363 29 44 52 58 73 29 40 39 49 55 77 91 104 148 152 164 167 186 226 5 8 6 6 5 4 5 5 4 5 4 5 3 4 .51 .68 .69 .75 .80 .98 .99 1.14 1.54 1.43 1.63 1.66 1.71 1.81 16 16 16 16 16 16 16 16 16 16 16 14 16 16 .216 .212 .205 .202 .188 .178 .189 .191 .177 .155 .221 Contracted. Free. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. ...do do do Transition Antidunes .. ..do [Antidunes) ...do do... .158 ...do do... ..do .545 5.8 16 35 54 68 81 104 111 132 169 242 271 304 6 9 5 8 6 6 5 5 5 5 3 3 3 .18 .21 .23 .46 .63 .74 .72 .87 .89 1.00 1.32 1.81 1.81 1.79 16 14 14 16 16 14 16 16 16 16 16 16 16 .471 .367 .291 .292 .258 .260 .220 .226 .230 .231 .472 Contracted. Do. Do. Free. Do. Do. Do. Do. Do. Do. Do. Do. Do. 16 34 64 67 79 108 105 do .47 .300 do .do Transition Antidunes do [ Antidunes) do ...do 40 TRANSPORTATION OF DEBRIS BY RUNNING WATER. TABLE 4 (B). — Observations on load, slope, and depth, with debris having 13,400 particles to the grim, or grade (B)— Con. Load. Slope. Dep th. Width. charge. Feed. Collec- tion. Period. Water surface. Bed. Dis- tance. By gage. By pro- files. Feet. Ft.'lsec. Om.leec. 9 9 Qm.jsec. 11 Minutes. 5 Per cent. Per cent. 0 61 Feet. 16 Feet. 0 106 Feet. Contracted. 9 9 13 5 .61 16 .102 Do. 28 6 .97 16 .073 Transition . . . Free. 39 5 1 15 16 .081 do Do. 51 5 1 29 16 073 Do. 98 4 1 87 16 .000 Do. 116 4 2.14 16 .067 do .. Do. 157 3 2.43 16 .066 .. do Do. 182 4 2.63 16 .070 [Antidunes] Do. 363 o o 0 030 044 60 .475 Contracted. 2 1 1 3 390 18 .16 60 .344 .323 [Dunes] Do. 1 9 1 6 360 17 .16 60 .321 .321 do Do. 6.4 6 2 60 .28 48 do Do. 6 8 6 8 44 .27 .31 44 .278 .271 do Do. 10 5 7.2 5 .23 16 .214 .244 do Do. 9 6 9 4 71 .33 40 .224 do Do. 8.7 9.7 6 .29 16 .208 ... do Do. 9 3 11 62 .31 44 .254 Do. 18 21 30 .42 .43 40 .188 .193 Do. 25 25 15 .43 .49 36 .195 .176 Do. 28 28 4 .54 16 .164 Do. 37 49 g .63 32 Transition Do. 46 4 .78 14 .154 Free. 55 9 .65 12 .150 do . . .. Do. 79 4 1.00 16 .120 Antidunes Do. 153 4 1.51 16 .107 .. do.. Do. 206 4 1.66 16 .129 do Do. 218 3 1.78 16 .112 . do . Do. 268 3 2.00 16 do Do. 305 3 2.14 16 [Antidunes] Do. 347 3 2.28 16 do Do. 545 107 5 .79 12 .194 Smooth Free. 117 5 .83 16 .167 Transition Do. 167 4 1.16 16 .157 Do. 199 3 1.31 16 .169 do * Do. 219 3 1.29 16 [Antidunes] . . .. Do. 274 3 1.46 16 Do. 306 3 1.63 16 .144 ... do . Do. 315 3 1.65 16 Do. 396 6 120 0.18 .16 a .20 48 .409 .324 0.416 (Dunes) Contracted. Do. Do. Do. Free. Contracted. Free. Contracted. Do. Free. Do. Do. Do. Do. Do. Do. ... do 0.35 .292 do... o .51 .210 do 6 .50 o.55 14 .214 .189 Dunes 5 .65 o.75 16 .188 .117 Transition o .73 .183 4 4 4 3 3 3 3 3 .87 1.05 1.15 1.37 1.62 1.72 1.82 1.90 16 16 16 16 16 16 16 16 .167 .163 .149 .143 Smooth. [Smooth] Smooth . . Transition . . [Antidunes] do.. do .... 1.96 .363 11 9.5 16 26 40 60 77 90 124 163 175 203 227 227 10 6 10 7 4 .45 .35 .59 .71 .90 .99 1.09 1.21 1.41 1.63 1.67 1.85 1.93 2.03 16 16 16 16 16 16 16 16 16 16 16 16 16 .128 .108 .105 .100 .089 .080 .078 .067 .063 .064 .067 .061 .060 .141 [Dunes]... . . . Contracted. Free. Contracted. Free. Do. Do. Do. Do. Do. Do. Do. Do. Do. Dunes 25 .73 .113 do. Transition.. do. do Smooth do Transition do... do.... do. do .545 30 55 79 101 138 165 210 241 2«6 321 331 355 386 8 5 5 6 5 4 4 4 4 3 3 3 3 .56 .70 .79 .92 1.07 1.10 1.34 1.38 1.58 1.85 1.85 1.96 2.06 16 16 16 16 16 16 16 16 16 16 16 16 16 .148 .127 .113 .115 .100 .098 .094 .094 .080 .071 Dunes Free. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. ...do do.. . . Transition . . Smooth. do Transition do.... do Antidunes do.... do. [Antidunes] .734 36 36 32 41 49 55 63 73 72 87 101 96 166 241 259 302 334 350 362 368 8 10 5 5 6 7 5 4 6 5 3 4 4 3 3 3 0 3 .42 .54 .45 .41 .37 .49 .60 .58 .61 .60 .70 .75 .92 1.17 1.22 1.31 1.53 1.49 1.45 1.50 16 18 16 12 16 16 16 16 16 16 14 16 •16 16 16 14 16 16 .205 .194 .177 .171 .149 .165 .155 .155 .120 .147 .139 .117 .113 .106 .105 .195 .209 [Dunes] Contracted. Do. Free. Do. Contracted. Do. Free. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. Dunes do... do 70 68 .53 .46 .161 .175 [Dunes]. Dunes do Transition 121 .67 .145 do Dunes . . Transition Smooth . . [Smooth] Transition Antidunes. [Antidunes] ...do do 1.119 90 112 131 138 140 149 158 215 2.53 331 3"iO 397 5 6 4 6 5 5 4 5 3 3 3 3 .53 .61 .64 .56 .65 .69 .71 .78 .78 .97 1.03 1.12 16 16 16 16 16 16 16 16 16 16 16 16 .222 .188 .184 .180 .196 .186 .189 .178 .171 .166 .152 .205 .202 .180 Free. Contracted. Do. Do. Free. Do. Do. Do. Do. Do. Do. Do. 117 130 133 .68 .60 .65 Smooth do.. . . . 129 132 .79 .79 .199 .192 Smooth do .. do do . do do Transition a Computed graphically from data in a notebook afterward lost. 46 TBANSPORTATION OF DEBBIS BY BUNKING WATER. TABLE 4 (D). — Observations on load, slope, and depth, with debris having 1,460 particles to the gram, or grade (D). Width. Dis- charge. Load. Slope. Depth. Character of bed. Outfall. Feed. Collec- tion. Period. Water surface. Bed. Dis- tance. By gage. By pro- files. Feet. 0.66 Ft.»/iec. 0.093 Om. /sec. Om./sec. 5.3 8.1 13 20 21 29 36 45 47 62 Minutes 6 6 5 4 5 4 6 4 4 4 Per cent. Per cent. 0.80 .94 1.19 1.39 1.68 1.83 1.88 1.98 2.25 2.47 Feet. 16 16 16 16 16 16 16 16 16 16 Feet. 0.077 .077 .075 .076 .063 .058 .061 .063 .056 .056 Feel. Dunes. . . Free. Do. Do. Do. Do. Do. Do. Do. Do. Do. do Transition . . . do Smooth . . do . do. . do do ....do .182 6.4 5.8 14 17 23 30 47 53 83 108 121 127 134 7 6 6 6 5 4 4 8 9 4 4 4 .39 .77 .82 .95 1.10 1.26 1.40 1.95 2.10 2.2fi 2.28 2.39 16 16 16 16 16 16 16 16 16 16 16 16 .166 .147 .135 .123 .105 .108 .108 .090 .090 090 .093 .090 Contracted. Free. Do. Do. Do. Do. Do. Do. Do Do. Do. Do. .do ...do Transition ..do ...do... do . ...do... do .. do. . do .545 8.9 8.9 18 32 33 6.9 9.5 17 27 37 64 92 145 172 203 209 218 231 331 310 6 5 5 5 4 5 4 4 4 4 4 3 3 3 3 .30 .28 .61 .61 .19 .20 .51 .57 .64 .81 .96 1.25 1.42 .55 .58 .57 .65 .98 2.02 16 16 14 16 16 12 16 16 16 12 12 12 14 12 12 .460 .422 .403 .318 .316 .267 .237 217 .211 .202 .209 .191 .177 .196 0.464 .425 .382 .322 Contracted. Do. Do. Do. Do. Free. Do. Do Do. Do. Do. Do. Do. Do. Do. Transition do . do do. . do do do do 1.00 .182 9.5 16 22 27 28 48 49 56 80 98 109 126 152 7 5 5 5 4 4 4 4 4 4 4 4 3 .69 .84 1.06 1.08 1.14 1.34 1.39 1.57 1.83 2.09 2.33 2.55 2.75 14 16 14 16 16 16 16 16 16 16 14 14 16 .110 .100 .094 .099 .088 .086 .082 .071 .066 .070 .066 .062 .064 Dunes... do Free. Do. Do. Do. Do. Do. Do Do Do Do Do Do Do Transition do do do do do do do do .363 0 1.8 1.8 5.9 12 12 23 0 2.3 2.0 5.4 12 12 23 28 35 86 112 130 141 170 181 -229 258 .037 40 23 24 24 20 20 24 16 16 16 16 14 16 14 16 14 14 .418 .292 .283 .245 .224 .235 .209 .170 .164 .136 .120 .116 .117 .115 .103 .104 .101 Contracted. Do. Do. Do. Do. Do. Do. Free. Do. Do. Do. Do. Do. Do. Do. Do. Do. 81 66 20 22 26 12 6 5 4 4 4 3 3 4 4 3 .16 .19 32 .27 .56 .53 .18 .18 .25 .35 .43 .58 .65 .80 1.14 1.32 1.44 1.49 1.65 1.75 1.94 2.11 .289 .286 .252 .224 .229 .199 [Dunes] ... do Dunes [DunesJ do do Dunes Transition do do do do do 1.85 2.00 .099 .098 do do .545 30 32 30 31 66 143 168 229 256 281 310 340 4 \ 4 4 3 3 3 3 .53 .53 .55 .78 .11 .26 .58 .61 .62 .74 .91 16 16 16 16 16 16 16 14 14 14 .236 .241 .187 .166 .162 .136 .136 .140 .136 .136 .240 Contracted. Do. Free. Do. Do. Do. Do. Do. Do. Do. do do do do do do THE OBSERVATIONS. 47 TABLE 4 (D). — Observations on load, slope, and depth, with debris having 1,460 particles to the gram, or grade (D) — Con. Width. Dis- charge. Load. Slope. Depth. Character of bed. Outfall. Feed. Collec- tion. Period. Water surface. Bed. Dis- tance. By gage. By pro- files. Feet. 1.00 Fl.'/tcc. 0.734 Gm.lstc. 0 5.8 19 20 49 Qm.lsec. 0 6.5 12 21 56 66 82 10S 170 189 193 265 293 354 377 Minutei. Per cent. 0.085 Per cent. Feet. 40 36 24 36 16 12 16 14 14 16 16 16 16 16 14 Feet. Feet. Dunes forming Contracted. Do. Do. Do. Do. Free. Contracted. Free. Do. Do. Do. Do. Do. Do. Do. 29 18 15 4 4 3 4 3 4 4 3 3 3 3 .18 .22 .26 0.21 .32 .37 .61 .61 .75 .83 .98 1.09 1.19 1.39 1.46 1.73 1.76 0.491 .423 .413 .285 .250 .241 .220 .206 .198 .195 .189 .185 .175 .162 0.482 411 .410 [Dllnfts] do.' do do 83 Transition . ... Smooth do ..do... do ...do... do . do 1.32 .363 8.9 10 25 59 60 100 102 154 166 169 195 202 11 11 23 53 57 89 97 135 136 161 236 247 10 10 8 5 2 4 3 3 3 3 3 3 .50 .40 .46 .30 .34 / .67 1.00 1.05 1.25 ,1.25 ll.54 1.57 1.67 1.89 1.91 16 16 16 16 16 16 16 16 14 14 16 16 .137 .171 .137 .124 '.129 .107 .111 .090 .094 .089 .088 .087 .140 .171 .154 [Dunes] Contracted. Do. Do. Do. Do. Free. Do. Do. Do. Do. Do. Do. do ( Dunes] Dunes . do [Smooth] Smooth . . . do ...do .734 17 22 48 50 53 51 65 68 86 90 99 172 202 192 192 15 25 43 49 47 60 72 59 72 83 84 170 174 209 200 10 10 10 8 8 5 5 5 5 5 4 3 3 3 3 .28 .33 .60 .51 .64 .34 .37 .52 .58 .61 .62 .55 .70 .79 .67 1.15 1.00 .98 1.15 1.17 16 16 14 16 16 14 16 16 16 16 16 12 14 12 12 .360 .272 .243 .224 .233 .202 .201 .206 .184 .181 .354 .304 .227 .248 .231 Contracted. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. do [Dunesl do .75 .66 .228 .192 1.28 1.00 .97 [Smooth] . . ........ .154 .160 .156 .152 .164 .168 do do . do 48 TRANSPORTATION OF DEBRIS BY RUNNING WATER. TABLE 4 (E). — Observations on load, slope, and depth, with debris having 142 particles to the gram, or grade (E). Width. Dis- charge. Load . Slope. Depth. Character of bed. Outfall. Feed. Collec- tion. Period. Water surface. Bed. Dis- tance. By gage. B£sr Feet. 0.66 Ft.'lsec. 0.182 Om.lsec. 25 25 31 31 31 Gm./sec. 21 21 28 31 30 Minutes. 15 19 15 13 15 Per cent. Per cent. 1.20 1.27 1.43 1.48 1.56 Feet. 16 16 16 16 16 Feet. 0.125 .117 .115 .115 .114 Feet. Contracted. Do. Do. Do. Do. Transition .363 38 38 48 48 48 31 31 42 46 47 9 9 9 10 10 1.04 1.04 1.27 1.26 1.29 16 10 16 16 16 .200 Transition (dunes to smooth). Contracted.' Do. Do. Do. .197 .200 .185 Transition .734 48 48 96 90 47 50 95 97 6 6 4 4 1.03 1.09 1.43 1.46 16 16 16 16 .372 .387 Contracted. Do. Do. Do. do do do 1.119 50 91 91 101 101 101 101 193 193 193 44 80 85 94 87 94 95 167 184 186 6 3 3 3 3 3 3 3 3 3 .56 1.23 1.31 1.11 1.24 1.31 1.29 1.38 1.28 1.41 14 16 16 16 16 16 16 16 16 16 .562 Contracted. Do. Do. Do. Do. Do. Do. Do. Do. Do. do do do 1.00 .182 6.4 19 19 47 50 50 78 5.2 19 19 44 44 50 75 21 10 16 8 8 9 5 .59 1.11 1.12 1.80 1.80 1.94 2.42 16 16 16 16 16 16 16 Contracted. (?) Do. Do. Do. Do. Do. Do. .094 .087 .081 .077 .093 .363 0.065 32 32 ' 48 48 48 32 16 16 16 16 .357 .314 .241 .242 .218 .212 .161 Contracted. Do. Do. Do. Do. Do. Free. Contracted. Do. Do. Tr. .11 Dunes formin? 2.1 2.1 6.7 6.6 24 50 95 142 2.9 2.2 7.7 7.9 22 53 104 168 61 CO 20 20 34 9 5 4 .23 .22 .25 .44 .44 .87 1.28 1.83 2.29 0.277 ""."215" .208 .107 [Dunes] Dunes .46 .39 .80 1.24 1.92 2.26 Transition . . . do .113 .110 do .115 do .734 .04 32 32 48 48 32 40 28 16 16 36 36 16 16 16 16 16 16 .618 .562 .447 .442 .411 .403 .364 .303 .301 - .324 .326 .273 .236 .223 .187 .189 Contracted. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. Tr. .07 2.1 2.1 6.9 6.9 21 21 24 41 41 47 87 95 142 142 203 1.7 3. C 4.5 10 20 20 26 44 47 45 86 100 135 150 220 70 63 31 25 8 18 16 10 6 8 6 4 3 3 2 .19 .18 .29 .26 .46 .48 .47 .73 .74 .83 1.19 1.23 1.46 1.50 1.61 do .17 .31 .31 .43 .459 .403 .407 .363 do ...do . do ...do.... do .61 Dunes do .82 Dunes Transition 1.17 1.50 1.47 1.80 Smooth 1.119 47 47 189 189 50 50 170 190 6 8 3 3 .52 .59 1.15 1.18 16 16 16 16 .380 .406 Contracted. Do. Do. Do. 1.32 .363 20 20 86 20 20 93 15 10 5 .67 .73 1.78 16 16 16 .131 .129 .097 Contracted. Do. Do. ....do .734 33 86 86 33 85 86 9 4 3 .60 1.10 1.12 16 16 16 . I'll!) .198 .211 Dunes Contracted. Do. Do. 1.119 50 50 172 172 46 50 163 171 6 4 3 3 .58 .62 1.13 1.22 16 16 16 16 .324 .317 Contracted. Do. do THE OBSERVATIONS. 49 TABLE 4 (F). — Observations on load, slope, and depth, with debris having 22.1 particles to the gram, or grade (F). Width. Dis- charge. Load. Slope. Depth. Character of bed. Outfall. Feed. ctio'r **w. Water surface. Bed. Dis- tance. By gage. By pro- files. Feet. 0.66 Ft.'/sec. a. 182 Qm.litt. 11 58 58 Gm.jsec. 12 52 53 Minutes. 17 23 11 8 Per cent. Per cent. 1.29 1.31 2.51 2.50 Feet. 16 16 16 16 Feet. 0.118 .129 .102 .106 Feet. Contracted. Do. Do. Do. [Smooth] Smooth . . do .363 26 26 71 71 26 26 80 71 17 12 6 10 1.12 1.13 1.89 1.96 16 16 16 16 .207 .204 .176 .170 Smooth Contracted. Do. Do. Do. do .7:(1 35 35 106 106 37 38 103 108 9 6 3 3 .97 1.00 1.68 1.75 16 16 16 16 .346 .340 Contracted. Do. Do. Do. ghines] unes do 1.00 .182 6.8 41 41 6.8 42 44 21 7 7 1.36 2.49 2.53 16 16 16 .090 .078 .080 Free. (?) Do. Do. do do .363 10 10 51 51 51 120 9.5 10 47 51 56 120 18 20 14 10 8 3 .85 .91 1.65 1.70 1.68 2.47 16 16 16 16 16 16 .160 .161 .128 .130 .140 Smooth Free. (?) Do. Do. Do. Do. Do. do [Smooth] do .734 26 26 104 104 25 25 98 114 14 12 5 5 .77 .77 1.52 1.60 16 16 16 16 .268 .265 .217 .211 Dunes . . Contracted. Do. Do. Do. do ...do .. do 1.119 52 52 207 207 53 56 197 209 5 5 3 3 .80 .85 1.65 1.67 16 16 16 16 .330 .343 Free. (?) Do. Do. Do. ..do.... ...do do 1.32 .363 21 21 70 70 21 21 68 72 7 21 9 5 1.16 1.21 2.05 2.07 16 16 16 16 .116 .118 .114 .108 Smooth Contracted. Do. Do. Do. do do .734 26 26 26 104 104 26 27 27 102 106 13 12 12 3 4 .83 .85 .86 1.48 1.58 16 16 16 16 16 .212 .209 .215 .176 .180 Dunes Contracted. Do. Do. Do. Do. do [Dunes] 1.119 58 58 58 212 212 50 50 59 188 208 12 9 6 2 2 .74 .78 .84 1.55 1.60 16 16 16 ID 16 .275 .288 .284 Dunes Contracted. Do. Do. Do. Do. do do g>unes] unes TABLE 4 (G). — Observations on load, slope, and depth, with debris having 5.9 particles to the gram, or grade (G). Width. Dis- charge. Load. Slope. Depth. Character of bed. Outfall. Feed. Collec- tion. Period. Water surface. Bed. Dis- tance. By gage. By pro- files. Feet. 0.66 Ft.'/sec. 0.363 Gm.liec. 10 25 25 51 50 100 100 Om. /sec. 11 25 28 47 50 100 105 Minutes. 15 12 15 6 8 3 3 Per cent. Per cent. 1.11 1.44 1.48 1.82 1.90 2.56 2.70 Feet. 16 16 16 16 16 16 16 Feet. 0.198 .186 .192 .175 .175 .160 .158 Feet. Contracted . Do. Do. Do. Do. Do. Do. .734 10 10 25 50 50 105 203 210 8.5 11 26 48 49 104 210 219 14 23 11 7 11 5 3 3 .68 .70 .95 1.19 1.19 1.71 2.43 2.35 16 16 16 16 16 16 16 14 .373 .364 .342 .322 .324 .292 .261 .264 Contracted. Do. Do. Do. Do. Do. Do. Do. 1.119 10 50 50 100 100 203 203 304 304 12 49 54 94 114 212 215 311 331 18 6 6 4 4 3 3 3 3 .62 .98 1.02 1.35 1.32 1.97 1.95 2.40 2.36 16 16 16 16 16 16 16 16 16 .558 .460 .451 .414 .411 .374 .378 .354 Contracted. Do. Do. Do. Do. Do. Do. Do. Do. l ! | ' 20021°— No. 80—14- 50 TRANSPORTATION OF DEBRIS BY RUNNING WATER. TABLE 4 (G). — Observations on load, slope, and depth, with debris having 5.9 particles to the gram, or grade (G) — Con. Width. Dis- charge. Load. Slope. Depth. Character of bed. Outfall. Feed. Collec- tion. Period. Water surface. •a., DlS- !ed- tance. By gage. Br{,er Feet. 1.00 Ftysec. 0.363 Gm.lsec. 10 20 25 25 34 51 102 Om./sec. 10 21 25 28 34 50 97 Minutes. 20 15 12 18 12 9 5 Per cent. Per cent. 1.27 1.48 1.61 1.62 1.76 2.09 2.74 Feet. 16 16 16 16 16 16 16 Feet. 0 143 .139 .136 .141 .132 .129 .114 Feet. Smooth . Contracted. Do. Do. Do. Do. Do. Do. Smooth do .. do... do .734 10 20 25 25 50 101 201 201 11 20 25 26 49 100 189 212 42 15 15 11 10 5 4 3 .78 .86 .95 .97 1.27 1.69 2.30 2.37 16 16 16 16 16 16 16 16 .272 .248 .248 .251 .235 .214 .190 .191 Contracted. Do. Do. Do. Do. Do. Do. Do. [Smooth] do (Smooth] 1.119 10 25 25 50 50 101 102 204 306 306 10 24 25 53 54 100 103 214 297 310 20 15 17 6 7 4 4 3 3 3 .64 .66 .67 .97 .90 1.22 1 31 16 16 16 16 16 16 16 16 14 14 .389 .357 .359 .324 .326 .308 .307 Contracted. Do. Do. Do. Do. Do. Do. Do. Do. Do. 1.78 2.04 2.21 .275 .252 .253 1.32 .363 25 25 51 51 98 98 24 26 49 47 95 123 12 9 10 10 3 4 1.90 1.97 2.25 2.34 3.02 3.10 16 16 16 16 16 16 .115 .104 .108 .105 .093 .097 Contracted. Do. Do. Do. Do. Do. .734 10 9 10 12 25 24 25 27 51 52 102 102 203 ' 200 21 18 13 13 6 4 3 .82 .82 1.08 1.14 1.41 1.82 2.44 16 16 16 16 16 16 16 .210 .210 .194 .200 .193 .171 .163 Contracted. Do. Do. Do. Do. Do. Do. 1.119 10 51 51 98 98 197 295 304 14 46 67 95 110 200 282 302 19 4 6 3 4 3 3 3 .71 .90 1.07 1.31 1.40 1.83 2.17 2.26 16 16 16 16 16 16 16 16 .297 .270 .261 .248 .245 .230 .201 .202 Contracted. Do. Do. Do. Do. Do. Do. Do. TABLE 4 (H). — Observations on load, slope, and depth, with debris having 2.0 particles to the gram, or grade (JJ). Width. Dis- charge. Load . Slope. Depth. Character of bed. Outfall. Feed. Collec- tion. Period. Water surface. Bed. Dis- tance. By gage. Bv pro- files. Feet. 0.66 Ft.*lsec. 0.363 Gm.lsec. 10 10 21 21 52 52 Om.lsec. 9.2 12 17 22 51 56 Minutes. 21 16 18 14 5 6 Per cent. Per cent. 1.49 1.58 1.80 Feet. 16 16 16 16 16 16 Feet. 0.184 .184 .183 .173 .167 .173 Feet. Contracted. Do. Do. Do. 1.84 2.43 2.47 .734 10 10 21 21 52 52 105 209 209 7.9 11 20 22 51 52 105 209 222 18 17 17 15 8 8 4 3 3 .90 .95 1.10 1.19 1.50 1.51 2.02 2.69 2.92 16 16 16 16 16 16 16 16 16 .345 .348 .333 .344 Contracted. Do. Do. Do. Do. Do. Do. Do. Do. .320 .310 .289 .250 .253 1 1.119 10 10 10 26 52 52 52 105 105 105 209 209 12 10 12 26 53 54 53 98 106 113 209 209 16 20 21 15 13 10 9 3 6 3 3 3 .74 .81 .89 1.03 1.26 1.28 1.33 1.65 1.63 1.62 2.31 2.38 16 16 16 16 16 16 16 14 16 16 16 16 .502 .510 .503 .470 .442 .437 .447 .391 .398 .392 | Contracted. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. Do. .340 .334 THE OBSERVATIONS. 51 TABLE 4 (I). — Special group of observations on load, slope, and depth, with debris of grade (f); for discussion of form ratio. Width. Dis- charge. Load. Slope. Depth. ( haracter of bed. Outfall. Feed. Collec- tion. Period. Water ,, , surface. Dis- tance, »™^- B&r- Feet. 1.00 Ft.'/sec. 0.734 (im./sec. 74 76 160 Om./sec. 84 45 194 Minutes. 4 4 3 Per cent. Per cent. 0.58 .58 .99 Feel. 16 14 16 Feet. 0.232 .241 .194 Feet. Transition Contracted. Do. Do. do .923 96 94 254 103 103 280 3 3 2 .58 .62 1.09 16 16 16 .270 .282 .233 Contracted. Do. Do. do 1.119 111 114 118 129 3 3 .57 .66 14 14 .312 .305 Transition Contracted. Do. 1.20 .734 66 67 193 199 74 74 218 223 3 3 3 3 .51 .60 1.05 1.07 16 16 16 16 .222 .215 .173 .175 Transition Contracted. Do. Do. Do. Transition {Transition] .923 99 249 246 104 266 274 3 3 3 .65 1.08 1.01 14 16 16 .232 .210 .201 Contracted. Do. Do. ...do... An 1.021 105 110 304 107 109 325 3 4 3 .57 .54 1.20 16 16 16 .250 .262 .197 [Transition] Contracted. Do. Do. Transition . . 1.119 108 111 322 339 114 123 292 326 4 4 3 3 0.56 .62 .61 1.00 1.03 12 16 16 14 .269 .269 .224 .231 .277 [Transition] Contracted. Do. Do. Do. Transition , [Smooth] Smooth 1.40 .734 76 76 220 216 74 75 215 226 4 4 3 3 .63 .59 1.04 1.07 16 14 16 16 .199 .196 .158 .151 Contracted. Do. Do. Do. . .do .do .923 72 82 271 283 79 74 279 297 4 3 3 .51 .57 1.03 1.04 16 16 16 14 .235 .237 .176 .175 [Transition] . Contracted. Do. Do. Do. Transition Smooth . [Smooth] 1.021 82 292 289 97 319 323 \ 3 .58 1.03 1.08 16 14 16 .248 .181 .180 Contracted. Do. Do. [Smooth] Smooth 1.119 105 123 292 284 292 104 130 325 337 312 4 4 3 3 3 .57 .61 .97 1.01 1.07 16 16 14 14 16 .254 .231 .213 .215 .203 Contracted. Do. Do. Do. Do. (Smooth] Smooth . . 1.60 .734 69 265 269 70 236 268 4 3 3 .60 1.14 1.12 12 16 16 .191 .121 .122 Contracted. Do. Do. Smooth . . . do .923 67 69 67 298 289 82 69 66 299 280 4 5 4 3 3 .45 .46 .50 1.05 1.07 12 12 12 16 16 .233 .202 .224 .161 .161 Contracted. Do. Do. Do. Do. do [Dunes] Smooth . . . do 1.021 67 63 66 310 66 73 80 311 4 4 4 3 .39 .47 .51 1.06 12 12 12 16 .243 .244 .232 .169 Contracted. Do. Do. Do. 1.119 82 96 85 336 83 91 98 332 4 4 4 3 .49 .47 .55 .99 16 12 12 12 .227 .234 .233 .188 Transition Contracted. Do. Do. Do. [Transition] do 1.80 .734 69 70 211 205 69 72 204 203 4 4 3 8 .61 .60 1.04 1.06 16 16 16 16 .174 .174 .137 .130 Contracted. Do. Do. Do. do ....do .923 70 67 260 260 75 74 258 273 4 4 3 3 .53 .58 1.04 1.04 14 14 16 16 .206 .204 .157 .152 Dunes Contracted. Do. Do. Do. do ... [Smooth] 1.021 70 69 70 292 296 69 76 76 323 308 4 4 4 3 3 .51 .52 .57 1.00 1.01 16 14 16 14 16 .224 .227 .225 .155 .158 Contracted. Do. Do. Do. Do. g)unes] Smooth . . . do 1.119 64 75 386 363 66 74 375 385 4 . 4 3 3 .42 .47 1.05 1.05 14 14 16 12 .240 .243 .160 .167 Dunes do Contracted. Do. Do. Do. An 52 TRANSPORTATION OF DEBRIS BY RUNNING WATER. TABLE 4 (I). — Special group of observations on load, slope, and depth, with dfbrin of grade (0) — Continued. Width. Dis- charge. Load. Slope. Depth. Character of bed. Outfall. Feed. Collec- tion. Period. Water surface. Bed. Dis- tance. By gage. By pro- files. Feet. 1.96 FtJ/sec. 0.734 Om.jsec. 85 84 234 222 234 Gm.jsec. 83 88 236 231 250 Mimttes. 4 4 3 3 3 Per cent. Per cent. 0.61 .69 1.05 1.09 1.12 Feet. 16 16 16 16 16 Feet. 0.147 .146 .117 .123 .122 Feet. [Transition] Contracted. Do. Do. Do. Do. do .923 96 246 97 279 3 3 .62 1.02 16 16 .176 .143 Transition Contracted. Do. 1.021 84 85 94 328 316 85 87 104 343 352 4 4 3 3 3 .50 .57 .54 1.09 1.09 16 14 16 16 16 .194 .200 .193 .141 .144 [Transition] Contracted. Do. Do. Do. Do. [Transition] do '. Transition 1.119 82 85 82 345 328 82 84 94 341 363 4 4 4 3 3 .45 .57 .52 1.00 1.00 12 16 16 16 16 .214 .222 .219 .150 .147 Contracted. Do. Do. Do. Do. Transition do Smooth (Smooth! TABLE 4 (J). — Observations on load, slope, and depth, with debris of two or more grades mixed. Designation of mixture, component grades, and percentages by weight. Width of trough. Dis- charge. Load. Slope of bed. Depth by gage. Character of bed. Feed. Collec- tion. Period. Per cent. Dis- tance. CAif.WCAISO * (C}50 Feet. 1.00 Ft.»lsec. 0.363 Om.lsec. 42 90 128 171 Qm.liec. 43 91 140 169 Minutes. 8 4 4 3 0.58 .92 1.23 1.47 Feet. 16 16 16 16 Feel. Transition. Do. Antidunes. Do. (A3Gi)=(A)75 ' (G)25 1.00 .363 20 90 80 175 175 IN 74 89 164 176 12 5 4 3 3 .51 .93 .95 1.38 1.36 16 16 16 16 16 Smooth. Transition. Do. Antidunes. Do. (AjGi)«=(A)67 • (G)33 1.00 .363 20 42 42 87 110 20 42 42 88 119 12 6 6 3 3 .53 .62 .64 1.00 1.18 16 16 16 16 16 Smooth. Do. Do. Antidunes. Do. (AiGi)— (A)50 ' (G)50 1.00 .363 22 46 91 22 43 93 8 6 4 .68 1.01 1.42 16 16 16 Smooth. Transition. Do. (AiG2)— (A)33 • (G)67 1.00 .363 42 40 6 1.36 16 Smooth. ( \iG<)— (A)22 • (G)78 1.00 .363 22 43 21 42 20 7 1.30 1.79 16 16 (B«Fi)=(B)78 • (F''22 1.00 .363 42 84 113 113 169 43 92 105 125 159 6 3 3 3 2 .59 .86 .99 1.05 1.38 16 16 16 16 16 Smooth. Do. Do. Do. Transition. (BjFi)=-(B)64 • (F)36 1.00 .363 39 78 105 157 42 80 109 157 7 5 4 2 .57 .85 1.11 1.49 16 16 16 16 Smooth. Do. Do. Do. (BiFi)=(B)47 * (F)53 1.00 .363 45 90 120 181 49 95 130 162 7 4 3 2 .73 1.14 1.41 1.61 16 16 16 16 Transition. Do. Smooth. Do. (BiFz)— (B)31 • (F)69 1.00 .363 28 41 81 35 40 82 11 6 4 .86 1.06 1.56 16 16 16 Dunes. Do. Do. (BjF()— (B)18 • (F)82 1.00 .363 16 27 40 17 29 43 15 10 6 .82 1.16 1.46 16 14 16 Dunes. Do. Do. (CiEi)— (C)79 • (E)21 . ... 1.00 .363 43 85 115 171 49 84 118 171 8 4 3 2 .70 16 16 16 16 Smooth. Do. Do. Do. .93 1.15 1.52 THE OBSERVATIONS. 53 TABLE 4 (J). — Observations on load, slope, and depth, with debris of two or more grades mixed — Continued. Designation of mixture, component grades, and percentages by weight. Width of trough. Dis- charge. Load. Slope of bed. Depth by gage. Character of bed. Feed. Collec- tion. Period. Per cent. Dis- tance. (C!E1)-(C)65 : (E)35 Feet. 1.00 Ftflsec. 0.182 Gm.jtec. 78 104 Gm./sec. 81 105 Minutes. 3 4 1.77 2.06 Feet. 16 16 Feet. Smooth. Do. .363 43 85 114 155 171 155 171 42 85 117 170 169 170 173 7 4 3 3 3 4 3 .69 1.03 1.27 1.57 1.65 1.60 1.74 16 16 16 16 16 16 16 Transition. Do. Smooth. Do. Do. Do. Do. (CiEj)— (C)48 ' (E)52 1.00 .182 26 40 40 53 51 53 51 81 77 81 28 40 41 50 60 52 61 67 78 87 12 10 10 6 .99 1.17 1.27 1.43 1.50 1.51 1.55 1.82 1.95 1.89 16 16 16 16 16 16 16 16 16 16 0.096 .090 .091 .081 Transition. Do. Do. Smooth. Do. Do. Do. Do. Do. Do. 6 .082 4 5 .363 39 39 78 104 104 104 157 40 41 78 93 115 119 158 6 7 5 3 3 3 3 .74 .76 1.07 1.37 1.40 1.43 1.85 16 16 16 16 16 16 16 .150 .151 Transition. Do. Smooth. Do. Do. Do. Do. (CiEj)=(C)31 • (E)69 1.00 .182 26 39 52 78 26 39 57 76 10 8 6 3 1.10 1.37 1.64 2.22 16 16 16 16 .093 .088 .082 Transition. Do. Smooth. Do. .363 39 39 78 108 157 39 39 81 112 158 8 9 3 3 3 .77 .84 1.25 1.54 2.10 16 16 16 16 16 Dunes. Do. Do. Transition. Do. (CiE()-(C)19 : (E)81. 1.00 .182 20 26 39 39 52 52 20 28 40 42 51 54 14 10 6 6 6 12 1.02 1.29 1.61 1.65 1.88 1.93 16 16 16 16 16 16 Dunes. Do. Do. Do. Transition. Do. .085 .079 .084 .078 .082 .363 40 53 78 109 40 56 76 109 8 7 4 3 1.00 1.14 1.45 1.75 16 16 16 16 Dunes. Do. Transition. Do. (C(Gi)— (C)80 : (O)20 1.00 .363 31 93 140 187 187 32 92 140 194 195 11 5 4 3 3 .62 1.07 1.30 1.48 1.62 16 16 16 16 16 ~Dunes. Transition. Do. Smooth. Do. (CjGi)-(C)67 : (G)33 1.00 .363 31 31 31 61 92 92 122 183 29 29 31 66 90 95 120 180 .59 .63 .62 .90 1.03 1.08 1.20 1.65 16 16 16 16 16 16 16 16 Smooth. Do. Do. Do. Do. Do. Do. Do. 12 10 7 4 5 3 2 (CiG,)=(C)50 : (G)50 1.00 .363 17 50 100 133 133 18 50 105 128 145 15 10 3 3 3 .71 .99 1.34 1.62 1.66 16 16 16 16 16 Smooth. Do. Do. Do. Do. (CiGj)-(C)33 : (G)67... . 1.00 .363 16 32 32 64 15 34 36 69 16 10 10 7 .95 1.17 1.27 1.59 16 16 16 16 Transition. Smooth. Do. Transition. (E4Gi)— (E)SO : (Q)20 1.00 .363 15 31 61 17 31 58 15 10 8 .72 ' 1.00 1.39 16 16 16 Transition. Smooth. Transition. (E,G])-(E)C7 : (G)33 1.00 .363 16 32 63 15 30 59 18 11 6 .76 1.03 1.45 16 16 16 Smooth. Do. Transition. (EiGi)-(E)50 : (G)50 1.00 .363 16 31 62 18 34 60 18 1 .87 10 1.12 s i.eo 16 16 16 Smooth. Do. Transition. 54 TRANSPORTATION OF DEBRIS BY RUNNING WATER. TABLE 4 (J). — Observations on load, slope, and depth, with debris of two or more grades mixed — Continued. Designation of mixture, component grades, 8nd percentages by weight. Width of trough. Dis- charge. Load. Slope of bed. Depth by gage. Character of bed. Feed. Collec- t.on- Period. Per cent. Dis- tance. (EiGi)— (E)33 : (O)67 Feet. 1.00 FtJjsec. 0.363 Gm./sec. 16 23 30 30 15 45 Gm./sec. 15 23 29 32 49 54 Minutes. 14 12 12 11 7 7 0.95 1.18 1.35 1.33 1.73 1.74 Feet. 16 16 16 16 16 16 Feet. Transition. Smooth. Transition. Do. (A!CiG!)=(A)25 : (C)25 : (G)50 1.00 .363 16 16 16 32 126 126 13 17 14 33 88 131 151 15 10 15 10 5 4 4 .77 .79 .82 .88 1.32 1.59 1.67 16 16 16 16 16 16 16 Transition. Do. Smooth. Do. Do. Transition. Do. (CDEFG)-(C)45 : (D)35 : (E)12 : (F)6 : (G)2. 1.00 .182 21 42 84 25 45 84 12 10 5 .82 1.23 1.72 14 16 16 Dunes. Transition. .363 67 67 84 84 84 113 113 169 169 69 69 80 83 83 120 118 172 173 8 6 5 5 5 5 5 4 4 .81 .85 .91 .93 .95 1.18 1.19 1.53 1.54 16 16 16 16 16 16 16 16 16 Smooth. Do. Do. Do. Do. .545 84 84 169 169 253 253 90 92 151 177 228 242 6 5 3 3 3 3 .65 .67 1.07 1.13 1.40 1.37 16 16 16 16 16 16 Smooth. Do. Do. Do. Do. TABLE 4 (K). — Observations on load, slope, and depth, with unassorted debris. a Width. Dis- charge. Load. Slope of bed. Depth by gage. Character of bed. Outfall. Feed. Collec- tion. Period. Per cent. Distance. Feet. 1.00 Ftfjsec. 0.182 Gm./sec. 19 19 38 75 145 Gm.isec. ' 18 18 38 76 147 Minnies. 15 15 8 3 2 0.74 .79 1.00 1.55 2.28 Feet. • 16 16 16 16 16 Feet. Dunes Contracted. Do. Do. Do. Do. -do Smooth . . . do do .363 22 22 38 38 77 154 20 27 34 42 74 150 10 12 6 6 5 3 .43 .50 .59 .67 .89 1.31 16 16 16 16 16 16 0.169 .176 .157 .158 Transition Contracted. Do. Do. Do. Do. Do. do Smooth ...do.... do ...do.... a The very coarsest particles were removed by passing the sample through a 6-mesh sieve, and the very finest by passing it over a 60-mesh sieve It retained the equivalents of grades (A), (B), (C), (D), (E), and (F). CHAPTER II.— ADJUSTMENT OF OBSERVATIONS. OBSERVATIONS ON CAPACITY AND SLOPE. THE OBSERVATIONAL SERIES. Each of the experiments in stream traction involved six quantities — (1) a fineness, or grade of d6bris, (2) a width of trough, (3) a discharge, (4) a slope, (5) a load, or capacity, and (G) a depth of current. The experiments were arranged in series, for each of which grade, width, and discharge were constant, while within each the magnitudes of slope, capacity, and depth were varied. There will be frequent occasion to mention these secondary units of the experimental work, and whenever the title series seems not sufficiently specific they will be called observational series. The number of such series recorded in Table 4 is 153. The factors of grade, width, and discharge, which are related to an individual series as fixed conditions, or constants, do in fact assume the character of variables when series is com- pared with series; but their modes of determi- nation and combination are not of such char- acter that their numerical values can be checked and adjusted by means of recorded relations. 300 Jzoo (0 a. 1) o 100 0 I 2 Slope FIGURE 13. — Plot of a single series of observations on capacity and slope. Capacity ingramsofdebrispersecond. Slope in percent. Themodesof traction are indicated. In each experiment the values of slope, load, and depth are mutually dependent ; within each series they form a triple progression, the depth decreasing while slope and load increase; but the laws of these interdependent varia- tions are partly masked by irregularities in the sequences. As a preliminary to the general discussion, the observational values were sub- jected to a process of adjustment, whereby the sequences were freed from irregularities. The irregularities are made manifest by the com- parison of the sequences of two variables, and first consideration will be given to those of capacity and slope. Figure 13 exhibits the relations of capacity to slope as observed in a single series of experi- ments (that for grade (C), with w=l.32 feet and Q = 0.363 ft.3/sec.). The ordinates indicate capacity, as measured by debris delivered at the lower end of the trough; the abscissas represent slope, as measured on the bed of the channel. The arrangement of the observa- tional dots suggests that if the observations were harmonious the dots would fall in a line of simple curvature. Such a line would express the law connecting capacity and slope. The departures of the dots from such linear arrangement represent irregularities, or errors, in the experimental data. The adjustment proposed is the replacement of the imperfectly alined dots by a generalized or representative line, or the replacement of the inharmonious 55 56 TRANSPORTATION OF DEBRIS BY RUNNING WATER. values of capacity and slope by a system of harmonious or adjusted values. ERRORS. As a first step in the treatment of the errors of the data they were studied with a view to the discrimination of the systematic and the accidental. The three modes of traction — the dune, the smooth, the antidune — although intergrading, are mechanically different. It was surmised that they might differ in efficiency, so that the capacity-slope curve might show a step in 400 200 100 5-80 | 60 40 20 7 .4 .6 .8 I 23 Slope FIGURE 14. — Logarithmic plot of a series of observations on capacity and slope. Compare figure 13. passing from one to another; and it was also surmised that the law connecting capacity with slope might not be the same for the several modes. A suggestive observation had shown that on very low slopes — slopes so low that capacity is minute — the current changes an artificially smoothed bed of debris to a system of dunes, and that with the develop- ment of dunes the load is notably increased without any change in general slope. To test the surmises all the series were plotted on log- arithmic section paper. Figure 14 shows a logarithmic plot of the same data which appear in figure 13; and it will be observed that the line suggested by the points has much less curvature in the logarithmic plot. Its ap- proximation to a straight line makes the study of its local peculiarities comparatively easy. The examination of the plots, while not dis- proving the surmises, showed that whatever diverse influences may be exerted by the modes of traction, they are too small to be discrim- inated from the irregularities due to other causes. Other sources of systematic error are con- nected with the methods of experimentation. INTAKE INFLUENCES. As the water entered the experiment trough from the stilling tank it was accelerated, the gain in mean velocity being associated with a quick descent in the surface profile. Beyond this descent the profile usually rose somewhat, and there was commonly a moderate develop- ment of fixed waves. This development was modified and the waves were on the whole reduced by the addition of the debris. As it fell into the water the debris had no forward momentum, and it therefore tended to retard the current. But the d6bris also accumu- lated on the bottom, reducing the depth of the water at that point, and this reduction neces- sitated an increase in mean velocity. In the immediate neighborhood of the place where debris was fed the slope of the water was affected by an abnormality distinguishable from the intake abnormality proper, and the joint abnormality faded gradually downstream. The nature of these features varied with the discharge and load, with the gradual develop- ment of the adjusted slope, and also with the mode of feeding. During the greater part of the experimental work the feeding was either automatic and continuous or else manipulated by hand in such way as to make it nearly con- tinuous, but for a minor part the feeding was intermittent, a measureful of debris being dumped into the water at regular intervals. OUTFALL INFLUENCES. In all the earlier work the trough had the same cross section at the lower end as else- where, and the water fell freely from its open end to the settling tank. As the resistance to its forward motion was less at the outfall than within the trough, the water flowed faster there. Its faster flow diminished the resist- ance just above, and thus the influence of ADJUSTMENT OF OBSERVATIONS. 57 outfall conditions extended indefinitely up- stream. An expression of this influence was found in the water profile, which was usually convex in the lower part of the trough, the degree of convexity diminishing upstream. Its effect on the profile of the bed is not readily analyzed, because that profile is adjusted through the velocity of water at the bottom of the current, and the bed velocity is not simply related either to mean velocity or to depth. ~~A FIGURE 15. — Diagrammatic longitudinal section of outfall end of experi- ment trough, illustrating influence of sand arrester on water slope. A second factor at the outfall end was the arrangement for separating the debris from the current. This included a well, ABCD, figure 15, which was sunk below the trough bed and into which the debris sank, while the current passed on to the outfall at E. In part of the work the space AD was entirely open; in another part a coarse screen was stretched across it. In either case the resistance of this part of the channel bed differed from the D resistance along the debris slope and may have been greater or less. From the well to the outfall, DE, the frictional resistance was loss than elsewhere. As the fixed part of the channel bed, DE, was horizontal and the debris portion, GA, was inclined, the profile of the bed changed at A. Projected forward, the slope GA passed below E to H, and when the debris slope, down which the transporting current flowed, was steep, the part DE was related to it somewhat as a dam. The tend- ency of the quasi-darn was to retard the cur- rent near the outfall and make the water profile concave, and in some of the experiments the profile actually became concave. Other outfall factors were recognized, but they are not here mentioned because they are believed to be of relatively small importance. In the reduction of observations on slope an attempt was made to lessen the effect of intake and outfall influences by omitting from the calculations the profile data obtained near the ends of the trough. The data from a con- siderable number of experiments were finally discarded altogether and do not appear in the tables. To replace the discarded data experi- ments were afterward made with a modified apparatus. FIGUKE 16. — Diagrammatic longitudinal section of der cent. Q-m./sec. 0.8 16.3 1.8 124 1.0 31.2 2.0 153 1.2 49.9 2.2 1X5 1.4 72.0 2.4 220 1.6 97.0 A number of tentative formulas were now compared with this empiric line, their param- eters being computed so that they would fit, as nearly as practicable, the values of C in Table 5. Certain functions, including the simpler functions of the circular arc and the exponen- tial function <7=e", could not be fitted, even approximately, to the data; but the fol- lowing functions yielded curves closely re- sembling that of figure 20: (6) (7) (8) (9) .(10) (11) Functions (6), (7), and (8) are special cases of the general formula of interpolation with integral exponents : No. (11) is a somewhat involved power function suggested by results of a preliminary discussion of the laboratory data. Nos. (9) and (10) are special cases of the general para- bolic function (z + a)" = .(13) and have the virtue of facilitating the graphic treatment of the material. Their logarithmic equivalents are, respectively, log (<7+/l)= log b + n log S _______ (14) log C"=log b + n log (S-a) _______ (15) and, as each of these is the equation of a straight line, the graphic derivation of the exponent, by means of logarithmic section paper, becomes a simple matter after the value of ^ or a has been determined. The adjustment of equations (6) to (11) to the specific data in Table 5 gives them the following forms, (6a) being derived from (6), etc.: C= -29.55 + 67.59^-7. 194S3 ______ (6a) C -- 12.865 + 44.08S'2 ______________ (7a) <7= - 19.25 + 16.945 + 34.48S2 _______ (8a) (7=-10.0+41.2S"-M _______________ (9a) 0-70.5(5 -0.39)1*-. . (lOa) (7-31.2* 2.68 „«•» -(Ha) When the curves corresponding to these equations are plotted for the region covered by the empiric line BD, they coincide very closely with that line. The greatest departure 62 TRANSPORTATION OF DEBRIS BY RUNNING WATER. is in the curve for (7a), but its divergence is not sufficient to throw it out of apparent harmony with the series of points representing the original observations. In order to exhibit further the properties of the formulas, their curves were extrapolated in both directions from the locus BD. Table 6 contains the numerical data used in plotting the extensions. Figure 20 gives the exten- sions of the curves for slopes greater than those of the experiments, and figure 21 the extensions for smaller slopes. TABLE 6. — Numerical data computed for the construction of curves in figures 20 and "21. (6a) (7a) (8a) (9a) ( lOa") ( lla^t Values of C corresponding to— S-0 Gm.lsee. 0 Om./wc. 0 Om./sec. 19 25 Gm.lsec. 10 0 Om./sec. Qm.jsfc. 0 S=0 1 845 9 57 S-0.2 —.810 — 8.30 do 073 S=0.3... .108 —11.07 — 6 20 do 490 S-0.4 1.908 — 6 97 3 28 039 1 59 S-0.5. .. 4.59 — 1.96 45 1 93 3 64 S-0.6 8.15 3 33 4 98 5 54 S— 07 12 6 9 50 10 33 S-O.B . 16.3 17.9 16 36 16 48 16 49 S-3.0 327 358 332 353 337 323 S— 4 0 504 654 589 631 S-5.0 645 1,039 920 990 849 S— 60 705 1 510 1 315 1 422 S-7.0. . 645 2 070 1 741 1 531 S— 80 409 2 718 2 313 2 519 S-9.0 —31 2,916 2 355 1 276 S=10.0 3,926 2,818 1.400 Per cent. \ ° .45 Per cent. 0 29 Per cent. -1.03 54 Per cent. -0.49 Per cent. 0.39 Per cent. 0 I 8.94 6 04 .22 Values of S corresponding to point of inflection 3.33 4 37 zpoo 1,500 1,000 0 I 2 3 4 5 6 7 8 Slope FIGURE 20.— Extrapolated curves of f=/(S) for tentative equations of interpolation, and for slopes greater than 2.4 per cent. The approximate range of this series of experiments is from a slope of 0.8 per cent to one of 2.4 per cent. The extrapolated curves pertain to slopes from 2.4 to 10 per cent and from 0.8 to 0 per cent. The prompt divergence of the lines as they leave the locus to which they were adjusted shows that they have widely different values for purposes of extra- polation, and therefore presumably for pur- poses of interpolation. Attention being given first to the curves for higher slopes (fig. 20), it will be observed that four of them ascend with progressively increas- ing rate. The curve of formula (1 1 a) ascends continuously, but its rate of ascent changes at the slope of 4.37 per cent from an increasing rate to a decreasing rate. The curve of formula (6a) exchanges its increasing rate of ascent for a decreasing rate at the slope of 3.33 per cent, attains a maximum at a slope of about 6 per ADJUSTMENT OF OBSERVATIONS. 63 cent, and crosses the line of zero capacity before reaching the slope of 9 per cent. The general characteristics of stream traction do not admit of a maximum in the relation of capacity to slope. Capacity for traction is clearly an increasing function of the stream's velocity, and the velocity is clearly an increas- ing function of the slope. There is reason also to believe that capacity increases at an in- creasing rate up to the slope corresponding to infinite capacity. There are three forces con- cerned in traction — first, the force of the cur- rent, of which the direction is parallel to the slope; second, a component of gravity, when gravity is resolved in directions parallel and normal to the slope; third, the resistance of the bed, which is a function not only of the others, but inversely of the slope. Within the range of experimental slopes the component of gravity is negligible in comparison with the force of the current, and the influence of slope on the resistance is relatively unimportant; but as the angle of stability for loose material is approached the resistance diminishes rapidly, and at the slope of instability (65 to 70 per cent for river sand) gravity is competent to transport without the aid of current, and the stream's capacity is infinite. All these factors depend on slope, and as the increment to capacity verges on infinity in approaching the -us + 8 (llaj /tea) .1 .2 .3 .4 .5 .6 .7 .8 .9 Slope FIGURE 21. — Extrapolated curves of C=/(S) for tentative equations of interpolation and for slopes less than 0.8 per cent. slope which limits variation, it is highly prob- able that capacity grows continuously with slope. This criterion suffices for the rejection not only of the specific formulas (6a) and (lla), but also of their types, (6) and (11). In for- mula (6a) the occurrence of the maximum value of C is determined by the negative coeffi- cient of S3; and it is true as a general fact that equations of the class indicated by (12) yield maxima whenever the coefficient of the highest power of the independent variable is negative. It is possible, or perhaps probable, that if each series of laboratory values were to be formu- lated under (7) or (8) the conditions for maxima would be found to occur. On the whole, the extrapolations for higher slopes tend to restrict choice to forms (9) and (10), with some reser- vation as to forms (7) and (8). Figure 21 gives extrapolated curves for slopes less than 0.8 per cent and represents the same equations as figure 20, except that the curve for (6a) is omitted. It will be observed that it magnifies greatly the space between 0 and B in figure 20, the scale of slopes being 10 times and the scale of capacities 100 times as large. The implications of the functions for low slopes are specially important because extrapolation from laboratory conditions to those of natural streams will nearly always involve the passage from higher to lower slopes. 64 TBANSPOBTATION OF DEBRIS BY BUNNING WATER. Curves (7a) and (11 a) reach the origin of coordinates — that is, their equations indicate that at the zero of slope there is no capacity for traction. Formula (11 a) gives small but finite capacities for very low slopes; but under formula (7a) finite capacities cease when the slope falls to 0.29 per cent, and for lower slopes there is indication of negative capacities. If the conditions of traction permitted, negative capacity might be interpreted as capacity for traction upstream ; but as this is inadmissible, the negative values may be classed as surd results arising from the imperfection of the correlation between an abstract formula and a concrete problem. The curves of (8a) and (9a) also intersect the axis of slope at some distance from the origin, and their extensions indicate negative capacity. The curve of (lOa) becomes tangent to the axis of slope at the point corre- sponding to a slope of 0.39 per cent and there ends, having no continuation below the axis. It is the real limb of a parabola of which all other parts are imaginary. It expresses to the eye the implication of formula (lOa) that trac- tion ceases when the slope is reduced to 0.39 per cent, and that its cessation is not abrupt but gradual; and also the implication of the general formula (10) that traction ceases when the slope is reduced to the value a. It is a matter of observation that when slope is gradually reduced, the current becoming feebler and the capacity gradually less, the zero of capacity is reached before the zero of slope. For each group of conditions (fineness, width, discharge) there is a particular slope corresponding to the zero of capacity. It is also a matter of observation that the change in capacity near the zero is gradual. Formulas (7a), (8a), (9a), and (lOa) therefore accord with the data of observation in the fact that they connect the zero of capacity with a finite slope; formula (Ha), which connects zero capacity with zero slope, is discordant. Also, formulas (lOa) and (lla) accord with the data of observation in that they make the approach of capacity to its zero gradual; while formulas (7a), (8a), and (9a), which make the arrival of capacity at its zero abrupt, are in that respect discordant. But one of the formulas (lOa), shows quali- tative agreement with both of the criteria applied through extrapolation to low slopes; and that formula is one of the two which respond best to the criterion applied through extrapolation to high slopes. That type of formula, or £=&,(£-»)"--- ..(10) was therefore selected for the reduction of the more or less irregular series of observational values of capacity to orderly series better suited for comparative study. In rewriting the formula the coefficient is changed from 6 to 6U because corresponding coefficients 62, 63, etc., are to be used in a series of formulas expressing the relations of capacity to various conditions. As slope is a ratio between lengths, (S — a)n is of zero dimensions and &! is of the unit C; it is the value of capacity when S — a=l. The slope which is barely sufficient to initiate traction has been defined (p. 35) as the competent slope. To whatever extent a represents the competent slope the formula has a rational basis. The local potential energy of a stream, or the energy available at any cross section in a unit of time, is simply proportional to the product of discharge by slope or, if the discharge be constant, is pro- portional to the slope. So long as the slope is less than that of competence the energy is expended on resistances at contact with wetted perimeter and air and on internal work occasioned by those resistances. When the slope exceeds the competent slope, part of the energy is used as before and part is used in traction. The change from competent slope to a steeper slope increases the available energy by an amount proportional to the increase of slope, and the increase of energy is associated with the added work of traction. Capacity for traction, beginning at competent slope, increases pari passu with the increase of the excess of slope above the competent slope, and there is manifest propriety in treat- ing it as a function of the excess of slope rather than of the total slope. It is of course also a function of the total slope; but an adequate formula for its relation to the excess of slope may reasonably be supposed to be simpler than a formula for its relation to total slope. If a represents competent slope, then the relation of capacity to S — a should be simpler than its relation to 8. Instructive information as to the relative simplicity of the two functions is obtained by ADJUSTMENT OF OBSERVATIONS. 65 comparing their logarithmic graphs. In figure 22 the curved line AB has been copied from the curve in figure 19. It is the graph of log C= f (log S) for grade (G), width 0.66 foot, and dis- charge 0.734 ft.'/sec. If for S we substitute S — 0.3, we modify the graph by moving each point of it to the left by an amount equal to log S — log (S — 0.3); and we produce the lino CD, which is the graph of log C=fa(\og (S — 0.3)). If in similar manner we derive the graph of log C=fm(\og (S-0.6)), the re- sult is the line EF. CD curves in the same direction as AB but less strongly; EF curves in the opposite direction. It is evident that the three curves belong to a continuous series, and that somewhere between CD and EF a member of the series is straight or approxi- mately straight. That straight line, GH, is the graph of log C=fIVQog (5-0.39)); but as it is straight, its equation may be written log <7=log 6, + n-log GS-0.39), in which log 6t is the ordinate of the inter- section of the line with the axis of log C, and n is the trigonometric tangent measuring the inclination of the line to the axis of log S. This is identical with equation (15) except that 0.39 appears in place of a; and in fact 200 100 80 00 20 7 .4 £ Slop* .8 1.0 Z.Q FIGCBE 22.— The relation of » in C=f(S-r) to the curvature of the logarithmic graph. the value of a in equation (lOa) was computed graphically by means of the logarithmic plot. The line AB, being the graph of log C— /.Gog S), is also the logarithmic graph of C=f(S). The line GE, being the graph of log C^/jvGog (8 — a)) is also the logarithmic graph of <7=/v(S — a). Their relation in re- spect to simplicity is that of the curve to the straight line. In view of these suggestions of harmony it is peculiarly pertinent to inquire whether a is actually representative of competent slope: and it will be convenient to make that inquiry in connection with the determination of values 20921°— No. 86—14 5 of a for the several series of observations on capacity and slope. THE CONSTANT a AND COMPETENT SLOPE. In the experimental data for graded debris — Table 4, (A) to (H) — are 117 series of values of capacity and slope. After these had been plotted and inspected in a comparative way, it was decided to restrict the main discussions to 92 series only, the discarded series being all short as well as somewhat discrepant among themselves. Of the 92 series retained, only 30 afford information as to the correspond- ing values of o; that is, only 30 of the loga- 66 TRANSPORTATION OF DEBRIS BY RUNNING WATER. rithmic plots exhibit curvature so definitely that the approximate magnitudes of the con- stants necessary to eliminate it can be inferred. In some of the remaining series the observa- tions are not so distributed as to bring out the curvature. For others the observational posi- tions on the plots are too -widely dispersed to give good indication of the character of the best representative line. On the 30 logarithmic plots the curves ap- proximately representing the observations were drawn, and the values of a necessary to replace the curves by straight lines were computed graphically in the manner just indicated. These values are given, to the nearest tenth of 1 per cent of slope, in Table 7, where they are arranged with reference to the conditions of the experiments. TABLE 7. — Values of a in C=bl(S—a)n, estimated from logarithmic plots of observations. Grade F Range of F W (feet). Values otffoi discharge (ft.3/sec.) of— 0.093 0.182 0.363 0.545 0.734 0.923 1.021 ; 1.119 (A) (B) (C) (D) (E) (F) (G) (H) 25,200 13,400 5,460 1,460 142 22.1 5.9 2.0 1.77 2.35 2.85 3.58 12.65 2.15 2.92 2.51 0.66 1.00 1.32 1.96 .23 .44 .66 1.00 1.32 1.96 .44 .66 1.00 1.32 t.96 .66 1.00 1.32 .66 1.00 1.32 .66 1.00 1.32 .66 1.00 1.32 .66 - - - 0.2 0.1 0.7 .5 .4 .0 i .1 — 0.1 .4 .2 .1 .3 .1 .1 3 0.1 .1 .1 — • — - — - — — - .0 .0 0 0.3 .3 .4 ! .; ! .6 .3 .1 fi NOTE.— The horizontal dashes indicate series of observations to which values of :M - (18) Assuming again that bed velocity is propor- tional to mean velocity, and again assuming the validity of the Chezy formula, we obtain from (18) 2 301 2.2 380 697 317 (C) .66 .734 .5 51 69.5 139 3.8 8.5 .6 68 93 154 .7 87 118 169 .8 107 146 182 .9 129 176 196 1 152 207 207 (C) 1.00 .182 .4 4.5 24.7 61.8 1 4.4 .5 7.6 41.8 83.6 .6 11.4 62.6 104 .7 15.6 85.8 123 .8 20.4 112 140 .9 25.6 141 157 1 31.2 171 171 1.2 44 242 202 1.4 57.7 317 226 1.6 72 396 247 1.8 90 495 275 2 106 582 291 2.2 125 686 312 2.4 145 797 332 2.6 166 912 351 2.8 189 1,040 372 ADJUSTMENT OF OBSERVATIONS. 81 TABLE 12. — Adjusted values of capacity, based on data of Table 4, with corresponding values of duty and efficiency — Con. Conditions. Adjusted data. Probable error (per cent) of— Or. w Q S C V E Adjusted data. Observa- tion. (C) 1.00 0.363 0.2 3.3 9.1 45.5 1.2 6.2 .3 9.6 26.4 88 .4 17.5 48.2 120 .5 26.6 73.3 147 .6 36.6 101 168 .7 47.5 131 187 .8 59 1(8 204 .9 72 198 220 1 85 234 234 1.2 113 311 259 1.4 143 394 281 1.6 175 482 301 1.8 210 579 322 2 245 675 338 2.2 281 774 352 2.4 320 882 367 2.6 3fiO 992 381 2.8 401 1,100 393 (C) 1.00 .545 .5 48.2 88.4 177 1.1 5.5 .6 64.9 119 19S .7 82 150 214 .8 100 183 229 .9 119 218 242 1.0 140 257 257 1.2 183 336 280 1.4 228 418 299 1.6 275 504 315 1.8 326 598 332 2.0 377 692 346 2.2 430 789 359 (C) 1.00 .734 .2 12.5 17.0 85 7.8 37.7 .3 27.3 37.2 124 .4 44.2 60.2 150 .5 63.2 86.2 172 .6 84 114 190 .7 106 144 206 .8 130 177 221 .9 155 211 234 1.0 ISO 245 245 1.2 235 320 267 1.4 294 401 286 1.6 352 480 300 1.8 416 567 315 2.0 483 658 329 2.2 550 750 341 (C) 1.00 1.119 .5 96 85.8 172 2.3 5.1 .6 128 114 190 .7 162 145 207 .8 198 177 221 .9 236 211 234 1.0 276 247 247 1.2 361 323 269 1 4 451 403 288 (C) 1.32 .182 .6 5.6 30.8 51.3 2.4 S. 4 .7 8.7 47.8 68.3 .8 12.3 67.6 84.5 .9 16.6 91.2 101 1.0 21.4 118 118 1.2 32.7 180 150 1.4 46.5 256 183 1.6 62 341 213 1.8 80 440 244 2.0 99 544 272 2.2 120 659 299 2.4 144 791 329 (C) 1.32 .363 .3 6.4 17.6 58.7 1.2 5.5 .4 13.6 37.5 93.8 .5 22.1 60.9 122 .6 31.9 87.9 146 .7 42.2 116 166 .8 53.8 148 185 .9 66 182 202 1.0 79 218 218 1.2 105 289 241 1.4 135 372 266 1.6 166 458 286 1.8 199 548 304 2.0 233 642 321 2.2 270 744 338 2.4 307 846 352 (C) 1.32 .545 .6 56.3 103 172 1.8 8.5 . 7 73.3 134 191 .8 91 167 209 .9 109 200 222 1.0 129 237 237 1.2 171 314 262 20921°— No. 86— 14 82 TRANSPORTATION OF DEBRIS BY RUNNING WATER. TABLE 12. — Adjusted values of capacity, based on data of Table 4, with corresponding values of duty and efficiency — Con. Conditions. Adjusted data. Probable error (per cent) of— Or. w Q S C V E \djusted data. Observa- tion. (C) 1.32 0.345 1.4 215 395 282 1.8 8.3 1.6 263 483 302 1.8 312 573 318 2.0 362 664 332 2.2 415 762 346 2.4 470 862 359 (C) 1.32 .734 .2 9.3 12.7 63.5 2.2 9.0 .3 25.0 34.1 114 .4 43.2 58.9 147 .5 63.8 86.9 174 .6 86 117 195 .7 109 148 211 .8 133 181 226 .9 160 218 242 1.0 187 255 255 1.2 243 331 276 1.4 304 414 296 1.6 367 500 312 1.8 432 589 327 2.0 500 681 340 2.2 870 777 353 (C) 1.96 .363 .5 10.5 28.9 57.8 3.5 15.5 .6 17.8 49.0 81.7 .7 26.5 73.0 104 .8 36.6 101 126 .9 47.9 132 147 1.0 60.0 165 165 1.2 88 242 202 1.4 119 328 234 1.6 154 424 265 1.8 193 532 295 2.0 235 648 324 2.2 280 771 350 (C) 1.96 .545 .5 25.7 47.2 94.4 2.0 7.0 .6 39.3 72.1 120 .7 55.0 101 144 .8 72 132 165 .9 91 167 186 1.0 111 204 204 1.2 156 286 238 1.4 205 376 268 1.6 259 475 297 1.8 316 580 322 2.0 375 688 344 2.2 440 808 367 (C) 1.96 .734 .4 34.0 46.3 116 2.8 13.2 .5 55 74.9 150 .6 78 106 177 .7 103 140 200 .8 131 178 222 .9 160 218 242 1.0 190 259 259 1.2 255 347 289 1.4 323 410 314 1.6 394 537 336 (("> 1.% 1.119 .5 88 78.7 157 2.2 P. 9 .fi 130 116 193 .7 177 158 226 .8 228 204 255 .9 283 253 281 1.0 343 307 307 1.2 480 429 358 (D) .66 .093 .6 2.7 29.0 48.4 2.6 8.1 .7 3.9 41.9 59.9 .8 5.4 58.1 72.5 .9 7.1 70.4 84.9 1.0 9.1 97.9 97.9 1.2 13.5 145 121 1.4 18.6 200 143 1.6 24.7 266 166 1.8 31.4 338 188 2.0 38.5 414 207 2.2 46.3 498 226 2.4 55.2 594 248 2.6 64.3 692 266 (D) .66 .182 .4 4.7 25.8 64.5 4.2 15.2 .5 7.7 42.3 84.6 .6 11.3 62.1 104 .7 15.3 84.1 120 .8 19.7 108 135 .9 24.6 135 150 1.0 29.8 • 164 164 1.2 41.0 235 188 1.4 53.4 293 209 ADJUSTMENT OF OBSERVATIONS. 83 TABLE 12. — Adjusted values of capacity, based on data of Table 4, with corresponding values of duty and efficiency— Con. Conditions. Adjusted data. Probable error (per cent) of— Or. w Q S C U E Adjusted data. observa- tion. (D) 0.66 0.182 1.6 67 368 230 4.2 15.2 1.8 82 450 250 2.0 98 538 269 2.2 114 626 284 2.4 132 726 302 2.6 150 824 317 (D) .6f> .545 .2 4.7 8.5 42.5 3.4 15.3 .3 11.7 21.5 71.6 .4 20.7 38.0 95 .5 31.2 57.2 114 .6 43.1 79.1 132 .7 56.2 103 147 .8 70.2 129 161 .9 86 158 176 1.0 101 1S5 185 1.2 137 251 209 1.4 175 321 229 1.6 217 398 249 1.8 261 479 266 2.0 308 568 284 2.2 357 655 298 (D) 1.00 .182 .7 11.3 62.1 88.7 2.5 9.1 .8 15.1 83 104 .9 19.2 106 118 1.0 24.0 132 132 1.2 34.5 190 158 1.4 46.4 255 182 1.6 59.8 329 206 1.8 74 407 226 2.0 90 495 248 2.2 106 582 264 2.4 124 682 284 2.6 144 792 304 2.8 164 902 324 (D) 1.00 .363 .3 6.9 19.0 63 3.3 13.2 .4 13.2 36.4 91.0 .5 20.9 57.6 115 .6 29.6 81.6 136 .7 39.0 107 153 .8 49.6 137 171 .9 61 168 187 1.0 73 201 201 1.2 99 273 228 1.4 127 350 250 1.6 158 435 272 1.8 190 524 291 2.0 225 620 310 2.2 261 819 327 2.4 299 824 343 (D) 1.00 .545 .5 27.6 50.6 101 1.2 3.9 .6 40.2 73.8 123 .7 55.0 101 144 .8 71 130 162 .9 89 163 181 1.0 108 ' 198 198 1.2 152 279 232 1.4 201 369 264 1.6 257 472 295 1.8 317 582 323 2.0 382 700 350 (D) 1.00 .734 .2 6.9 9.4 47.0 2.5 11.5 .3 17.5 23.8 79.3 .4 31.0 42.2 106 .5 47.0 64.0 128 .6 65.0 88.6 148 .7 85 116 166 .8 106 144 180 .9 129 176 196 1.0 153 208 208 1.2 208 282 236 1.4 268 365 361 1.6 332 452 283 1.8 388 542 302 (D) 1.32 .363 .5 11.9 32.8 65.6 7.0 24.3 .6 18.8 50.9 84.8 .7 27.0 74.4 106 .8 36.6 101 139 .9 47.2 130 144 1.0 59.2 163 163 1.2 87 240 200 1.4 118 325 232 1.6 154 424 265 1.8 194 534 297 2.0 237 653 326 84 TRANSPORTATION OF DEBRIS BY RUNNING WATER. TABLE 12. — Adjusted values of capacity, based on data of Table 4, with corresponding values of duty and efficiency — Con. Conditions. Adjusted data. Probable error (percent) of — Gr. VI Q S C U E Adjusted data. Observa- tion. (D) 1.32 0.734 0.3 10.2 13.9 46.3 5.3 21.4 .4 20.9 28.5 71.2 .5 34.0 46.3 92.6 .6 49.2 67.0 112 . 7 67 91.3 130 .8 87 118 148 .9 108 147 163 1.0 131 178 178 1.2 185 252 210 (E) .06 .363 1.0 24.8 68.3 68.3 0.9 2.1 1.2 36.2 99.7 83.1 1.4 49.4 136 97.2 (E) .66 .734 1.0 40.0 54.5 54.5 1.2 59.0 80.4 67.0 1.4 82 112 80 1.6 108 147 92 (E) 1.00 .182 .4 2.4 13.2 33.0 1.9 5.0 .5 3.8 20.9 41.8 .6 5.5 30.4 50.7 .7 7.4 40.6 58.0 .8 9.6 52.7 65.0 .9 12.0 66.0 73.3 1.0 14.8 81.4 81.4 1.2 20.7 114 95 1.4 27.6 152 108 1.6 35.3 194 121 1.8 43.5 239 133 2.0 52.5 288 144 2.2 62.4 343 156 2.4 73 401 167 2.6 84 461 177 2.8 96 527 188 (E) 1.00 .363 .2 1.3 3.6 18.0 5.1 14.6 .3 3.3 9.1 30.3 .4 6.0 16.5 41.2 .5 9.2 25.3 50.6 .6 13.1 36.1 60.1 .7 17.5 48.2 68.9 .8 22.5 62.0 77.5 .9 28.0 77.2 85.8 1.0 33.8 93.1 93.1 1.2 47.2 130 108 1.4 62.0 171 122 1.6 78.5 216 135 1.8 96 264 147 2.0 115 317 158 2.2 137 378 172 2.4 159 438 182 (E) 1.00 .734 .2 4.2 5.7 28.5 5.0 19.5 .3 9.1 12.4 41.3 .4 15.2 20.7 51.8 .5 22.4 30.5 61.0 .6 30.8 42.0 70.0 .7 40.0 54.5 77.9 .8 49.8 67.8 84.8 .9 60.5 72.4 91.6 1.0 72.0 98.1 98.1 1.2 98 134 112 1.4 126 172 123 1.6 156 213 133 1.8 188 256 142 2.0 222 302 III (E) 1.00 1.119 .5 44 39.3 76.6 3.2 6.4 .6 60 54 89 .7 77 69 98 .8 97 87 108 .9 117 105 117 1.0 138 123 123 1.2 188 168 140 1.4 242 216 154 (E) 1.32 .363 .6 15.3 42.2 70.3 .7 20.3 55.9 79.9 .8 25.4 70.0 87.5 .9 30.7 84.6 94.0 1.0 36.3 100 100 1.2 49.3 136 113 1.4 63.5 175 125 1.6 78.8 217 136 1.8 95 262 145 2.0 112 309 154 (E) 1.32 .734 .5 24.7 33.7 67.4 .6 33.0 45.0 75.0 .7 42.0 57.2 81.7 .8 51.7 70.4 8S.O .9 62 84.5 93.9 1.0 73 99.5 99.5 1.2 97 132 110 ADJUSTMENT OF OBSERVATIONS. 85 TABLE 12. — Adjusted values of capacity, based on data of Table 4, with corresponding values of duty and efficiency — Con. Conditions. Adjusted data. Probable error (per cent) o(— Or. w Q 8 C U M Adjusted data. Observa- tion. (E) 1.32 1.119 0.6 49.4 44.2 73.7 1.8 3.6 .7 65.0 58.1 83.0 .8 82.5 73.7 92.1 .9 102 91.1 101 1.0 123 110 110 1.2 171 153 128 (F) .66 .182 1.2 9.3 51.0 42.5 .8 1.7 1.4 13.7 75.3 53.8 1.6 18.9 104 65.0 1.8 24.7 136 75.6 2.0 31.4 172 86.0 2.2 38.7 213 96.8 2.4 46.8 257 107 2.6 55.7 306 118 (F) .66 .363 1.0 20.5 56.5 56.5 2.3 4.6 1.2 29.9 82.4 68.7 1.4 41.0 113 80.8 1.6 53.2 147 91.9 1.8 67.0 185 103 2.0 81.5 225 112 (F) .66 .734 .8 31.6 43.0 47.8 .4 .8 1.0 39.0 53.2 53.2 1.2 55.3 75.3 62.8 1.4 73.0 99.5 71.0 1.6 93.5 127 79.4 1.8 115 157 87.2 (F) 1.00 .182 1.2 4.2 23.1 17.6 1.4 7.3 40.1 28.6 1.6 11.3 62.1 38.8 1.8 16.4 90.1 50.0 2.0 22.5 124 62.0 2.2 29.8 164 74.6 2.4 38.2 210 87.5 2.6 47.7 262 101 (F) 1.00 .363 .8 7.4 20.4 25.5 2.0 4.8 .9 10.5 28.9 32.1 1.0 14.0 38.6 38.6 1.2 22.8 62.8 52.3 1.4 33.4 92.0 65.7 1.6 46.0 127 79.4 1.8 60 165 91.7 2.0 76 209 104 2.2 94 259 118 2.4 112 309 129 2.6 133 367 141 (F) 1.00 .734 .7 20.0 27.3 39.0 .5 1.0 .8 27.3 37.2 46.5 .9 35.1 47.8 53.1 1.0 42.8 58.3 58.3 1.2 63 85.8 71.5 1.4 85 116 82.9 1.6 109 148 92.6 1.8 136 185 103 (F) 1.00 1.119 .7 37.5 33.5 47.9 .9 1.8 1 .8 51.0 45.6 57.0 .9 64.8 57.9 64.3 1.0 79.5 71.0 71.0 1.2 112 100 83.3 1.4 150 134 95.8 1.6 191 171 107 1.8 236 211 117 (F) 1.32 .363 1.2 21.4 59.0 48.2 l72 2.4 1.4 31.0 85.4 61.0 1.6 41.5 114 71.3 1.8 53.0 146 81.1 2.0 66.0 182 91.0 2.2 80 220 100 (F) 1.32 .734 .8 22.9 29.6 ?7.0 3.2 7.2 .9 31.0 40.1 44.6 1.0 40.2 52.0 52.0 1.2 61.6 79.7 66.4 1.4 87.0 119 85.0 1.6 115 149 93.2 (F) 1.32 1.119 .8 43.3 38.7 48.4 3.2 7.2 .9 56.0 50.0 91.1 1.0 70.0 62.6 62.6 1.2 101 90.2 75.2 1.4 138 123 87.9 1.6 178 159 99.4 1.8 221 197 109 86 TRANSPORTATION OF DEBRIS BY RUNNING WATER. TABLE 12. — Adjusted values of capacity, based on data of Table 4, with corresponding values of duty and efficiency — Con. Conditions. Adjusted data. Probable error (per cent) of— Gr. w Q S C V X Adjusted data. Observa- tion. (G) 0.66 0.363 .0 8.5 23.4 23.4 2,2 5.8 .2 14.8 40.8 34.0 .4 23.0 63.4 45.3 .6 32.7 90.1 56.3 .8 43.7 120 66.7 2.0 56.0 154 77.0 (G) .66 .734 . 7 10.3 13.3 19.0 1.5 4.3 .8 16.0 20.7 25.9 .9 22.8 29.5 32.8 .0 30.4 39.3 39.3 .2 48.3 62.5 52.1 .4 69 89.2 63.7 .6 93 127 79.4 .8 120 155 86.2 2.0 149 193 96.5 2.2 181 234 106 2.4 216 280 117 2.6 253 327 126 (G) .66 1.119 .6 11.9 10.6 17.7 1.2 3.6 .7 19.0 17.0 24.3 .8 27.7 24.7 30.9 .9 37.6 33.6 37.3 1.0 49.0 43.8 43.8 1.2 75.0 67.0 55.8 1.4 107 95.6 68.3 1.6 143 128 80.0 1.8 182 163 90.6 2.0 227 203 102 2.2 276 247 112 2.4 325 290 121 2.6 382 341 131 (G) 1.00 .363 1.2 9.7 26.7 22.2 1.9 5.0 1.4 16.3 44.9 35.0 1.6 24.4 67.2 42.0 1.8 34.1 94.0 52.2 2.0 45.2 125 62.5 2.2 57 157 71.4 2.4 71 196 81.7 2.6 86 237 91.2 2.8 102 281 100 3.0 120 331 no (G) 1.00 .734 .7 8.0 10.9 15.6 1.9 5.5 .8 13.3 18.1 22.6 .9 19.5 26.6 29.6 1.0 26.9 36.6 36.6 1.2 44 60.0 50.0 1.4 65 88.6 63.3 1.6 89 121 75.6 1.8 115 157 87.8 2.0 145 198 99.0 2.2 177 241 110 2.4 212 289 120 2.6 249 339 130 (G) 1.00 1.119 .6 13.5 12.1 20.1 4.3 13.6 .7 22.7 20.3 29.0 .8 33.6 30.0 37.5 .9 46.0 41.1 45.6 1.0 61.0 54.4 54.4 1.2 92 82.2 68.5 1.4 128 114 81.5 1.6 171 153 95.6 1.8 217 194 108 2.0 268 239 120 2.2 321 287 130 2.4 380 340 142 (G) 1.32 .363 1.8 18.1 49.5 27.7 2.4 4.8 2.0 28.3 78.0 39.0 2.2 39.9 101 45.9 2.4 53.8 148 61.7 2.6 69.8 192 73.9 2.8 88.0 242 86.5 3.0 109 300 100 3.2 134 389 115 (G) 1.32 .734 .8 7.9 10.8 13.5 2.2 5.8 .9 12.8 17.4 19.3 1.0 18.9 25.8 25.8 1.2 33.8 46.1 38.4 1.4 52.0 70.8 50.6 1.6 74.0 101 63.2 1.8 98.0 134 75.4 2.0 125 170 85.0 2.2 155 211 95.9 2.4 187 255 106 2.6 222 303 117 2.8 260 354 126 ADJUSTMENT OF OBSERVATIONS. 87 TABLE 12. — Adjusted values of capacity, based on data of Table 4, with corresponding values of duty and efficiency— Con. Conditions. Adjusted data. Probable error (per cent) of— Or. • Q S C U E Adjusted data. Observa- tion. (0) 1.32 1.119 0.7 17.9 16.0 22.9 3.1 8.7 .8 28.0 25.0 31.2 .9 39.5 35.3 38.1 1.0 52.2 46.6 40. 6 1.2 81 72.4 60.3 1.4 115 103 73.6 1.6 153 137 85.6 1.8 194 173 96.2 2.0 238 213 108 2.2 285 255 116 2.4 335 299 125 2.6 387 346 133 (H) .66 .363 1.2 3.1 8.5 7.1 1.0 3.5 1.4 6.9 19.0 13.6 1.6 12.4 34.2 21.4 1.8 19.4 53.5 29.7 2.0 28.0 77.1 38.6 2.2 38.1 105 47.8 2.4 49.8 137 57.1 2.6 63 174 66.9 2.8 78 215 76.8 (H) .66 .734 .8 4.5 6.8 8.5 1.6 4.9 .9 8.2 12.4 13.8 .0 12.8 19.4 19.4 .2 24.6 37.3 31.1 .4 39.2 59.4 42.4 .6 57.0 86.4 54.0 .8 77.0 117 65.0 2.0 100 152 76.0 2.2 125 189 85.9 2.4 151 229 95.6 2.6 181 274 105 2.8 213 323 115 3.0 246 373 126 3.2 282 427 133 (H) .66 1.119 .7 6.0 5.4 7.7 11.2 .8 11.1 9.9 12.4 3.2 .9 17.6 15.7 17.4 1.0 25.2 22.5 22.5 1.2 43.5 38.8 32.3 1.4 65.3 58.3 41.6 1.6 91.0 81.3 50.8 1.8 120 107 59.5 2.0 152 136 68.0 2.2 186 166 75.5 2.4 223 199 83.0 2.6 263 235 90.4 OBSERVATIONS ON DEPTH. MODE OF ADJUSTMENT. To adjust the observations of depth it is necessary to deal with their relations to another variable, and either slope or capacity might be used. The selection of the particular variable for comparison was a matter of convenience only, because adjustment to either one would bring the depth values into orderly relation to the adjusted values of the other also; but the question of convenience was not unimportant. A fairly thorough preliminary study was there- fore made, in which the depth measurements for many series were plotted in relation, sever- ally, to measurements of capacity and measure- ments of slope. In figure 27 the horizontal scale is that of slope, the vertical of depth. The round dots represent observations made with grade (C), width 1.32 feet, and discharge 0.734 ft.3/sec. Despite irregularities, the grouping suggests as + 1^ \^^ * IS. + \ \ s*. \ \ *"u-« - '-•— • 0.5 1.0 1.5 Slope FIGUKE 27.— Observations of depth of current, in relation to slope, plotted as dots. The crosses show logarithms of the same depths and slopes. the representative lino some such curve as that drawn. On taking account of the physical con- 88 TRANSPORTATION OF DEBRIS BY RUNNING WATER. ditions, it is evident that as the slope is flat- tened, the current is slowed and the depth in- creased, and that zero slope gives infinite depth. The theoretic curve, therefore, has an asymp- tote in the vertical line corresponding to zero slope. Similarly the depth is reduced by in- crease in slope but remains finite for very high slopes. The theoretic curve has as asymptote a horizontal line corresponding, exactly or appioximately, with the (horizontal) line of zero depth. These asymptotes relate the curve d=f(S) to the hyperbola. In the same figure, but with use of a different scale, are a series of crosses which show the same observations as they appear when plotted on logarithmic sec- tion paper. They are the plot of observations ou log d =/, (log 8) ; and their arrangement sug- gests that the representative line may be straight. Plots were made to show the relations of depth observations to associated capacity, and these also suggest the hyperbola and the straight line. If, however, the locus of d =/u( f) is a hyperbola it differs materially from that of d=f(S), for as depth increases and current slackens, capacity becomes zero when current reaches the value of competence, and depth is not then infinite. So the line of zero capac- ity is not an asymptote to the curve. It is to be observed also that the represen- tative lines for log d =/, (log S) and log d =/„ Gog (7) can not both be straight, for if they were there could be derived from them a straight line representing log C=fv (log S), and it has already been found that that line is curved. It is, indeed, probable that neither of the loga- rithmic plots involving depth is straight; yet there are cogent practical reasons for assuming one or the other to be so. One reason is the very great convenience of the straight-line func- tion, and another that the relatively small range of the depth data renders impracticable such a discussion of the curvature of logarithmic loci as was made in the case of capacity versus slope. Accordingly, the most orderly plots of log d =fL (log S) and log d =/re (log C) were com- pared with special reference to curvature. For the function of log C the plots were found to indicate curvature in one direction only, while for the function of log S they indicated slight curvatures in both directions, with the straight line as an approximate mean. The function log d =/! (log S) was accordingly selected for the adjustment of the depth observations; and the representative line on the logarithmic plot was assumed to be straight. In accordance with that assumption, the adopted formula of inter- polation was *-£-- -<21) with its logarithmic equivalent, log d = log b' — nt log S. .(22) The coefficient 6' is a depth, the depth cor- responding to a slope of 1 per cent. The data were all plotted on logarithmic section paper. The notation was made to distinguish depth measurements made at a single point by means of the gage (see p. 25) from those based on full profiles of water sur- face and bed of debris. The former were used exclusively in the drawing of the representative lines, but not because they were regarded as of higher authority. It was thought best not to combine data which in certain cases were known to be incongruous ; and the gage observa- tions covered the whole range of the work, while the profiles did not. The measurements by profile were used in criticising the measure- ments by gage, and they determined the accept- ance or rejection of certain gage measurements. It was noted that in some series of observa- tions the depth measurements by the two methods were in close accord, while in others there was a large systematic difference; and certain series were rejected because of such large differences. The plots were made to distinguish also the observations associated with different modes of traction — the dune, smooth, and antidune modes. The observations with the smooth mode were assumed to be best, as a class; and these, together with the observations connected with the transitional phases of traction, were used to fix an initial point of each representa- tive line. The direction of the line was then adjusted by eye estimate to make it repre- sentative of all the points of the particular series. In this adjustment consideration was given to the conditions affecting the measure- ments of both depths and slopes. The lines were first drawn for those obser- vational series which appeared from the plots to be most harmonious, and for these there was ADJUSTMENT OF OBSERVATIONS. 89 discovered a tendency toward parallelism, within each group, of lines associated with a particular sand grade and channel width but differing as to discharge. For numerous other groups, involving very irregular observational positions, such parallelism was assumed; and under that assumption the bettor series of a group were made to control the directions of the lines for the poorer. For three groups the directions were interpolated by use of the directions found in affiliated groups. For each observational series the direction of the representative line gave the value of n^ in equations (21) and (22), and the intersection of the line with the axis of log d gave the value of I'. TABLE 13. — Value* ofnt in d——^. Grade of debris. Value of «i for trough having width (in feet) of— 0.23 0.44 0.66 1.00 1.32 1.96 i? 3.11 1 3.51 3.89 0.203 .138 .110 .094 .083 .075 0.108 1.63 .135 2.04 .115 2.38 .102 2.70 .092 2.30 .085 2.26 (I 086 .li 069 .14 059 .11 052 .11 047 0.163 .143 .130 .120 .112 .106 .101 .28 .46 .62 .75 .87 2.00 2.10 0.370 0.240 1.73 .328 .213 1.96 .298 .192 2.16 .275 .177 2.34 .256 .165 2.50 . 242 . 157 2. 66 230 0.546 .480 .436 .400 .375 .355 043 I - ADJUSTMENT OF OBSERVATIONS. 91 TABLE 14. — Adjusted values of depth of current (d), with values of mean velocity ( Vm] and form ratio \^=^1) — Con. Grade.. (C) .66 (C) .66 (C) .66 (C) .66 a Q... . .093 .182 .363 .545 .734 »i p.e.... .56 0.8 .56 1.8 .56 1.4 .56 1.3 .56 1.9 S | Vm R d Vm R \ (n 0.70 23.8 0.93 0.60 33.2 0.99 0.50 29.7 1.15 0.10 34 3 1.63 0.30 19.4 1.64 0.10 39.8 1.64 0.08 91.4 1.45 0.07 132.8 1.38 0.12 37.6 1.69 0.08 97.5 1.54 0.06 219.3 1.52 S Values of d. 0. 2 2.13 1 80 2.57 2 10 2.17 1.89 1.78 1.72 .69 .66 .64 .62 .61 .60 .58 .58 .57 3 2.46 2.19 2.05 .97 .92 .88 .85 .82 .79 .77 .75 .74 .73 .72 4 1.68 1 93 5 2.08 1.73 .67 .64 .61 .59 .58 .55 .54 .53 .52 .51 1.61 1.57 1.54 1.51 1.50 1.49 1.47 1.46 1.45 1.44 1.43 2.22 2.11 2.04 1.98 1.95 1.92 1.88 1.85 1.83 1.81 1.80 1 79 1.84 1.78 1.74 1.71 1.69 1.68 1.65 1.63 1.62 1.61 1.60 1 60 6 1.96 7 6.96 3.98 2.98 2.49 1.99 1.74 1.59 1.49 4.04 3.08 2.60 2.31 1.98 1.80 1.68 1.60 1.54 1.90 1.86 1.83 1.81 1.78 4.76 1.74 1.73 2.87 2.63 2.46 2.34 2.19 2.09 2.02 .97 .93 .90 .8 7.41 4.17 3.10 2.22 1.85 1.65 1.52 1.43 .65 .63 .62 .61 .59 .58 1.58 1.57 1 57 9 1 o 1.2 14 1.6 1 8 2 0 2 •> 1.36 1.50 2. 4 1.46 .88 .71 1.78 1 77 1.59 2 6 1.43 .86 .71 2. 8 1.84 .70 1.76 3 0 1.82 .70 1.76 3.2 .69 3.4 .69 3 6 1 .69 3 8 .69 4. 0 .68 4. 2 1.68 Grade. . (B) (C) 1.32 1.96 0.44 0.66 Q 0.182 0.363 0.545 0.734 0.363 0.545 0.734 1.119 0.093 0.182 0.093 0.182 0.17 39.5 1.61 0.12 96.5 1.54 0.10 165.4 1.55 0.08 233.5 1.57 0.18 93.9 1.64 0.14 153.5 1.60 0.12 228.2 1.46 0.10 363.3 1.43 0.40 22.6 1.43 0.20 34.4 1.50 0.16 18.2 1.58 0.11 38.6 1.54 Parameters of adjusting equation . . fti S Values of ii. 0 2 2.86 3 2.56 2.14 1.96 2.44 2.15 2.43 2.12 1.97 1.88 1.83 1.79 1.75 1.73 1.70 1.67 1.65 1.64 1.63 1.62 1.61 1.61 1.60 1.60 1.59 1.59 1.59 1.59 4 2.80 2.44 2.25 2.13 2.05 1.99 1.94 .88 .84 .80 .78 2.20 2.02 1.92 1.85 1.81 1.77 1.75 1.71 1.68 1.66 1.64 2.99 2.09 1.93 1.83 1.77 1.72 1.69 1.66 1.63 1.60 1.58 1.57 1.91 1.79 1.73 1.68 1.64 1.62 1.60 1.57 g 1.94 1.86 1.81 1.78 1.75 1.73 1.70 1.67 1.66 1.65 1.87 1.81 1.78 1.75 1.72 1.71 1.68 1.67 1.65 ' 1.64 2.57 2.35 2.21 2.12 2.06 2.01 .94 .89 .85 .83 2.23 2.09 2.01 .94 .90 .86 .82 .78 .76 .74 2.33 2.16 2.05 1.98 1.92 1.87 1.82 1.79 1.76 1.74 1.72 1.70 1.69 1.69 1.68 1.67 1.66 1.66 1.65 1.65 1.65 6 4.29 3.39 2.86 2.57 2.39 2.15 2.00 1.91 1.84 2.22 2.10 2.00 1.93 1.88 1.81 1.76 1.72 1.69 1.67 1.65 8 9 1 o 1 2 1 4 1.6 1.8 2.0 2 2 .76 .75 1.63 1.62 .81 1.79 1.56 1.79 1.75 2 4 74 2 6 2 8 3 0 | 32 3 4 3 6 3 8 4 0 4 2 1.64 102 TRANSPORTATION OF DEBRIS BY RUNNING WATER. TABLE Jo. — Values of ilt the index of relative variation for capacity in relation to slope — Continued. Grade. . (C) Conditions of experimentation w 0.66 1.00 1.32 Q 0.363 0.545 0.08 122.8 1.48 0.734 0.182 0.363 0.545 0.734 1.119 0.182 0.363 0.545 0.734 {t 0.08 82.9 1.46 0.04 162.1 1.50 0.15 40.4 1.59 0.11 100.1 1.41 0.09 159.2 1.33 0.07 199.2 1.35 0.06 301.6 1.39 0.22 33.7 1.85 0. 16 0. 13 99.9 156.1 1.40 1.35 0.11 218.5 1.30 bi n... S Values of fi. 0 2 2.12 86 3.14 2.18 1 76 2.90 2.12 1.80 1.67 1.60 1.55 1.51 1.49 1.47 1.44 1.42 1.40 1.39 1.38 1.37 3 2.23 3.00 4 1.83 1.74 1.68 1.65 1.62 1.60 1.59 1.56 .75 .69 .65 .62 .60 .59 .58 .56 2.54 2.27 2.12 2.02 .95 .90 .87 .81 .78 .75 73 1.95 1.81 1.73 .68 .64 .61 .59 .55 .53 .51 50 1.64 2.33 5 1.63 .61 .59 .58 .57 .57 1.63 1.57 .53 .50 .48 .47 .44 .43 .42 41 1.57 1.53 1.50 1.48 .46 .45 .48 .42 .41 .41 1.58 1.55 .52 .50 .49 .48 .46 .45 2.06 .6.... 2.92 2.70 2.55 2.45 2.37 2.27 2.20 2.14 2.11 2.08 2.06 2.04 1.91 1.82 .75 .70 .67 .61 .58 .55 .54 .52 .51 .50 1.72 1.65 1.61 1.57 1.55 1.51 1.49 1.47 1.45 1.44 1.43 1.42 7 . g 9 . . 1 0 1.2.... 14.. ... 1.55 1.54 1.53 1.52 1.52 1 51 .55 .54 .54 .53 1.53 1 6 1 8 2 0 .71 .70 69 .49 .49 48 .40 1.39 1.40 1.39 2 2 2 4 2 6 1.51 .68 .47 28 .68 .47 3 0 3 2. 3 4 36... 3 8 Grade. . (C) (D) 1.96 0.66 0.182 1. 0.363 00 Q 0.363 0.545 0.734 1.119 0.093 0.182 0.545 0.545 0.734 Parameters of adjusting equation S:::::: 0.24 93.4 1.62 0.20 155.7 1.50 0.17 245.1 1.34 0.14 438.3 1.59 0.19 13.2 1.80 0.14 37.2 1.55 0.08 0.17 115.2 32.8 1.51 1.67 0.12 87.9 1.49 0.10 129.4 1.68 0.08 174.6 1.52 S Values of ii. 0.2 2.52 2.54 2.12 1.90 1.81 1.75 1.72 1.6» 1.67 1.65 1.63 1.61 1.60 1.59 3 d 0.734 1.119 0.363 0.734 1.119 0.363 0.734 1.119 0.363 0.734 1.119 XT 0.36 64.5 1.69 0.28 85.0 1.78 0.58 23.6 1.85 0.41 65.7 1.70 0.33 115.5 1.63 0.71 15.39 2.37 0.50 62.3 1.71 0.40 114.5 1.54 0.80 19.38 2.01 0.56 52.9 1.72 0.48 75.1 1.67 n s Values of u. 0.2... | 1 .3 i 4 .5 g 3 23 3.63 .7.. 3.49 3 08 2.89 2 68 4.11 3.49 3.12 2.88 2.58 2.40 2.29 2.20 2.14 2.09 2.05 2.02 3.09 2.78 2.58 2.44 2.25 2.14 2.06 2.00 1.96 1.92 1.89 3.61 5.31 4.17 3.58 3.21 2.78 2.54 2.38 2.27 2.20 2.14 2.09 2.05 8 4.58 3.86 3.43 2.94 2.64 2.49 2.38 2.29 2.22 2.17 2.12 2.08 3.09 2.78 2.57 2.32 2.16 2.06 .99 .93 :S .83 5.75 4.56 3.92 3.23 2.88 2.65 2.50 2.40 2.32 2.25 2.20 2.16 2.12 .9... 2.82 2.65 2.42 2.28 2.19 2.12 2.07 2.03 1.99 1.96 2.54 2.43 2.29 2.20 2.14 2.09 2.05 2.02 2.00 1.98 1 0 1 2 3.57 3.15 2.90 2.71 2.60 2.51 2.43 2.38 2 33 6.03 4.69 4.02 3.62 3.35 3.16 3.01 2.90 2.81 1.4 1.6 1.8 3.91 3.67 3.50 3.36 3.26 3.18 3.11 2.0 .... 2.2.. 2 4 2.6. .. 2 8 3.0... 2.29 3.2.. 3.05 2 09 3 4 3 00 VARIATION OF THE INDEX. Each column of the table contains a set of values of tt which pertain to the same grade, fineness, width, and discharge and of which the changes are related to slope only. In figure 31 a number of these sets are plotted in relation to slope. The curves have a strong family like- ness, arising from the fact that the data were all adjusted by the sigma formula; but the likeness would not altogether disappear if the assumptions of that formula were abandoned. The general relations of -the index to slope are as follows: (1) It varies decreasingly with slope. (2) Its rate of change is greater for low slopes than for high. The upper group of curves all pertain to grade (C) and width 1.00 foot, but represent different discharges. They show (3) that the rate of change for similar slopes is greater for small discharges than for large. The second group of curves all pertain to grade (C) and discharge 0.363 ft.3/sec., but represent different widths. They show (4) that the rate of change for similar slopes is greater for broad channels than for narrow, or, as the depth varies inversely with the width, that the rate of change is greater for shallow streams than for deep. The third group of curves all pertain to width 1.00 foot and discharge 0.363 ft.3/sec., but represent different grades of debris. They show (5) that the rate of change is greater for coarse debris than for fine. In the third group the curves for grades (A), (B), (C), (D), and (E) lie close together, while those for the coarser grades (F) and (G) are well separated. This is probably connected with the fact that the range of fineness grad- ually increases from (A) to (E) and then drops abruptly from (E) to (F). The influence of increasing range approximately neutralizes that of decreasing fineness, and the inference is (6) that the rate of change in the index is greater for small range than for large. Consider now the variations of the index in relation to width. In figure 32 (p. 106) the ordinates, as before, represent values of \ and the abscissas represent width of channel. The points fixed by the data are shown by the dots. (7) The upper group of curves all pertain to grade (C) and discharge 0.182 ft.3/sec., but rep- resent different slopes. Their common char- RELATION OF CAPACITY TO SLOPE. 105 actor is a distinct minimum. From the neigh- borhood of width 0.66 foot there is increase of it in the direction of greater width, and also in the direction of less width. (8) The position of the minimum is appar- ently the same for low slopes as for high. (9) The minimum is most strongly marked in case of the gentler slopes. The second group of curves all pertain to grade (C) and a slope of 1.0 per cent but differ in respect to discharge. Each of them shows a minimum, except the curve for discharge Grade Grade (CJ 'O W-lft. W-lft 0. 363 ft.^ec. I Z Slope FIGURE 31.— Variations of i'i in relation to slope. 0.093 ft.3/sec., which has but two fixed points. They show also that— (10) The position of the minimum is related to discharge. For large discharges it is asso- ciated with relatively large widths, for smaller discharges with smaller widths. (11) The minimum is more pronounced, or the associated rates of change in the index are higher in case of small discharge than of large- Various analogies, which appear in another part of this paper, render it probable that all the preceding inferences are of a general char- acter; but those in regard to width are not sus- tained by all the data. The curves of the third group are based on observations with grade (B) and are drawn, like those of the first group, to contrast the relations of the index to width for different 106 TRANSPORTATION OF DEBRIS BY RUNNING WATER. slopes. With the gentler slopes they give indications of a minimum, but not with the steeper. The character of the discrepancy is such as to suggest that the values of the index computed for width 0.23 foot vary too rapidly with slope; and this result might be brought about by assigning too large a value to a. A critical review of the data, however, failed to find warrant for any material change in that constant. It is believed that a group of discrepancies which this instance illustrates are connected with the history of the experimental work. The first grade to be investigated was (B), and the methods of manipulation were subject to various minor changes, which were not always recorded; but the range of conditions was large. Grade (C) was next taken up, and again the range of conditions was large. Other grades followed, with less elaborate range of condi- tions; but the work on grade (G) was somewhat expanded, in order to learn the influence of coarser debris on various factors. The work on (G) also had the advantage of the fullest .66 1.32 1.00 Width FIGURE 32.— Variations of ii in relation to width of channel. 1.96 development of experimental method as well as that of uniformity of method. Because of this history it is believed that the results for grades (C) and (G) are of higher authority than those of other grades; and the belief is strengthened by the general symmetry and internal con- sistency of the (C) and (G) results. The infer- ences, given in preceding paragraphs, from data of grades (C) and (G) are therefore ac- cepted, and the discordance of data for grade (B), while not specifically explained, is ascribed in a general way to unrecorded differences in laboratory methods. The curves of the fourth group of figure 32 all pertain to grade (G) and discharge 0.734 ft.3/sec. but differ in respect to slope. Com- pared with the first and second groups they are seen to be consistent with the inference as to a minimum value of %, but the minimum falls outside the range of widths for which data were obtained. With grade (C) and discharge 0.734 ft.3/sec. the minimum falls between widths 1 foot and 1.32 feet, but nearer to the former. With grade (G) and the same discharge it ap- parently falls with some width less than 0.66 foot. This indicates that — BELATION OF CAPACITY TO SLOPE. 107 (12) The position of the minimum is related to fineness. For the finer debris it is associated with relatively great width; for the coarser, with smaller width. The curves of the fourth group support the ninth inference, that the minimum is most strongly marked for the gentlest slope. In the study of the data many other com- parisons of the influence of width were made, but they are not here illustrated. Their chief service was in indicating the comparative value of different divisions of the body of data. The general fact brought out — and one emphasized in various other ways — is that the measures of .093 .183 .363 .734 .545 Discharge FIGURE 33.— Variations of ij in relation to discharge. 1. 119 precision derived from discrepancies of observa- tions within a single series by no means cover the whole field. The discrepancies discovered when properties of different series are compared are quite as important and must be given con- sideration in connection with the broader gen- eralizations. Let us now consider the relations of the vary- ing value of ij to discharge. These are illus- trated by figure 33. The curves of the upper group all pertain to grade (C) and slope 1.0 per cent but differ in respect to width of chan- nel. Those of the second group pertain to grade (C) and slope 1.8 per cent; those of the third group to grade (G) and slope 1 .8 per cent. The general fact is that — (13) As discharge increases the value of it di- minishes. There are three exceptions, of which 108 TRANSPORTATION OF DEBRIS BY RUNNING WATER. two do not exceed the computed probable errors of the data, and the third is connected with a value of \ to which the lowest weight is ascribed. (14) The rate of change in the index is greater for small discharges than for large. (15) For the same discharges the rate of change in the index is greater for wide chan- nels than for narrow, and is therefore greater for shallow streams than for deep. FIGURE 34.— Variations of ii in relation to fineness of de"bris. The curves of the lowest group all pertain to width 0.66 foot and slope 1.8 per cent, but differ as to grade of debris. They indicate that— (16) The rate of change in the index is greater for coarse debris than for fine. The peculiarities of spacing, as in a previous in- stance, may show the influence of the factor of range in fineness within the several grades. To consider now the relations of the values of \ to the fineness of debris, the comparison is made with linear fineness — instead of bulk fine- ness, as in discussing a, and it is found con- venient to plot the logarithms of the quantities instead of the quantities themselves. In figure 34 the curves of the upper group are derived from experiments conducted with a trough width of 0.66 foot, and each one pertains to a particular combination of slope and discharge. Those of the second group are derived from ex- periments with a trough width of 1 foot. (17) In the main they show decrease of it with increase of fineness, but the finer grades give the opposite indication. The data are not sufficiently harmonious to determine whether the law of change is continuous or involves a reversal. If it is continuous, i{ is an inverse function of F. In view of the fact that the double variation of \ in relation to width is a complicating factor and of the further fact that that variation is less pronounced with high slopes than with low, two curves (the lowest group of fig. 34) were constructed from data pertaining to the highest practicable slope, 2.4 per cent. Each curve belongs to a particular width of trough, and each is a composite with respect to discharge. Their indication is practically the same as that of the other groups.1 The character of the material has not seemed to warrant a quantitative discussion of the variations of the index of variation, and a sum- mary of the qualitative discussion is neces- sarily limited to generalities. The index of relative variation or the sensitiveness of capac- ity for traction to change of slope is a decreas- ing function of the slope, the discharge, the fineness of debris, and the range of fineness and is a minimum function of width of channel. In symbols, »!=/$,& F,&,w) --- ..... (39) If we assume tentatively that the function re- placing it in the exponent is the product of functions of the individual conditions — that is, if we write then we must also recognize that in /,(£),/, is itself a function of Q, F, H, and w, that fu is a function of F and w, and that/v is a function of i Tn the data on flume traction the relation of capacity to fineness exhibits peculiarities quite analogous to those here found in the relation of ti to fineness. The capacity is larger for very fine and very coarse d<5bris than for intermediate grades. A tentative explanation (see Chapter XII) connects the larger capacity for fine debris with a tradi- tion in process from traction to suspension. RELATION OF CAPACITY TO SLOPE. 109 S, Q, and F. Parallel complexities would also arise if attempt were made to formulate the relations by means of such an expression as From equation (10) (p. 96), The sensitiveness of capacity to slope appears to be a function of the conditions jointly rather than severally. The development of complexity within com- plexity suggests that the actual nature of the relation is too involved for disentanglement by empiric methods, but that conclusion does not necessarily follow. Just as a highly complex mathematical expression may be the exact equivalent of a fairly simple expression of a different type, so a physical law may defy for- mulation when approached in a certain way yet yield readily when the best method of approach has been discovered. FORMULATION WITH CONSTANT COEFFICIENT. For the relation of capacity to slope the formula equivalent to (34) is <7=V 0.4 .6 2.26 1.97 1.97 1.97 1.97 1.97 1.97 1.97 2.49 1.99 1.78 1.66 1.58 1.53 1.49 3.01 2.26 1.93 1.76 1.64 .8 1.81 2.23 2.02 1.79 1.67 1.60 1.54 1.51 1.48 1.45 1.55 1.55 1.55 1.55 1.55 1.55 1.55 1.55 1.0 1.62 1.51 1.44 1.39 1.36 1.60 .60 .60 .60 .60 .60 1 60 1.86 .68 .56 .49 .44 .40 1.75 1.68 1.63 1.60 1.57 1.55 1.54 1.90 1.77 1.69 1.63 1.59 1.56 1.2 1.4 1.6 1.78 1.78 1.78 1 8 2.0 2.2 1.78 (Grade.. w Q:::::. 1.00 0.363 1.00 0.363 (EjG,) 1.00 0.363 1.00 0.363 (E,G2) 1.00 0.363 (A,C,G,) 1.00 0.363 (CDEFG) , Natural. 1.00 1.82 1.00 0.363 1.00 0.545 1.00 0.182 1.00 0.363 Parameters of interpolatian equation ••-•{•" 0.60 1.47 0 1.87 0.30 1.90 0.40 1.28 0 2.03 0.48 0 1.67 1.63 0 1.48 0 1.22 0.30 1.43 0 1.84 S Values of ii. 0.4 .84 .84 .84 .84 .84 .84 .84 .6 1.87 22 .8 .... 1.87 1.87 1.87 1.87 1.87 1.87 3.04 2.71 2.53 2.42 2.34 2.56 2.13 1.92 1.79 1.70 1.64 4.19 3.22 2.79 2.55 2.39 2.28 .63 .63 .63 .63 .63 .63 1.48 1.48 1.48 1.48 1.48 1.4$ .22 .22 .22 .22 1.22 1.22 2.29 2.04 1.90 1.82 1.76 1.72 1 0 3.68 2.94 2.58 2.36 2.21 2.03 2.03 2.03 2.03 2.03 2.03 1 2 1.4 ... 1 6 1.8 . ... 2.0 2.4 1 65 2.6 2.8 ! BELATION OF CAPACITY TO SLOPE. 115 The adjusted values of capacity may be found in Table 12, together with a variety of other data. The values of it are contained in Table 17, having been computed for the range of slopes covered by the observations. In order to compare these with the indexes of relative variation for the single grades from which the mixtures were made, Table 18 has been prepared, containing indexes correspond- ing to the uniform slope of 1.2 per cent. An analysis of this table shows that 3 mixtures are more sensitive to variation of slope than are components, 13 are less sensitive, and 18 show sensitiveness of intermediate rank. A general average shows the mixtures 1 1 per cent less sensitive than the (means of) components, but the contrast is much more pronounced for natural river debris and for the most complex artificial combination than for the simpler mixtures. TABLE 18. — Comparison of values of i^ for mixtures and their components. [S-1.2; w-l.OO.J Component grades. a Value of it when ratio of finer to coarser is — Finer. Coarser. 1:0 4:1 2:1 1: 1 1:2 1:4 0: 1 (A) C) .. 0.363 .363 .363 .182 .363 .363 .363 .363 I .182 { .363 [ .545 1 .182 \ .363 .81 .81 .65 .81 .55 .55 .79 .34 .81 .55 .44 2.02 1.81 1.34 1.55 3.57 2.54 1.85 1.79 3.57 3.57 3.57 (A) . G) 1.84 1.36 .84 .21 " .78 .51 .66 2.53 al.98 2.59 ol.60 1.68 1.93 1.92 2 79 o2.09 2.72 1.68 1.67 2.94 02.03 02.21 2.75 ol.55 1.77 (B) F) 1C) E) (C) E) 1.47 01.97 "1.87 (C) G) .. (E) G) ... ( A\ ir\ (G) Mixture of (C), (D), (E), (F), Natural combination ranging i ind (G) ol 63 ol.48 3.57 ol 22 1.90 3.65 2.54 ol 84 o For these, i under condition of uniform mode of trac- tion ( T, T, T), and under condition of uniform slope (S, S, S). Ver- tical distances represent i\, with a common scale but different zeros. Horizontal distances represent, for the upper pair of curves, discharge; for the second pair, logarithms of fineness; for the lower pair, width of channel; the values Increasing from left to right. greatly to be preferred to that of uniform slope as a basis for the extension of laboratory gen- eralizations to large natural streams. TABLE 20. — Relations o/i, to discharge, debris grade, and channel width, when conditioned (1) by a constant mode of traction and (2) by a constant slope. Grade, (C) w, 0.66 foot. w, 0.66 foot. Q, 0.363 ft.s/sec. Grade, (C) Q, 0.182 ft.'/sec. Q Values of n with— Grade. Values of ii with— w Values of ii with — Smooth traction. Constant slope 1.0 per cent. Smooth traction. Constant slope 1.4 per cent. Smooth traction. Constant slope 1.2 per cent. 0.093 .182 .363 .545 .734 1.79 1.70 1.59 1.59 1.58 1.87 1.73 1.59 1.58 1.57 (B) (C) (E) (G) H) 1.61 1.59 2.13 2.41 3.35 1.54 1.55 1.99 2.71 4.69 0.44 .66 1.00 1.32 1.88 1.70 1.81 2.20 1.81 1.70 1.81 2.27 IN CHANNELS OP SIMILAR SECTION. THE CONDITIONS. It will be shown in the following chapter that one of the important conditions affecting capac- ity is the relation of stream depth to stream width, or the form ratio E. The matter has, in fact, already received some attention in con- nection with the variation of a. Now, in each observational series the width is constant while the depth varies, so that the form ratio is a variable. Its variations accompany and are inseparable from those of slope; and the varia- tion of capacity (within an observational series), which up to this point has been treated as if it were purely a function of slope, is in reality a function of slope and form ratio jointly. To separate the two factors and there- by discover the relation of capacity to slope for streams of similar section, it is necessary to bring together data obtained by use of troughs with different widths, selecting points of two or more adjusted series which are characterized by the same ratio of depth to width. In every RELATION OF CAPACITY TO SLOPE. 117 such comparison the capacity and slope asso- ciated with the narrower trough are relatively large, while for the wider trough they are rela- tively small. The opportunities for comparison are not abundant, because in the main the observational series which are of like conditions except as to width do not overlap in respect to form ratio. SIGMA AND THE INDEX. In the records of the main body of experi- ments 24 cases of overlap are found, all asso- ciated with the finer grades of debris, from (A) to (D). In each of these the comparison in- cludes two widths only, no instance occurring in which it can be extended to three. There is, however, a special group of experimental series, planned in part for this particular pur- pose, in which the trough-width interval is so small that triple overlaps occur. The special experiments were made with debris of grade (C); the trough widths were 1.0, 1.2, 1.4, 1.6, 1.8, and 1.96 feet; and the experiments yield nine triple overlaps. In six of these the extent of overlap is such that numerical comparisons have been made for more than one value of R. With the aid of the computation sheets described on page 95, a table was compiled in which adjusted capacities and slopes were arranged with respect to form ratio, and in this table, which has not been printed, the matter of overlaps was canvassed. Table 21 contains the data involved in the triple overlaps. TABLE 21. — Selected data, for grade (C), shewing the relation of capacity to slope when the form ratio is constant. Q- 0.734 Q-0.923 Q-1.021 Q-1.119 1C S C w S C w S C 1C S C 0.07 .no 1.38 354 .SO 1.17 262 .96 .75 118 .08 .60 1.08 226 .HO .89 151 .96 .49 48 .09 1.60 1.29 413 1.80 .94 210 1.96 .58 88 .10 1.40 1.33 346 1.60 1.19 405 1.00 .72 102 1.80 .82 195 1.80 .55 57 1.96 .50 75 .11 1.40 1.06 224 1.40 1.44 522 1.60 1.07 380 1.60 .60 71 1.60 .83 185 1.80 .74 193 1.80 .45 35 1.80 .60 83 1.96 .50 88 .12 .. 1.40 1.17 357 1.60 .87 261 1.60 .69 130 1 SO 62 134 1.80 .50 56 1.96 .41 55 .13... 1.20 1.43 382 1.40 .96 243 1.40 1.16 380 1.40 .72 103 1.60 .58 94 1.60 .65 135 1.60 .44 37 1.80 .42 36 1.80 .47 58 .14 1.40 1.20 470 1.60 .60 130 1.80 .45 66 .15 1 40 99 331 1.60 .51 95 1 80 39 .16 1.40 .83 236 1.60 .44 71 1.80 .34 To illustrate the use made of such data, the case of #=1.119 ft.3/sec. and 5 = 0.15 is selected. In that example the values of slope and capacity are given for the trough widths 1.4, 1.6, and 1.8 feet. These values come from three adjusting equations, which are, for the widths severally, £=386 (S-0.08)1-72, (7=400 (S-O.IO)1-60, and (7=430 (S-0.12) 1-67. The graphs of the equations are shown in figure 37. On each graph is a dot indicating the point which corresponds to the tabulated values of C and S, and for each of these points the ratio R is 0.15. The curves represent the relations of C to S under the condition of uniform width. The three dots are points on an undrawn curve to express the relation of C to S under the condition of constant form ratio. 118 TRANSPORTATION OF DEBRIS BY RUNNING WATER. It is convenient, in discussing this undrawn curve, to assume that its equation involves a and is otherwise of the same type as the equa- tions used in discussing the data for constant width. In this case, moreover, the assump- tion is countenanced by the fact that traction is limited by the competent slope. By making the assumption, it is possible to compute all the parameters of the curve from the coordi- nates of the three known points and write its equation : £=469 (S -0.23) '-25 (43) In the present connection the most significant of the parameters is o\ and the values of a have been computed for each of the cases of Table 21. They are assembled in Table 22. TABLE 22. — Values of a corresponding to data in Table SI. R Value of » when Q is— 0.734 0.923 1.021 1.119 0.07... .08... .09 f ° \ o —.44 j""+."36" { +.33 +.31 +.04 . 10. . . .11... .12... .13... .14... / +.20 \ +.20 ........ /"+'." 6s" \ +.11 ""+."i3" I +.21 < +.24 1 +.24 .15 . 16 400 a to , in which vl is variable, the equation C=clSil, in which Cj is constant for each series of observa- tions. Formulation by means of ct and jl is more nearly analogous to the contour map, but the variability of the exponent jt is no less formidable than that of \ while the definition and derivation of y\ are less simple and its significance is less clear. Further utilizing the analogy of the map, we may think of the capacity-slope relation as an undulating topography, in which the vertical 0 f(C) element is -^ or s,c,\ and the horizontal ele- ments are qualifying conditions. Formulation is a mode of representing this topography, the hills and valleys of which do not depend on the mode but are real. Two modes have been tried, each with limitations, but the ideal mode is not known. The contour map or the relief model would serve admirably if the qualifying conditions were two only, but as they number at least four, a graphic or plastic expression is possible only in space of n dimensions. By reason of the complexity of the relation of capacity to slope and because of the lack of a mechanical theory of flow and traction, the laboratory data do not warrant inferences as to the quantitative relations of capacity to slope for rivers. Of various attempts to evade the complexity, two are thought worthy of record. In each series of experiments the mode of traction changes with increase of slope, first from dune to smooth, then from smooth to antidune, but the critical slopes are not the same for different discharges or widths or degrees of fineness. It appeared possible to gain in simplicity by treating separately the data associated with a single mode of traction, and data for the smooth mode were accordingly segregated and discussed. Greater simplicity was not found, but the range of variation is somewhat smaller for the single mode of traction. The second attempt was connected with the form of cross section of the current. Within a single series of experiments the width was con- stant and the depth varied, so that the capacity was conditioned not only by slope but by form ratio, R = ~. By comparing one observational series with another it is possible to obtain data conditioned by difference in slope, but without difference in form ratio. The discussion of such data developed only moderate modifica- tion of the results previously obtained and no reduction in complexity. DUTY AND EFFICIENCY. For the purposes of this paper duty has been defined as the ratio of capacity to discharge: n U= -Q. Combining this with C=f(S) , the most general expression for the relation of capacity to slope, we have U= Under no form dis- covered for/(£) is this expression reducible to simpler terms. For each value of discharge, duty is simply proportional to capacity; and the entire discussion of this chapter applies to duty as well as capacity. The parameters, n, \, ji, !«,, and a may be transferred, without modification, to formulas for duty. Efficiency has been defined as the ratio of capacity to the product of discharge by slope: -(44) The combination of this with C=v1S-1 .(45) The transformation of the exponent is im- portant. While capacity varies as the it power 122 TRANSPORTATION OF DEBRIS BY RUNNING WATER. of slope, efficiency varies as the i, — 1 power. The index of relative variation is 1 less for efficiency than for capacity. Table 15 could therefore be adapted to efficiency by diminish- ing all its values by unity. As the lowest known values of \ are greater than unity — the lowest in Table 15 is 1.31— it follows that effi- ciency is an increasing function of slope under all tested conditions. The combination of (44) with vields -(40) -(46) showing that in passing from the field of ca- pacity to that of efficiency the exponent asso- ciated with a constant coefficient also is reduced by unity. Following the form of equation (35), we have log c"- log a ,35 , 7' = logS"-logS'--- where C' and C" are specific values of capacity corresponding severally to the slopes S' and S". Designating by Ie the synthetic index of effi- ciency in relation to slope, and by E' and E" the efficiencies corresponding to C' and C" , we have "" log ff"-l E' log S' '-log S' .(35b) As C= ESQ, log C' = log E' + log S' + log Q, and log C"' = log E" +log S" +log Q. Substituting these values in (35a) and reducing, we have Subtracting the members of this expression from those of (35b) and transposing, we have /.-/,-!- (47) That is, the synthetic index of relative varia- tion of efficiency with reference to slope is less by unity than the corresponding index for capacity. It is evident that at competent slope, when capacity is zero, efficiency also is zero. Like capacity, it increases with increase of slope. Under the assumption that its law of increase is of the same type, its value varying with a power of (S — a), two expressions have been derived, but neither has been found reducible to simple form. The algebraic work being omitted, they are n — , — ?W— S-ir logS-2 log (S-»)-2 In the first of these expressions the coeffi- cient is variable, being a function of S; so that the exponent may be regarded as the index of relative variation for efficiency in relation to r* (S — a). As — s— falls to zero when S falls to the limiting value a, and as it approximates unity when S is indefinitely large, the values of the exponent He between n and n — 1 for all practical cases. In the second expression the coefficient is constant, with respect to slope, but the exponent is transcendental and intract- able. Thus it appears that the derived expression for efficiency as a function of S — a is not simply related to the coordinate expression for capacity and is not available for practical purposes; but it does not necessarily follow that the actual relation of efficiency to slope can not be formulated for practical purposes by an equation of the sigma type. All that is really shown is that if capacity and efficiency are both formulated in that way, the results are not consistent. Formula ( 10) was adopted for the capacity-slope relation, not because it expresses a demonstrated law of relation, but because it so far simulates the real law of rela- tion as to be available for the marshaling of the observational data. It seems quite possible that had the data been first translated from terms of capacity into terms of efficiency, the type of formula would have been found equally available. By way of testing the matter a few com- parative computations were made, observa- tional series being selected for the purpose from those which in the adjustment gave small probable errors. From the original data in Table 4 values of efficiency were computed, and these were plotted on logarithmic paper in rela- tion to S — a, the values of a being those em- ployed in the adjusting equations. In four of the nine cases treated the locus indicated was a KELATION OF CAPACITY TO SLOPE. 123 straight line; and the drawing of the line gave values of B and na in E = B (S-aJ"". In each of the remaining five cases the indicated locus was a curve, and the curvature was such as to indicate a larger constant in place of a. This larger constant, a1} was determined graph- ically, and the other parameters were com- puted as before. The results are given in Table 23a, together with comparative data from Table 15. TABLE 23a. — Comparison of parameters in the associated functions of capacity and efficiency, C=6,(S— a)n and E=B (S— a,)"". Grade. w Q 61 S • »i »i— » n "ll n-n,, (B) 0.66 0. 182 1 39. 8 196 0.10 0.10 0 .64 0.67 0.97 (B 1.00 .363 i 97.5 268 .08 .18 .10 .54 .56 .98 f! .66 .182 38.6 187 .11 .11 0 .54 .63 .91 r .66 .545 123 193 .06 .06 0 .48 .56 .92 o i.oo .363 100 280 .11 .31 .20 .48 .37 1.11 c 1.32 .363 100 240 .16 .22 .06 .40 .54 .86 .66 .734 64.5 78 .36 .56 .20 .69 .74 .91 o 1.00 .734 65.7 63 .41 .41 0 .70 .96 .74 (H) .66 .734 52.9 58.6 .56 .76 .20 .72 .80 .92 Mea .09 .92 The proximate inferences from these plots and comparisons are, first, that efficiency may be formulated, with sufficient accuracy for practical purposes, as proportional to a power of S — a1 ; second, that, when it is thus formu- lated, the approximate values of al are in gen- eral larger than the values of a obtained in the formulation of capacity; and, third, that the values of the exponent are smaller than the equivalent values for capacity, the differences usually being somewhat less than unity. The field of these inferences was also tra- versed by a mathematical inquiry, of which the results are more definite. If the relation of efficiency to slope be formulated by E=B(S-ol)n» -(47a) the exponent «„ is always less than n, but never so small as n— 1. For the range of conditions covered by the experiments, it is little greater than n — 1 . The value of at is always greater than the corresponding value of a, the differ- ence being usually small. The difference is greater when the value of the exponent is relatively small. Equation (47a) is incom- patible with the corresponding equation for capacity, (10). If the locus of E=f(S) be separately plotted by means of the two equa- tions, the resulting curves are not coincident, but they intersect at three points and lie close together elsewhere (in the practical field) unless the difference between a and al is large. On the whole, it appears entirely feasible to formulate efficiency by means of equation (47a). CHAPTER IV.— RELATION OF CAPACITY TO FORM RATIO. INTRODUCTION. Details of channel form in a natural stream are highly diversified. In connection with the bendings to right and left the current is thrown to one side and the other, with the result that the cross section is not, for the most part, sym- metric about a medial axis but shows greater depth on the side of the swifter flow. In the straight channels of the laboratory there was little departure from bilateral symmetry and the cross section was approximately rectangu- lar. For this reason those relations of traction to form of cross section which are found to ex- ist in the laboratory can not, in general, be in- ferred of natural streams. Nevertheless there is probably an approximate correspondence be- tween the two types when the tractional prop- erties of a broad, shallow channel are compared with those of a narrow, deep channel; and to that extent the discussion of form ratio CR=— ) w' is pertinent to the problems of natural streams. In connection with the study of the labora- tory data the form ratio is a factor of great im- portance, for not only is capacity for traction directly conditioned by it, but it affects every law of relation between capacity and another condition. In the discussion of capacity in relation to slope the effects which might have been referred to form ratio were treated instead as due to width, while small account was taken of the co- ordinate influence of depth. For many pur- poses the choice of viewpoint is indifferent, but when large and small channels are to be com- pared there is decided advantage in taking ac- count of form ratio. The form ratios of labo- ratory channels and river channels, for exam- ple, are of the same order of magnitude, but the widths are not. SELECTION OF A FORMULA. MAXIMUM. When identical discharges are passed through troughs of different width and are loaded with d6bris of the same grade, and the loads are 124 adjusted so as to establish the same slope, it is usually found, not only that the capacity varies with the width, but that some intermediate width determines a greater capacity than do the extreme widths. That is, the curve of capacity in relation to width exhibits a maxi- mum. The form ratio varies inversely with the width; and the same maximum appears when the capacity is compared with form ratio. The curves in figure 38, introduced to illustrate this fact, show data from Tables 12 and 14 for grade (C), with Q= 0.363 ft.3/sec. and £=1.0 percent. In the upper curve capacities are compared with widths; in the lower one the same capaci- ties are compared with form ratios. Width t • u 0 ^ O.I 0.2 Form rati o FIGURE 38.— Illustration of the relation of capacity to width of channel and to form ratio, when slope and discharge are constant. The formula for the discussion of such rela- tions must be one affording a maximum. It must also satisfy various physical conditions, as will presently appear. The explanation of the maximum, so far as its main elements are concerned, is not difficult. The phenomenon was in fact anticipated in the planning of the experiments, and certain courses of experimentation were arranged with special regard to the discovery of the form ratio of highest efficiency. Conceive a stream of constant discharge and flowing down a constant slope but of variable width. The field of traction is determined by the width, and the evident tendency of this factor is to make the capacity increase as the width increases. The rate of traction for each unit of width is determined by the bed velocity in that unit, and the bed velocity is intimately BELATION OF CAPACITY XO FORM KATIO. 125 associated with the mean velocity. Velocity varies directly with depth, and, inasmuch as increase of width causes (in a stream of con- stant discharge) decrease of depth, the tend- ency of this factor is to make capacity decrease as width increases. Velocity is also affected by lateral resistance, the retarding influence of the side walls of the channel. The retardation is greater as the wall surface is greater, therefore as the depth is greater, and therefore as the width is less. As capacity varies inversely with the retardation, and as the retardation varies inversely with width, it fol- lows that the tendency of this factor is to make capacity increase as width increases. Thus the influence of width on capacity is threefold : Its increase (1) enlarges capacity by broadening the field of traction, (2) reduces capacity by reducing depth, and (3) enlarges capacity by reducing the field of side-wall resistance. Now, without inquiring as to the laws which affect the several factors, it is evident that when the width is greatly . increased a condition is in- evitably reached in which the depth is so small that the velocity is no longer competent and capacity is nil. It is equally evident that when the width is gradually and greatly reduced the field of traction must become so narrow that the capacity is very small, and eventually the current must be so retarded by side-wall friction that its bed velocity is no longer competent and capacity is nil. For all widths between these limits capacity exists, and some- where between them it attains a maximum. The forms of algebraic function which afford a maximum are many; but no general examina- tion of them is necessary, because the physical conditions of the problem serve to indicate the appropriate type. As just observed, the varia- tion of form ratio (when discharge and slope are constant) involves simultaneous variations of width and depth. To develop an expression for the relation of capacity to form ratio, it is convenient first to determine separately the relations of capacity to width and to depth, and then to combine the two functions. CAPACITY AND WIDTH. To consider separately the response of capacity for traction to variation of width it is necessary to relinquish, for the time being, the assumption of constant discharge and variable depth, and substitute for it the assumption of constant depth and slope, with variable dis- charge. That is, we are to conceive a stream of constant slope, of which the width is pro- gressively increased or diminished and of which the discharge is varied in such way as to maintain a constant depth. Figure 39 repre- sents the cross section of such a stream, whether natural or of the laboratory type. Near the sides the current is retarded by side friction. Also, the freedom of its internal movements is restricted by the sides, just as it is everywhere restricted by the upper surface and the bed. These lateral influences diminish with distance from the sides and finally cease to be perceptible. We may thus recognize, in a broad stream, two lateral portions, AB, in which capacity is affected by the sides, and a medial portion, AA, in which capacity is not thus affected. In the medial portion total capacity is strictly proportional to the distance AA; or, in other words, the capacity per unit r S D A A. Z> B; ~~~~ ___— "'" FIGURE 39.— Cross sections of stream channels; to illustrate the rela- tion of capacity to width. distance, Ct, is uniform. In a lateral portion the capacity per unit distance diminishes as the side is approached. Whatever the law of diminution, the total capacity of a lateral por- tion is equivalent to the capacity per unit dis- tance in the medial portion, multiplied by some distance AD, less than AB. Therefore the total capacity for the whole stream is 2DB) . -(48) C=Cl(AA + 2AD) = Cl> It is evident that for a shallow stream the distances AB and DB are less than for a deep stream; and while the assumption may not be strictly accurate, it must be approximately true that DB is proportional to the depth. Making that assumption and introducing the numerical constant «, I replace 2DB by ad, and write As we are here concerned only with the law of variation of C, we may conveniently replace this by the proportion C*xw — ad. 126 TBANSPOBTATION OF DEBEIS BY BUNNING WATEB. Substituting for d its equivalent Cxw(l-aR). d As d is by postulate constant, and as w = -^ H w oc-g. We may therefore substitute -= for w in K ti the proportion above, obtaining 1-aR Cx- R -(49) This expression gives the relation of capacity to form ratio, so far as that relation depends on variation of width. Eventually it is to be complemented by an expression similarly dependent on variation of depth. The preceding analysis involves the assump- tion that the stream is so broad, in relation to its depth, that its medial portion is unaffected by lateral influences. The resulting proportion, (49), is not necessarily applicable to narrower streams. It is quite conceivable that when the channel is so narrow that the reaction of the sides affects all parts of the current the variation of capacity follows a different law. The analytic consideration of the case of narrower channels has not been attempted, but some information has been obtained from the experiments. The following examination of experimental data is directed toward this question and also toward that of the magnitude of the constant a. TABLE 24. — Relation of capacity for traction to width of channel, when slope and depth are constant. Grade. Slope (per cent). Depth (feet). Value of C when width (feet) is— 0.23 0.44 0.66 1.00 1.32 l.H« (B) (C) (B) and (C). 0.8 1.2 .8 1.2 0.8 and 1.2 0.08 .10 .12 .15 .20 .08 .10 .12 .10 .12 .14 .20 .10 .12 .14 .08 .10 .12 .14 .15 8.4 15 21.0 36.7 62.5 29 46 65 12.5 19.2 27 52 45 [i? [17.4] 26.6 38.0 60.0 113 50 74 103 22.0 34 51 111 80 115 156 20.4 35.4 57.5 112 43 66 95 145 8.6 15.2 32.0 2.8 88 156 H2 211 290 42 76 135 22.1 32.0 25.0 [5.3) 78 109 151 221 171 [300] 480 20.0 Geometric means. 15.6 29.6 36.1 43.1 29.4 50.7 62.5 78.2 42.4 75.2 78.0 115 158 211 \. 11.7 124 f The assumptions of the present section include constant slope and constant depth, with discharge and capacity adjusted to varia- tion of width. The experiments involve con- stant width and constant discharge, with automatic adjustment of slope and depth to variation of load. In order to check the analysis by means of the laboratory data it is necessary to employ some method of inter- polation. Two methods were tried, but only one need be described. Attention being first restricted to a par- ticular grade of d6bris and a particular slope of channel, the computation sheets (p. 95) for the different discharges were entered with a particular depth as argument, and the associated values of capacity and slope were taken out. These values were plotted on logarithmic section paper as a series of points. Through these points was drawn a curve — the locus of log (7=/(logS), under the condition that d is constant. By means of this curve values of C were interpolated, corresponding to selected values of S. The process was then repeated with other depths, other widths, and other grades ; and in this way were obtained sets of values of capacity in relation to width, under the condition of constant depth and slope. Such interpolated values of capacity are presented in Table 24. It was found that the data for grades (B) and (C) only are full enough to serve the present purpose. The tabulated capacities are also plotted, in relation to width, in the upper and second divi- RELATION OF CAPACITY TO FORM RATIO. 127 sions of figure 40. If the data were precise, ami if equation (48) were strictly accurate, the 300 zoo 100 400 300 200 100 200 100 GracLe Grade Means S) C) \ 2 FIGURE 40.— Capacity for traction in relation to width of channel, when depth and slope are constant. Scale of capacities, vertical; widths, horizontal. oblique lines of the figure would all be straight and would all intersect the line of zero capacity somewhere to the right of the origin. The irregularities of the lines are of such distribution as to indicate that they are occasioned chiefly by the imperfection of the data, and so far as may be judged by their inspection the formula is substantially correct. As the individual lines do not well indicate the points of intersection with the horizontal axis, a set of composites were prepared, each combining the data for a particular depth, without distinction as to grade of debris or slope of channel. In the computations for these a few interpolations were first made, and then the capacities were combined by taking their geometric means. The numerical results appear at the bottom of Table 24, and these are represented by dots in the lower division of figure 40. The indications of the dots were then generalized by drawing straight lines among them, and the intersections of these lines with the line of zero capacity gave points corresponding to D in figure 39. More strictly, the distance of each intersection from the origin gave an estimate of the quantity 2 DB in equation (48). As each estimate is asso- ciated with a particular depth, and as « = — -3 — , the intersections give also values of the con- stant oi in (49). d 2DB , but different values of m. The logarithmic equivalent of (54) is log (7= log &2 + log (1 - aR) +mlogR Differentiating, we have d log C=*d log (l-aR)+md log R Dividing by d log R, we have dlog C d log (1 - aR) d log R ~ d log R Making substitutions from dlog C d log (1 - aR) •• -adR " 1 - aR and and reducing, we have dR aR (60) BELATION OF CAPACITY TO FOEM RATIO. 131 It is evident that increases with increase of R; therefore i2 decreases with in- crease of R. Also i2 is positive when m > ^— — -& aR approached but not reached, the positive values of the index are all less than m. When aR = 1 , or R = -, the condition limiting traction I .2 .4- .6 FIGURE 45.— Relation of capacity, C, to form ratio, K. The variation of the function C— 6s (1— aR) R"> withflneness of debris. Scale of C vertical; scale of B horizontal. vary inversely with fineness, while Z>2 varies directly with fineness. The variations of m and p are more pronounced than that of a. TABLE 29. — Observed and computed quantities illustrating the influence of the fineness of debris on the relation of capacity to form ratio. Observational data. Constants of equations. Slope (per cent) and dis- Grade i, charge (ft.'/sec.). and fineness Width (feet). Form ratio. Capacity (gm./sec.). m f a !>, (Fi). 0.8 (B) 1.00 0.217 138 \ -0.09 0.734 13,400 1.32 .120 139 \ 0.20 0.16 1.04 242 + .06 1.96 .064 130 1 + .13 (C) .66 .440 107 - .48 5,460 1.00 1.32 .214 .134 130 133 1 •" .126 .85 191 - .10 - .01 1.96 .070 131 ] + .06 (G) .66 .540 16 -1.42 8.9 1.00 .263 13.3 (1.55 .44 1.38 170 +1.07 1.32 .164 7.9 +1.26 134 TRANSPORTATION OF DEBRIS BY RUNNING WATER. SPECIAL GROUP OF OBSERVATIONS. The special group of observations recorded in Table 4 (I), page 51, were arranged largely for the purpose of defining p, the optimum form ratio. They differed from the main body of observations in that the interval between the discharges employed and the interval between the widths employed were both smaller; and they were restricted to a single grade of debris. They had the advantage of an experimental method believed to be the best developed in the laboratory; and in view of this advantage their series were constituted of fewer individual ob- servations than those of the main body of ex- periments. The results have not been satis- factory, and attempts at formulation in the present connection have developed marked in- congruities. In figure 46 three curves are given, and the corresponding numerical data appear in Table 30. Each curve is based on five observational points, but they are so irregu- larly placed that their control is feeble. On comparing the two cases having the same discharge, it is seen that the greater slope is associated with the smaller values of m, p, and a and with the greater value of &2, the difference being most strongly marked for m and p. On comparing the two cases having the same slope, it is seen that the greater dis- 300- 300 S'l.l q '1.H9 FIGUBE 46. — Relation of capacity, C, to form ratio, R. Variation of the function C=b?(l— aK)Em with slope and discharge. Data from special group of experiments with debris of grade (C). Scale of C vertical; scale of R horizontal. charge is associated with the smaller values of m and p and with the larger value of 62, while the values of « are nearly equal. TABLE 30. — Observed and computed quantities illustrating the influence of slope and discharge on the relation of capacity to form ratio. Observational data. Constants of equations. Grade. Discharge (ft. "/see.). Slope (per cent). Width (feet). Form ratio. C m t> (T h * l! (C) 0.734 1.1 1.0 0.190 230 1 -0.25 1.2 .143 240 - .07 1.4 .108 240 \ 0.3 0.123 1.88 586 + .05 1.6 .079 231 + .12 1.8 .072 230 I + .14 1.119 .6 1.2 .216 114 - .72 1.4 .180 125 - .35 1.6 .140 131 .5 .129 2.58 546 - .06 1.8 .122 125 + .05 1.96 .102 130 + .15 1.119 1.1 1.2 .185 350 - .34 1.4 .145 400 - .28 1.6 .108 400 .1 .048 1.90 643 — .16 1.8 .091 418 - .11 1.96 .078 420 - .07 SUMMARY AS TO CONTROL BY CONDITIONS. The treatment of the observational data by means of a formula specially designed to show the relation of capacity for traction to the proportions of the cross section develops incongruities. These are of such distribution as to indicate that they are due in chief part to the observations and their methods of adjustment. Discrepancies which manifestly pertain to the data are so large that it is not practicable to determine whether the imper- fections of the formula are important. The exponent m varies inversely with slope, with discharge, and with fineness. Thus all the conditions which tend to increase capacity tend also to make capacity less sensitive to changes in form ratio. The optimum form ratio, p, varies inversely with slope and with fineness. As to its varia- RELATION OF CAPACITY TO FORM RATIO. 135 tion with discharge the evidence is not unani- mous; either it varies inversely under some circumstances and directly under others, or else its proper variation is inverse and data of contrary import are erroneous. The latter view is thought more probable, because in many other connections the controls of slope and discharge follow parallel lines. The constant ft, which represents the resist- ance of side walls or banks to the flow of the stream, is also a decreasing function of slope, discharge, and fineness. The constant &2, which is of the unit of capacity, varies directly with slope and in- versely with fineness, and the evidence as to its variation with discharge is conflicting. As 62 is the value of capacity when - aR) R™ = m+1 it corresponds to a rather complicated relation between E, m, and a, or R, m, and p; and this relation makes the interpretation of the lack of order among its tabulated values a difficult matter. If 62 is kft out of the account, it is possible to generalize by saying that the constants of equations (54) and (58) vary decreasingly wTith the conditions which affect capacity increas- ingly- (61) (62) THE OPTIMUM FORM RATIO. The ratio of depth to width which gives to a stream its greatest capacity for traction is of importance to the engineer whenever he has occasion to control the movement of debris. The title optimum ratio is especially appropriate when his desire is to promote that movement. The range of values for the ratio, under lab- oratory conditions, is from 1:2 to 1 : 20. One effect of this wide range, when taken in connec- tion with the variety of conditions by which the ratio is controlled, is to complicate the formulation of practical rules; but this diffi- culty is not insuperable. It is qualified to an important degree by the consideration that capacity, in the region of its maximum, changes very slowly with change of form ratio, so that an approximate determination of the ratio has practical value. The values of the ratio given in Table 31 are appropriate to the conditions of the Berkeley laboratory — that is, they pertain to troughs a few inches or a few feet wide, with smooth vertical sides. It is important to note also that they apply only to transportation of debris over a bed of debris, and not to flume traction, which has a different law. (See p. 213.) TABLE 31. — Estimated ratios of depth of current to width of trough, to enable a given discharge, on a given slope, to transport its maximum load. Material transported. Slope (per cent). Ratio for discharge (ft.8/sec.) of — 0.25 0.50 0.75 1.00 [ 0.5 ] 1.0 I 2.0 ( 1.0 \ 2.0 \ 3.0 1:4 1:6 1:10 1:6 1:9 1:16 1:8 1:12 1:20 1:25 1:3 1:4 1:9 1:15 Coarse sand or fine gravel 1:3 1:5 1:7 ""i:2" 1:25 1:3 No way has been found to extend the quan- titative results to rivers. It can hardly be questioned that the optimum ratio for rivers varies inversely with slope, discharge, and fineness of debris, but its absolute amount can not be inferred from the experimental results. River slopes are relatively very small and river discharges are relatively very large, and the two differences affect the ratio in opposite ways. To compute the joint result we should have definite and precise information as to the laws of dependence, but our actual knowledge is qualitative and vague. In this connection it is of interest to record a single observation on river efficiency. Where Yuba river passes from the Sierra Nevada to the broad Sacramento Valley its habit is rather abruptly changed. In the Narrows it is nar- row and deep; a few miles downstream it has become wide and shallow. Its bed is of gravel, with slopes regulated by the river itself when in flood, and the same material composes the load it carries. In the Narrows the form ratio during high flood is 0.06 and the slope is 0.10 per cent. Two miles downstream the form ratio is 0.008 and the slope is 0.34 per cent. Thus the energy necessary to transport the load where the form ratio is 0.008 is more than three times that which suffices where the form ratio is 136 TRANSPORTATION OF DEBBIS BY RUNNING WATER. 0.06; and it is evident that the larger ratio is much more efficient than the smaller. The data do not serve to define the optimum form ratio, but merely show that it is much greater than 0.008. In this instance the slopes and fineness are of the same order of magnitude as those realized in the laboratory, but the dis- charges are of a higher order. When water without detrital load is con- veyed by an open rectangular conduit, the form ratio of highest efficiency is that which yields the highest mean velocity. It is ap- proximately 1:2. This corresponds to the maximum value of the optimum ratio for trac- tion, and the correspondence might have been expected on theoretic grounds. The two fac- tors which, in ultimate analysis, determine capacity for traction are velocity of current along the bed and width of bed. When dis- charge and slope are such as barely to afford competence with the most favorable form ratio, that ratio is one giving the highest velocity, namely, 1:2. The other factor, width of bed, is evidently favored by lower values of R; and therefore, as the conditions recede from the limit of competence, the opti- mum form ratio becomes smaller. This line of reasoning might, in fact, have been used to show a priori — what has actually been shown by the experiments — that the value of p varies inversely with slope, discharge, and fineness. SUMMARY. Capacity for traction varies with the depth of the current, being approximately (though not precisely) proportional to a power of the depth. Capacity varies also with the width of the current, being approximately propor- tional to the width less a constant width. This constant width is equivalent to the prod- uct of the depth by a numerical constant. When the discharge is constant, any change of width causes a change of depth and also a change of form ratio, R = -. A formula for the variation of capacity in relation to form ratio, when the discharge is constant, is based on the above-mentioned properties and takes the form C=l2(l-aR)Rm ..(54) in which a is a numerical constant; or in which p is the optimum form ratio, or the form ratio giving the highest capacity. That capacity should have a maximum value corresponding to some particular value of form ratio is made to appear from theoretic con- siderations, and the fact of a maximum is shown by the experimental data. The same data show that the optimum form ratio has different values under different conditions, its values becoming smaller as slope, or discharge, or fineness increases. The sensitiveness of capacity to the control of form ratio is indicated in the formulas by the exponent of R, and that also varies with conditions. It becomes smaller as slope, or discharge, or fineness increases. It is believed that all the generalizations from the laboratory results may be applied to natural streams, but only in a qualitative way; the disparity of conditions is so great that the numerical results can not be thus applied. CHAPTER V.— RELATION OF CAPACITY TO DISCHARGE. FORMULATION AND REDUCTION. As a condition controlling the capacity of a stream for the traction of debris, discharge is the coordinate of slope. Each of the two fac- tors is proportional to the potential energy of the stream, on which traction depends; and the control of each is exercised through the control of velocity. Their fundamental difference in relation to traction is connected with depth of current. When velocity is augmented by in- crease of slope, the depth is reduced; when it is augmented by increase of discharge, the depth is increased. Notwithstanding this difference, the relations of capacity to discharge parallel those of capacity to slope to a remarkable extent. Thanks to this parallelism, the discus- sions of the present chapter may be based in considerable part on those which have preceded. The data for the comparison of capacity with discharge are contained in Table 12. In each division of that table assigned to a grade of d6bris are a number of subdivisions pertaining severally to particular widths of channel. Each column of such a subdivision pertains to a par- ticular discharge and contains a series of ad- justed capacities corresponding to an orderly series of slopes. The values of capacity con- nected with the same slope and comprised in the same subdivision constitute a group illus- trating the relation of capacity to discharge. Table 32 contains in its upper part a number of such groups, selected and arranged for the pres- ent purpose. So far as practicable they per- tain to the same slope, but it was not possible to secure absolute uniformity in this respect and at the same time make the representation in- clude data of all the grades of debris. For grades (A) to (E) the slope is 1.0 per cent; for grades (F) to (H), 1.2 per cent. TABLE 32. — Data on the relation of capacity to discharge, with readjusted values of capacity, Cr, and values of the index of relative variation, ia. {Grade (A) 1.0 .66 (A) 1.0 1.00 (A) 1.0 1.32 (A) 1.0 1.96 (B) 1.0 .23 (B) 1.0 .44 (B) 1.0 .66 (B) 1.0 1.00 (B) 1.0 1.32 (B) 1.0 1.96 Slope (p sr cent) . . reel) Width ( Data Q C C C C C C C C C C 0.093 .182 .363 .545 .734 1.119 [14.8] 39.5 7.8 13.3 13.2 28.7 10.8 33.5 81 120 37.5 100 36.8 104 30.1 85 143 199 29.3 79 140 204 96 67.6 120 190 313 140 231 250 240 359 ( "• I ft, .028 1.08 290 .048 1.05 370 .067 1.09 389 .105 1.00 367 .002 .79 49.5 .017 1.00 170 .033 1.13 270 .057 1.12 313 .080 1.06 313 .125 1.08 312 Readjusted capacities « Cr Cr Cr Cr Cr Cr Cr Cr Cr Cr 0.093 .182 .363 .545 .734 1.119 15.1 38.5 89 142 7.8 13.3 13.2 28.7 11.1 31.6 77 127 37.5 99 167 231 37.0 103 174 250 30.3 83 140 201 28.5 83 140 201 96 159 234 370 67.6 123 185 311 Index of relative variation Q fa fa fa fa fa is k >, k i> 0.093 .182 .363 .545 .734 1.119 1.54 1.28 1.17 1.14 0.81 .80 1.21 1.10 1.75 1.38 1.24 1.20 1.43 1.21 1.15 1.12 1.72 1.33 1.24 1.20 1.63 1.33 1.25 1.21 1.89 1.36 1.24 1.19 1.41 1.24 1.17 1.10 1.64 1.40 1.30 1.21 Probable error (per cent) / C \ Cr 4.8 2.4 1.4 0.7 2.4 1.2 1.4 0.7 137 138 TRANSPORTATION OP DEBKIS BY RUNNING WATER. TABLE 32. — Data on the relation of capacity to discharge, with readjusted values of capacity, CT, and values of the index of relative variation, % — Continued. {Grade (C) 1.0 .44 (C) 1.0 .66 (C) 1.0 1.00 (C) 1.0 1.32 (C) 1.0 1.96 CO) 1.0 .66 (D) 1.0 1.00 (D) 1.0 1.32 (E) 1.0 .66 Slope (per cent).. Width (feet) Data Q C C c C C C c C C 0.093 .182 .363 .545 .734 1.119 10.9 24.7 13.8 32.1 73 112 152 9.1 31.2 85 140 180 276 21.4 79 129 187 29.8 24.0 73 108 153 60 111 190 343 59.2 24.8 101 131 40 1 '• \ »J .020 1.02 157 .041 .94 213 .071 1.00 285 .099 1.05 305 .156 1.14 354 .059 .90 195 .102 .88 224 .143 .81 201 .108 Q Cr Cr Cr C, Cr Cr Cr Cr Cr 0.093 .182 .363 .545 .734 1.119 10.9 24.7 13.2 33.9 74 112 151 9.2 29.6 67 102 33.0 84 135 187 m 22.0 75 130 190 24.3 69 110 151 59 120 190 340 59.2 97 131 Index of relative variation Q 's (| k k is k d is is 0.093 .182 .363 .545 .734 1.119 1.30 1.14 1.68 1.21 1.06 1.02 1.00 2.46 1.64 1.24 1.15 1.11 1 07 2.30 1.44 1.28 1.21 1.33 1.07 1.01 2.00 1.22 1.08 1.02 2.00 1.60 1.45 1 32 1.33 1.10 1.01 / C \ c. 2.2 1.0 3.6 1.6 2.7 1.3 2.9 1.5 2.4 Probable error (per cent) 1.2 (Grade (E) 1.0 1.00 (E) 1.0 1.32 (F) 1.2 .66 (F) 1.2 1.00 (F) 1.2 1.32 (G) 1.2 .66 (0) 1.2 1.00 (G) 1.2 1.32 (II) 1.2 .66 Slope ( percent)., (feet) Width Eata « c C C C C C C C C 0.093 .182 .363 .545 .734 1.119 14.8 33.8 9.3 29.9 4.2 22.8 36.3 21.4 14.8 9.7 3.1 72 138 73 123 55.3 63 112 61.6 101 48.3 75 44.6 92 33.8 81 24.6 43.5 ! • i (>s .188 .263 .139 .242 .339 .199 .94 .345 .433 .233 1.24 .54 .83 Q Cr Cr Cr Cr Cr Cr Cr Cr Cr 0.093 .182 .363 .545 .734 1.119 15.1 4.2 46 76 22.5 46 Index of relative variation <3 is <3 is is is ll k fa l| 0.093 .182 .363 .545 .734 1.119 1 2.04 3.46 1 29 1.82 1.57 1 22 Probable error (per cent) 1 C 1 Cr BELATION OF CAPACITY TO DISCHARGE. 139 The same data are plotted in figure 47, where horizontal distances represent logarithms of dis- charge and vertical distances logarithms of ca- pacity. For each of the above-mentioned " groups " the plotted points are connected by a series of straight lines, and each of the broken FIGURE 47. — Logarithmic plots of the relation of capacity to discharge. The horizontal scale is that of log Q, the vertical of log C. The zeros of log C for the different plots are not the same. lines thus produced is the rough logarithmic graph of an equation C=*f(Q). The graphs are arranged according to grades of debris, and secondarily according to widths of channel, their order from left to right corresponding to the sequence from narrower to broader channels. In effecting this arrangement graphs were moved bodily up or down but not to the right or left. It appears by inspection that the graphs bend toward the right as they ascend. To this rule there are a few exceptions, but the only strongly marked exceptions are connected with grade (E), the data for which have previously been recognized as anomalous. As the incli- nation of each line indicates the sensitiveness of capacity to the control of discharge, the general bending to the right, or the reduction of inclina- tion in passing from lower to higher discharges shows that sensitiveness diminishes as dis- charge increases. This feature is similar to one observed in studying the relation of capacity to slope, and as that feature was found to be con- nected with competence, the resemblance leads at once to the suggestion that here also is a con- nection with competence. If a very small discharge be made to flow over a sloping bed of debris and the discharge be gradually increased, transportation of ddbris wiU commence when the competent discharge is reached and will increase with further in- crease of discharge. It is a plausible hypothe- sis that the capacity is more simply related to the excess of discharge above the competent quantity than to the total discharge. Follow- ing the procedure in the case of capacity and slope we may assume that capacity is propor- tional to a power of the excess of discharge above a constant discharge, the constant dis- charge being closely related to competent dis- charge. In the following formula, constructed on the plan of equation (10), C=13(Q-K}° ...(64) K is a constant discharge and &3 is a constant numerically equal to the value of capacity when the discharge equals x+1, although it is not strictly a capacity. The dimensions of capacity are M+l T'1, of discharge L+3 T'1, and of (Q-K)° L+3° T-°; and these values give to 63 the dimensions L'30 M+l T°~l. As a preliminary to the adjustment of the observational data of Table 32 by this formula, values of « were graphically computed by the method previously employed in connection with a. The computations were applied to all groups of values of capacity in the table, except such as comprise less than three capaci- ties and except also the aberrant data of grade (E). In Table 33 the results are arranged 140 THANSPORTATION OF DEBRIS BY RUNNING WATER. with reference to grade of debris and channel width. TABLE 33. — Values of the discharge constant, K, computed from the discharges and capacities of Table 32. Grade. Value of K when width of channel (in feet) is — 0.66 1.00 1.32 1.9C i C I,' ii 0.070 .095 .100 .130 .130 .160 0.075 .035 .030 0.215 .000 .060 0.056 .025 .060 .149 .280 .326 .180 The tabulated values of K show great irreg- ularity. It is not difficult, however, to recog- nize a tendency to grow larger in passing from finer debris to coarser; and there is also, though it is ill defined, a tendency to increase with increasing width. The cause of the first- mentioned tendency is readily understood, because relatively coarse debris requires a relatively swift current to move it; and the reality of the second also finds support when the conditions affecting competent discharge are considered. Postulating, initially, a channel of some particular width, containing a stream of which the discharge is barely competent, let us assume that the width is increased. In spreading to the new width the stream loses depth and velocity, and its velocity is no longer compe- tent. That the velocity may again become competent the discharge must be increased. Thus it is in general true that competent dis- charge increases with increase of width. In the cross section of a broad laboratory current, figure 48, a medial portion, AA, is unaffected FIGURE 48.— Ideal cross section of a stream in the experiment trough, illustrating the relation of competent discharge to width. by side-wall resistance and has competent velocity. Two lateral portions, AB, have less than competent velocity. Let us imagine these lateral portions replaced by narrower divisions, AD, in which velocities are the same as in AA. The effective width for the main- tenance of competent velocity is then DD = w — 2 BD. If now width and discharge be increased or diminished, with maintenance of competent velocity, the quantity 2 BD is unaffected, so that it may be regarded as a constant. Velocities being, by hypothesis, uniform through the whole space DD, the dis- charge is proportional to the width of that space. Qc , (68) DUTY AND EFFICIENCY. The variation of capacity with discharge is indicated in general terms by an equation of the type of (33) : ff-v.e*-.- (69) Duty being the quotient of capacity by dis- charge, this gives ^=-. %L = v3Qi*-1-- - (70) and, as efficiency is the quotient of capacity by discharge and slope, 3i - _? O'*-1 • S V (71) That is, the index of relative variation for both duty and efficiency, in relation to discharge, is less by unity than the corresponding index for capacity. Therefore the values of the index in Table 32 need only to be reduced by unity to apply to duty and efficiency. Under ordinary conditions the index for duty and efficiency falls between unity and zero ; or, in other words, duty and efficiency increase with increase of discharge, but their increase is less rapid than that of discharge. Exception- ally the increase is much more rapid, the excep- tions being associated with discharges little above the limit of competence. On the other hand, there appear to be conditions under which the index falls below zero, so that duty and efficiency diminish with increase of discharge. The diminution indicated by the figures in the column (of Table 32) for grade (D) and width 0.66 foot is only of the order of magnitude of the probable error; but a pronounced diminu- tion would be inferred from the values of the index for grade (B) and width 0.23 foot. As the results from the last-mentioned group of observations stand by themselves in various respects, some reservation is felt in regard to them, and there is at least room for doubt whether the diminution is actually demon- strated. With respect to all conditions the variations of the index for duty and efficiency follow the same laws as the index for capacity; but, as a consequence of the uniform reduction by unity, the rates of variation are higher. If, for ex- ample, in passing from a smaller to a larger dis- charge, the index for capacity falls from 1.40 to 1.20, a reduction of one-seventh, the index for efficiency falls from 0.40 to 0.20, a reduction of one-half. Lines of reasoning strictly parallel to those employed in the last section of Chapter III yield the following conclusions: The synthetic index of relative variation for the duty of water in relation to discharge is less by unity than the corresponding synthetic index for capacity in relation to discharge. The synthetic index of relative variation for efficiency in relation to discharge is less by unity than the corresponding synthetic index for capacity in relation to discharge. If the duty of water, or if efficiency, be as- sumed to vary as some power of Q — K, the ex- ponent of that power (expressing the instanta- 0 — K neous rate of variation) equals o — ~ • charge increases from K toward infinity, the exponent diminishes from o toward o — 1 . If the relation of efficiency to discharge ( and similarly for the relation of duty to discharge) be expressed by E^BAQ-Kj'n ---------- (71a) As dis- the value of on is always less than the corre- sponding value of o, the difference approaching but not exceeding unity. The value of /q is RELATION OF CAPACITY TO DISCHARGE. 145 always greater than the corresponding value of K, usually much greater. It was found by trial that, within the range of conditions real- ized in the laboratory, the difference between values of efficiency computed directly by means of (71a) and values computed indirectly by means of ( 64) is not large, its order of magnitude being that of the probable errors. The control of duty and efficiency by dis- charge is further considered in the following section. COMPARISON OF THE CONTROLS OP DIS- CHARGE AND SLOPE. CONTROLS OF CAPACITY. We are now in position to compare the in- fluences exerted by slope and discharge, sever- ally, on capacity. The general fact brought out by the comparison is that capacity is more sen- sitive to changes of slope than to changes of discharge, but the difference in sensitiveness is not the same for all conditions. TA BLE 36. — -Comparison of the index of relative variation.i, . for capacity and slope, with the index, i3./or capacity and discharge. [Foi1 grades (A) to (D) the data are for S— 1.0; for grades (G) and (II) for S— 1.2.] Grade. Q 10=0.44 w— 0.66 W=»1.00 w=1.32 w=1.96 »i i> ii is ll k ii k ii is (A) (B) (C) (D) (0) (H) 0.182 .363 .545 .734 1.119 .093 .182 .363 ..545 .734 1.119 .093 .182 .363 .545 .734 1.119 .093 .182 .363 .545 .734 .363 .734 1.119 .363 .734 1.119 1.99 1.28 2.08 1.83 1.43 1.21 2.17 1.70 1.72 1.33 1.90 1.41 1.48 1.14 1.79 1.12 1.71 1.20 1.55 1.31 1.17 1.10 2.31 1.81 1.21 1.10 2.34 1.82 1.58 1.49 1.75 1.38 1.24 1.20 1.92 1.67 1.62 1.61 1.63 1.33 1.25 1.21 1.94 1.75 1.73 1.71 1.89 1.36 1.24 1.19 2.01 1.87 1.66 1.60 1.64 1.40 1.30 1.21 2.39 1.88 1.30 1.14 1.88 1.73 1.59 1.58 1.57 1.68 1.21 1.06 1.02 1.00 1.87 1.59 1.47 1.45 1.48 1.64 1.24 1.15 1.11 1.07 2.37 1.67 1.55 1.47 2.30 1.44 1.28 1.21 2.14 1.87 1.62 1.85 2.00 1.60 1.45 1.32 2.22 1.80 2.46 2.33 2.01 1.69 1.87 1.65 2.00 1.22 1.08 1.02 2.11 1.33 1.64 1.01 1.85 1.01 2.98 2.42 2.29 6.03 3.23 2.78 2.04 1.29 1.22 3.46 1.82 1.57 | 1 1 In Table 36 values of the index of relative variation are brought together from Tables 15 and 32. The selection includes all such as cor- respond in respect to debris, trough width, slope, and discharge, with the exception of those of trough width 0.23 foot, which appear to be anomalous. There are 64 pairs of values. Of the 64 comparisons, 62 show capacity as more sensitive to slope, 2 as more sensitive to discharge. The two exceptional cases are from experiments with debris of grade (D) and with channel width 0.66 foot; and the data from those experiments were reexamined in search for an explanation of what seems an anomaly. No explanation was found, and, as the observa- tions are supported by the estimates of pre- 209210— No. 86 — 14 10 cision, it remains probable that there are real exceptions to the general rule. The means of the 64 values of \ and i3 are, severally, 1.93 and 1.42; and the ratio of the first to the second is 1.36. On the average, the sensitiveness of capacity to slope is one- third greater than the sensitiveness of capacity to discharge. • To ascertain the variation of the ratio -»* H with discharge, the values in Table 36 were specially grouped for the taking of partial averages. The first group gave comparative ratios for discharges of 0.093 and 0.182 ft.'/sec., by means of four sets of index values, each set agreeing as to all conditions other than dis- 146 TRANSPORTATION OF DEBRIS BY RUNNING WATER. charge. The second group gave comparative ratios for discharges of 0.182, 0.363, and 0.734 ft.3/sec., by means of five sets of index values; and two other groups made other comparisons, as shown in Table 37. The upper division of the table gives mean values of it and ia; the that within each group the values of the indexes decrease as discharge increases, while the values of the ratio, as a rule, increase with the increase of discharge. To the first rule there are no exceptions; the exceptions to the second are not so important as to leave the principle in doubt. lower division, their ratios. It will be observed TABLE 37. — • Variations of the indexes it and 13, and their ratio, in relation to discharge. Croup. Sets. Q-0.093 Q-0.182 Q-0.363 Q-0.543 Q-0. 734 Q- 1.119 ii b ii 13 ii I, 1] is (l is ii is 1 2 3 4 4 5 g 7 2.23 1.48 1.81 2.02 1.21 1.74 1.73 1.76 2.67 1.30 1.41 1.97 1.65 1.16 1.69 1.26 1.59 1.97 1.19 1.32 1.85 1.22 ii/t. «!/>! >:/•» 8 Wfc ii/i> 1 2 3 4 1.50 1.50 1.16 1.33 1.25 1.35 1.42 1.34 1.49 1.35 1.52 The variations of the indexes with discharge as a factor controlling capacity, is more pro- have already been illustrated in another way nounced for large discharges than for small, (pp. 107, 143). The new fact brought out is under like conditions of slope, width, and that the superiority of slope over discharge, fineness. TABLE 38. — Variations of the indexes it and i3, and their ratio, in relation to width of channel. Group. Sets. 10-0.44 tc-0.66 w-1.00 10-1.32 w-1.90 I'l {3 ii k I] k h h il 13 1 2 3 4 8 8 2.10 1.19 1.94 1.67 1.50 1.17 1.71 1.63 1.35 1.20 1.83 1.66 1.55 1.28 1.83 1.50 Ji/ii ii/is iilh ii/is ii/ij 1 2 3 1.77 1.29 1.40 1.27 1.36 1.18 1.30 1.22 A different grouping of the index values, but similar in principle, gave the means and ratios of Table 38, which is related to channel width just as Table 37 is related to discharge. With a single exception, the mean values of indexes increase with width; thus illustrating general facts previously noted on pages 104 and 143. Without exception, the ratios of ij to is decrease with increase of width. The new fact brought out is that the superiority of slope over discharge, as a factor controlling capacity, is more pronounced for narrow chan- nels than for wide, under like conditions of slope, discharge, and fineness. As form ratio varies inversely with width, it follows that the superiority of slope is more pronounced, under like conditions, when form ratio is large than when it is small. A third grouping of the index values, making a similar comparison of their averages and ratios with fineness of debris, is reported in Table 39 (p. 147). The mean values of indexes increase on the whole in passing from finer to coarse grades, but there is much irregularity. The same irregularity was encountered when these relations were examined in other con- nections. (See pp. 108 and 143.) The ratios decrease on the whole from finer to coarser, and there is but one discordance among the sequences. To compare the variations of the indexes with changes in slope, the 32 pairs of values of RELATION OP CAPACITY TO DISCHAKGE. 147 ia mentioned on page 143 were contrasted with of the three groups indicates that the ratio corresponding values of i,. The results are of the indexes increases with increase of summarized in Table 40, below, in which each slope. TABLE 39. — -Variations of the indexes it andi3, and their ratio, in relation to fineness of debris. (J 0 (I ») (< ') (] >) « '•) (I r) 1] fa ii i, ll ij ii h ll !3 ll k 1 11 1.77 1.28 1.74 1.39 1.76 1.45 2 8 1.79 1.38 1.65 1.32 1.87 1.54 3 3 2 70 1 66 4 63 2 64 I'l h !l ia til '3 111 k III k ill ii 1 1. 38 I. 25 1. 21 2 1. 30 1. 26 1. 21 3 1 59 1 76 TABLE 40. — Variations of the indexes it and tj, and their ratio, in relation to slope. Group. Sets. S-0.5 S-1.0 S-1.2 S-2.0 S-2.4 ll fa ii fa ll is ii fa il ia 1 2 3 18 10 4 1.90 1.56 1.6S 1.87 1.23 1.45 1.66 1.12 2.82 1.47 2.13 1.08 ll/fa ill's Jl/fa ii/ia ii/is 1 2 3 1.22 1.35 1.29 1.49 1.91 1.98 To sum the results of the preceding para- graphs: The sensitiveness of capacity for trac- tion to changes of slope, as measured by the exponent ilt is in general greater than its sen- sitiveness to changes of discharge, measured by ig. The superiority of the control by slope persists through nearly (or perhaps quite) the entire range of conditions realized in the laboratory. If the superiority is measured by m the ratio ^, its average value (based on 64 com- ** parisons) is 1 .36, and it increases with increase of slope, discharge, form ratio, and fineness of debris. (72) Another mode of comparing the controls by slope and discharge is by means of the syn- thetic index of relative variation (p. 99), and in some respects this mode is more satisfactory than the one given above. The synthetic index, 73, of the relative variation of capacity and dis- charge was computed for 21 cases, in each of which the greater discharge was approximately double the lesser. The synthetic index /„ of the relative variation of capacity and slope, was computed for 21 pairs of cases, the greater slope being double the lesser. Each value of 7S was joined to two discharges and a slope and associated with a pair of values of /,. Each of the two values of 7t was joined to one discharge and two slopes, the slopes being so chosen, whenever possible, that their mean coincided with the slope of the associated 73. The mean of the 42 values of 7t is 1.92; that of 21 values of 73 is 1 .42 ; and the ratio of these means is 1 .35. The result is practically identical with that obtained by the discussion of values of it and i3. CONTROLS OF DUTY. The index of relative variation of the duty of water in relation to slope (p. 121) is ilr the same as the index for capacity and slope. The corresponding index for the duty of water in relation to discharge (p. 144) is i3— 1. The rt ratio of these indexes, - — —, is evidently greater i3 — n than the ratio -r, which has just been consid- 148 TRANSPORTATION OF DEBRIS BY RUNNING WATER. ered — that is, the superiority of the control by slopes, as compared with the control by dis- charge, is more strongly marked in the case of duty than in the case of capacity. i. 1.93 For general averages, - = r-j-= 1.36, and . *t _L93^1.3. ^ In Table 41 the values of . _^ ^ correspond to those of t1 in Tables 37 to 40— that is, they are based on the partial means of those tables and are arranged under the same groups, the purpose being to show how the dominance of control by slope, as expressed by a ratio, varies with certain conditions. TABLE 41. — Variations of the ratio . ' in relation to dis- charge, width of channel, fineness of debris, and slope. Values of ^—. for group — 1 2 3 4 Q- 0. 093 .182 .363 .545 .734 1.119 w— 0. 44 .66 1.00 1.32 1.96 Grade (A (B (C (D (0 (I) S- 0.5 1.0 1.2 2.0 2.4 4.6 8.6 2.7 5.8 4.3 6.5 2.8 10.3 8.4 6.1 8.4 11.2 3.9 9.8 4.9 3.3 8.1 5.9 3.7 6.3 4.5 3.9 8.0 8.1 3.5 1 4 9 3.1 3.4 7.3 4.2 6.0 13.8 26.6 In comparing this table with the tables of -^, the most conspicuous feature noted is that the variation of — with all conditions is much more pronounced than the variation of A *3 Associated with this is the fact that the excep- tions or apparent reversals observed in Tables 37 and 39 are not repeated in Table 41. In verbal generalization of the tabulated results it is to be borne in mind (1) that the alphabetic order in which the grades are arranged is the order from fine to coarse, and (2) that variation with respect to form ratio is the inverse of variation with respect to width of channel. The general fact is that the dominance of control by slope, as com- pared with control by discharge — a dominance always pronounced — is notably increased by increase of slope, discharge, fineness, or form ratio. (73) CONTROLS OF EFFICIENCY. The index of relative variation of efficiency in relation to slope (p. 144) is il— 1; and the corresponding index for efficiency in relation to discharge is i3—l. The ratio of these indexes n "| n ^r is greater than •=*, the corresponding ratio in the case of capacity, with exception of those doubtful cases in which il= 0. 44 .66 1.00 1.32 1.96 Grade (A) (B) (C) (D) (0) (H) S= .05 1.0 1.2 2.0 2.4 1.4 2.4 1.8 2.7 3.1 1.7 4.1 3.0 3.9 5.5 1.9 3.9 2.0 1.5 3.1 2.4 1.7 2.7 1.9 1.7 3.6 2.0 1.6 3 0 2.4 1.6 2.9 1.9 3.9 . ..1 fi-fi 14:1 EELATION OF CAPACITY TO DISCHARGE. 149 i — 1 In Table 42 the values of -^ — =- correspond to *3~ n those of . • 1 ,- in Table 41 and are similarly \~ derived from data of Tables 27 to 40. The pur- pose of the table is to show how the dominance of control by slope, aa expressed by a ratio, varies with certain conditions. A comparison of tabulated values for the several ratios shows that the ratios associated with efficiency vary with conditions more rap- idly than those associated with capacity, but less rapidly than those associated with duty. In Table 42, just as in Table 41, there are no exceptions as to the direction of the trend of variation. Bearing in mind that the alphabetic order in which the grades are arranged is the order from fine to coarse, and that variation with respect to form ratio is the inverse of variation with respect to width, we see that the general fact shown by the table is that the dominance of control by slope — a dominance always pro- nounced — is notably increased by increase of slope, discharge, fineness, or form ratio. - (74) SUMMARY. With debris of a particular size and a chan- nel bed of a particular slope, there is a particu- lar discharge which is barely competent to cause transportation. With increase of dis- charge above this barely competent disc harge, there is a proportional addition to the stream's potential energy. The relation of capacity to discharge is formulated on the assumption that the capacity is proportional to some power of the added energy, and therefore to the same power of the added discharge. As each grade of debris is somewhat heterogeneous as to the size of its grains, this assumed principle can not be applied strictly; the practical assumption is that capacity varies with a power of the differ- ence between the discharge and a constant discharge, the constant being so chosen as best to harmonize the data. By means of such formulation the data were readjusted and the rate of variation of capacity with discharge, or the index of relative varia- tion, i,, has been computed for a variety of conditions. It is found to be greater as the slope of channel, the discharge, the fineness of debris, and the form ratio are less. The aver- age of the values computed lor laboratory con- ditions is 1 .42 and the ordinary range is from 1.00 to 2.00. The rate at which the efficiency of the stream and the duty of the stream's water vary with discharge is denoted by an index which is less by unity than that for capacity. Its average is 0.42 and its ordinary range is from 0 to 1.00. It has previously been shown that the corre- sponding indexes showing the relation of capacity to slope are larger. In other words, ' capacity is more sensitive to changes of slope than to changes of discharge. If relative sensi- tiveness to the two controls be expressed by a ratio, the average value of that ratio is 1.36. The ratio varies with conditions, being rela- tively large when slope, discharge, fineness, and form ratio are relatively small. The primary adjustment of observations of capacity, described in Chapter II, was an ad- justment with respect to slope. The probable errors computed from differences between ad- justed and unadjusted values were influenced by only a portion of the observational errors. In readjusting values of capacity with respect to discharge, another division of the observa- tional errors was encountered and its import- ance was estimated. The probable errors computed from the results of the two adjust- ments are believed to represent with sufficient approximation the order of precision of the ad- justed values of capacity, which constitute the main body of data for the discussions of the report. The order of precision is expressed by saying that the average probable error of the adjusted values is a little more than 3 per cent. CHAPTER VI.— RELATION OF CAPACITY TO FINENESS OF DEBRIS. FORMULATION. To study the laws affecting the control of ca- pacity for traction by fineness of debris, capaci- ties should be compared which are subject to like conditions in all other respects. For this purpose data from Table 12 were arranged as in Table 43, where the capacities in each hori- zontal line are conditioned by the same slope, discharge, and width of channel. Ah1 the data of that table pertain to a slope of 1.0 per cent; but similar tables were constructed for slopes of 0.5, 0.7, 1.4, and 2.0 per cent. The same data were also plotted on logarith- mic paper; and, after a preliminary examina- tion, five sets were selected for special investi- gation. The plots of these appear in figure 50, where ordinates are logarithms of capacity and abscissas are logarithms of linear fineness. It is to be noted that the zero of log C is not the same for the different graphs. The graphs were moved up or down, so as to avoid confusion through intersection. The first law illustrated by the plots is that capacity increases as fineness increases; the second, that it increases more rapidly for small fineness than for great fineness. Despite ir- regularities of the data it is evident that the locus of log C=f (log F) is a curve, and that the function has a general resemblance to those fjfj (G> (F) (B)fA) FIGUKE 50.— Logarithmic plots of capacity for traction in relation to fineness of de'bris; corresponding to data in Table 44. found in comparing capacity with slope and discharge. TABLE 43. — Values of capacity for traction, arranged to illustrate the relation of capacity to grades of debris. Conditions. Value of C for grade— S w Q (A) (B) (C) (D) (K) CD (G) (H) 1.0 0.66 1.00 1.32 1.96 0.093 .182 .363 .545 .734 1.119 .093 .182 .363 .545 .734 1.119 .093 .182 .363 .545 .734 1.119 .093 .182 .363 .545 .734 1.119 10.8 33.5 81 120 13.8 32.1 73 112 152 9.1 29.8 39.5 24.8 20.5 8.5 140 101 40 39 30.4 49 12.8 25.2 37.5 100 30.1 85 143 199 31.2 85 140 180 276 24.0 73 108 153 14.8 33.8 14.0 231 72 138 42.8 79.5 26.9 61 36.8 104 29.3 79 140 204 21.4 79 129 187 59.2 36.3 250 131 73 123 40.2 70 18.9 52.2 96 67. « 120 190 313 60.0 111 190 343 240 359 150 KELATION OF CAPACITY TO FINENESS OF DEBRIS. 151 For any velocity, as determined by slope, discharge, and width, there is a competent fineness, marking the limit between traction and no traction ; and to this extent, at least, the relation of fineness to traction is analogous to the relations of slope and discharge. It is not easy to carry the analogy further, because slope and discharge are conditions of active force, and fineness is a condition, of reactive force, or resistance; but an experiment in formulation reveals a parallelism quite as striking as that between the capacity-slope and capacity-dis- charge functions. Assuming -(75) in which 0 is a constant fineness and bt a con- stant of the numerical value of capacity when F= <£ + !,' the five sets of data in Table 44 were treated graphically for the determination of and p. The methods were such as have already been described (pp. 65 and 139). The formulas were then used to compute read- justed values of capacity, Cr, and values of the index of relative variation, it> and probable errors were also computed. TABLE 44. — Numerical data connected with the plots in figure SO, and illustrating the relation of capacity for traction to grades of debris. 1 2 3 \w Conditions of experiments 41 0.62 4.5 55 0.60 2.4 49 0.58 3.7 p f C 6.8 2.8 7.6 2.9 4.9 1.9 C C, tt ft C ft '. ii C Cr i« ii Grade (\) 121 110 106 85 40 20 8 140 123 102 76 42 21.7 7.4 0.63 .64 .66 .70 .87 1.42 5.45 1.83 1.66 1.50 1.72 1.68 2.35 4.11 185 149 143 127 62 33.4 16.3 196 173 143 108 62 34.4 16.3 0.61 .62 .63 .66 .80 1.18 2.80 1.79 1.63 1.53 1.63 1.78 2.42 3.15 Grade(B). Grade (C). 137 145 0.65 124 123 . 67 .69 1.54 1.55 Data and computed results .... Q^e ( E) Grade (F)! Grade (G). Grade (H). 49.4 59 .81 41 37 1.08 23 22 1.82 6. 9 7. 1 13. 50 2.10 1.99 2.71 4.69 1 4 \w Conditions of ex periments < Q Is 1.32 0.734 1.0 1.32 0.363 2.0 Parameters of equations f* fe::::: 48 0.61 3.8 55 0.50 11.4 1C (Or 1.7 0.6 6.2 2.4 C C, i. ii C C, i, k Gra Gra Gra le(A). le(B). 3e (C) . 3e(D). ie(E). le(F). ie(G). ie(IJ).. 250 204 187 131 73 40.2 18.9 250 219 180 133 74 40.2 19 0.64 .65 .66 .70 .84 1.22 5.20 1.71 1.71 1.47 1.85 1.54 2.40 3.43 328 255 233 237 112 66 28.3 319 285 242 190 115 67 27.4 0.53 .54 .55 .58 .72 1.19 4.55 .97 .63 .52 .92 .60 2.03 3.67 Gra Gra Gra PRECISION. The average probable error of the read- justed capacities was found to be ±2.1 per cent. This error is to be ascribed in part to discordance of the data among themselves, and in part to discordance of the formula with the data; but the distribution of the residuals is not such as to imply important discordance of the formula. i The dimensions of capacity are -M+i T-', and of fineness £-*. The constant 6(, being equal to C/ (F—*t>)p, hag dimensions L+'P J/+' T~l. 152 TRANSPORTATION OF DEBRIS BY RUNNING WATE&. There are, however, important discordances among the data. Considered as errors, the discordances constitute a group which were not detected in the adjustments of capacities to slopes and to discharges, but which escaped those tests because related peculiarly to the grades of debris. From the residuals of the present readjustment the average probable error of capacities before readjustment is esti- mated at ±5.4 per cent, whereas the average probable error of the body of once-adjusted capacities from which these were selected was estimated at 2.5 per cent. On the assumption that the estimate for the whole body of values applies to the selected group, the share of error associated with the grades is estimated as V5.42-2.52= ±4.8 per cent. Inspection of the logarithmic plots suggested that part of the discordance of the data is syste- matic. To bring out the systematic element the original values of capacity in Table 44 were divided by the readjusted values, and means were taken of the quotients. The means are listed below, and in figure 51 they are plotted logarithmically in relation to fineness. The plotted points conspicuously out of line are those for grades (B) and (D), the capacities determined for grade (B) being relatively too small and those for grade (D) too large. The same result was obtained from a canvass of a wide range of data. Ratios oj original values of capacity to adjusted values. Grade . .. (A) (B> (C) (D) (E) (F) (G) 1 002 812 602 388 178 95.9 43.4 0.96 0.89 1.01 1.13 0.98 0.97 1.02 It is surmised that these errors arise in part from variations of experimental method, and this suspicion attaches especially to grade (B), which was the first to be treated in the labora- tory. It attaches much less to grade (D), for (G) (E) CD) (C) CB) (A) FIGURE 51.— Average departures of original values of capacity from the system of values readjusted in relation to fineness of d. VARIATIONS OF THE CONSTANT . The laws which control the variation of 0 have not been developed from the observations, but their general character may be inferred deductively by considering the relations of com- petent fineness to various conditions — it being assumed that is intimately related to com- petent fineness. Postulate a current of which the velocity is determined by a particular slope, a particular discharge, and a particular width. For this current a certain fineness is competent. Increase of slope or discharge increases the velocity and makes a lower fineness competent. Decrease of width, which corresponds to in- crease of form ratio, increases velocity and makes a lower fineness competent. Thus com- petent fineness, and therefore , varies inversely with the slope, discharge, and form ratio. A) ________ --(76) capacity and fineness. The formula for the index (cf. pp. 100 and 141) is -(78) INDEX OF RELATIVE VARIATION. Framing an equation of the type of (33) — C=viF{< ............. (77) in which it is the index of relative variation for With this formula, values of it were computed from the data in Table 44, and they are given in the lower part of that table. By inspection it appears that the index in- creases as fineness diminishes, its growth being at first slow but becoming rapid as competent fineness is approached. Because of the dis- cordances of the data it is not easy to derive a body of values of the index for discussion in relation to other conditions, but it is relatively easy to obtain comparative values of the syn- thetic index, It, and the variations of these values may be assumed to show the same trends as the variations of it. Values of 74 were com- puted between corresponding data of grades (C) and (G) by the formula j log Cl- -logtf- in which (7, and Ca are specific capacities corre- sponding to the finenesses F, and Fn; and the results are given in Table 45. From these results it is inferred (1) that the index varies inversely with the slope, (2) that it varies inversely with discharge, and (3) that it varies directly with width, and therefore inversely with form ratio. The response is in general of a very pronounced character, but to this there is exception in one of the compari- sons with width. It is possible that the index is a maximum function of width and a mini- mum function of form ratio. With some reser- vation on this point, we may generalize: 1, P, &)-. -.(79) If equation (79) be compared with equations (39) and (68), it will be seen that the variation of the capacity-fineness index observes the same laws of trend as the variations of the capacity- slope index and the capacity-discharge index. In view of this general parallelism of variation, it is thought that the relative magnitudes of average it, average i,, and average i3 may be adequately discussed by means of a moderate number of comparisons. Accordingly only those values of it computed from the five equa- tions of Table 44 are used. The corresponding values of i3 are not available, but those of i, 154 TRANSPORTATION OF DEBRIS BY RUNNING WATER. are given in Table 15. They have been copied into Table 45, so as to be conveniently com- pared. Inspection shows that it is in general much smaller than \ but that it becomes greater as the limit of competence is approached. As to the first of these generalizations there can no question, but the second is not equally satis- factory. In the vicinity of competence the value of it is highly sensitive to the influence of ; and in the same region \ is highly sensi- tive to a. The features of the table might be produced by slight overestimates of or by slight underestimates of a. In view of this consideration it is probably best to leave the higher values of the index out of the account and base a computation of averages wholly on the lower values. Including only the 28 values of each index associated with grades (A) to (F), the means are, for it 0.77, for it 1.79; and the ratio of the second to the first is 2.4; that is, the sensitiveness of capacity to control by slope is estimated to be 2.4 times as great as its sen- sitiveness to control by fineness. The ratio of sensitiveness for slope and discharge iji3 having been estimated at 1.36, it follows that the ratio 2 4 for discharge and fineness is =-55 — 1.8. Mean it : mean is : mean it : : 2.4 : 1.8 : 1.0. It is to be understood that these estimates are of the most general character. The ratios doubtless vary notably with conditions. The property with which capacity has been compared in this chapter is linear fineness, F, defined as the reciprocal of diameter, or as the number of grains to the linear foot. Bulk fine- ness, F2, defined as the reciprocal of volume, is proportional to the cube of linear fineness. It follows that the index of relative variation when capacity is compared with bulk fineness is one-third the corresponding index, it, for capacity and linear fineness; and the same factor applies to synthetic indexes. If bulk fineness were substituted for linear fineness in equations of the form of (75), the values of would be quite different and the values of p would be uniformly one-third as great. TABLE 45. — -Values of It, the synthetic index of relative variation for capacity and fineness, compared with slope, discharge, and width of channel. Fixed conditions Coarser grade (G) (C) (G) (C) if (G) & (G) (C) 2.0 0.363 (G) (C) 2.0 0.734 Discharge (ft.s/sec.) . . Width (feet) 0.363 1.32 0.734 1.32 1.32 1.32 8 /, S It Q It Q A w /( w /, 1.8 2.4 1.05 .76 1.0 2.0 0.99 .61 0.363 .734 1.05 .65 0.363 .734 0.92 .61 0.66 1.00 1.32 0.59 .74 .92 1.00 1.32 0.57 .61 DUTY AND EFFICIENCY. The relations of duty and efficiency to ca- pacity involve discharge and slope but are in- dependent of fineness. Fineness, therefore, has exactly the same control of duty and efficiency that it has of capacity, and the conclusions of this chapter apply without qualification to duty and efficiency. SUMMARY. Capacity for traction is greater for fine d6bris than for coarse — that is, capacity increases with fineness. The law of increase admits of formu- lation in a manner strictly analogous to that employed in comparing capacity with slope and discharge — that is, it is found that ca- pacity varies approximately with a power of the fineness less a constant fineness. The value of the constant finenes3 varies with conditions, being greater as slope, discharge, and form ratio are greater. The rate at which capacity varies with change of fineness, or the index of relative variation, is not the same for all 'con- ditions, being greater as slope, discharge and form ratio are less. Under similar conditions the rate is less than the corresponding rate for capacity and slope, the average ratio between them being as 1 to 2.4. The arrangement of capacities in accordance with the assumed law of increase develops dis- crepancies which are believed to be of the na- ture of systematic errors. The largest of these have a magnitude of about 10 per cent. They are tentatively ascribed to peculiarities of the <16bris used in experiments and to imperfectly developed laboratory methods. CHAPTER VII.— RELATION OF CAPACITY TO VELOCITY. PRELIMINARY CONSIDERATIONS. The work of stream traction is accomplished by the movement of water along the bed of the channel. For that reason the system of water movements and water velocities near the bed is intimately related to the load or capacity for load. In certain parts of this paper and in the writings of some other investigators use is made of the term "bed velocity," or its equiva- lent, but the term has no satisfactory defini- tion. The difficulties which are encountered in this connection have to do also with the vertical velocity curve. In all the streams with which we are here concerned the flow is eddying or turbulent. At any point the direction of motion and the FIGITRE 52.— Vertical velocity curve, drawn to illustrate its theoretic character near the stream's bed. OD is the origin of velocities. velocity are constantly changing. If a mean be taken of the instantaneous forward com- ponents of velocity — the components parallel to the axis of the stream — it gives for the point a mean velocity coordinate with the mean velocity for the cross section obtained by dividing the discharge by the sectional area. It will bo observed that the mean at a point is a mean with respect to time, while the sec- tional mean is primarily a mean with respect to space. The mean at a point, as thus defined, being called Vp, the vertical velocity curve may be defined as the curve obtained by plot- ting the values of Vp for any vertical of the current in their relation to depth. As commonly drawn by hydraulic engineers, it terminates downward at some distance from the origin of velocities, OD — say at B in figure 52 — connoting a finite velocity for the water in actual con- tact with the bed. This implication contra- venes a theorem of hydrodynamics that the velocity at contact with the wall of a con- duit is either zero or indefinitely small. The theorem is believed to have been established experimentally by the work of J. L. M. Poiseuille * and is generally accepted. In the direct study of the velocities of streams instru- mental observation is not carried from surface to bed, but ceases at some point, C, and the drawing of the curve below that point is a matter of inference. The inference accordant with the hydrodynamic principle is that the curve changes its course below C and reaches the origin at or near O.1 This inference accords also with our observations in connection with the study of saltation (see p. 29); and those observations suggest likewise that the curve is materially modified by the resistances to the current involved in the work of saltation. It thus appears that in the region with which traction studies are specially concerned the range of Vp is great. The work of traction depends on a system of velocities and nob on a single one, and there is no individual value of Vp with special claim to the title "bed veloc- ity." It would be possible to define bed velocity as the value of Vp at some particular distance from the bed or at a distance consti- tuting some particular fraction of the depth of current; but such a definition would be hard to apply. However smooth a stream bed of debris may be in its general aspect, it is never smooth as regards details. Figure 53 gives an ideal pro- FIGURE 53.— Ideal profile of a stream bed composed of debris grains. tile, the intersection of a bed by a vertical plane. Not only are there salients and reentrants, but some of the reentrants communicate with the voids within the mass of dfibris. In many of > See Lamb's Hydrodynamics, 3d ed., p. 544, 1906. ' See Cunningham, -Allan, Hydraulic experiments at Roorkee, p. 46, 1S75, and Inst. Civil Eng. Proc., vol. 71, p. 23, 1882, where he discusses t he horizontal velocity curve; and Von Wagner, idem, p. 90. 155 156 TRANSPORTATION OF DEBRIS BY RUNNING WATER. the reentrants are doubtless stationary eddies, with reversed currents where the value of Vp is negative. It appears equally difficult to give definition to the bed as a datum from which to measure upward, and to select and define a locus for bed velocity. There is reason to sus- pect also that the problem as thus stated is unduly simplified by the assumption that the bed is a stable entity, clearly separate from the zone of saltation above. It did indeed so ap- pear when the process of saltation was studied through the glass wall of the observation trough, but what was witnessed was the phase of the process at the edge of the channel bed, where the current was retarded by the resist- ance of the channel wall. At a distance from that wall, in the region where the cloud of sal- tatory particles effectually precludes visual ob- servation, the passage from stability to mobil- ity may be less definite. I am led to this sug- gestion by the observations, quoted byMcMath,1 of a civil engineer who descended in a diving bell to the bottom of the Mississippi at a point where the depth was 65 feet and the bottom of sand. Stepping to the bed, he sank into it about 3 feet, and then thrusting his arm into the yielding mass, could feel its flowing motion to a depth of 2 feet, the velocity diminished down- ward. In interpreting these phenomena, allow- ance must be made for the fact that the pres- ence of the diving bell created an abnormal condition and if it rested on the bed put a stop to saltation. The flow of the sand is then to be ascribed to the difference in water pressure on the two sides of the bell. But the fact of the flow seems to indicate an antecedent state of mobility, a laj'-cr of the bed being supersatu- rated so as to have the properties of quicksand. If such a layer exists, then the transition from the bed to the saltation zone is not abrupt but gradual. The difficulties in attempting to define bed velocity are supplemented by those which affect the measurement of velocities near the bed while traction is in progress (p. 26), and to- gether they have served to prevent the use of bed velocity as a factor for quantitative com- parison with capacity. This result has been regretted because the forces which accomplish traction are applied directly through the veloc- ities of water near the bed, and it was admitted i McMath, R. E., Van Nostrand's Mag., vol. 20, p. 227, 1879. only after the failure of repeated attempts to obtain serviceable estimates of bed velocity. In the present chapter observed or interpo- lated capacities are compared with mean veloc- ities of the stream, mean velocity being com- puted as the quotient of measured discharge by measured sectional area. The measurements of discharge and width being relatively simple and accurate, the determinations of mean velocity have the same degree of precision as the meas- urements of depth. (See p. 26.) In comparing capacity with mean velocity, it is convenient always to treat fineness of debris and width of channel as constants, but it is also advantageous to recognize three separate points of view as to the status of discharge, slope, and depth. First, We may treat discharge as constant, in which case slope and depth vary, along with velocity and capacity. Each of the observa- tional series (Tables 4, 12, and 14) conforms to this viewpoint. When discharge is con- stant, the increase of power necessary to increase velocity is given by increase of slope, and the increase of velocity causes the un- changed discharge to occupy less space. As velocity and capacity increase, slope increases and depth decreases. Second, we may treat slope as constant. With slope constant, the increase of power necessary to increase velocity is given by increase of discharge, but the rate at which discharge is increased is greater than the rate of increase given to velocity, and the increased discharge therefore requires more space. As velocity and capacity increase, both discharge and depth also increase. Third, we may treat depth as constant. To increase velocity by increasing slope will, as we have seen, reduce depth. To increase velocity by increasing discharge will, as we have seen, increase depth. To increase velocity without changing depth, it is necessary to enlarge both slope and discharge. No experi- ments were conducted with fixed depths, but the data for this comparison are readily obtained by interpolation. It is proposed to examine the relation of capacity to velocity from each of these view- points, developing the results so far as neces- sary to give a basis for a comparison of the viewpoints. RELATION OF CAPACITY TO VELOCITY. 157 A preliminary remark applies to all. For each4grade of d6bris and width of channel, and for each specific assumption of a constant discharge, slope, or depth, there is necessarily a competent mean velocity, below which no traction takes place. The conception of such a competent velocity has underlain all the discussions of competent slope, competent dis- charge, competent fineness, and competent form ratio. A broad analogy therefore points to the propriety of formulating the capacity- velocity relation as other relations of capacity have been formulated. And the inference from analogy finds support in logarithmic plots of C=f( Vm) under each of the three above- mentioned conditions. It may fairly be as- sumed, therefore, that the index of relative variation for capacity and velocity itself varies with velocity, being relatively small for high velocities, being relatively large for low veloci- ties, and becoming indefinitely large as com- petent velocity is approached. For the purposes of this chapter, however, it has seemed best to employ a simpler method, using only the synthetic index of relative variation — characterized by the symbol I. Calling the synthetic index for the variation of capacity with respect to mean velocity Iv, we may conveniently distinguish by Irj the formula ..(80) log <7- log (7" logFm'-logTV in which C' and C" are specific capacities, and Vm' and Vm" are the corresponding mean velocities. Graphically, Iv is the inclination of a line connecting two points of which the coor- dinates are, for the first, log C' and log Vm', and for the second, log C" and log Vm". Where the available data serve to place more than two points on the logarithmic plot of C=f( Vm), defi- nite suggestion may thereby be made that the line connecting the extreme points does not constitute the most probable location of the chord theoretically corresponding to IY', and in such cases a line is drawn with regard to all the data, and its inclination is measured on the plot. The subject of competent velocity, which is of interest independently of the formulation of capacity and velocity, will be considered at the end of the chapter. THE SYNTHETIC INDEX WHEN DISCHARGE IS CONSTANT. In Table 14 are 73 series of values of Vm, each value corresponding to a stated value of S. The coordinate series in Table 12 contain values of C corresponding to the same values of S. From each pair of series were taken the highest and lowest values of Vm and the corresponding values of C, and from these four quantities was computed a value of Iv ..(G) IE)- FIGURE K.— Tractional capacities of components of mixed grades, in relation to the percentages of the components in the mixtures. Curves at left show capacities for coarser component, at right for finer. Katios at left show percentages of finer component , at right of coarser. half of (A) there is substituted an equal amount of (G), which is 16.2 times coarser, 89 the capacity is changed in the ratio When for half of (G) there is substituted an equal amount of (A), which is 16.2 times finer, 89 the capacity is changed in the ratio .-^ = 5.56. The geometric mean of these ratios, 1.64, may plausibly stand for the effect of substituting an equal mixture of (A) and (G) for a grade symmetrically intermediate between (A) and (G). The method is easily criticized, and its assumptions will certainly not bear close scrutiny, but it nevertheless yields a sort of composite which is of use in showing that the general effect of diversifying a stream's trac- tional load is to enlarge capacity. Corre- sponding composite ratios have been obtained from the other examples of Table 61 and are given below. Ratio of Grade. Ratio of fineness. capacity change attributed to mixture. (AG 10.2 1.64 (CG 9.7 2.22 (BF 8.5 1.72 (CE 3.4 1.17 (EG) 2.9 1.56 EXPERIMENTS WITH MIXED GRADES. 175 The composite ratios have been arranged in the order of the ratio of fineness, but the com- parison shows no correspondence. They prove equally inharmonious when compared with the fineness of the .finer component, the fineness of the coarser component, or the fineness of the mixture. Their irregularities must be as- cribed to observational errors and to causes not at present to be discriminated from observa- tional errors. It is of interest, however, to note that the large measure of capacity change as- sociated with the (CG) combination might be inferred also from a comparison which involves a different viewpoint and also some practically independent data. If we think of grades (A), (C), and (E) as modifiers of capacity for grade (G), we may compare their efficiencies by means of the following quantities, taken from Table 61 : Grade. Capacity. Grade. Capacity. (AiOO (C,G,) (EiG,) 89 106 49 (AiG,) (C.Gj) (EiGs) 42 50 33 The superiority of grade (C) as a modifier for (G) is thus brought out without making use of the capacities for the uncombined grades (A), (C), (E), and (G); and the result from mix- tures of 1 : 1 is supported by that from mixtures of 1 : 2. CONTROL BY SLOPE AND DISCHARGE. The preceding comparisons are conditioned by a discharge of 0.363 ft.3/ sec., a slope of 1.4 per cent, and a channel width of 1 foot. With a different set of conditions a different set of quantitative relations would be found, and the qualitative also would doubtless be modified. The observations on mixtures included no other width, and there was but a single set of ex- periments using a different discharge, but the range in slope was coordinate with that for the separate grades. Figure 57 shows the capacity-slope curves for the (AG) set of experiments, figure 58 for the (BF) set, and figure 59 for the (CE) set. In figure 57 the curves for mixtures form a graded series between those for the component grades, and there is almost perfect harmony of form and attitude. As the points representing capacities associated with a slope of 1.4 per cent all lie in the same vertical line, and as similar points for another slope lie in some other vertical line, it is evident by inspection that inferences from data of any other available slope would be practically identical with those from the slope of 1 .4 per cent. It is also evident 200 o ID Q- 1:1 0 I 2 Slope FIGURE 57.— Curves of capacity in relation to slope for grade (A), grade (G), and mixtures of those grades. The ratios of components in the mixtures are indicated. that indexes of relative variation, \ or /„ for the mixtures constitute, with those for the components, an orderly system. The same remarks apply also to the (CE) groups of 200 100 ifs> 1:1 1:2 0 I 2 Slope FIGURE 58.— Curves of capacity in relation to slope for grade (B), grade (F), and mixtures of those grades. The ratios of the components in the mixtures are indicated. curves, but they do not apply to the (BF) group in figure 58. The attitudes of the curves for mixtures are there out of harmony with the attitudes of the (B) and (F) curves. The curves for the mixtures seem to belong to a 176 TBANSPORTATION OF DEBBIS BY RUNNING WATER. different system, intersecting or tending to intersect the curves for the components. Evi- dently the indexes of relative variation are inharmonious, and evidently the inferences drawn from data for the slope of 1.4 per cent would not be duplicated by a discussion of data from a slope of 1.0 per cent or 1.6 per cent. 200 o (0 Q. - .4- .8 1.2 Discharge FIGURE 01.— Capacity-slope curves for related mixtures and capacity-discharge curves for mixture and component grades. further supported by the facts brought to- gether in the left-hand diagram of figure 61, which shows the capacity-slope curves of mix- tures (C4Et) and (C4GJ along with that for the five-part mixture. The right-hand diagram of figure 61 is a group of plots of capacity as a function of dis- charge. These plots pertain to the complex mixture and its five components, and all are conditioned by a slope of 1.2 per cent. They show that the fractional superiority of the mixture is not confined to the use of a particu- lar discharge, and they indicate also that the capacity-discharge relation is essentially the same for the mixture as for separate grades. The locus of C =f(Q) for the mixture is approxi- mately a straight line, and if produced it inter- sects the axis of Q to the right of the origin. It might be expressed by an equation in the form of (64) with an exponent near unity. 20021°— No. 80—14 12 A NATURAL GRADE. Two series of experiments were made with an alluvium in its natural condition, except that the very finest constituents had been removed by passing it over a 60-mesh sieve. The obser- vations are recorded in Table 4 (K), and the adjusted capacities in Table 60. In figure 62 the capacity-slope curves are plotted, and each is accompanied, for comparison, by the corre- sponding curves for grades (A) and (C). The approximate mechanical analysis of this material, stated in terms of the separated grades of tho laboratory series, is as follows: Per cent. (A) (B) (C) 13 27 Per cent. (D) 42 (E) 10 Coarser than (E) 2 The fact that the capacities (fig. 62) are greater than those for grade (C), notwithstand- ing the dominance of a component correspond- 178 TRANSPORTATION OF DEBRIS BY RUNNING WATER. ing to the coarser grade (D), testifies again to the advantage for traction of a mixture as compared to a grade of narrow range in fineness. It will be observed also that each of the curves from the natural grade resembles closely its neighbors from artificial grades. So far as their evidence goes the type is the same for both, and the tendency of their evidence is to show that the laws connecting capacity with slope, as developed by the study of sorted debris, apply also to unsorted stream alluvium. 300- 200 100- fA) 0 I 2 Slope FIGURE 62.— Capacity-slope curves for a natural grade of de'bris, com- pared with curves for sieve-separated grades. Table 63 (p. 180) gives computed finenesses for various mixtures and for this natural grade of de'bris. The fineness of the natural grade is nearly identical with that of grade (C), and the two thus afford a direct comparison between the capacities of a natural grade and a narrowly limited grade. For the same discharge and slope their tabulated capacities are respectively 168 and 143 gm./sec., the ratio of advantage to the natural grade being 1.17. The computed fineness of grade (CDEFG) does not corre- spond to that of any simple grade, but a com- parison made by means of interpolation gives 150 and 131 gm./sec. as corresponding capaci- ties for the mixture and a simple grade of the same mean fineness, and the ratio of the first to the second is 1.15. These ratios are smaller than those estimated from data for binary mixtures (p. 174), but are coordinate in value. The question of relative authority will be con- sidered later. CAUSES OF SUPERIOR MOBILITY OF MIXTURES. When a finer grade of debris is added to a coarser the finer grains occupy interspaces among the coarser and thereby make the sur- face of the stream bed smoother. This quality of smoothness appealed to the eye during the progress of the experiments. One of the coarser grains, resting on a surface composed of its fellows, may sink so far into a hollow as not to be easily dislodged by the current, but when such hollows are partly filled by the smaller grains its position is higher and it can withstand less force of current. In other words, the larger particles are moved more readily on the smoother bed, and this fact also was a matter of direct visual observation. The promotion of mobility applies not only to the starting of the grain but to its continuance in motion. It encounters less resistance as it rolls or skips along the bed, and it is less apt to be arrested. When a single large particle travels along a bed composed wholly of grains much smaller it rarely leaps, but rolls instead, and it must in general be true that the larger particles in mixtures roll more and skip less than their smaller companions. The admixture of finer debris thus changes the mode of traction for the coarser, and it is believed that the enhanced capacity is due mainly to this change. Capacity for the coarser is increased because the new condition reduces its resistance to the force of the current. The fact that under some conditions the capacity for fine material is slightly increased by the addition of coarser is not so easily ex- plained. The coarser grams do not make the bed smoother but rougher. The rougher bed retards the current. Even while rolling the larger grains are holding back the water, and the larger grains reduce the area of bed on which the traction of the smaller takes place. In these ways the presence of the coarse mate- rial tends to reduce the capacity of the current for the fine, and these factors certainly seem adequate to explain the general fact that capacity for the finer debris is reduced by ad- mixture of the coarser. Two factors may be named with the opposite tendency. The first is the impact of the coarser particles. In rolling and leaping they disturb the finer, tending thus to dislodge them from their resting places and either start them forward or else give them new positions from which they may be more easily swept. The second is the production of diversity in the EXPERIMENTS WITH MIXED GRADES. 179 current. Every obstruction diversifies the current. The deflection necessary to pass it both constitutes and causes diversity of direc- tion, and diversity of direction necessitates diversity of velocity. If a pebble is placed on the sandy bed of a small stream, the trans- formation of the adjacent parts of the bed by the diversified current is obvious. The build- ing up of the bed in the lee of the pebble testi- fies to lowered velocity, and the scouring at the side to heightened velocity. In the same way each coarser gram of a heterogeneous stream load diversifies the current about it and gives to such of the filaments as are accelerated greater power for the traction of finer grains. With reference to the transportation of the finer debris, the coarser grains have the func- tion of obstructions whether they are partly embedded or lie on the surface or are rolled along. If the factors concerned in the traction of the finer components of mixtures have been cor- rectly stated, it is not difficult to understand that under most conditions the net result of their influences will be a reduction of capacity, and also that special conditions may deter- mine an increase. VOIDS. The packing together of larger and smaller grains which tends toward a smooth stream bed tends also toward the reduction of inter- stitial spaces within the bedded d6bris, thus reducing the percentage of voids. It was sug- gested in the laboratory that the percentage of voids, used inversely, might serve as a sort of index of mobility, and estimates of the voids were accordingly made for most of the mate- rials employed in the experiments with mixed grades. Partly for this purpose, and partly to obtain the factors needed to correct the weigh- ings of load for interstitial water, a series of special weighings were made. A vessel holding 535 cubic centimeters was filled with saturated debris and weighed. Af- terward the same debris was weighed in a dry condition. The computation was made by the formula W— W Percentage of voids = — -„. in which W is the weight of saturated d6bris in grams and W the weight of dry d6bris. Table 62 contains the estimated voids for the binary mixtures. In each series the percent- ages are smaller for the mixtures than for the component grades; and when the percentages were plotted in relation to the proportions of component grades (after the manner of the capacities in figs. 55 and 56) each series was found to indicate a minimum. The positions of the minima correspond to mixtures with 30 to 40 per cent of the finer grade of debris. In comparing voids with capacities, the minima of the void curves are to be considered in rela- tion to the maxima of the capacity curves. Capacity Percentage of voids Linear fineness Bulk fineness FIGURE 63.— Curves showing the relations of various quantities to the proportions of fine and coarse components in a mixture of two grades of ddbris, (C) and (G). The horizontal scale, when read from left to right, shows the percentage of the coarser component in the mixture. Those maxima, however, are associated with mixtures having 60 to 90 per cent of the finer grade; and the attempt at correlation therefore fails. A single void curve is reproduced in figure 63, together with the corresponding capacity curve. TABLE 62. — Percentages of voids in certain mixed grades of debris, compared with the percentages in the component grades. Percent- age of finer grade in mixture. Percentage of voids in grade — (AC) (AG) of which the largest is 0.70; and for grades similar to ABCD a single weak estimate of 0.57. If we conclude that the sensitiveness of capacity to fineness is less for natural grades than for the narrowly limited grades, we must base the inference almost wholly on the data from the mixtures of two grades, connecting the latter with natural grades by aid of the analogy outlined above. This I am willing to do, but at the same time I would record my recognition of the weakness of the evidence and reasoning. It is estimated that, on the average, the capacity of streams for natural grades of debris varies with the 0.60 to 0.75 power of linear fineness. This is equivalent to saying that capacity varies with the 0.20 to 0.25 power of bulk fineness, or with the fifth or fourth root of bulk fineness. While the curve in figure 65 is based on the mechanical analysis of material which consti- tuted the tractional load of a river current, there is no reason to believe that its form pre- sents a dominant type. Inspection of other analyses, in fact, suggests that such curves ex- hibit much variety and may sometimes even present two maxima. The load which a natu- ral current carries is determined not only by the two limits of competence, but by the char- acter of the material within its reach. Neigh- boring affluents of a river may bring to it strongly contrasted grades of debris, or their tribute may at one time be much finer than at another. Moreover, a river is not a simple cur- rent, but a complex of currents, which vary in competence and in the character of their loads. It is true that, the channel being considered as a whole, its load at one point is essentially the same as just above or just below, but the mode of movement involves a continual remodeling of the bed and a sorting and re-sorting of the material. The load at any particular point and time is conditioned by many factors of the complex. For this reason a representative sample of a river's load is not easy to define or to collect. In view of this complexity it is difficult to apply even a simple formula to problems in river engineering, and refinement in formula- tion would be of little avail. For the same reason it is not practicable to derive a formula directly from river data, and the product of the laboratory is the best available, despite the artificial simplicity of its conditions. DEFINITION AND MEASUREMENT OF MEAN FINENESS. The term "mean fineness/' as here used, is not free from the possibility of misapprehen- sion. As the fineness of d6bris is a property depending on the size of component particles, it is not unnatural to think of fineness as a property of the particles — and there is, for that matter, a fineness of particles. To obtain the mean fineness of particles, one would first de- termine the finenesses of the individual parti- cles, and then the mean of those finenesses. The basal unit would be the particle. In de- riving the mean fineness of a body of debris the basal unit is some unit by which quantity EXPERIMENTS WITH MIXED GRADES. 183 of debris is measured. It may be a unit of weight or a unit of volume. In this report a body of debris is conceived to be composed of equal volume units, each of which has a deter- mined or determinable fineness, and its mean fineness is the mean of the finenesses of the volume units. In dealing with bulk fineness the mean computed is the arithmetical mean of the finenesses of units. In dealing with linear fineness the mean computed is the cube root of the arithmetical mean of the cubes of the finenesses of units. The intricacy of the definition of mean linear fineness arises from the relation of linear fine- ness to bulk fineness. The fundamental con- cept is that of bulk fineness, and the definition of linear fineness rests upon it. Linear fineness is essentially a derivative of bulk fineness, and mean linear fineness is an exactly similar de- rivative of mean bulk fineness. To pass from an assemblage of linear finenesses to their mean, it is necessary to pass through bulk fineness, and that passage involves cubes and cube root. Bulk fineness is defined as the number of particles in a unit volume (1 cubic foot), it being assumed there are no voids. It is the reciprocal of the volume of the particle — which might be called bulk coarseness. There are two practical modes of measuring it. If the specific gravity of the debris be known (or assumed), measurement includes a weighing and a counting. Then, W being the weight, N the number of particles, G the specific gravity, TP0 the weight of a cubic foot of water, and Ft the bulk fineness, W.GN ~~ .(86) If the specific gravity be not known, measure- ment includes two weighings and a counting. Then, W being' the weight in air and W, the weight in water, .(87) This procedure determines bulk fineness when all particles have the same volume; when they are of different volumes it determines mean bulk fineness. As a matter of fact, all our measurements in the laboratory were of mean fineness. It is not possible by any method of sorting with which I am acquainted to separate from a natural alluvium a grade which is really uniform in fineness. When the mean fineness of a sample of debris is desired, there is no need to separate it into grades, because the process for measur- ing the mean fineness of the whole is identical with that for measuring the fineness of a grade, When bodies of d6bris of known finenesses are mingled, the mean fineness of the mixture is computed by a formula (85), which sums the finenesses by unit volumes (or weights) and then divides by the number of unit volumes. Linear fineness is defined as the reciprocal of the mean diameter of the particles of the d6bris. Like bulk fineness, it is treated as a property of the body of debris and not as a property of the particle. Mean diameter is defined as the diameter of a sphere having the same volume as the particle. Defined thus, linear fineness is a function of volume of parti- cle, and as bulk fineness is also a function of that volume, the two have a fixed relation: F-(5 .(88) Substituting in (88) from (86) and (87), we have The computations of fineness for this report used (89) or (90) ; or, what is equivalent, they first determined bulk fineness by (86) or (87), and then derived linear fineness by (88). The computations of mean linear fineness applied (88) to mean bulk fineness. It would have been possible to formulate fine- ness in such a way that the definition of linear fineness would be direct and comparatively simple, but any such formulation would en- counter complexity in some of its parts, pro- vided it established a logical relation between linear fineness and bulk fineness. Its adoption would also -involve the sacrifice of simplicity in the measurement of fineness. Any system re- quiring the direct measurement of diameters would be inferior for practical purposes to the one here used. The subject of scales of fineness has been elaborated because nearly all the results as to 184 TRANSPORTATION OF DEBRIS BY RUNNING WATER. the control of capacity by fineness would be quite different if a different scale were used. If, for example, mean linear fineness had been defined as the arithmetical mean of linear fine- nesses, the curve for linear fineness in figure 63 would be a straight line, while the line for bulk fineness would be a curve similar to that shown for linear fineness but turned through 180°. When the fineness of the tractional load of a stream is to be determined by means of a sam- ple of the d6bris const ituting its bed, account must be taken of another factor. Omitting considerations affecting the selection of a sam- ple, which belong to Chapter XIII, let us as- sume that the sample hi hand is representative of the stream's tractional load. In addition to the debris which was carried along the bed, it inevitably includes finer material which was carried in suspension. Suspended particles are arrested along with the coarser and form part of every stream deposit. Once lodged in the interstices of coarser particles, they are shel- tered from the current and are not again dis- turbed so long as the coarser material remains. If the deposition of the coarser debris is very rapid the amount of entangled finer stuff may be small, but when deposition is slow the inter- stices act continuously as traps and catch sus- pendible debris until they are filled. The lat- ter is the usual condition, and the tractional sample therefore ordinarily contains a consid- erable percentage of suspensional material. To separate the two it is necessary to draw an arbitrary line, for the graduation in fineness is complete. As regards interstitial space, the tractional part of the sample is comparable with the more complex mixtures of the laboratory, and its voids may be estimated as 25 per cent of the whole space. The suspensional d6bris packed in these voids may be assumed itself to include 25 per cent of voids, so that the net volume of its particles is three-fourths that of the containing voids, or 18.75 per cent of the whole space. The net volume of the trac- tional particles being 75 per cent of the whole space, the two divisions of the sample bear the relation, by net volume or by weight, of 75 to 18.75, or of 4 to 1. This gives a practical rule for separation. The sample should be divided, with aid of sieves and scales, into a coarser four- fifths and a finer one-fifth, and only the coarser part should be used in estimating mean fine- ness. In figure 65 the entrapped suspensional material is represented by the triangular area ODE. SUMMARY. The purpose of the experiments with mix- tures was to bring the results from work with separate grades into proper relation with phenomena of unsorted natural material. The indications given by these experiments are in part direct and in part conditioned by the principle adopted in framing a scale of fineness. The adopted principle makes the conception of bulk fineness fundamental and that of linear fineness derivative. The capacities for traction observed in the experiments with narrowly limited grades are less than for equivalent grades with greater diversity in fineness. A study of data from mixtures of two narrow grades indicates that the ratio of advantage for diversified debris is from 1.17 to 2.22, the mean of five estimates, from different groups of data, being 1 .66. Two comparisons of results from highly diversified grades, with results from nearly homogeneous grades of the same fineness, give as estimates of the ratio of advantage 1.15 and 1.17. The larger estimates were made by an indirect method but are independent of the scale of fineness. The smaller estimates were made by a direct method but involve the theory of the scale of fineness. In combining the two groups of estimates, greater weight is assigned to the smaller, not because they are of recog- nized higher authority, but because the same scale of fineness will almost necessarily be used in applying the results of the investiga- tion to practical questions. The compromise value of 1.2 is adopted, as a correction to be applied to values of capacity in Table 12 in estimating capacities for diversified grades of like fineness. The advantage of diversification appears to arise largely from the fact that the finer particles, by filling spaces between the coarser, make a smoother road for the travel of the coarser, and it is not proved that a highly diversified debris gives higher capacity than one containing only two sizes of particles. It is especially notable that when fine ma- terial is added to a previously homogeneous coarse material not only is the total capacity increased, but the capacity for the coarser part EXPERIMENTS WITH MIXED GRADES. 185 of the load is increased, and it may even be enlarged several fold. The general effect of adding coarse to fine is to reduce the stream's capacity for the fine, but under some conditions there is a slight increase. The general relations of capacity to slope, discharge, and fineness (and presumably to form ratio also) are the same for natural and other complex grades of de'bris as for the sieve- sorted grades of the laboratory, but some of the constants are not quite the same. The sensitiveness of capacity to slope is on the average the same for both classes of d6bris grades, but the variation of sensitiveness in relation to slope, as determined by the con- stant a, is somewhat less for natural grades. As to the relation of capacity to discharge comparison was limited to a single example, and that suggested no modification of the constants derived from work with laboratory grades. The sensitiveness of capacity to fineness is somewhat less for natural grades than for the the laboratory grades. No values of the constant were obtained for complex grades, and comparisons of sensitiveness were made only by means of the synthetic index of relative variation. The average value of that index, for natural grades of de'bris transported under laboratory conditions, is estimated at 0.20 to 0.25 for bulk fineness. CHAPTER X.— REVIEW OF CONTROLS OF CAPACITY. INTRODUCTION. In the preceding seven chapters the relations of capacity for stream traction to a variety of factors have been examined one at a time. It is now proposed to bring together some of the discovered elements of control. The experi- mental data thus far considered pertain to straight channels, and the factors of control connected with bending channels have not re- ceived attention. Those factors must be in- cluded when the attempt is made to bring laboratory results into relation with river phe- nomena, but as they constitute a category by themselves it is convenient to leave them out of the account in correlating the results from straight -channel work. The immediate determinants of capacity are (1) the velocities of the current adjacent to the channel bed, (2) the widths of channel bed through which those velocities are effective in moving debris, and (3) the mobility of the debris constituting the bed and the load. It was not found practicable to measure bed ve- locity, but measurement was applied to its two chief determinants, slope and discharge, and also to its ultimate associate, mean velocity, and these have been discussed separately. Width has entered into the discussion chiefly as an associate of depth in the determination of form ratio. By reason of these and other inter- relations the six controls of capacity which have been discussed — slope, discharge, fineness, depth, mean velocity, and form ratio — are not independent, and not all should appear in a general equation. Slope, discharge, and fine- ness being accepted as of primary importance, it is feasible to add but one of the others, and choice has been made of form ratio. FORMULATION BASED ON COMPETENCE. The functions used in discussing the relations of capacity to slope, discharge, and fineness are similar, and each involves a conception of com- petence. Competence enters also the theoiy of 186 the relation of capacity to form ratio, but it enters in a different way. It is convenient to omit at first the form-ratio function and con- sider together the three which are similar. They are: (10) -(64) (75) Each of these equations expresses the law of variation of capacity with respect to one con- dition when the other two conditions are con- stant, and in that sense they are independent ; but there is a mutual dependence of parameters which is of so complete a character that they are essentially simultaneous. The dependence of parameters is more readily stated by means of a specific instance than in general terms. In equation (10) 6,, a, and n are constant so long as Q and Fhold the same values; they do not vary with variation of S. But when the values of Q and F are changed those of llf a, and n are modified. Through this control of its parameters the equation involves the relation of capacity to discharge and fineness. The coefficient 6, is the value of capacity when (S-o) = \;la when (Q-K) = l; &< when (F—<}>) = 1. Replacing them by Z»5, as the numerical value of capacity when (S — a) = \, (Q-K) = 1, and (F- <£) = !, we may combine the three equations into The constant J5 is not of the same unit with either &„ b3, or &4. Its dimensions, derived from those of the variables of (91), are L""'30 M+l T°~l. From the experimental data have been com- puted 92 values of n, 20 values of o, and 5 values of p. (See Tables 15, 32, and 44.) All these are positive. The following statistical summary gives a general idea of their relative magnitudes. Its figures are not based on the same range of observational data; but the REVIEW OF CONTROLS OF CAPACITY. 187 ranges for o and p correspond approximately with the middle part of the range for n. Number Expo- nent. of deter- mina- Mean value. Range of values. tions. » 92 1.59 0. 93-2. 37 0 20 1.02 . 81-1. 24 P 5 .58 .50- .62 It will be recalled that while the forms of the equations involving a,K, and were based on the conception of competence, it was not found possible to correlate those parameters strictly with competent slope, discharge, and fineness. The correlations were obstructed by phenomena of dune rhythms and of diversified fineness and could not be completed, but the forms of equa- tion were found to be well adapted to the com- bined expression of observational data above the region of competence. Their relation to competence is not absolute but intimate, and it is so intimate that certain properties of the parameters may properly be inferred from the physical theory of competence. When the swiftest velocity on the bed is barely able to move d6bris, there is a threefold condition of competence. For the particular discharge and fineness, the slope is competent; for the particular slope and fineness, the dis- charge is competent; for the particular slope and discharge, the fineness is competent. The conditions of competence for the three factors controlling capacity are thus not only similar but simultaneous and coincident. Neither factor can sink alone to the limiting level of competence, but the three arrive together. This is an important principle and lies at the foundation of the systematic interdependence of parameters and variables. .(92) In equation (91) the quantities (S-a), (Q-K), and (F-& be- come zero simultaneously. When S = a, then also Q = K, and F= ; and vice versa. As capacity varies directly with (S — a), (Q — K~), and (F—), it is also true that each of these varies directly with capacity. Any change of condition which affects capacity affects those three quantities in the same sense. For example, suppose discharge to be increased. This not only increases (Q — K) and thereby increases capacity, but it also increases (S — a) and (F—). One mode of expressing this fact is to say that capacity measures the remoteness of each controlling factor from the initial status of competence, and all recede or approach together. Let us now make a more definite assumption, that discharge is increased while slope and fineness remain the same. The resulting in- crease of (S — a), as S is unchanged, implies a diminution of a; and the increase of (F—cf>) implies a diminution of . That is, a and vary inversely with discharge. Parallel reason- ing shows that a and K vary inversely with fineness, and that K and vary inversely with slope. These relations are here developed deduc- tively from the theory of competence. They have been developed inductively from the ob- servational data, for equations (26), (66), and (77) include No way has been found in which to study the exponents deductively. The only evidences of order discovered by comparison of observa- tional data pertain to n, which has been found (equation 27) to vary inversely with discharge and fineness. The question whether o and p follow similar trends could not be answered by the adjusted data because of the cumulative effect of accidental errors. There is, however, considerable force in analogic reasoning, based not only on equations (93), but on other ele- ments of symmetry in the relations of capacity to the several factors — elements to be noted later. The state of the evidence may be ex- pressed by .(94) n =/,„(& [o =/„($, [P =fy (S, It is convenient to have a name for the group of constants designated by Greek letters, and as they define the conditions of competence, they may be called competence constants. The exponent n and the associated compe- tence constant a, as they vary with Q and F 188 TRANSPORTATION OF DEBRIS BY RUNNING WATER. and do not vary with S, are controlled by Q and F. As they both vary inversely with Q and with F, it follows that they vary directly one with the other. It is evidently true in general that each exponent varies directly with the associated competence constant. --(95) ?-/l»(0J With a constant, the variation of S — a is determined by variation of S. Considered as additive, their variations are identical; but if we regard the changes as ratios, the changes in (S — a) are proportional to o_ and those in S to «•. The ratio between these fractions, which is o t; , is a measure of the sensitiveness of (S — a) o — a to changes in S. It is evident that as S in- creases the sensitiveness diminishes. As ca- pacity varies with a power of (S — a), the sensi- tiveness of capacity to slope becomes less as the slope increases. Any change in S causes, according to (92) and (93), a change of opposite character in K and o. When S is increased, K and o are re- duced. As the sensitiveness of (Q — «) to change in Q is measured by Q_K, it is evident that the reduction of K lessens the sensitiveness. This has the effect also of lessening the sensi- tiveness of capacity to discharge ; and that sen- sitiveness is further lessened by the reduction of o. Parity of reasoning shows that increase of slope lessens the sensitiveness of capacity to fineness, so that the effect of increasing slope is to reduce the sensitiveness of capacity to all three of its controlling factors. It is evi- dent also that a similar result would be reached if the analysis began by assuming an increase of discharge or fineness. It is a general principle that any change in one of the control factors, slope, discharge, and fineness, causing capacity to increase, has the effect also of making capacity less sensitive to changes in each and all of the control factors; and the inverse proposition is of course equally true. The statement being phrased to include both, the sensitiveness of capacity to the three controlling conditions varies inversely with capacity. The term "sensitiveness," as used in the pre- ceding paragraphs, is equivalent to the more specific "index to relative variation," for which the symbol i has been used; and by reference to various studies of the control of the index by conditions it may be seen that the entire scope of the general principle just stated has been covered by essentially in- ductive generalizations. From equations (39), (68), and (79), .(96) *<-/»($,& F) This checking of deductive by inductive results helps to establish the second and third equa- tions of (94), which were inferred from anal- ogies. Very little is known of the nature of the functions in (93) to (96), beyond the fact that those of (95) are increasing and the others decreasing. Deductive reasoning has not been successfully applied, and induction has escaped the entanglement of accidental errors in only a single instance and to a limited extent. The exceptional instance is that represented by the first equation of group (96). The symbols being translated into words, that equation reads: The index of relative varia- tion for capacity in relation to slope varies inversely with slope, with discharge, and with fineness. There are in fact three distinct propositions, and each of these might be expressed by a separate equation. As to the first proposition, that the index varies in- versely with slope, it was found, inductively, that the rate at which it varies with slope is itself a decreasing function of slope and also of discharge and fineness; and knowledge of similar character was gained as to the second and third propositions -(pp. 104-108). Repre- senting by diis, diig, and diiF the rates of varia- tion of the index in relation to slope, discharge, and fineness, severally, we have (97) di1F=f3(F) These fragmentary determinations are all of one tenor, and in view of the remarkable symmetries already discovered among the ele- REVIEW OF CONTROLS OF CAPACITY. 189 .(98) ments of equation (91), they render probable the general proposition : The rate at which capacity varies inversely with each of the three controlling conditions, slope, dis- charge, and fineness, itself varies inversely with each of the condi- tions. Returning to (92) and (93), we may indicate certain corollaries. Starting from the status of competence, let us assume that slope is increased, with destruc- tion of the status, and that the status is re- stored by reducing discharge. In the restored status IT is greater than in the original, K is less, and is unchanged. It is evident that the nature of the result does not depend on the particular assumptions, and that we may pass to the general proposition: When capacity is zero, the compe- tence constants are so related that a change in any one of them } (99) involves a change of contrary sign in some other. Starting from a status characterized by a particular value of capacity, we may first break it by increasing slope and then restore it by decreasing discharge (fineness remaining unchanged). The first change reduces K and <£; the second increases a and <£. It does not appear whether the net result for (j> involves change in its value, but if so the change is probably small in relation to the increase in a and the decrease in «. It is evident that the nature of the result does not depend on the particular assumption, and that we may pass to two general propositions, each of which includes (99) as a special case: Under the condition that capac- ity is constant, the competence constants are so related that a change in any one of them in- volves a change of contrary sign in some other. Under the condition that capacit- is constant, the values of slope, discharge and fineness are so related that a change in any one of them involves a change of contrary sign in some other. ..(100) (101) .(102) It follows also that Under the condition that capacity is constant, the value of each controlling condition (S, Q, or F) is so related to the corre- sponding competence constant (a, K, or ) that the two vary in same sense. Propositions (100) and (102) are deduced from equations (93). By parity of reasoning equations (94) yield (103) and (104), but these two propositions share whatever uncertainty attaches to (94). Under the condition that capacity is constant, the exponents n, o, p are so related that a change in any one of them involves a change of contrary sign in some other. Under the condition that capacity is constant, the value of each controlling condition (S, Q, or F) is so related to the corre- sponding exponent (n, o, or p) that the two vary in the same sense. ..(103) (104) As capacity can not be increased under (91) without increasing S, Q, or F, and as the increase of one of these involves under (93) the decrease of two competence constants, without any change of the third, it follows that the competence constants, collectively, vary in- inversely with capacity. The same reasoning, if applied to (91) and (94), yields a similar conclusion as to the exponents. To combine the two in a single statement: The competence constants a, K, and , taken as a group, and the exponents n, o, and p, taken as a group, vary inversely with capacity. I find it not easy to bring into combination the laws of internal relation between para- meters of a group and the laws which connect the groups with capacity; but if these laws be regarded as conditions, it is possible to frame more comprehensive theorems of tentative character. Equations (106) are of this class 190 TRANSPORTATION OF DEBRIS BY RUNNING WATER. and are thought worthy of examination, al- though the data at hand do not suffice for their testing. .... (106) (91) The equation under discussion, is an expression of relation between capacity and three of its controls, namely, slope (S), discharge (Q), and fineness (F). It involves seven parameters, of which six are functions of the independent variables, S, Q, and F. It is thus a bare framework, and the completion of the structure calls for the replacement of the six parameters by their values in terms of the variables. The laws contained in the equa- tions and propositions numbered (92) to (98), with their corollaries, (99) to (105), are con- tributions toward the completion of the struc- ture, but they are largely of the nature of re- strictions. They impose conditions to be sat- isfied by the perfected equation. Some of the conditions are already embodied in the form of (91), and with reference to such conditions it is important that the origin of that form be not overlooked. The form assumes that the three competence constants are the values of the corresponding variables when ca- pacity is zero, whereas their identification with those values is by no means complete. The definition and recognition of the status of com- petence are so obstructed by the complicating conditions of nonhomogeneous de'bris and dune rhythm that no more can be asserted than an indefinitely representative relation. For most purposes, however, we are little concerned with conditions in the immediate neighborhood of the competence limit, so that this qualification is of small practical moment. Outside of the neigh- borhood of competence the support of the form is empiric ; it has served well as a scheme for the marshaling of the observations. The support is qualified, in turn, by the fact that the obser- vations are not of such harmony and precision as to discriminate nicely among formulas of adjustment. In view of these qualifications, the possibility has been recognized that some of the laws above enumerated might emanate from the form of the equation and have no other basis; and in view of this possibility the foundations of each conclusion have been scru- tinized. I believe that all the inferred laws, from (92) to (105), are essentially inductive. It is easy to understand that any construc- tive effort which should hang all supplementary conditions on the framework of (91) would re- sult in a formula so unwieldly as to be useless. It is a matter of faith with me that if our data were so precise as to substitute definite quanti- tative relations for the fascicle of trends and indefinite parallelisms they have actually fur- nished, some way would be found leading from complexity to simplicity. I am not without hope that the presentation here made may sug- gest to the mechanist, familiar with the aspects of solved problems of similar difficulty, a ra- tional theory under which the data may advan- tageously be recombined. In an effort to discover unities among the complexities of the capacity relations, equation (91) was given the following form: The three factors making the second division of the second member, being independent of the units of measurement, seemed well adapted to the expression of comprehensive harmonies, if such exist. The following negations were demonstrated: SO F The quantities -, -, and -7 are not equal, nor are the ratios between them constant. The quantities - - 1, ^ - 1, and -, - 1 are not (f A. G) equal, nor are the ratios between them con- stant. The quantities f--l), (~-l), and /F V ( -7 — 1 ) are not equal, nor are the ratios be- tween them constant. It was also found that the symmetric factors in equation (91), namely, (S — e)", (Q-K)°, and (F—)P, are not equal, nor are the ratios be- tween them constant. . THE FORM-RATIO FACTOR. In its relation to form ratio capacity has two zeros, one corresponding to a high ratio, the other to a low. Each of these corresponds also to a competent bed velocity, so that into REVIEW OF CONTROLS OF CAPACITY. 191 a perfect formula competence would enter twice. The formula adopted, however, ignores the element of competence, chiefly because its recognition, which would add a complication, was not seen to be of advantage for the expres- sion of the control of capacity in the more im- portant regions outside the vicinity of compe- tence. The accepted formula is The quantity p is that value of E which corre- sponds to the maximum value of 0 — the maxi- mum standing between the two zeros — and 62 is a capacity constant. The function as a whole qualifies capacity by means of a numer- ical factor and may be combined with (91) by multiplication of the factors: (109) The coefficient b, replacing 65 and b2, is a quan- tity of the same unit with 66 (see p. 186), but numerically independent. The function now added is of distinct type from the others, for instead of advancing by a continuous law from zero to infinity it first rises to a finite maximum and then returns to zero. The first three factors are harmonious; the fourth discordant. At every stage in the investigation the discussion of the laboratory data has been hampered by this discordance. In order to treat adequately the relation of ca- pacity to either slope, discharge, or fineness, it was necessary to isolate that relation by equaliz- ing other conditions, and slope or discharge or fineness could readily be equalized; but the form-ratio factor was intractable. By means of interpolation it was possible to assemble varied data characterized by the same form ratio, but that did not meet the difficulty. It was necessary to take account of the relation of the particular ratio to the optimum ratio, p; and the value of ,0. varies with all other condi- tions. The sensitiveness of capacity to form ratio, as measured by the index of relative variation, is less than its sensitiveness to other conditions. The average of 48 values tabulated in Chapter IV is 0.24, while similar averages for fineness, discharge, and slope are three, five, and seven times as great. The distribution of sensitive- ness, in relation to values of the independent variables, is illustrated by figure 66, where four curves are plotted, each representing a particu- lar instance, selected as typical. The vertical scale is the same for all ; and the ordinates rep- resent values of the index of relative variation. The horizontal scale is that of slope for the curve SS, of discharge for the curve QQ, of fineness for the curve FF, and of form ratio for the curve BpR. The vertical cc represents the competence constants and is an asymptote to three of the curves. The horizontal line mm gives the value of the exponent m correspond- ing to the form-ratio index. For values of R greater than p the index is negative, but its curve is drawn above the zero line to represent sensitiveness, which is not affected by sign. -q . FIGURE 66.— Typical curves illustrating the distribution of the : tiveness of capacity for traction to various controlling conditions. Ordinates represent values of the index of relative variation; abscissas, to four different scales, represent values of slope, discharge, linear fineness, and form ratio. The two parameters of the form-ratio factor have laws of variation similar to those of the other parameters; each varies inversely with values of all independent variables except its own, P=f(&,Q, h m=/, (&,Q,h (6D -62) It follows that each varies directly with each of the other parameters (a, K, are severally equal to the competent values of 8, Q, and F. Now change width and depth so as to increase the form ratio and capacity becomes finite. That ca- pacity may be finite, a, K, and cj> must be less than S, Q, and F, and as the latter have not changed the competence constants have been reduced by the increase of form ratio. By parity of reasoning it can be shown that if the initial form ratio be so large as to make the bed velocity competent a reduction of form ratio will cause a reduction of a, K, and <£. Somewhere between the two form ratios of competence lies p, the form ratio of maximum capacity, and between the same limits lie mini- mum values of the competence constants. The greater the capacity induced by adjust- ment of form ratio, the greater the reduction of slope, for example, necessary to reduce capacity to zero, and as this reduction varies directly with the depression of a below the initial value of S, it follows that the minimum value of a (and similarly of « and ) coincides with the maximum of capacity. The conclusion that the competence con- stants vary inversely with capacity is therefore true for the case in which changes in capacity are caused by changes in form ratio. It can be shown also that the exponents, n, o, and p, follow the same law. The extension of this principle to the domain of form ratio gives assurance that the conclu- sions embodied in equations and propositions (99) to (108), conclusions which were reached from phenomena of slope, discharge, and fine- ness, are not vitiated by the traversing phe- nomena of form ratio. The function in the form-ratio factor of (109) being characterized by a maximum, the varia- tions of parameters with respect to form ratio are characterized by a minimum. This laia may be so combined with those of (93), (94) (61), and (62) as to yield the following sys tern of equations for the trends of changes in parameters consequent on changes in the foia independent variables of equation (109): P = n =f, o =/„ P =/7 »»=/. P, R) , P, R) ..(110) In the development of the form-ratio factor of equations (58) and (109), detailed in Chap- ter IV, the factor first appeared as (1 - nE)Rm, the quantity a being a numerical coefficient in- troduced to represent the resistance to the cur- rent occasioned by the sides of the channel. It was afterward shown thataiiss&s, M&n. et doc. ,5th ser., vol. t, pp. 381- 431, 1871. 20921°— No. 86 — 14 13 accessions. So long as the velocity along the bed exceeds a certain value the current trans- ports sand. Below that limit, the sand is undisturbed. The discharge, Q, and width, w, being known, the mean depth, d, the slope, s, the mean velocity, Vn, and the bed velocity, Vb, are given by the following three equations, of which (1) and (3) are from Darcy and Bazin. The constants are in meters. (1) (2) (3) The sand travels (1) by rolling, (2) by sus- pension. A particle of water impinging on the bottom gives motion to a sand grain, the motion having a direction which depends on the impact and on the positions of adjoining particles, solid and liquid. The grain is pro- jected free from the bottom or is rolled along it, the particular result depending on the inclination and force of the impact and on various conditions which affect the resistance. Suspension corresponds especially to impacts associated with high velocities. Suspension is rare below a certain critical velocity for each density and size of sand grain. Transporta- tion is slow at low stages of a variable stream, rapid and by suspension at high stages. The grains describe trajectories analogous to those of the water particles, but shorter; and there are frequent returns to the bottom, as well as restings between excursions. Larger grains are lifted less high, or are rolled only, or remain at rest. Small grains afford a better hold (prise) in relation to their weight. The smallest of all are carried in the body of the current. The amount by which the pressure on the upstream face of a grain immersed in a current exceeds the pressure on the downstream face is proportional to the square of the velocity. Represent it by a Vb2, the coefficient a depend- ing on size, form, and position. For the sand of the Loire, the resistance developed equals a 0.252, as that sand is immobile when Vb < 0.25. "The difference is equal to the product of the mass of the grains by then- velocity, projected on the same axis as F6 — that is to say, on the axis of the stream. This product, being pro- 194 TRANSPORTATION OF DEBRIS BY RUNNING WATER. portional to the discharge [load] of sand, C, may be represented by the expression hC. We have then : 0= £( Vb2 - 0.252) = m( Fb2 - O.oe)- - - (4) The value of the m will be sought from obser- vation, which will correct in a measure for the introduction of F$ into the equation without allowance for the speed of the grains, etc. Subtraction of 0.06 ceases when the sands are prevented by suspension from rubbing on the bottom; therefore the formula becomes C=mVb2 for velocities above a certain limit. (It is readily understood that m, like a, is only approximately constant.)" The rate at which dunes advance has been measured, in the French rivers, in relation to the velocities of the associated currents. It rises with F6 until the critical velocity is reached, and then drops as the change is made from rolling to suspension. The advance of dunes, depending on the fall of grains into the eddy (fig. 10) when they have been rolled to the crest, is affected by the introduction of suspension because then only a part of the traveling grains are received by the eddy. Observations made on the Loire give as the limiting bed velocity for transportation [com- petent velocity for transportation], F6<, = 0.25 meters per second. According to an engineer who has discussed those observations [H. L. Partiot ?], the corresponding surface velocity, V,, is equal to ^JO.ll; and the formula for the rate of advance of the dunes, Rate of advance = 0.00013 (FS2-0.11) .(5) is good for all values of Vs up to 1.016. One might base on this a formula for load in rela- tion to surface velocity, but the formula would be incomplete unless developed so as to take account of the depth; and it is best for the present to adhere to equation (4), which con- nects load with bed velocity. Lechalas, however, for a temporary purpose, uses formulas of Darcy and Bazin to connect Fb with Vs, under certain assumptions as to depth, and with their aid computes for the Loire the critical bed velocity, Vbcc, at which suspension of the sands begins. Fi)0(, = 0.55 meters per second. This is the velocity cor- responding to Fs = 1.016, the surface velocity which limits the applicability of formula (5). The following table contains the observa- tional data on dunes of the Loire and compares the observed rates of dune advance with rates computed by formula (5). TABLE 63a. — Data on subaqueous dunes of the Loire. Rate of dune advance. Surface ve- Height of locity. dunes. Observed. Computed. Met./sec. Meters. Met./sec.XW-* Met.lsec.XW-* 0.58 0.900 3.0 3.0 .64 .300 3.3 3.9 .73 .300 5.1 5.5 .75 .782 6.3 5.9 .81 .967 6.7 7.1 .81 .967 7.5 7.1 .83 .760 7.6 7.5 1.00 .953 10.5 11.6 1.016 .920 12.4 12.0 1.016 .580 12.0 12.0 1.03 .487 6.2 12.35 1.05 .612 7.0 12.9 1.11 1.198 5.8 14.6 1.13 .650 8.7 15.2 1.33 .950 5.6 21.6 In later passages Lechalas recognizes the variations of velocity in passing from one vertical to another of the same stream section and makes (4) the formula for a division, one unit wide, of the cross section. Thus modified, it is applied in a variety of ways to practical engineering problems of the Loire. DISCUSSION. Lechalas's classification of transportation processes differs from that adopted for our work in that he makes saltation, at least verbally, a part of suspension. I am led, how- ever, by a study of the more detailed descrip- tions of his colleague Partiot, to believe that the line practically drawn between rolling and suspension differs in small measure only from the line we have drawn between traction and suspension. The lower critical velocity of Lechalas is the exact equivalent of our velocity competent for traction, and his upper critical velocity corre- sponds approximately to our velocity compe- tent for suspension. The two attempts at formulation likewise agree in giving promi- nence to the factor of competence. They differ in the mode of using that factor, and they are actuated by different preconcep- tions. KEVIEW OF CONTROLS OF CAPACITY. 195 In the first sentence of the passage (pp. 193- 194) which has been inclosed in quotation marks to indicate its literal translation, Lechalas ap- pears to equate the velocity of a particle of the load with the difference between the forward pressure of the current and the resistance given by the particle. Hooker1 suggests that the ac- celeration of the particle instead of its velocity is intended ; but with or without such emendation the author's reasoning is obscure to me, for I see no necessary physical relation between the number or mass of debris particles moved and the pressure of the current. The load may be defined as the product of the mass of particles by their average speed; and their speed, being produced by the pressure of the current, may be simply related to it, but any relation of the mass to the pressure is necessarily indirect and presumably involved. Whatever the strength or weakness of the postulates on which the formula is based, the manner in which it incorporates the principle of competence gives it a rough resemblance to those we have developed, while the char- acterization of its constant m gives to it a large empiric factor; and it is- in order to inquire whether, as an empiric formula, it finds support in the Berkeley data. As the Berkeley observations do not include bed velocities, the most direct comparison is impracticable; but an indirect relation may readily be established. The difficulty we have found in defining bed velocity may be avoided, for the purpose of 1 Hooker, E. H., Am. Soc. Civil Eng. Trans., vol. 36, p. 256, 1896. the present comparison, by accepting the definition used by Lechalas in - (3) and by assuming depth to be constant. Ac- cording to the Chezy formula this assumption makes Vm approximately proportional to T/S~, so that loads', in (3), is proportional to Vm. It follows that Vb is proportional to Vm, and this permits us to substitute Vm for F6 in equation (4) by changing the constants: -l<) __________ (6) This expression implies that capacity for trac- tion varies with mean velocity at a rate which diminishes as mean velocity increases but is never so low as that of the second power of mean velocity. The corresponding data from our experiments, namely, the data for capacity in relation to mean velocity under the condi- tion of constant depth, are in accord with this, except that they indicate a limiting index of relative variation somewhat less than 2. In Table 51 the values of the synthetic index, Ird, range from 2.03 to 7.86; and a value of 2.03 for the synthetic index implies smaller values of the instantaneous index. This discrepancy is not important, and the formula of Lechalas, regarded as empirical, is probably adequate for the discussion of a body of observations on capacity and velocity. It could not, how- ever, be used in connection with the Berkeley data unless both K and Tc (or m and 0.06 in equation (4)) were permitted to vary with conditions. CHAPTER XI.— EXPERIMENTS WITH CROOKED CHANNELS. EXPERIMENTS. In order to study the influence which bends in the channel exert on capacity for traction, a short series of experiments were made with channels having angular bends and others with channels having curved bends. Each of these channels had a width of 1 foot and was shaped by means of partitions within a trough 1.96 feet wide. (See fig. 67.) Above and below the bends were straight reaches of the same width. All the experiments were made with de"bris of grade (C) and with a discharge of 0.363 ft.3/sec. The loads were measured. In some experi- ments the head lost in the region of the bends was measured by means of level readings on the water surface above and below. After each experiment the profile of the bed was determined by levelings at intervals of 1 foot, and in several cases the region of the bends was covered by such levelings and sketches as to make it possible to construct a contour map of the bed. I 7 n ? m T iv FIGURE 67.— Plans of troughs used in experiments to show the influence of bends on traction. SLOPE DETERMINATIONS. The profiles of the channel beds as shaped by the current were plotted. Through that part of the profile corresponding to the straight channel above the bends was drawn a straight line representing the mean slope for that region, and a similar straight line was drawn below the region of the bends. From a point on the first line near the position of the first bend to a point on the second line corresponding to a position several feet below the last bend a straight line was drawn, and this was assumed to represent the mean slope of the channel in the region affected most by the bends. The profiles above the bends showed evidence of the rhythms com- monly observed in the straight^channel experi- ments. The profiles below the bends showed steeper undulations, which were ascribed to 196 the influence of the strong agitation of the water in passing the bends. In estimating the slopes in the region of the bends the dis- tance used was the length of the medial line of the channel. The observations for head were made at points 2 feet and 4 feet above the first bend and 3 feet and 5 feet below the last bend, and the slopes were computed for the distances, on the medial line, between the points. Check estimates of slope for straight chan- nels were obtained by interpolation from Table 12, the determinations of load being used as arguments. The data are assembled in Table 64, where the stronger and weaker determinations of the slope of the debris surface are severally indi- cated by the letters a and &. The measure- ments of water slope are thought to be coordi- EXPERIMENTS WITH CROOKED CHANNELS. 197 nate in value with those of debris slope. The slopes computed from load measurements are probably of less weight. TABLE 64. — Comparison of slopes required for straight and crooked channels, respectively, under identical conditions of discharge, fineness, width, and load. [DeT>ris of grade (C); discharge, 0.363 fU/sec.; width, 1 foot.] Shape of crooked channel. (See fig. 67.) Load. Slope in straight channel. Slope in crooked channel. Computed from load by Table 12. Profile of de'bris above first bend. Average slope of de'bris. Average slope of water surface. I Gm.iiec. 19 75 56 60 61 60 60 63 63 66 64 64 Pmccnt. 0.42 .92 .77 .80 .82 .80 .80 .83 .83 .85 .84 .84 Per cent. 0. 30 6 .896 .746 .70 a .946 .76 a .86 a .726 .626 .876 .87 a .86 a Per cent. 0.456 .89 o .80o .816 1.15 a .80a .856 .836 .806 .81 0 .77o .SO (i Per cent. 0.46 .93 .78 .72 II Ill IV V NOTE.— Values marked n are given greater weight than those marked 6. FORMS AND SLOPES. The combinations of bends in the experiment channels are shown in figure 67. In three of the channels the bends were angular; in two curved. In channel I the angle of deflection was 10.5°; there 'was a single group of four bends, returning the course to its original direc- tion ; and the short reaches were approximately 5 feet long. In channel II the arrangement was the same, -with reaches of about 2.5 feet and deflection angles of 21.5°. In channel III were three groups of four deflections each, the angles being of 40.9° and the length of reach about 1.4 feet. Channel IV had the same pro- portions as No. II, with the substitution of curves for angles. The radius of curvature for the medial line was 6.55 feet. Channel V con- tained two groups of curves, each similar to the group in No. IV. From the data (Table 64) connected with channels I, II, and III, it appears that with angular bends a greater slope is necessary to transport the load than when the channel is straight — that is, the capacity is reduced by angular bends. The reduction is greatest when the angle of deflection is greatest, and it is so small for an angle of 10° as to bave doubt whether it might not disappear altogether with a somewhat smaller angle. The single group of curves (IV) appears to reduce capacity slightly (increase of slope for same load): but the double group (V) gives slopes indicating an increase of capacity. As the bends of alluvial streams are curved, the curved experiment channels may be as- sumed to represent them better than do the angular channels, and it is possible that mean- dering channels have a greater capacity for traction than straight channels of the same length. There are, however, certain elements of incompleteness in the representation which make definite inference hazardous. The course of a stream which shapes its own channel through an alluvial plain is made up of bends and reaches. In passing from reach to bend there is a gradual increase of curvature until the radius of curvature, for the medial line, is between twice and three times the width of the channel, and the change from bend to reach is also gradual. The forms are automatically adjusted to the system of accelerations and velocities within the current. The angular change of direction in the bend may be one of a few degrees only, but in meandering streams it is commonly from 90° to 180°. In the arti- ficial channels all curves were circular arcs with a radius of 6.55 times the channel width; there was no graduation in the radial acceleration due to deflection; the change from right-hand deflection to left-hand deflection was abrupt, without the intervention of a reach, and the changes of direction were through angles of 21.5° and 43°. That such differences are com- petent to affect transportation to a material extent is indicated by the relations of deeps and shoals (crossings) to bends. Fargue,1 from a discussion of an artificially adjusted portion of the Garonne, reached the conclusion that the distance downstream from the apex of a curve to the deepest point of the associated deep and the distance downstream from a point of inflec- tion to the associated crossing are each nor- mally one-fourth the length of a stream unit — defined as the portion between two points of inflection. In our experiments (see fig. 68) each of these distances is one-half the length of the stream unit instead of one-fourth. 1 Fargue, L., La forme du lit des rivieres a fond mobae, Paris, 1908. 198 TRANSPORTATION OF DEBRIS BY RUNNING WATER. rUour interval O.O2 foot Width oftroufffiJfoot FIGURE 68. — Contoured plot of a stream bed, as shaped by a current. In view of these facts, the apparent results of the experiments with crooked channels must be received with caution, and probably nothing more should be claimed for them than a general indication that the capacity of a moderately bent channel does not differ greatly from that of a straight channel. FEATURES CAUSED BY CURVATURE. Incidentally the experiments illustrated sev- eral consequences of curvature in addition to the influence on slope and capacity. At each turn the swiftest part of the current was thrown to the outer or concave side of the channel, and the slower parts moved toward the opposite side, the transfers giving to the current as a whole a twisting motion. The action on the debris became exceptionally strong near the outer side and exceptionally weak near the inner. A result of the strong action was that part of the load was thrown upward, so as to be temporarily suspended, and a result of the diversity of velocity was the maintenance of deep places near the outer wall and of shoals near the inner. Associated with the twisting motion were many whirls or eddies; and the general obliquity of motion had the effect of reducing the mean velocity in the direction of the general flow. The reduction of mean velocity was recorded in an increase of mean depth, which amounted, in the average of all examples, to 7 per cent and ranged from 2 to 14 per cent. CHAPTER XII.— FLUME TRACTION. THE OBSERVATIONS. SCOPE. That which distinguishes flume traction from stream traction is the fixity of the channel bed. In stream traction the shapes of the bed are adjusted to the rhythms of the mode of trans- portation, and its texture is that of the debris in transit. In flume traction the bed is unre- sponsive, but its texture, being independently determined, has an important influence on the mode of transportation. The experiments were arranged to determine the influences of different textures of bed on mode of traction and capacity for traction and were otherwise varied in respect to slope, width, discharge, and the character of debris transported. GRADES OF DEBRIS. The material transported in the experiments included most of the grades already described (see Table 1 and Plate II) and also several mixtures not previously mentioned. In order conveniently to show the relations of the mix- tures to their components, all the grades of debris used in the flume experiments are listed below in Table 65, the data of Table 1 being repeated so far as necessary. The elements of the table are defined at page 21. The material of the coarse grades (I) and (J) differs from that of the finer, being about 2 per cent less dense. Its particles also are somewhat less thoroughly rounded, their journey from the parent rock bed having been short. TABLE 65. — Grades of debris . Grade name. Sieves used in separation (meshes to 1 inch). D. Mean di- ameter of particles (foot). F. Number of par- ticles to linear foot. Ft. Number of particles to cubic foot. (B) . 40-50 0.00123 812 1.023.000,000 (C)... 30-40 .00168 602 417,000,000 (E) 10-20 .00561 178 10,770,000 (G)... 4-6 .0162 61.8 451,000 H) 3-4 .0230 43.4 156,000 i)..:: 1-2 .0547 18.3 11,900 (j) J-l .110 9.1 1,440 (EjGi)... .00698 143.2 5,610,000 (EjHiIj) .00706 141.7 5,430,000 (EjHjIjJj).. .00836 119.6 3,266,000 APPARATUS AND METHODS. The experiment trough, a modification of that represented in the frontispiece, was 60 feet long and 1.91 feet wide, with vertical sides. The sides and bottom were of wood, planed and painted. For a portion of the experiments the bottom was covered by a false bottom, specially prepared to present a definite character of roughness, the sides re- maining smooth. The trough was so arranged that it could be given various determinate slopes up to 3 per cent, and by means of an inclined false bottom a slope of 4.5 per cent was made. By means of a partition the width was reduced, for the greater part of the work, to 1.00 foot. The width of trough at the outfall end was regulated by a contractor, as described on page 25. The debris was delivered at the outfall to a settling tank, which had two divisions; and a deflecting apparatus was so arranged that the delivery could be instanta- neously diverted from one division to the other. Above the trough near its head was a sloping platform on which measured units of debris were dumped at regularintervals, determined by a watch, and from which the debris was fed to the current by hand, with the aid of a scraper. The rate of £eed was modified by changing the interval between dumpings, and by successive trials it was adjusted to the capacity of the current. In accelerating the debris, as it fell into the water, the current was retarded, so that close to the feeding station it was slower than else- where. When the load was approximately adjusted to the general capacity of the current it constituted an overload in this particular tract, with the result that a portion was de- posited. The load would then traverse an upper division of its course on a bed of debris, while in the lower and principal division it was in direct contact witn the bottom of the trough. A tendency of the stream of debris to clog near the upper end of the trough, although moving freely beyond, was the ordinary criterion of the 199 200 TRANSPORTATION OF DEBRIS BY RUNNING WATER. proper adjustment of the feed, and when this condition existed the load delivered at the out- fall was assumed to represent the capacity of the stream. The outfall was then directed for a measured interval of time to a reserved division of the settling tank, and the d6bris thus separately received was weighed. Besides the deposit connected with the feed- ing of debris there were transitory deposits of a rhythmic character, as described later. Five characters of channel bed were used, namely, a planed and painted wood surface; a rough-sawn, unplaned wood surface; a surface of wood blocks, with grain vertical; a pavement of sand grains, set in cement; and a pavement of pebbles. (See PI. III.) PROCESSES OF FLUME TRACTION. MOVEMENT OF INDIVIDUAL PARTICLES. Flume traction differs from stream traction in its extensive substitution of rolling for sal- tation and in the important place it gives to sliding. The relative importance of these modes of particle movement is determined (1) by the texture of the bed surface in relation to the size of the particle, (2) by the velocity of the water in relation to the size of the particle, and (3) by the shape of the particle. In stream traction the order of roughness of the bed is given by the fineness or coarseness of the material of the load, and this fact deter- mines saltation as the dominant process. In flume traction the bed may be much smoother. On a smooth stream bed any particle with a broad facet is apt to slide, and a well-rounded particle to roll. Rolling is determined (rather than sliding) not only by the fact that the pro- pulsive force of the current and the resistance given by the bed constitute a couple, but also by the fact that the current applies a greater force to the upper part of the particle than to the lower. The less smooth the bed surface, the greater its resistance and the more effective the couple in causing the particle to roll. With any particular texture of bed, the sizes of par- ticles may determine their modes of progress the largest sliding, those of smaller size rolling, and the smallest leaping. Increase of velocity tends to increase saltation at the expense of rolling and to increase rolling at the expense of sliding. A particle rolled slowly is in continuous con- tact with the bed. A round particle may roll rapidly on a smooth bed without parting from it. Roughness of the bed causes changes of direction in the vertical plane, and such changes combined with high velocity cause leaps. If the particle is not round its rolling involves rise and fall of the center of mass, and such changes combined with high velocity cause leaps. Shape of particle may thus be a deter- minant between saltation and rolling, as well as between rolling and sliding. A flattish particle, which may either slide or roll, travels faster when rolling. This is due partly to the fact that when it rolls it rolls on edge and thus projects farther into the current, and partly to the fact that the resistance at contact with the bed is greater for sliding than for rolling. It is also true that rolling par- ticles as a class outstrip sliding particles as a class, the difference in speed being marked. For particles of similar size those which domi- nantly roll outstrip those which dominantly leap. This is part of a more general fact that the better-rounded particles travel faster than the more angular. In the traction of mixed debris, where rolling is characteristic of larger particles and saltation of smaller, the larger travel faster than the smaller. There are thus two important ways in which rolling gives greater speed than saltation. It was not learned whether a particle which alternately rolls and leaps travels faster in one way than in the other. A suspended particle, having the same speed as the water, outstrips all others. It is there- fore possible that as saltation approaches the borderland of suspension its speed exceeds that of rolling. When samples of different grades are fed tc the same current in succession, it is found that the coarser travel the faster, whatever the mode of progression (except suspension). In the fol- lowing record of experiments the speeds of grade (J) constitute an apparent exception, but their slowness is ascribed to the fact that the particles of that grade were relatively angular. TABLE 66. — Relative speeds of coarse and fine debris in flume traction. Depth of water. Mean veloc- ity of water. Average speed of d<5bris particles. Grade (E). i Grade (H). Grade (J). Foot. 0.127 .182 .110 .182 Ft.lsec. 4.35 3.04 2.52 1.S2 Ft.lsec. 3.2 2.3 1.85 .70 Ft./sec. 3.9 2.4 2.0 .85 Ft.lsec. 3.8 2.0 U. S. GEOLOGICAL SURVEY PROFESSIONAL PAPER 86 PLATE III nut ROUGH SURFACES USED IN EXPERIMENTS ON FLUME TRACTION. FLUME TRACTION. 201 It may be observed in passing that the re- corded speed of the particles is on the average 75 per cent of the mean velocity of the water, the ratio being gi eater as the velocity is greater and as the depth is less. A higher ratio would of course be found if the speed of particles were to be compared with that of the lower part of the current. The percentage might be less than 75 if the stream were fully loaded. In watching the traction of mixed debris it was observed, as already mentioned, that the larger particles traveled faster than the smaller. As it is difficult in such an observation to avoid giving attention largely to the more active par- ticles, the observation applies especially to those which roll, but there is probably a similar con- trast between the speeds of less active parti- cles also. In an experiment with glass balls of different sizes it was found that the larger were rolled by the current somewhat faster than the smaller. In attempting to understand the more rapid propulsion of the larger particles it is natural to compare the phenomenon with the more familiar fact that in the absence of water a large stone or ball will descend an incline with greater speed than a small one, but there is an important difference between the two cases. The object descending a dry incline is impelled by gravity, acting directly, and part of the resistance comes from the fluid in which it is immersed, whereas the rolling debris pebble is impelled chiefly by the moving fluid which surrounds it. The advantage in speed accruing to the larger pebble in flume traction is prob- ably connected with the fact that the velocities of the current increase from the bottom upward. If the velocities were all the same the pressures applied to similar pebbles of different diameter would be proportional to their sectional areas, or to the squares of their diameters; but because of the gradation of velocities the increase of pressure in conse- quence of increase of diameter is more rapid than the increase of the square of the diameter. The chief resistance to the forward movement of the pebble is engendered at its contact with the bed and is of the nature of rolling resistance. For uniform speed, the rolling resistance of a wheel is proportional to its downward pressure divided by its diameter; and since in the case of the pebble the pressure is proportional to the cube of the diameter, the rolling resistance is proportional to the square. Increase in size of the rolling pebble is thus accompanied by increase of both propulsive force and rolling resistance, the increment to propulsive force being somewhat the larger; and the equality of force and resistance is restored by an increase of speed, which has the effect of reducing the propulsive force and increasing the resistance.1 The reduction of propulsive force is connected with the fact that that force is determined at each instant by the velocities of the water, not as referred to the fixed bed, but as referred to the moving pebble. The analysis of forces might be further de- veloped, but the foregoing brief outline serves to indicate a mechanical principle underlying the observed fact that a current rolls large particles more swiftly than small. The princi- ple is of fundamental importance in account- ing for certain contrasts between the laws of flume traction and those of stream traction. The discrimination of traction and suspen- sion, usually easy in the experiments with stream transportation, was difficult in the flume work. The zone of saltation grew deeper with progressive increase of slope until it occupied the whole depth of the stream. Further increase of slope, with increase of load, made the cloud of particles denser, but there seemed no way of telling when the condition became that of a flowing mixture of water and sand. COLLECTIVE MOVEMENT. To the general fact that in flume traction the particles of the load either roll or slide in continuous contact with the fixed bed or else skip from point to point along it, there are two noteworthy exceptions. When the transported debris includes par- ticles which are small compared to the projec- tions constituting the roughness of the bed, some of the debris finds lodgment among the projections. The roughness of the bed is thus diminished and the process of transportation is modified. The bed comes to be constituted in part of the fixed summits of projections and in part of mobile debris, and the process be- comes a blending of flume traction proper and stream traction. When such conditions existed in the experiments it was found that the capacity was essentially that due to stream traction. i The formula given by W. J. M. Rankine, in his " Applied mechanics " and elsewhere, for the rolling resistance of a wheel is R— -|a+6(t)— 3.28)} where R is resistance, Q gross load, r radius of wheel, v velocity in ft. /sec., and a and b constants. The experimental values of a and b are such as to give velocity only a moderate influence on the resistance. 202 TRANSPORTATION OF DEBRIS BY RUNNING WATER. The second exception is a phenomenon of rhythm. It is probable that the rate of flume traction is always affected by rhythm, just as is that of stream traction, and the rhythm in load implies a rhythm in efficiency of current. Under certain conditions, of which the most important is small slope, the rhythm is mani- fested by the making of a local deposit from the load, a patch of debris appearing on the bed of the trough. Such a patch travels slowly down- stream, being succeeded after an interval by another. With suitable variation of conditions the patches become more numerous, are regu- larly spaced, and occupy a greater share of the bed surface. They may even exceed the in- terspaces in area. When wide apart they are usually shaped in gentle slopes, but as their ranks close they assume the profiles of dunes, with steep frontal faces. The advance of the debris patches is like that, of typical dunes, in that the upstream slopes are eroded while the downstream slopes are aggraded; and in this way then- travel is a factor in debris transportation. It does not, however, become the dominant factor, as in the dune phase of stream traction. There is always a large share of the load which passes over the deposits without being arrested and continues its journey across the intervening bare spaces. In this rhythmic process there is a combina- tion of elements belonging distinctively to flume traction and to stream traction, and it is possible that the process constitutes a transition from one system to the other; but, so far as de- veloped by the experiments, it appears as a phase, a rhythmic phase, of flume traction. The slopes with which it was associated were much flatter than those necessary to carry the same load by the method of stream traction. TABLE OF OBSERVATIONS. In the following table the data are arranged primarily by character of bed surface and sub- ordinately by width of channel, discharge, grade of debris, and slope of channel bed. The characters of bed surface are illustrated in Plate III. Load is given in grams per second, width in feet, discharge in cubic feet per second, and slope in percentage. TABLE 67. — Observations on flume traction, showing the relation of load to slope and other conditions. a. Over a surface of wood, planed and painted. w Q S Value of L for grade — (C) (E) (G) (II) (I) (J) (EiGO (EjHJj) (E,H2l3J,.) 1.00 1.91 0.363 .734 .303 .736 0.32 .50 . 75 1.00 1.28 1.50 1.93 2.00 2.50 2.77 3.00 3.41 3.93 4.01 4.05 4.50 .32 .50 .75 1.00 1.28 1.50 1.93 2.00 2.50 2.77 3.00 3 41 3.93 4.01 4.05 4.50 .75 1.28 1.93 2.77 3.44 .75 1.28 1.93 2.77 3.44 9.1 6.1 6.1 28 58 50 55 95 138 117 154 107 126 248 166 294 294 ' 183 202 199 243 268 453 384 438 524 334 271 342 397 406 318 366 444 360 386 479 566 636 625 484 783 793 1,092 1 025 850 1,095 622 660 20.5 685 18.1 801 22.8 68 120 98 110 179 315 335 261 212 219 297 372 381 348 449 474 899 721 814 976 561 660 1,248 778 570 427 909 949 590 640 771 580 726 765 1,075 1,170 1,595 790 1,557 1,730 2,360 1 980 2,480 1,278 1,081 41.6 103 193 324 439 114 222 368 641 857 1,127 58 117 222 392 471 147 248 430 725 846 1,468 30 137 255 463 569 115 325 501 799 1,028 49 110 230 134 282 432 FLUME TRACTION. 203 TABLE 67. — Observations on flume traction, showing the relation of load to slope and other conditions — Continued, b. Over a surface of wood, rough-sawn, unplaced, and unpatnted. w Q S Value of L for grade — (E) CO) (H) (I) (J) (EsHiI,) 1.00 0.363 .734 1.07 1.88 2.11 2.20 2.88 3.09 3.81 4.15 1.07 1.88 2.11 2.20 2.88 3.09 3.81 4.15 186 133 141 178 355 304 247 284 342 515 609 358 390 526 813 930 881 302 269 313 343 646 632 461 513 545 982 1,558 1,550 673 720 772 1,605 2,140 1,665 c. Over a surface of rectangular wood blocks, with grain vertical. Value of L for grade — (E) (H) (I) (J) (EjHiIs) (EjHAJj) 1.00 0.734 2.00 315 337 561 750 620 789 3.00 580 ii!H 962 1,397 1,089 l,3fiO 4.00 911 1,026 1,497 2,060 1,660 2,020 d. Over a pavement of sand grains, grade (G), set in cement, the debris being also of grade (G). w Q s L 1.00 0.363 1.00 12.1 2.00 71 3.00 175 4.00 317 .734 1.00 61.2 2. SO 214 3.00 413 4.00 656 e. Over a pavement of pebbles, grades (H) and (I), set in cement. w Q S Value of L for grade — (E) (Q) (H) (D (J> (E,Gi) (EjHiIs) (EaHiWi) 1.00 0.734 2.00 3.00 4.00 225 425 630 144 316 531 125 273 463 190 483 714 228 636 992 196 401 605 1,059 471 ADJUSTMENT OF OBSERVATIONS. FORMULATION. In flume traction, as in stream traction, there is a finite slope — competent slope — cor- responding to the zero capacity. An inspec- tion of the observational data by plotting served to show that they could advantage- ously be adjusted by means of the formula based on the theory of competent slope: Forty-two observational series were found to give information as to the value of a. Of these, 26 indicated positive values, three negative, and the remainder values so small as to be of uncertain sign. The mean of the 42 values is + 0.29 per cent of slope. Eleven could be compared directly with values adopted in the adjustments of stream traction data, the mean of the eleven values being, for flume traction on a smooth surface +0.14 per cent, and for stream traction +0.28 per cent. This differ- ence is consonant with observed differences in competent slope for the two modes of traction. With the aid of this information, and with use of considerations connected with modifications 204 TRANSPORTATION OP DEBRIS BY RUNNING WATER. of the mode of flume traction by rough sur- faces, but here omitted, a scheme of values of a for the observational series of Table 67 was made out. The adjustments were then made, by the graphic methods described in Chapter II; and their results appear in Table 68. The same table records the parameters of the adjust- ing equations, and also the probable errors. No adjusted values are given for grade (I) in the first division of the table, for the reason that the data, although apparently based on good observations, are strongly discordant. The observations were retained in the record because an aberrant fact, if established, may prove peculiarly valuable; but in this case the interpretation has not been discovered. TABLE 68. — -Values of capacity for flujne. traction, adjusted in relation to slope of channel. a. Traction over a surface of wood, planed and painted. w Q S Value of C for grade— (C) (E) (C) (H) (I) (J) (E,G,) (E,H,I,) (EtHJM 1.00 0.363 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 28 93 186 300 435 590 760 945 25.5 74 134 205 283 366 453 553 654 25 87 158 234 315 398 486 575 665 29 93 170 255 350 450 558 666 138 266 410 568 732 910 1,085 117 254 408 580 763 955 1,155 254 350 451 558 666 780 625 860 1,085 Probable error (per cent) 2.9 3.7 0. 8 ' 1.1 1.3 1.8 1.2 Parameters of adjusting!*' ' ' equations. 0.04 1.63 100 0.06 1.40 80 0.25 1.16 122 0.35 1.22 138 1.80 .96 520 0.15 1.30 114 0.25 1.29 199 0.40 1.28 223 :::::::::: w Q a Value of C for grade — (C) (E) (G) (H) en (J) (Eid) (EjH.Ij) (E,HJ»T,) 1.00 0.734 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 46 177 345 548 790 1,050 1,360 1,700 54 145 258 382 518 665 - 820 9S5 1,150 50 151 268 393 527 668 815 965 1,11D 65 179 310 453 610 770 940 1,108 336 592 870 1,160 1,465 1,790 2,120 335 635 970 1,310 1,675 2,060 2,450 338 470 645 830 1,027 1,235 1,450 790 1,220 1,630 2,030 2,430 653 910 1,170 1,440 Probable error (per cent) 2.1 1.8 1.7 0.7 0.7 1.7 2.7 0.7 Parameters of adjusting!' • • ' equations. jj~" 0. 03 0. 04 1.60 1.35 185 \ 154 0.16 1.20 187 0.22 1.29 221 0.65 1.01 470 1.20 .90 965 0.10 1.24 202 0.16 1.20 415 0.28 1.21 500 w Q a Value of C for grade — Q S Value of C for grade— (C) (E) (G) (H) (C) (E) (G) (H) 1.91 0.363 1.0 1.5 2.0 2.5 3.0 3.5 77 145 224 69 130 200 281 370 464 85 75 1.91 0.734 1 160 171 1 241 275 2 328 381 2 415 491 i 3 507 602 3 0 197 158 5 324 281 0 459 415 5 560 185 204 320 378 460 550 607 730 755 915 910 1,095 0 . 720 5 880 Probable error (pe 0.5 3.0 1.5 Probable error (per cent) 1.4 3. 3 2. 4 Parameters of adjusting 1°, • •• equations 1 ?"" 0.07 1.44 86 0.10 1.42 80 0.30 0.55 „__ 1.16 1.09 130 182 meters of adjusting!' ,... 0.05 0.08 0.20 0.35 »... 1.16 1.30 1.12 1.06 il..210 176 238 322 FLUME TRACTION. TABLE 68. — Values of capacity for flume traction, adjusted in relation to slope of channel — Continued. b. Traction over a surface of wood, rough-sawn. 205 w Q S Value of C for grade— (E) (0) (H) (I) (J) (E,H,I.) 1.00 0.363 1.5 2.0 2.5 3.0 3.5 4.0 97 145 200 259 331 385 120 195 279 366 460 560 229 341 460 582 710 840 160 220 285 355 427 249 374 500 630 755 895 Probable error (pe • cent)... 4.6 Parameters of adjusting!" • • • equations. j^'" 0.10 1.34 62 0.30 1.12 82 0.40 1.28 106 1.00 1.01 248 0.30 1.16 184 w Q S Value of C for grade — (E) (G) (H) (I) P) (E,HiIt) 1.00 0.734 1.5 2.0 2.5 3.0 3.5 4.0 450 660 885 1,115 1,360 1,615 290 390 495 600 712 331 434 540 648 758 354 462 570 680 786 538 752 970 1,100 1,325 1,490 1,800 2,100 Probable error (pe 2.3 Parameters of adjusting]'' • • equations. 0.08 1.25 128 0.20 1.11 173 0.30 1.02 207 0.65 1.06 390 1.20 .76 950 0.20 1.18 325 c. Traction over wood blocks with grain vertical. d. Traction over pavement of sand grains, grade (O). Value of C. Q-0.363 Q- 0.734 1.00 1.0 11 61 1.5 36 131 2.0 72 215 2.5 119 310 3.0 175 413 3.5 240 525 4.0 311 650 Parameters of ad- (a ... justing equa-^n... tions. \b\ . . 0.50 1.70 36 0.30 1.72 101 Probable error (per 1.1 0.1 Value of C for grade — (E) (H) (I) (J> (EiH.Ij) (E.HO.J,) 1.00 0.734 2.0 313 354 540 730 613 780 2.5 442 507 770 1,090 860 1,075 3.0 583 673 1,008 1,415 1,110 1,390 3.5 740 850 1,245 1,740 1,375 1,700 4.0 900 1,026 1,500 2,040 1,660 2,010 Parameters of adjusting I*' • • equations. 0.10 1.48 123 0.45 1.48 201 0.75 1.05 430 1.20 .81 880 0.50 1.16 385 0.60 1.06 545 206 TRANSPORTATION OF DEBRIS BY SUNNING WATER. TABLE 68. — Values of capacity for fume traction, adjusted in relation to slope of channel — Continued, e. Traction over pavement of pebbles, grades (H) and (I). w e S Value of C for grade — (E) (G) (H) (I) (EiGO (EsHA) (E3H2I3J2) 1.00 0.734 2.0 2.5 3.0 3.5 4.0 220 313 416 529 650 143 220 310 412 522 122 192 272 363 460 200 325 460 600 740 250 420 605 800 1,015 209 290 393 605 830 1,060 Parameters of adjusting j* • ' • equations. (ft 0 10 1.50 84 0.40 1.60 67 0.55 1.52 68 1.00 1.65 65 1.00 1.19 200 1.00 1.26 250 1.20 1.28 280 PRECISION. Probable errors were computed from the residuals of 25 series, the residuals being measured on the plots as percentages. The greatest probable error computed for a series of adjusted values of capacity is ±4.6 per cent, and their average is ±1.3 per cent. The aver- age of the 25 determinations of the probable error of an observation is ±3.8 per cent. The residuals number 139, and their average value, which also is a measure of the precision of the observations, is ±4.3 per cent. A comparison of these measures with those obtained from the data for stream traction shows that the flume-traction data are de- cidedly the more harmonious. The average residual is more than twice as great for stream traction as for flume. Part of this difference may be due to the fact that the experiments with flume traction came last and had the benefit of previous experience, but it is to be ascribed chiefly to the fact that in flume trac- tion the slope is constant, while in stream traction it is subject to rhythmic fluctua- tions. DISCUSSION. CAPACITY AND CHANNEL BED. Data illustrating the influence of the character of the channel bed on the quantity of debris which a stream can transport have been as- sembled in Table 69. They are taken chiefly from the preceding table, but a few items are from Table 72 and Table 12. A single item, marked as interpolated, is based on a combi- nation of data from Tables 68 and 72. They pertain to all the simple grades and mixed grades with which experiments were made in flume traction; and stream traction is repre- sented, so far as possible, by coordinate data. Comparisons are made for slopes of 2 and 3 per cent and discharges of 0.363 and 0.734 ft.3/sec. The greatest capacity is in each case asso- ciated with the smoothest of the tested channel beds, the surface being that of a plank, planed and painted, with the grain running parallel to the current. TABLE 69. — Comparison of capacities for flume traction associated with different characters of channel bed. [Width of trough, 1 foot.] Q S Character of channel bed. Value of C. Simple grades. Mixtures. (B) (C) (E) (G) (H) (I) (J) (EiGO (E2HiI3) (E3H2I3J2) 0.363 .363 .734 .734 2.0 3.0 2.0 3.0 Planed wood 388 300 205 145 234 160 72 45 393 285 175 254 195 364 249 410 341 Sawn wood. . . Sand pavement Debris 266 245 115 366 259 Planod wood 451 366 600 500 732 £82 Sawn wood Sand pavement. . . . Debris 120 393 331 Planed wood 548 382 290 313 470 354 354 711 533 540 790 453 870 660 613 970 Sawn wood. . . Woodblock 733 780 Sand pavement 215 143 145 668 540 Gravel pavement. . . 220 222 665 495 583 122 200 250 Debris 483 Planed wood 830 570 673 [1,060] 970 1,008 1,630 1,490 1,415 770 1,465 1,150 1,110 1,675 Sawn wood Woodblock... 1,390 Sand pavement 413 310 Gravel pavement 416 272 209 460 605 605 FLUME TRACTION. 207 Next in order are two varieties of unplaned wooden surface; the first being that of boards or planks, paiallel to the current, retaining the roughness left by the saw; the second a pave- ment made by sawing planks of Oregon pine into short equal blocks and setting them on edge. Both these surfaces, as well as those described below, are illustrated in Plate III. These two varieties proved to have approxi- mately the same properties in respect to trac- tion, and the capacities associated with them are 23 per cent less than those for planed lumber. The range in ratio is not large for the different experiments, and the value given may be taken as a constant representing the difference in efficiency between new unplaned and planed wooden flumes. The difference tends, however, to diminish with wear, the unplaned lumber becoming smoother and the planed rougher. The next grade of roughness was given by coarse sand — d6bris of grade (G) — set in cement, so as to constitute a pavement re- sembling sandpaper. The only material run over this was debris of the same grade, the special purpose being to compare flume trac- tion with stream traction — the condition of fixed bed with that of mobile bed — when the degree of roughness is the same. The experi- ments gave the streams 50 per cent greater capacity when sweeping the debris over the fixed bed than when moving it at the same slope by the method of stream traction. The sand pavement gives capacities half as great, on the average, as the surface of planed lumber, but the contrast is stronger for the smaller discharge and lower slope and less marked for the larger discharge and steeper slope. The roughest surface used, a pavement of pebbles prepared by setting in cement a mix- ture of grades (H) and (I), gave still lower capacities. These range from 20 to 62 per cent of the corresponding capacities given by planed lumber. The obstructing influence of the rough bottom is most strongly manifested when the material transported has a coarseness corresponding to the texture of the pavement. For finer material its roughness is mitigated by the lodgment of debris, which has the effect of establishing a pavement of the finer material. The word "debris" in the table indicates a channel bed composed of loose debris, the debris in transit, and the associated process is that of stream traction. The available data afford comparison only for the four finer grades, (B), (C), (E), and (G), the grades which would be designated sand. Each com- parison, with an apparent exception to be considered immediately, shows stream trac- tion to be less efficient than flume traction. When stream traction is compared with flume traction over a smooth surface, the observed ratio of efficiency ranges from 19 to 88 per cent, the smaller ratios being associated with the coarser grades of debris. The exception occurs when capacity over a bed of debris is compared with capacity over a pavement of pebbles, the two capacities being found to be the same. The cases which afford this comparison are for grades (E) and (G), and these fine materials, by filling the hollows of the pavement, create a condition of bed in which stream traction dominates. The comparison is really between normal stream traction and stream traction modified by the appearance of crests of fixed pebbles in the channel bed. In harmony with tliis interpreta- tion is the fact that capacities for stream trac- tion and for traction over the gravel pavement, when compared severally with capacity for trac- tion over smooth wood, both show contrasts which increase with coarseness of the load. The important general facts brought out by the comparisons are (1) that with a given dis- charge, channel width, and slope, the process of flume traction is able to transport more debris than that of stream traction, and (2) that a stream's capacity for flume traction varies inversely with the roughness of the flume bed. The first of these principles serves to explain certain phenomena of clogging. When there is fed to a flume a load greater than its stream is able to transport, a portion is deposited. This changes the character of the bed in such a way as to substitute stream traction for flume traction. Stream traction, being less efficient, can carry still less load, and a larger fraction is deposited. If the conditions re- main unchanged the bed is built up until its slope becomes that necessary to carry the en- tire load by stream traction. Unless the trough is deep or short, overflow results. When clogging is initiated by a temporary overloading, the stream loses power to carry its normal fractional load, and deposition con- 208 TRANSPORTATION OF DEBRIS BY RUNNING WATER. tinues unless the load is reduced considerably below the normal. In practical operations the first step toward the abatement of a clog is te stop all feeding of load above the deposit, that the stream may be able to take on load and thereby reduce the deposit. CAPACITY AND SLOPE. The rate at which capacity for flume traction is increased by increase of slope is contained implicitly in the values of n and a assembled in Table 68; for n gives the rate of variation of capacity with 8 — a, and the instantaneous rate of variation of capacity with slope is given by ij = -q . All the tabulated values of n except two are greater than unity; and in each of the cases where n is less than unity all values of \ computed for the range of slopes covered by the experiments are greater than unity. The gen- eral fact is thus indicated that, within the prac- tical range of conditions, capacity increases with slope in more than simple ratio. Effi- ciency also increases as slope increases. The sensitiveness of capacity to changes in slope varies with changes of condition; and this variation might be illustrated, as in treating of stream traction, by the tabulation and discus- sion of values of the index of relative variation, *!- It will suffice, however, in this case to make comparisons by means of the synthetic index, 7t. Table 70 contains values of that in- dex computed between the limits of S = 2.0 and 8 = 3.5. They represent 42 of the 51 series of values given in Table 68, the other nine series not having sufficient range for the computa- tion. The arithmetical mean of the 42 values of 7t is 1.46, and the range is from 1.08 to 2.08. TABLE 70. — Values of /, for flume traction, computed between the limits S—2.0 and S=3.5. w Q Character of channel bed. Value of /i. Simple grades. Mixtures. (C) (E) (G) (H) (I) (J) (E,Gi) (EjHJ,) (EiHjIjJ:) 1.00 1.00 1.91 1.91 0.363 .734 .363 .734 Planed wood 1.66 1. It 1.43 .37 .30 .54 1.30 1.42 1.30 1.20 .41 .54 .40 .17 .56 1.40 .42 .32 .29 1.52 Sawn wood 1.66 Planed wood 1.66 1.69 1.30 1.35 Sawn wood 1.27 1.49 Woodblock 1.55 .44 1.39 Sand pavement 1.60 Gravel pavement .56 .50 .34 1.89 1.32 1.22 .95 .40 1.23 1.96 2.08 Planed wood do Inspection of these data shows, first, that the values are always greater for Q = 0.363 than for Q = 0.734. The experiments deal with no other discharges, but it is probably true in general (as in case of stream traction) that increase of discharge is accompanied by decrease of the sensitiveness of capacity to slope. If the index varies in a systematic way with fineness of d6bris, its increase is connected with decrease of fineness, but the finest debris of the table, (C), carries large values of the index. The apparent conflict of evidence has its parallel in the fuller data for stream trac- tion (see p. 108 and fig. 34), and it is possible that the sensitiveness increases in two direc- tions from a minimum value. Its variation might in that case be connected with the law relating capacity to fineness, as brought out in a later section. The relation of the index to roughness of bed does not follow a simple law. Its values are in general least for the bed of rough lumber and progressively greater for planed lumber, wood blocks, and gravel pavement. The greater sensitiveness of capacity to slope when the channel bed is a coarse pavement may be connected with the fact that the mode of transportation over such a bed is approxi- mately stream traction; and this suggests that in flume traction the sensitiveness may be less than in stream traction. Direct comparison can not be made with use of the values of 7, in Table 70, because the slopes used in stream traction experiments have less range; but spe- cial computations were made, so far as the data were found to overlap. The results are con- tained in Table 71 and indicate that the sensi- tiveness is greater for stream traction than for flume traction over a smooth bed, in case of grades (E) and (G), but less in case of grade (C). FLUME TRACTION. 209 TABLE 71. — Comparison of values of Itfor flume traction over a bed of planed wood, with corresponding values jor stream traction. Width of trough (feet). Grade of debris. Dis- charge (ft.a/sec.). Limiting values of slope (per cent). /ifor flume traction. /i for stream : traction. 1.00 (C) 0.363 0.5-2.0 1.71 1.60 .734 .5-2.0 1.79 1.46 ' (E) .363 .5-2.0 1.50 1.99 .734 .5-2.0 1.41 1.-65 (G) .363 2. 0-3. 0 2.27 2.55 .734 1.0-2.0 1.38 2.42 Various qualifications and doubts being omitted, the preceding paragraphs may be generalized by saying that the sensitiveness of capacity to slope is somewhat less in flume traction than in stream traction. It varies in both directions from a mean value expressed by the exponent 1 .5, being greater as the slope is less, as the discharge is less, as the fineness is less, and as the channel bed is rougher. Efficiency for flume traction increases with slope. CAPACITY AND DISCHARGE. Special series of experiments were made to determine the variation of capacity with dis- charge. In each series the conditions of slope, width, and grade of debris were kept constant and the discharge was varied. The observa- tions are given in Table 72, and the same table contains the adjusted values of capacity, to- gether with the parameters of the adjusting equations. Inspection of logarithmic plots showed the propriety of adjusting by means of the formula used with the data for stream traction, <7=&3«2-K)° (64) and the computations were graphic. TABLE 72. — Observations and adjusted data illustrating the relation of capacity for flume traction to discharge , for a rectangular flume of planed wood 1 foot wide. [L, observed load; C, adjusted value of capacity; Q, discharge.] (B) (E) (H) (I) (EjHiIi) (E) (H) (I) (Ejllilj) Q L L L L L L L L L 0 039 28.0 25.0 .093 54.5 79 128 50.6 87 .182 151 96 136 110 144 169 228 272 325 .545 .734 619 322 446 417 546 528 738 617 907 580 785 635 914 922 1,170 718 1,147 1,550 Parameters of adjusting equations .. K 0 63 0.002 1.34 1,430 0.9 0.007 1.09 630 0.2 0.040 .88 760 0.4 0.140 .68 1,000 0.9 0.000 1.05 1,345 0.5 0.004 1.10 1,125 0.3 0.025 .88 1,175 1.0 0.080 .79 1,650 0.7 0.-040 .88 2.220 1.4 Q C C 0 C C C C C C 0.039 28 27 093 58 79 108 53 123 .182 145 95 134 116 146 169 230 270 325 .363 .545 .734 371 640 205 322 445 415 549 548 711 «27 890 572 795 450 670 870 600 880 1 170 730 1,130 1 550 Several values of capacity in Table 72 agree as to conditions with values in Table 68, and these values would be identical if the experi- ments were homogeneous. A comparison shows that the values given by the experiments comparing capacity and discharge are in gen- eral the greater, the average difference being 6 per cent. This is evidently of the nature of systematic error and is probably connected with some change in apparatus or in detail of 20921°— No. 8&— 14 14 experimental method which occurred between the making of the two groups of experimental series. The sensitiveness of capacity to changes of discharge varies with conditions. It is greater as the discharge is less, as the slope is less, and as the channel bed is rougher. It is relatively great for the coarsest and finest of the de'bris used and less for intermediate grades. Un- der similar conditions it is less for flume trac- 210 TRANSPORTATION OF DEBRIS BY RUNNING WATER. tion than for stream traction. Expressed as an exponent, 73, its average value for the range of the experiments recorded in Table 72 is 1 .26. Values of 73 were also obtained from data in Table 68 by comparing the capacities for Q — 0.363 ft.3/sec. with those for <> = 0.734 ft.3/sec.; and the mean of such values is 0.97. These mean values are not necessarily inconsistent, for the synthetic index varies with the range in discharge for which it is computed and is lower as the discharges are higher; but a study of in- dividual values shows that under identical con- ditions the data of Table 68 give the lower esti- mates of sensitiveness. The data as a whole indicate that, for the range of conditions real- ized in the experiments, the average value of the exponent expressing sensitiveness to dis- charge is 1.2. There can be no question that for the larger discharges used it falls below unity. As the sensitiveness of the duty of water, and also the efficiency, to discharge is expressed by an exponent which is less by unity than the corresponding exponent for the sensitiveness of capacity to discharge, it follows that duty and efficiency vary little with discharge. In general they gain slightly with increase of discharge, but they lose when the discharge or slope is relatively large. This accords with a result obtained by G. A. Overstrom/ who found from experiments with launders that duty rose with increase of depth to a limited extent only. CAPACITY AND FINENESS. In Tables 68 and 72 the values of capacity standing in any horizontal line constitute a series illustrating the variations which are re- lated to grades of debris, and if those in the columns for mixtures be excepted they illus- trate the relations of capacity to fineness. Table 73 contains a selection of data from those tables, together with a single line taken from Table 69. TABLE 73. — -Values of capacity for flinne traction, illustrating the control of capacity by fineness of debris. Character of channel bed. w Q S Valu» of C for grade — (B) (C) (E) (G) (H) (I) (J) .91 .00 .00 .00 .00 .00 .00 .00 .00 0 734 363 363 734 734 734 734 734 363 2.0 2.0 3.0 2.5 3.0 3.0 3.0 459 415 202 366 518 665 495 583 416 115 460 550 268 451 645 830 570 673 272 388 383 590 790 1,050 398 527 668 540 625 1,220 1,630 1,490 1,415 653 910 970 1,008 200 Wood block 310 45 De'bris (stream tra ction) 2.0 266 245 As the tables are examined, one of the fea- tures arresting attention is that in most of the series the smallest value of capacity does not appear at one end of the line but at some inter- mediate point. The occurrence of a minimum is in fact characteristic of all tested varieties of flume traction except that in which the bed is a pavement of pebbles. To give the feature graphic expression the data of the last five lines of Table 73 are plotted in figure 69, where the horizontal scale is that of linear fineness, F. The plotted points are far from regular, but the general character of the representative curves is unmistakable, and freehand lines have been drawn. On another sheet, not reproduced, the same data were plotted in relation to mean diameter of particles — the reciprocal of F— with similar result, except that the lower two curves became concave upward. These curves illustrate the most important difference between the laws of flume traction and those of stream traction. In stream trac- tion capacity increases continuously as fineness increases. In flume traction capacity increases with fineness when the grades of debris com- pared are relatively fine but increases with coarseness when the grades are relatively coarse. So far as these experiments show, the minimum of capacity corresponds to a coarse sand, but its position on the scale of fineness may be assumed to vary with slope, discharge, and roughness of bed. The curve for flume traction over a bed paved with gravel shows no minimum but is of the same type as the curve for stream trac- tion. This fact is confirmatory of an inference 1 Quoted by R. H. Richards in Ore dressing. FLUME TRACTION. 211 already drawn (p. 207), that the transportation of fine debris over a fixed bed of coarser debris particles is essentially of the nature of stream traction. It may fairly be inferred that if we were able to extend this curve into the region of debris coarser than the gravel of the bed, a minimum would be developed. If the curves were to be traced toward the right, by means of additional experiments with 1,400 200 Linear fineness 600 FIGURE 69. — Curves illustrating the relation of capacity for flume trac- tion to fineness of de'bris. Data from bed of planed wood are recorded by crosses; from wood-block pavement by circles; other data by dots. finer de'bris, there can be little doubt that they would be found to continue their ascent ; but eventually, as curves of traction, they would come to an end with the passage of the process of transportation from traction into suspension. In the opposite direction they may be conceived to attain a maximum and then drop suddenly to the base line; for despite the law of increase of capacity with coarseness, there must be a degree of coarseness for winch the force of the current is not competent, and when that is reached the ordinate of capacity becomes zero. The position of this limit, which I have in earlier pages called competent fineness, evidently depends on slope and discharge, as determining the force of the current, and on the degree of rounding of the de'bris. The double ascent of the curve of flume traction is susceptible of plausible explanation, by means of considerations connected with the process of rolling. The process of rolling in- volves a question of space. Each rolling pebble occupies an area of the channel bed somewhat larger than its sectional area, even if the pebbles are arranged in the closest possible order. If we conceive the channel bed to be occupied by rolling pebbles of a particular size, separated by spaces which bear a definite ratio to the diameters of the pebbles; and again conceive it to be occupied by rolling pebbles of a larger size, with the same ratio between interspace and diameter; it is evident that the total volumes or masses of pebbles in the two cases will be proportional to the diameters. If the larger pebbles have twice the diameter of the smaller, then a given area of bed will contain twice as much rolling load of the larger pebbles as of the smaller. It is also true, as stated on an earlier page, that the rolling speed is somewhat greater for larger pebbles than for smaller. The tendency of these two factors is the same, to make the load greater for large particles than for small, when the process of transportation is rolling. The analysis is doubt- less too simple — the degree of crowding on the bed, for example, may not be the same for different sizes, and the degree of crowding may affect the speed of rolling— but qualifying factors can hardly impair the qualitative inference that the rolling load increases with coarseness. It is a matter of observation that, under similar conditions determining force of current, the dominant process in flume traction is for coarse debris rolling and for fine debris salta- tion. When the process is rolling, as just shown, capacity increases with coarseness of de'bris. When it is saltation, as illustrated by the body of experiments on stream traction, capacity increases with fineness. With the passage from saltation to suspension the effect is even heightened, and it is probable that in a number of the recorded experiments the process was largely that of suspension. Thus the double ascent of the capacity-fineness curve is determined by the distinctive properties of two (or three) modes of propulsion. If the preceding explanation is well founded, the nature of the law connecting capacity with the degree of comminution of the debris in any particular case depends on those conditions which determine the dominant process of con- veyance. If the channel bed is smooth, and if 212 TRANSPORTATION OF DEBRIS BY RUNNING WATER. slope and discharge are so adjusted as to give a moderate velocity, the progression of sand may be by rolling, and in that case the capacity for different sands will vary inversely with their fineness. But if over the same smooth bed the current runs swiftly, sand will be made to travel by saltation, or by saltation and suspen- sion, and then the capacity for different sands will vary directly with their fineness. The par- ticular velocity with which the function re- verses will depend on the quality of the bed, being lower if the bed is somewhat rough, because roughness changes rolling to saltation. The critical velocity will be higher for gravel than for sand, because higher velocity is needed to make coarser debris leap. The experiments which have been made were not sufficiently varied to afford test for these inferences, and as there is no present oppor- tunity for continuance of laboratory work the inferences must be regarded as largely hypo- thetic. TABLE 74. — Capacities for flume traction of mixed grades and their component simple grades. Character of bed. Q Mixed^grade. Capacity for mixture (gm./sec.). Capacity for single grade (gm./sec.). (E) (G) (H) (I) (J) 0.363 (F,GO 450 770 1,465 1,675 582 1,115 1,110 1,390 460 605 605 225 366 385 665 586 665 502 665 233 259 446 495 444 583 417 583 230 416 242 416 181 416 225 398 385 668 Do .734 .734 (EiGi) Do (EsHiIj) 293 830 335 830 116 366 223 570 222 673 278 673 586 910 502 910 233 500 446 970 444 1,008 417 1,008 Do .734 335 1,630 .363 (EsHiIs). .. Do Wood block .734 .734 .734 .734 734 (EjHiIj) (EjHiIz) Do (E,H,I|Jt) (EiGi) 230 310 278 1,415 Do 121 272 121 272 242 209 181 209 Do .734 (E,H2ISJ!) 121 MIXTURES. Three mixtures of simple grades of debris were treated in the laboratory. The compo- nents of one, (E^), were a medium sand and a coarse sand. The others, (E2HjI2) and (E3H2I3J2), combined medium sand with fine gravel and coarser gravel. All three were tested in relation to slope on the smooth chan- nel bed and on the gravel pavement — one on the rough-sawn bed and two on the bed of wooden blocks. One entered into the experi- ments on capacity in relation to discharge. The results are contained in Tables 67, 68, and 72. The data from those tables which pertain to a channel width of 1 foot and a channel slope of 3 per cent are assembled in Table 74. The table contains also the capacity quota for each constituent, computed from the capacity for the mixture; and beneath each of these quotas is printed the capacity for the constituent when the entire load is composed of it. A general fact brought out by this table is that the current can transport more of a mixture than it can of any one of the constit- uent grades. The table records a single excep- tion, the capacity for the mixture being 1,390 gm./sec., while that for its coarsest constituent grade is 1,415 grams. Another general fact shown is that the ca- pacity for each component as part of a mixture is less than the capacity for the same compo- nent if transported separately. To this also there is a single apparent exception, but as it FLUME TRACTION. 213 occurs among the data for traction over a gravel pavement it illustrates a feature of stream trac- tion rather than flume traction. The two principles may be illustrated to- gether by saying that if a stream is carrying its full load of a grade narrowly limited in range of fineness, and a different grade of debris is added, the total load is thereby increased, but this increase is accompanied by a diminution of the quantity carried of the first-mentioned grade. In contrast with this is the law found for stream traction — that the load of the ini- tially transported grade is increased by the moderate addition of other debris, provided the added d6bris is relatively fine. The difference between the two cases is thought to be connected with rolling. In flume traction over a smooth bed the path for rolling particles is roughened by the presence of smaller particles. In stream traction the pathway for larger particles is smoothed by the presence of smaller particles and rolling is promoted. In stream traction the capacity for a mixture is determined chiefly by the capacity for its finer components, and as mean fineness also depends chiefly on the fineness of the finer componejits, mean fineness is a serviceable gage of capacity. In flume traction the rela- tion is quite different. Because of the double ascent of the curve of capacity and fineness, it may readily occur that the capacity for a mixture is most nearly related to that for the coarsest component — in fact, that is true of the three mixtures tested in our experiments — and when that is the case there is no parallelism between capacity and mean fineness. CAPACITY AND FORM RATIO. The data bearing on the relations of capacity to the depth and width of current, and their ratio, are meager. Most of the experiments were conducted with a single trough width, 1 foot. The only other width used was 1.91 feet, and its use was associated with but four grades of debris and a single character of channel bed — the smoothest. Depths of current were not in general meas- ured during the passage of loads, because the surfaces of load-bearing currents were usually so rough as to make good determinations im- possible. Good measurements were made of unloaded streams, and the results are here tabulated. Attempts to measure depths of loaded streams yielded one result thought worthy of record. With a discharge of 0.734 ft.3/sec., a width of 1 foot, a slope of 4 per cent, and a full load of debris of grade (E3IT2I3J2), the depth was 9 per cent greater than for the corresponding unloaded stream. Table 76 compares the capacities found for a trough width of 1.91 feet with corresponding capacities for a width of 1 foot. By aid of Table 75 it brings capacities into relation also with depths. TABLE 75. — Depths and form ratios of unloaded streams, in troughs of wood, planed and painted. to— 1.91 w-1.00 Q S d R d R 0.363 1 0.076 0.040 0.120 0.120 2 .062 .032 .096 .096 3 .050 .029 .082 .082 .734 1 .119 .002 .194 .194 2 .098 .051 .154 .154 3 .086 .045 .130 .136 TABLE 76.— Capacities for flume traction in troughs of different wullhs. Q S Grade (C). Grade (E). Grade (G). Grade (H). w— 1.91 w-1.00 w-1.91 to- 1.00 tt=1.91 1C- 1.00 to- 1.91 W-1.00 Values of C. 0.363 .734 .363 .734 1.0 2.0 3.0 1.0 2.0 3.0 1.0 2.0 3.0 1.0 2.0 3.0 n 93 224 300 69 200 370 158 415 720 74 205 366 145 382 665 85 241 415 185 460 755 87 234 398 151 393 668 275 491 254 451 197 177 459 543 550 915 470 830 C...I ClM 0.83 .75 0.93 .98 1.01 1.09 1.09 1.08 0.98 1.03 1.04 1.22 1.17 1.13 1.08 1.08 1.11 .89 1.17 1.10 214 TRAKSPORTATION OF DEBRIS BY RUNNING WATER. Simple combinations of the quantities, into which it is not necessary to enter, show (1) that with constant depth, capacity increases with width and more rapidly than width, and (2) that with constant width, capacity increases with depth and more rapidly than depth. The rate of increase with depth, if expressed as an exponent, may be as low as 1.2 or as high as 2.0. From these premises, a line of reasoning parallel to that of Chapter IV shows that for flume traction, as for stream traction, the function C=f(R) increases to a maximum and then decreases. The value of R corre- sponding to maximum capacity — the optimum form ratio — can not be determined from obser- vations involving but two channel widths, but its limits can in some cases be indicated. For example, when the capacities conditioned by the same slope and discharge are approximately the same for the two trough widths, it may be inferred that the optimum falls between the values of R associated with the two capacities. Thus for grades (E), coarse sand, with a slope of 2 per cent, the capacities are about the same for the two widths, while the form ratios are 0.032 and 0.096, or one-thirtieth and one- tenth; and it is inferred that the optimum form ratio falls between those fractions. From inferences of this sort, used in combi- nation with the general principles of flume traction, a number of tentative conclusions have been drawn. They are of so hypothetic a character that the reasoning connected with them is not thought worthy of record, but the conclusions themselves may perhaps be of some service, until replaced by others of more secure foundation. They are: The ratio of depth to width giving a current the highest efficiency for flume traction (1) is greater for gentle slopes than for steep, (2) is greater for small discharge than for large, (3) is greater for fine debris than for coarse, (4) is greater for rough than for smooth channel beds, and (5) is in general less than for stream traction. The first and second propositions apply also to stream traction. The third is the reverse of the relation determined for stream traction, and its applicability may be limited to condi- tions under which rolling is the dominant mode of transit. By aid of the fifth and fourth Table 31 may be roughly applied to practical problems of flume traction. TROUGH OF SEMICIRCULAR CROSS SECTION. A few experiments were made with a trough of galvanized sheet iron, 1 foot wide, having a semicircular section. It was given a slope of 1 per cent, and in it were tested four grades of debris. The observational data appear in Table 77. The object of the experiments was to determine whether a channel with curved perimeter is more efficient or less efficient than one with rectangular section; the capaci- ties obtained are compared, in the table, with those determined for a flat-bottomed trough of the same width. As the discharge used did not fill the semicylindric trough, the width of water surface was less than 1 foot. The width of channel bed occupied by the load ranged from one-fifth to one-half of the width of water surface. The medial depth of water was greater than in the rectangular channel, and the mean velocity was higher. The higher velocity is a factor favorable to the develop- ment of capacity; the narrower field of traction is an unfavorable factor. The resultant of the two was unfavorable, the capacities for the semicylindric trough being only half as great as for the rectangular. The result is qualified by the fact that the troughs compared were not of the same material, but the disparity of capacities is too great to be ascribed to that factor. TABLE 77. — Data on flume traction in a semicylindric iron trough of 0.5 foot radius: with comparative data for a rec- tangular wooden trough 1 foot wide. Slope (per cent). Discharge (ft.«/sec.). Grade of debris. Capacity (gm./sec.). In semi- cylindric trough. In rec- tangular trough. 1.00 0.363 .734 (C) (F (]•: (F (G) 43 88 37 64 66 74 93 74 145 151 Depth ( stream ( Width of unloade< Width of (foot) if unloaded IQ- 0.363... foot). \Q=0.73» 0.181 .257 .770 .874 .17 to. 37 3.85 4.60 0.120 .194 1.00 1.00 1.00 3.02 3.78 water surfaceJQ™ 0.363... lstream(foot)\Q= 0.734... be occupied by oad Mean velocity of un-/Q-0.3R3... loaded stream(ft./sec.)\Q=0.734... The fact that the doubling of the discharge did not double the observed load indicates that the duty of water diminishes as discharge FLUME TRACTION. 215 increases, but the data are too few to give confidence to the inference. The three capaci- ties with a discharge of 0.363 ft.3/sec. indicate a minimum in the curve of C=f(F), thus sup- porting the generalizations already made from the results with rectangular troughs. The low efficiency here found for an open channel having a circular arc for its perimeter suggests that the cylindric form commonly given to closed conduits for the hydraulic con- veyance of debris may not be the most efficient. A second suggestion is connected with the fact that the semicylindric trough, while it nar- rows the field of traction, at the same time gives a high velocity to the water. It thus concen- trates the available force and energy on the narrow field. Though the result is not favor- able to capacity, it may be favorable to com- petence. When but a small load is to be trans- . ported, the practical problem may be one of competent velocity ; and such a trough appears well adapted to the production of competent velocity with economy of discharge and slope. SUMMARY. Iii the transportation of d6bris in flumes much of the movement is usually by rolling and sliding. This is especially true if the cur- rent is gentle or the debris coarse. With a very swift current or with fine debris the par- ticles travel by a series of leaps, and with the finest debris the load is suspended. When the conditions are such that the principal move- ment is by rolling and sliding, the capacity of the current increases with the coarseness of the debris transported, this law holding good up to the limit of coarseness at which the current is barely competent to start the particles. When the conditions are such that the princi- pal movement is by saltation, the capacity of the current increases with the fineness of the debris, the law holding good up to and prob- ably beyond the critical fineness at which the current is competent to carry the debris in suspension. Under all conditions the capacity is increased by steepening the slope, and the increase of ca- pacity is more rapid than the increase of slope. The capacity may vary with a power of the slope as low as the 1 .2 power, or with one higher than the second. A general average for the experimental determinations is the 1.5 power. Under all conditions the capacity is increased by enlarging the discharge. It may be in- creased in the same ratio, in a higher ratio, or in a somewhat lower ratio. The highest capacity is associated with the smoothest channel bed. Progressive increase of roughness reduces capacity progressively un- til the texture of the bed becomes coarser than the debris of the load. The mode of transpor- tation then passes from flume traction to stream traction. Under like conditions of slope, dis- charge, and character of debris, flume traction gives higher capacities than stream traction. Rectangular or box flumes have higher ca- pacity than semicylindric flumes of similar width. Up to a limit, which varies with con- ditions, the capacity is enlarged by increasing the width of channel at the expense of depth of current. The ratio of depth to width which gives highest efficiency has not been well cov- ered by the experiments, but it is believed to be rarely greater than 1:10 and often as small as 1 : 30. For large operations the determina- tion of width will usually represent a compro- mise between efficiency and the cost of con- struction and maintenance. As most of the experiments were made with sorted debris, each grade being nanowly limited as to range in the size of its particles, and as most practical work is with aggregations having great range in size, the loads and capacities here reported need qualification. By experiments with mixtures of the laboratory grades it was found that the load carried of a mixture is greater than the load of any one of its important components taken separately. It is in general true that the capacities for com- plex natural grades of debris are greater than the tabulated capacities for the laboratory grades they most nearly resemble. COMPETENCE. The experiments in flume traction were prac- tically limited in their range by phenomena of competence, and these limitations were of use in determining values of a, K, and , but no effort was made to observe competence di- rectly and precisely. There are, however, a few observations by others, which may properly be assembled here, although it is not practicable to use them as checks on our work. Our in- definite data pertain to slope and discharge, 216 TRANSPORTATION OF DEBRIS BY RUNNING WATER. while the observations of others pertain to velocity. The experiments of Dubuat * (1783) have been assumed, both by him and by others, to pertain to stream traction, but his account of apparatus and methods makes it probable that what he really investigated was chiefly com- petence for flume traction. He used a trough of plank, with the grain lengthwise, and meas- ured the velocity of the current by observing the speed of balls slightly heavier than water as they were swept along the bottom. In a current of a particular velocity he placed successively various kinds of debris and noted their behavior, then changed the velocity and repeated. His results are as follows: Competent bed velocity (ft./sec.). Potter's clay Between 0.27 and 035 0.7 1.1 0.35 0.62 1.1 2.1 3.2 0.53 0.7 1.55 3.2 4.0 Coarse angular sand River gravel: Size of anise seed Size of peas Size of common beans Rounded pebbles, 1 inch in diame- ter Angular flints, size of hen's eggs. . . J. W. Bazalgette, in discussing the flushing of sewers and therefore presumably consider- ing flume traction rather than stream traction, quotes the following results of experiments by Robison : 2 Competent bed velocity (ft./sec.). Fine sand 0. 5 Sand coarse as linseed 67 Fine gravel 1.0 Round pebbles, 1 inch in diameter 2. 0 Angular stones, size of eggs 3. 0 In 1857 T. E. Blackwell3 conducted elabo- rate experiments to determine the vel >cities necessary to move various materials in sewers. His channel was of rough-sawn plank, 60 feet long and 4 feet wide, with level bottom. Ve- locities were measured by a tachometer, but the relation of the velocity measurements to the bed is not stated. The tests were applied to natural d6bris of various kinds and also to types of artificial objects likely to enter sewers. The objects were treated singly and in aggregates, with the general result that an aggregation re- 1 Dubuat-Naneay, L. G., Principes d'hydraulique, vol. 1, p. 100; vol. 2, pp. 57, 79, 95, Paris, 1786. 2 Inst. Civil Eng. Proc., vol. 24, pp. 289-290, 1865. 3 Accounts and papers [London], Sess. 2, 1857; Metropolitan drainage, vol. 36, Appendix IV, pp. 167-170, Pis. 1-5. quires higher velocity to move it than does a single object. It is evident that the experi- ments on single objects pertain to flume trac- tion and some of those on aggregations to stream traction. From his tabulated results the subjoined data are selected as representing or illustrating the velocities competent for natural debris, the column of mean diameter being added by me. He infers from the ex- periments that (1) for objects of the same character competent velocity increases with the mass; (2) for objects of the same size and form it increases with the specific gravity; (3) for objects of different form it is greater in propor- tion as they depart from the form of a sphere; and (4) for objects in motion the rate of travel increases with the velocity of the current. TABLE 78. — Observations by Blackwell on velocity competent for traction. Material. Volume. Mean di- ameter. Competent velocity. Single objects (illustrating flume trac- tion) : Brickbat (roughly cuboid).. Cubic inches. 18.5 Feet. 0.27 Ft./sec. 1 75-3 00 Do 12.98 24 2 25-2 50 Do... 13.6 .25 2 00-2 25 Do 7 33 20 2 00-2 25 Do... 4.76 .17 2 25-2 50 Do 2 59 14 1 75-2 00 10.37 .22 3 00-3 25 Do 6 05 19 2 00-2 25 Do 4.11 16 2 25-2 50 Do 1.95 .13 2 50-2 75 20 1 50-1 75 Do .16 2 25-2 50 13 2 50-2 75 Aggregations(illustrating stream trac- tion): Gravel .042 2.25-2.50 Do .021 1 25-1 SO Sand 0 -1 00 WORK OF OVERSTROM AND BLUE. Certain experimental work on the capacity of currents for flume traction has for its specific purpose the determination of dimensions for launders, the flumes in which pulverized ore is conveyed. R. H. Richards's "Ore dressing" 4 contains an abstract of results obtained by G. A. Overstrom, accompanied by the statement that the experimental data are extensive but as yet unpublished. The troughs employed were flat-bottomed and probably of wood. For each slope, width of flume, and grade of transported material he found (1) that the duty of water varies with the discharge and that some par- ticular discharge is associated with a maximum duty, so as to be the most economical; (2) that < Vol. 3, pp. 1592-1594, 1909. FLUME TRACTION. 217 the most economical discharge is sensibly pro- portional to the width, so that for each slope and grade of material there is a particular dis- charge per unit of width giving a maximum duty; and (3) that the most economical dis- charge is greater for low slopes than for high. The first of these results is in fair accord with our own. Five of the nine values of the expo- nent o in Table 72 are loss than unity. For the corresponding series the values of ia range both below and above unity and the corre- sponding values of the variable exponent for duty in relation to discharge, i3— 1, range below and above zero. The value zero evidently cor- responds to a maximum value of duty. His third result is in strict accordance with ours; his second can not be compared without fuller details. A diagram exhibiting his determinations for the traction of crushed quartz sized by 40-mesh and 150-mesh sieves shows for different dis- charges per unit width the variation of duty with slope. For the larger discharges duty varies as the first power of slope ; for the small- est discharge with the second power. This cor- responds to a variation of capacity with the second to third power of slope. Our most available data for comparison are those of grade (C), the capacity for which varies with the 1.66 power of slope (Table 70). As grade (C) was separated by 30-mesh and 40-mesh sieves, it is considerably coarser than the crushed quartz, and, being stream worn, it is less angular. The marked difference in the observed laws of variation is evidently suscepti- ble of more than one interpretation, but it is thought to be connected with difference in fineness, as more fully stated on a following page. F. K. Blue l made a series of experiments in which the trough was of sheet iron, 50 feet long, 5 inches deep, and 4 inches wide, the bottom being semicylindric with 2-inch radius. It was so mounted that it could be set to any slope up to 12 per cent. Two materials were used as load, the first a beach sand of 60-mesh aver- age fineness, the other a sharp quartz sand of about 80-mesh fineness, containing about 10 per cent of slime from a stamp mill. With each material the discharge and load were varied; and for each combination of discharge and load i Eng. and Min. Jour., vol. 84, pp. 530-539, 1907. the slope was adjusted to competence, and mean velocity was determined by means of a measurement of depth. Discharge and load were not measured directly but in certain com- binations. Instead of discharge, the total vol- ume of water and load was measured. This quantity was used chiefly in the computation of mean velocity, for which purpose it is better fitted than is discharge alone. Load was meas- ured as a volume, the volume of the transported material as collected in a settling tank, and is reported only through a ratio, q, which is the quotient of the volume of load by the volume of discharge plus load. This is essentially a duty but differs materially from duty ( U= £. J as defined in the present report. Representing by W the weight in grams of a cubic foot of debris, including voids, and by v the percentage of voids, it follows from the definitions that U qW 2= w + m=v)' and = 1-gd-t;) From the discussion of his data Blue finds (1) that q varies as the square of the slope and (2) that it varies as the sixth power of the mean velocity. He does not specifically con- sider the relation of q to discharge, but exami- nation of his tabulated data shows that q is but slightly sensitive to variations of discharge plus load. As Blue's coarsest material, the beach sand, has approximately the fineness of our grade (A), while the finest we treated in flume traction is of grade (C), the most definite comparison of results can not be made, but there is neverthe- less interest in such comparison as is possible. Computing values of £7 from his data for beach sand, and plotting them in relation to slope, I obtained UxS2-02 This gives for capacity and slope, C Annales des ponts et chaussees, M&n., Sth ser., vol. 1, p. 270, 1871. a Quoted, with some of the data, by Partiot, idem, pp. 271-273. The coefficient is there erroneously given as 0.0013. < Given by Lechalas in the same volume, pp. 387-388. bars by their migration downstream. In some of the "regularized " streams of Europe-, where the main channel is artificially restricted to curves of large radius, they are developed in systematic alternation at the two sides, and the thalweg winds between them.5 In the Missis- sippi they sometimes appear in the reaches. Their progress downstream is accomplished by deposition on forward slopes and erosion of rear slopes, but the forward slopes are not steep, like those of dunes, and their material is not wholly deri ved from the traction al load. Blasius 8 re- gards them as essentially dunes, correlating them specifically with dunes of reticulated pat- tern. My own view, not necessarily inconsist- ent with his, connects them with the fixed bars separating the deeps of a meandering stream. A free stream does not tolerate a straight chan- nel. If a straight channel of moderate width be given to a stream, the current swings rhyth- mically to right and left, and if the banks yield it develops meanders. The meanders then migrate, according to laws of their own, and the bars are fixed in relation to the mean- ders. If the banks do not yield, the system of shoals and deeps established by the swinging current migrates slowly downstream. It is evident that the migration of these shoals is one of the factors — and may be an important factor — in the work of transportation ; and also that every measurement of the migration of a shoal is a partial measurement of load. Pilots of Mississippi steamboats observe that the bars at crossings are built up by floods, and such changes have been measured by engineers. The generalization has sometimes been made that deposition is a specific function of floods, but a more satisfactory interpretation is given by McMath,7 who maintains that the rising river scours from the deeps to deposit on the shoals, and the falling river scours from the shoals to deposit in the deeps. The transfers are the joint work of traction and suspension. As such changes of the stream bed are measurable they afford quantitative data as to load, and it was from their observa- tion that Hider, as previously quoted, inferred that the dune movement in the Mississippi includes but a small fraction of the tractional load. t Engels, H., Zeitschr. Bauwesen, vol. 55, pp. 604-680, 1905. s Idem, vol. CO, pp. 4(3-472, 1910. i Mississippi River Comm. Kept, for 18X1, p. 252. APPLICATION TO NATURAL STREAMS. 233 The dune movement, the migrations of greater bars, and the transfers of debris from deep to shoal and shoal to deep are all compe- tent to give information as to tractional load, but the estimates they give are minimum estimates, to be supplemented by estimates of the material which at flood stages is swept steadily along without contributing to any of the temporary deposits in such way as to be accessible to measurement. AVAILABILITY OP LABORATORY RESULTS. THE SLOPE FACTOR. Wo are now ready to inquire whether, in view of the diversities and complexities affect- ing traction by natural streams, the formula for tractional capacity derived under the comparatively simple conditions of the labora- tory is of practical value in connection with natural streams. The four factors of the formula may first be considered separately. The general slope of a stream is the quotient of fall by distance, the distance being taken along the stream's course. It is best measured at high stage, because the chief work of grading the channel is accomplished by floods. With reference to variations in capacity at a single locality, slope does not enter, the varia- tions being referred to discharge; but account must be taken of slope in comparing different divisions of the same stream and in comparing one stream with another. In all such cases the stream's slope is as definite a quantity and is susceptible of as precise measurement as is the slope of the laboratory channel. It differs as to its repre- sentative character. The laboratory slope is connected with a single discharge and a single grade of debris of determinable fineness. The slope of the natural stream does not represent the adjusting work of a determinable discharge but is a compromise product of the work of many discharges, and it is usually true that the velocities associated with these discharges have determined equally diverse mean fine- nesses of debris. The work of the natural stream, moreover, has been characterized by greater diversity, from point to point, of the bed velocities, and its system of velocities has been regulated in part by suspended load. These difficulties would prove insuperable if attempt were made to infer the capacity of a natural stream from that of a laboratory stream, but they are not necessarily important in transferring a law of variation from a group of laboratory streams to an equally harmonious group of naturalstreams. If the diversification of discharges and finenesses is of the same type for the examples of natural streams between which comparison is made, it may well be that the slopes are comparable, one with another, in the same sense in which they are comparable in laboratory work, and that their relations to capacity should follow the same law. THE DISCHARGE FACTOR. Discharge differences must be considered when the tractional capacities of different streams are to be compared, and also in com- paring the capacities of the same stream at different tunes. In making comparison between different streams it is important that the discharges used be coordinate — that is, that they represent equivalent phases of stream work. If co- ordination be not secured, allowance must be made in one stream or the other for the varia- tion of capacity with stage. In case the prob- lem is such that the choice of phase is optional, preference should be given to flood phases, because the general slope and the details of channel shape are approximately adjusted to such phases. The greatest known discharge is probably less representative of the channel conditions than is the mean of annual maxima of discharge. It is believed that with use of discharges that are both representative and co- ordinate the discharge factor of the empiric formula may be applied. The result of such application will be the more satisfactory in proportion as the streams compared are allied in type and will be relatively unsatisfactory for streams in different climatic provinces or for comparison of a direct alluvial stream with one which meanders. It is to be observed that in ah1 studies of allu- vial streams the discharge of which account should be taken in connection with traction is the discharge flowing in the channel proper. That which passes the banks ceases to con- tribute of its energy to the work of traction, and the portion of load diverted with it is not tractional. The case of variation of discharge in the same stream is complicated by simultaneous variations of fineness and competence. In the 234 TRANSPORTATION OF DEBRIS BY RUNNING WATER. experiments with sieve-separated grades of debris fineness and competence were constants with reference to discharge; but in a natural stream, where the tractional load may have great range in fineness, the mean fineness of the load varies with discharge, and the reason of its variation is that competence varies with discharge. The two competences which limit the range in fineness move up and down as discharge changes, and the mean fineness moves with them. Therefore the response of capacity to discharge can not be considered by itself. For convenience in analyzing the conditions, let us assume first a discharge which is adjusted to the details of channel form, to the deeps and shoals. If, now, the discharge be increased, and with it the whole system of velocities, trans- portation will be everywhere stimulated, part of the tractional load will join the suspended, and the scouring of the deeps will bring into the tractional load a greater proportion of the coarser elements of the load. The mean fine- ness of that load will be reduced, and the ca- pacity, while enlarged by increase of discharge, will be somewhat reduced by loss of fineness. The increase with discharge will be less than if the fineness were constant. If, on the other hand, the discharge be re- duced, some of the coarser material comes to rest, while finer debris is added from the suspended load. So the reduction of capacity from diminished discharge is qualified by the effect of increased fineness. But before the change in discharge has gone far the deeps become pockets for the reception of deposits, and traction is restricted to the intervening shoals, where it causes erosion. The erosion is selective, leaving an ever-increasing assem- blage of residuary coarse material, which tends to protect the finer. The current on the shoals no longer obtains a full supply of the material for which it is competent, and the load and capacity part company. Or we may say that as the erosion of the shoals progresses the mean fineness of the accessible debris is reduced until a grade is reached for which the current is not competent. In either case the decadence of traction follows a law which is not well repre- sented by the discharge term of the laboratory formula. . The above analysis postulates a wide range and somewhat equable distribution of fineness in the debris of the stream bed, a condition not always found. It might not apply, for ex- ample, to a stream which drains a district of friable sandstone and is therefore supplied with nothing coarser than sand. Nor would it apply well to a stream supplied with very coarse and very fine debris but not well supplied with intermediate grades. In most alluvial streams, and probably in all meandering streams, the work of traction which is accomplished on the shoals at low stage and midstage is almost negligible in comparison with the high-stage traction. Not only is the rate of traction slow, but the field of traction is restricted. If a single formula will not fit both low and high stages, the one adjusted to high-stage variations will have the greater practical value. Yet another consideration enters here, and one of peculiar importance. When discharge is reduced, and the competence of the current for traction is thereby changed, the coarse material eliminated from the tractional range ceases to be transported; but when discharge is increased, and the competence of the current for suspen- sion is thereby changed, the fine material eliminated from the tractional range continues to be transported. It is, in fact, transported more rapidly, so that a greater amount passes a given section each second. For most or all practical purposes the change in mode of trans- portation is of no moment, and those purposes would be served by a formula which should include the material shifted and ignore the change in mode. In the system of reactions set up by change of discharge the two modes of transportation are so interwoven, in fact, that the practical discrimination of the sus- pended and tractional loads is impossible. Even in the laboratory experiments devised specially for the study of traction a certain amount of interplay was tolerated, for tem- porary suspension appeared over the crests of some of the antidunes and also in the bends of the crooked channels. If the purely tractional point of view is to be exchanged for another, what shall be sub- stituted ? One natural suggestion is to include- in a single view the entire load, suspended and tractional; another to include along with the tractional only that part of the suspended load which for part of the time is tractional also. That which would be included in one view and APPLICATION TO NATURAL STREAMS. 235 excluded from the other is the finer part of the suspended load, the part that does not sink to the bottom so long as the current is sufficiently active for traction. Being purely suspensional, its quantity is peculiarly a function of supply and is connected with discharge only through the association of discharge with rain. Wher- ever discharge is largely a matter of tribute from snowbanks, the suspended load is con- spicuously independent of discharge. If we exclude from view the purely suspen- sional material, a natural criterion for inclusion is the finest debris which low-stage discharge moves by traction, and as low-stage traction is limited to the bars, or interpool shoals, it is the finest tractional debris of those shoals. If we consider a gradual increase of discharge from least to greatest, we have at first no traction. Then for a particular discharge, which may be called the competent discharge, traction begins on the shoals, only the finest of the de'bris being moved. Gradually coarser and coarser mate- rial is included, the range in fineness and the load increasing together; but in this phase of action the load is not necessarily the equiv- alent of the capacity, for it may be limited by the supply of de'bris of requisite fineness. After a time another critical discharge is at- tained, which initiates the loaning of de'bris from traction to suspension, and thereafter a constantly increasing share of the traveling de'bris is suspended. As the suspended parti- cles travel faster than the saltatory, and as capacity is the ability of the stream, measured in grams per second, to move de'bris past a sectional plane, the transfer from traction to suspension is an important factor in the en- hancement of capacity. The relation of capacity to discharge, con- templated from this viewpoint, has two ele- ments in common with the discharge factor of the laboratory formula. It includes a compe- tent discharge, corresponding to the zero of capacity, and it associates continuous increase of capacity with continuous increase of dis- charge. It differs, however, in important ways, and the possibility of expressing it by a definite formula is not evident. In the pool and rapid phase of activity the supply of de'bris suitable for traction is usually limited, and in many streams it is exhausted during each recurrence of the phase. In the phases of greater dis- charge, when traction occurs in the deeps as well as on the shoals, the sequence of capacities depends not only on discharge but on the rela- tive proportions of debris of different grades of fineness in the material of the load. It is prob- able that for most streams the load-discharge function is discontinuous at the limit of the pool and rapid phase. Because of this presumable discontinuity, because the tractional work while the pools exist accomplishes only a local transfer of de'bris, and because the work performed is usually of negligible amount in comparison with the work of larger discharges, it is prob- ably better to ignore altogether the pool and rapid phase in any attempt at general formu- lation. If that be left out of account and if the general features of the laboratory formula be retained, the constant « becomes the dis- charge which initiates traction in the deeps, and thus initiates through transportation of bottom load. If we accept that as a starting point, the material so fine as to be suspended by that discharge may be classed as purely sus- pensional, and other material suspended by larger discharges may be grouped with the tractional load. For the tractional load thus enhanced, or the amplitractional load, as it may conveniently be called, the rate of varia- tion with discharge is evidently higher than the rates found for simple grades in the labora- tory, and it may be much higher, for the de'bris diverted from traction to suspension, instead of lagging behind the lowest and slowest threads of the current, now speeds with the current's mean velocity. It is possible that a practical formula for the fluctuations of an alluvial river's load may fol- low these lines, taking the form where Ca is the capacity for amplitractional load, and K, is the smallest discharge competent to establish a continuous train of traction through deeps and shoals; but the suggestion as to form has no better basis than analogy, and no data are known tending to determine the magnitude of the important parameter o. THE FINENESS FACTOR. When the work of two natural streams is compared and the streams are of the same type, it i-i believed that the fineness factor of the 236 TRANSPORTATION OF DEBRIS BY RUNNING WATER. laboratory formula is applicable. It is true that fineness enters in a relatively complex way into the determination of the loads of natural streams, but for the comparison indi- cated the elements of influence are severally represented by the experiments, and their to- tals should follow a law of the same type. For small discharges this inference is subject to certain qualifications, which will appear from what follows. When the work of the same stream is com- pared under different discharges, a difference in fineness is developed under the laws of compe- tence. With larger discharge the mean fine- ness is less than with small discharge, and the difference in fineness conspires with the differ- ence in discharge to determine capacity. For reasons explained in the last section, however, capacity can not always be considered synony- mous with load when the discharge is small. THE FORM-RATIO FACTOR. In the reaches of a direct alluvial stream there is approximate uniformity of depth at high stage, and the conditions involving form ratio are essentially like those realized in the labora- tory. To such cases the principles developed in the laboratory studies should be applicable. It is true in a general way, as already men- tioned on page 223, that at a high stage of a natural stream the sectional area is about the same for the reaches as for the bends, and so too is the width. It follows that the mean depth is about the same, although the maxi- mum depth may be very different. The high- stage capacity is also the same at every sec- tion, after the channel form has been adjusted to the discharge. If these generalizations are correct, the principle involved in the form-ratio factor of the laboratory formula is applicable to curving streams, provided form ratio is inter- preted as the ratio of mean depth to width, and not as the ratio of maximum depth to width. In the analysis of conditions determining the relation of capacity to form ratio (Chapter IV) an important role was ascribed to the resistance of the banks; and the quantity of that resist- ance was represented in one of the parameters of the formula, ot. The optimum form ratio, p, was found to vary inversely with at and, there- fore, to vary inversely with the resistance of the banks. The resistance afforded by river banks is greater than that given by the smooth walls of laboratory channels, and this element tends to make the optimum form ratio relatively small for rivers. Its influence, however, is over- shadowed by those of slope and discharge. As the optimim ratio varies inversely with slope, and as most rivers have lower slopes than the experimental streams, the general tendency of the slope element is to make the ratio large for rivers. As the optimum ratio varies in- versely with discharge, and as the discharges of natural streams are relatively large, the tend- ency of this element is to make the ratio small for natural streams. The rates of variation being unknown, the net result of the three in- fluences can not be inferred deductively. The data from Yuba River, cited in Chapter IV (p. 135), show that for one case of a natural stream the optimum ratio is decidedly larger than that established by the stream in its alluvial phase and is of the order of magnitude of the determi- nations made in the laboratory. THE FOUR FACTORS COLLECTIVELY. The results of the preceding discussions ad- mit of a certain amount of generalization. When different streams of the same type are compared, and especially when the type is al- luvial, the law of their relative capacities at high stage may be expressed by the laboratory formula (109). The ability of that formula to express the variation of capacity with discharge in the same stream is problematic. It has not been shown that the system of numerical parameters determined for laboratory conditions can be used in extending the appli- cation of the formula to natural streams. If the formula were rational, the result of an ade- quate mathematical treatment of the physical principles involved, the constants measured in the laboratory would be of universal application (with moderate qualification for the conditions imposed by the curvature of natural channels) ; but the constants of an empiric formula afford no basis for extensive extrapolation. THE HYPOTHESIS OF SIMILAR STREAMS. When the Berkeley experiments were planned it was assumed that the relations of capacity to various conditions would be found to be sim- ple, and that the laboratory streams were rep- resentative of natural streams except as to tie APPLICATION TO NATURAL STREAMS. 237 characters associated with bending channels. Because of the discovered complexity of the l:;ws affecting capacity it is now apparent that the laboratory formulas can not be applied to streams in general. It is, however, probable that among the great variety of natural streams there is a more or less restricted group which is in such respect similar to the laboratory group that the empiric results of the laboratory — or at least the results embodied in exponents — may properly be applied to it . The criteria of similarity between large and small have been discussed to some extent by others in connection with the investigation of hydraulic problems by means of models. William Froude inferred from theoretic con- siderations that if the speed of a ship and the speed of its miniature model " are proportional to the square roots of the dimensions, their re- sistances at those speeds will be as the cubes of their dimensions,"1 and he afterward veri- fied this result by experiments. T. A. Hear- son, in projecting a model river for the inves- tigation of various hydraulic problems, dis- cussed separately the resistance to flow by the wetted perimeter, the influence of varying sec- tional area, and the influence of bends. He concluded that if the linear dimensions were kept in the same proportion, so that the river channel and its model were similar in the geo- metric sense, the velocities would be related f.s the square roots of the linear dimensions, and the discharges as the 2.5 powers of the linear di- mensions. It would be necessarythat thorough- nesses of the channel surfaces have the same dif- ferences as the linear dimensions, and that the movable debris of the bed r.lso follow the laws of linear dimensions.2 His deductions were not tested by the construction and use of a model, but they derive a large measure of support from the verification of Froude's analogous theorem. So far as I am aware, r.ll the models actually constructed to represent rivers and tidal basins have been given an exaggerated vertical scale.3 O. Reynolds 4 made a series of models of tidal basins in which the scales of depth and of tidal 1 These words are quoted from Inst. Naval Arch. Trans., vol. 15, p. 151, 1874. I have not seen Froude's original discussion of the subject. 2 Inst. Civil Eng. I'roc., vol. 146, pp. 21G-222, 1900-1901. • See Fargue, L., La forme du lit des rivieres a fond mobile, pp. 57, 128, 1908. Fargue recommended for a model river a vertical scale of 1:100 and a horizontal scale of 1:20, from which he deduced a discharge ratio of 1:3,200 and a velocity ratio of 1:16. * British Assoc. Adv. Sci. Repts. 1887, pp. 555-502; 1889, pp. 328-343; I860. pp. 512-534; 1891, pp. 386-404. amplitude were greater than the scale of length, the ratios ranging from 31:1 to 105:1. No ad- justment was made as to size of debris, the re- quirements of his investigation being met by any material fine enough to be moved by the currents. A tidal oscillation was communi- cated to water resting on a level bed of sand, with the result that the bed was gradually molded into shapes more or less characteristic of estuaries. From general considerations a "law of kinetic similarity" was deduced: = constant Lt in which p is the tidal period, h the depth of water (proportional to the amplitude of the tide), and L the length of the estuary. Under this law the results were generally consistent, but there was found to be a limit to the range of suitable conditions, and this limit was formu- lated by Jt3e = constant in which e is the exaggeration of the vertical scale. Eger, Dix, and Seifert,5 making a model of a portion of Weser River for the purpose of studying the effect of projected improvements, adopted 1 : 100 as the scale of horizontal dimen- sions and depths, and 1 : 6.7 as the scale of mean diameters of debris particles composing the channel bed. It was then a matter, first of theory and computation but finally of trial, to select scales for discharge and slope. The main condition to be satisfied was that for discharges corresponding to high and low stages the depths of water should be properly related, according to the scale of linear dimensions. For the scale of discharges 1 : 40,000 was finally adopted, and for slopes 650 : 1 . The resulting ratio of veloci- ties was 1:4; and this ratio, combined with the ratio of debris sizes, was found to give a time ratio (for the accomplishment of similar changes in the bed of the stream) of 1:360. The scale of velocities being only 1:4 while the scale of distances was 1:100, there was an exaggeration of velocities in the ratio of 25: 1.6 The quantities of debris moved being in the ratio of 1:100 3, the distances moved in the « Zeitschr. Bauwesen, vol. 56, pp. 323-344, 1906. • So stated by the authors. An allowance for the general principle that velocities are proportional to the square root of the hydraulic mean depth, and therefore to the square root of linear dimensions, would indicate 1:10 as the normal ratio of velocities, and give 2.5:1 as the exaggeration. 238 TRANSPORTATION OF DEBKIS BY RUNNING WATER. ratio of 1 : 100, and the times consumed in the ratio of 1:360, the ratio of loads (per second) wasl:Q™ or 1:280,000. The results were ooU satisfactory; it was found that the successive forms given to the river bed by variations of discharge were repeated in the model. The exaggerations of the vertical scale by Reynolds and of the slope by Eger, Dix, and Seifert had the important effect of shortening the time necessary to produce the desired re- modeling of the mobile bed. The absolute proportionality adopted by Froude and recom- mended by Hearson would have entailed a prohibitive consumption of time and might have added a serious complication in connec- tion with the use of very fine debris.1 The similarity controlled by Reynold's law was a relation between the wave periods and dimensions of tidal basins and is not closely related to similarity in the control of trac- tional load. The similarities obtained in con- structing the model of the Weser are more in point, because they involve an average rate of movement of debris; but they throw no light on the laws of variation of debris movement* which is the important matter in bridging the interval between our experiment streams and natural streams. After attempting to use various suggestions which came from the adjustments of the Weser model, I have re- turned to the principles of geometric similarity employed by Froude and Hearson. Let us assume, as possible or plausible, that the principles developed in the laboratory, together with all parameters which are of the nature of ratios, are independent of the scale of operations and may be applied to streams of far greater magnitude, provided all linear fac- tors are magnified in equal degree. If width and depth are enlarged in the same ratio, the form ratio is unchanged. If longitudinal dis- tance and loss of head are enlarged in the same ratio, the slope is unchanged. If the dimen- sions of the transported particles are enlarged in the same ratio as the linear elements of chan- nel, the linear coarseness is increased, or the linear fineness is reduced in that ratio. The natural streams which may be consid- ered as similar to the experimental streams constitute a class of moderate size. The form ' Such considerations as these affected the selection of materials for the Berkeley experiments land prevented the employment of very gentle slopes. ratio for rivers ranges lower than for experi- mental streams, but there is some overlap. The smaller of the form ratios of the laboratory are representative of a considerable number of rivers. The slopes of rivers range lower than for laboratory streams, but here again there is overlap, and the natural streams which are similar in respect to slope are in general such as have coarse debris, so that there may be correspondence in that regard also. The simi- lar natural streams to which hypothesis extends the laboratory results are those of large form ratio and steep slope, carrying coarse debris. The primary difference between a large stream and a small one being one of discharge, and our general inquiry being directed to the valuation of capacity for traction, let us seek an expression for the relation of capacity to discharge when similar streams of different size are compared. The laboratory data determine control of capacity by slope, discharge, fineness, and form ratio. In similar streams slope and form ratio are constant, and we need consider here only discharge and fineness. As we are comparing the laboratory streams as a group with similar natural streams, also taken as a group, it is advisable to employ a mode of formulation which lends itself to the use of averages, and the most convenient is that of the synthetic index. In lia and 7to are average values of the synthetic index and may be estimated, from data in Tables 32 and 43, as 1.32 and 0.77. Designat- ing elements of the larger and smaller streams severally by subscripts „ and , , we have, from the above, (113) If we designate by L the ratio between a linear dimension of the larger stream and the corre- sponding dimension of the smaller. d, L Calling mean velocity V, bearing in mind that the hydraulic mean radius is a linear dimension of channel, and recalling that the Chezy formula APPLICATION TO NATUKAL STREAMS. 239 makes V vary as the square root of the hy- periments treat with confidence is from 0.5 to draulic radius, we have 3.0 per cent. Direct appli cation is limited to •„ streams having slopes within that range. By -W-' = Z°-5 postulate, Then, since discharge is the product of width, depth, and velocity, 5_ T2.5 M141 Q, F, Q,~w, whence and Substituting this value of the fineness factor in (113), and reducing, wo have .. .(115) The result indicates that whence - C, Q, ^ Q,, Q, C or, since -Q= U, the tractional duty of water, That is, for similar streams the tractional duty of water is the same. As the exponents connecting capacity with discharge, capacity with fineness, and mean velocity with hydrauh'c radius are all averages of low precision, the result is far from being so secure as might be inferred from equation (115). Its best support is really found in the plausi- bility of its conclusion. Our experience with a variety of physical laws makes it easy for us to believe that with suitable parity of conditions a unit of discharge will accomplish the same work as part of a large stream that it will ac- complish as part of a small stream ; and so the conclusion is plausible. The fact that an at- tempt to test the hypothesis of similar streams by combining it with experimental data has led to a plausible result is a fact favorable to the hypothesis. Let us now assume the hypothetic law to be a real law and draw such inferences as may be warranted. The range of slopes which the ex- A/"_ r ' ._ T D, ~ V,,~ Substituting in equation (114), we have ef~\E7. whence 0.5 1.0 2.0 3.0 That is to say, for similar streams, the ratio ^-5 may be regarded as constant. This relation affords a criterion for the discrimination of those natural streams which are similar to the laboratory streams, provided they are also similar in slope and form ratio. The following limiting values for j^ for different slopes are all estimated on the assumption of a form ratio of 0.05: Limiting values of " 3,000,000-40,000,000 2,000,000-30,000,000 1,500,000-10.000,000 500,000- 4,000,000 The form ratio 0.05 is considerably below the average of the ratios developed in the labora- tory, and it is also much above the average for alluvial rivers at flood stage. Any allow- ance which might be made for this discrepancy would have the effect of increasing the estimate of limiting values of the ratios of Q to .D2-5. Subject to this qualification the ratios indicate the types of natural streams which are "similar" to the laboratory streams and to which various laboratory results may be applied. The streams are in general either small creeks or else rivers transporting very coarse debris. As the slopes are determined by flood discharges, such dis- charges should be used in the classification. For the streams thus classified as similar to laboratory streams the duty of water is of the same order of magnitude, and so are the rates of variation of duty with the several conditions of slope, discharge, and fineness. The rates of variation apply especially to comparisons of one stream with another. For the estimation of variation with discharge in the game stream something should bo added to the laboratory rate to allow for the varying assistance which 240 TRANSPORTATION OF DEEMS BY BUNKING WAITER. suspension gives to the work of traction (p. 234) . For estimation of optimum form ratio some- thing should be deducted from the laboratory indication to allow for the greater resistance of the channel walls. SUMMARY. Natural streams of alluvial type differ from the streams used in the laboratory in ways con- nected with the bondings of their courses and with variations of discharge. The differences affect forms of cross section, the distribution of velocities within the section, and the partition of the load between suspension and traction. The two portions of load are carried at the expense of the stream's energy, each reduces the velocity, and the reduction of velocity de- termines the limit of carrying power. The whole burden of the stream includes not only two divisions distinguished by mode of trans- portation but as many minor divisions as there are grades of debris, and the load carried of each grade reduces the capacity for all the grades. These and other complexities make it diffi- cult to apply the laboratory results to natural streams. It is probable that the forms of the laboratory formulas are applicable, with limi- tations, to the comparison of one stream with another, but the availability of the exponents is problematic. There are special difficulties in attempting to use the formulas for the compari- son of capacities of the same stream at different stages, and in such comparisons the tractional load can not be considered by itself, because much material "which is swept along the bed at lower stages is lifted by flood velocities into the body of the current. It is thought that the laboratory formulas may be applied to natural streams which are geometrically similar to the laboratory streams — -that is, to streams having the same slopes and form ratios and carrying d6bris of proportionate size. The class of streams to which the formulas apply by reason of simi- larity is necessarily restricted, being character- ized in the main by high slopes and coarse debris. It can include few large streams. CONCLUSION. It was a primary purpose of the Berkeley in- vestigation to determine for rivers the relation which the load swept along the bed bears to the more important factors of control. As a means to that end it was proposed to study the mode of propulsion and learn empirically the laws con- .nectirig its output with each factor of control taken separately. The review of results in the present chapter shows that the primary purpose was not accomplished. In the direction of the secondary purpose much more was achieved, and a body of definite information is contrib- uted to the general subject of stream work. A valuable outcome is the knowledge that the output in tractional load is related to the con- trolling conditions in a highly complex manner, the law of control for each condition being qual- ified by all other conditions. With the aid of the Berkeley experience it would be possible to avoid certain errors of method and arrange experiments which should yield more accurate measurements of the same general class — and it is natural that the experi- menter should feel the desire to do his work over in a better way — but I am by no means sure that adequate advantage would reward a continu- ance of work on the same or closely related lines. The complex interactions could be given better numerical definition, but it may well be doubted whether their empiric definition would lead to their explanation. It is possible that the chasm between the laboratory and the river may be bridged only by an adequate theory, the work of the hydromechanist. It is possible also that it may be practically bridged by ex- periments which are more synthetic than ours, such experiments as may be made in the model rivers of certain German laboratories. (See p. 16.) The practical applications for results from experiments in stream traction belong almost wholly to the field of river engineering. For the transportation of detritus and related mate- rials by artificial currents, stream traction will rarely be used, because flume traction is more efficient. Our results in flume traction have therefore an immediate practical application, and as they were limited in range the advantage of extending them can hardly be questioned. The report on the Berkeley investigation properly closes with this chapter, but there are several by-products which seem worthy of rec- ord. Some of them are presented in the fol- lowing chapter, and others are contained in appendixes. CHAPTER XIV.— PROBLEMS ASSOCIATED WITH RHYTHM. RHYTHM IN STREAM TRANSPORTATION. This chapter is concerned with certain prob- lems upon which the Berkeley investigation touched, but which were not seriously attacked. The low precision of the observations on stream traction, a precision characterized by an average error of about 11 per cent and an. aver- age probable error after adjustment between 2 and 3 per cent, had for its chief cause the failure to eliminate from the experiments the influ- ence of rhythm. The slope of the water sur- face, the slope of the channel bed, and the load of debris transported were all subject without intermission to rhythmic fluctuations. If ma- terially better observations of the same sort are to be made, this difficulty must be successfully dealt with, and the first step toward mastering it is to understand it. The removal of a diffi- culty, however, is neither the sole nor the most important result to be expected from the study of fractional rhythms. Underlying them are physical principles which are of importance in the dynamics of rivers, and their study con- stitutes one of the available lines of approach to the broader subject. For their empiric study the general plan of the Berkeley apparatus is well adapted, but our experience indicates that certain details should be modified. The use of a long trough is ad- visable, with contraction at the outfall, and with delivery of the load to a settling tank be- yond the outfall. The appliances and methods should be such as to secure uniformity in dis- charge, in character of d6bris, and in rate of feed. The apparatus for regulating discharge, de- scribed on pages 20 and 257, was one of the most satisfactory parts of the Berkeley equipment. Its most important feature, as affecting pre- cision, was the delivery of the water through an aperture under a considerable head. Uniformity of debris can hardly be secured without the employment of an artificial, nar- rowly limited grade, and the available means of sorting is the sieve. It is to be observed, however, that after a grade has been separated by sieves it is still subject to sorting by current, the current recognizing differences of form and density which the sieves ignore. When d6bris that has once been handled by the stream is to be used a second time, remixing may be ad- visable. None of the devices we employed to feed debris to the current achieved uniformity. Those which depended on the flow of wet de- bris through an aperture failed because the proportion of water could not be kept constant. The others depended on handwork and ex- perienced the irregularity usual to handwork. An apparatus planned near the end of the ex- perimental work, but never tried, is of such promise that its essential features are here described. 20921°— No. 86—14- -16 FIGPBE 72.— Suggested apparatus for automatic feed of debris. A drum, A in figure 72, is turned slowly by power, its rate being regulated by clockwork. Its position is above the trough containing the experiment stream, B. The surface of the drum is uniformly roughened. Above it is a vertical rectangular shaft, filled with moist debris. The shaft does not touch the drum. The width of the separating space at D is con- trolled by some suitable device. As the drum turns, a debris layer of uniform thickness is carried with it, and this falls into the stream. To prevent irregularities due to adhesion, a 241 242 TRANSPORTATION OF DEBRIS BY RUNNING WATER. detaching device is placed at some point E, the device possibly consisting of an open-rank comb of elastic wire. The shaft should be smooth and of uniform section, so that the particles of de'bris near the drum may be brought into actual contact with one another by pressure of the debris above. That the debris may not be caused to flow by excess of moisture, it should be thoroughly drained be- fore use. In a rough construction designed to test the practicability of the apparatus the drum surface was roughened by covering with a wire screen, and this was found to secure the delivery of the de'bris. Uniformity of feed having been provided, the rhythms of transportation may be observed as oscillations in the de'bris delivered at the out- fall. Rhymths of slope may be studied, at least initially, by observing changes in the profile of the water surface. In the record of the Berkeley observations it is not practicable fully to discriminate rhyth- mic inequalities from those occasioned by irregularities of de'bris feeding, but there is reason to believe that several rhythms of differ- ent period coexist. The shortest rhythms are those connected with the dunes and antidunes, and these are evidently associated with rhythms of the flow of water. RHYTHM IN THE FLOW OF WATER. Reynolds,1 treating of the flow of water through tubes, distinguishes two modes of flow as direct and sinuous. They are otherwise called steady and turbulent. In direct flow the filaments of current have simple lines, which are straight and parallel if the walls of the con- duit are straight and parallel. In sinuous flow the filaments of current are neither simple nor parallel and may be intricately convoluted. The flow in tubes is direct for low velocities and sinuous for high, the critical velocity vary- ing inversely with the diameter and the rough- ness of the tube and directly with the viscosity of the water. It would follow from his gener- alizations that the flow of such streams as were used in our experiments would be sinuous, and i Reynolds, Osborne, An experimental investigation of the circum- stances which determine whether the motion of water shall be direct or sinuous and of the law of resistance in parallel channels: Eoy. Soc. I'hilos. Trans., London, vol. 174, pp. 935-982, 1883. Also, The two manners of motion of water: Roy. Inst. Great Britain Proc., 1884. its actual sinuosity was a matter of observa- tion. To a large extent the curvature of the flow lines was shown by the motions of minute suspended particles, and when this evidence was lacking it was still possible to infer diver- sity of current from continual changes in the configuration of the water surface. It is probable that all the diversities of flow were rhythmic, for that is the nature of in- equalities of motion developed by the inter- action of constant forces; but in many cases the rhythms were so numerous and so related in period and other characters that their com- bination gave the impression of irregularity. In many other cases, however, some one rhythm was dominant, and these cases appear especially worthy of study. The cycle of movements constituting a rhythm unit may be definitely related to space, or to time, or to both space and time. The pulsations of the water surface which ob- structed our observations of water profile were manifestations of time factors. The regular sequence of dune crests was a manifestation of a space factor. The two phenomena co- existed. In the space interval from dune crest to dune crest certain elements of motion were constant. There was a large stationary vortex in the hollow between the crests (fig. 10, p. 31), and there might be stationary elements in the configuration of the water surface. At the same time the flow lines of the water at all points swayed in direction and were affected by variations of velocity, and these changes were rhythmic with respect to time. Doubt- less many of the fluctuations belonged to or were associated with vortices which traveled with the general current. It is conceivable, or even probable, that the stationary features and the traveling features were coordinated, so that within the rhythmic space unit corres- ponding to the dune interval there was a cycle of variations of motion which was rhythmic with respect to time. The rhythmic dunes marched slowly down the trough and with them marched the intercrest vortices and associated motions. To that ex- tent the features of the bed controlled the fea- tures of the current. Nevertheless the dunes were not essential to the existence of a water rhythm characterized by a definite space inter- val. There was positive evidence that the dune PEOBLEMS ASSOCIATED WITH RHYTHM. 243 interval was determined by a preexistent water rhythm. Xot only was the dune interval a function of depth and velocity of current,1 but the creation of dunes by water rhythm was repeatedly observed. In certain experiments a slow current, moving over a bed of debris arti- ficially smoothed and leveled, was gradually quickened until transportation began. The movement of particles did not begin at the same time all along the bed but was initiated in a series of spots separated by uniform intervals, and the first result of the transportation was the creation of a system of dunes. In certain experiments on flume traction a slow current, moving over a smooth channel bed of wood, swept along a small quantity of sand. With increase of the load of sand local deposits were induced, which took the form of thin straggling patches, similar to one another in outline and separated by approximately equal bare spaces. These moved slowly down- stream, the mode of progress being similar to that of dunes, and with further increase of load they acquired the typical profile of dunes. In both groups of experiments it was evident that the primary rhythm pertained to the water and was independent of the work of transpor- tation. In the second group, where sand swept steadily forward over the whole area of the bed, it was evident that the water rhythm did not involve reversed currents along the bed and therefore did not include such stationary vor- tices as accompany dunes. In the experiments on stream traction the development of dunes was conditioned by the three factors of velocity, depth, and load, be- sides an undetermined influence from fineness i Measurements of dune interval in the laboratory were too few to dem- onstrate the factors of control, but comparison of laboratory data with data from other sources leaves little question that control is exercised by depth, velocity, and fineness. In the laboratory, where depths were a matter of inches, the dune interval rarely exceeded 2 feet. In Mississippi River depths measured in scores of feet are associated with intervals measured in hundreds of feet. In tidal estuaries, where dunes of a special type are exposed at low tide, the depths of the formative cur- rents are intermediate between those of the laboratory and those of the Mississippi, while the intervals are measured in feet or tens of feet. For data of the Mississippi see Johnson, J. B., Kept. Chief Eng. U. 8. A., 1879, pp. 1963-1967, and Eng. News, 1885, pp. 68-71; Powless, W. H., Mississippi River Comm. Kept, for 1881, pp. 66-120; and especially Hider, Arthur, Mississippi River Comm. Rept. for 1882, pp. 83-88. For data on the dunes of tidal estuaries see Cornish, Vaughan, Geog. Jour., vol. 18, pp. 170-202, 1901. Hider finds that the dune interval is greater at high stages of the river than at low, the depth and velocity both decreasing with the change from high to low. Cornish's observations show that the interval varies directly with depth of current under condi- tions which make it probable that the velocity varies inversely with depth. of debris.2 It is probable that load and fine- ness enter only as factors of resistance, so that the essential conditions are velocity, depth, and bed resistance. Within certain rather wide ranges of value for these controlling factors the bed is molded into dunes. When the limit is exceeded by increase of velocity or resistance, or by decrease of depth, the dunes disappear and the bed becomes smooth and plane. At the same time the oscillations and other dis- turbances of the water surface are reduced; but as they do not altogether disappear it is to be inferred that the flow is still characterized by internal diversity. With still further increase of velocity or resistance, or with further reduction of depth, another critical point is passed, and the process of traction becomes again rhythmic, but in an antithetic way. The bed is molded into anti- dunes, which travel upstream (pp. 31-34), and the water surface also is molded into waves, which copy the forms of the antidunes and move with them. The internal move- ment of the water is again characterized by a dominant rhythm, but the type of rhythm is different from that associated with dunes. The rhythm is also less stable, and its intensity exhibits a cycle of change. With low intensity the waves are nearly equal in height and length, but sooner or later inequalities develop and the higher waves overtake the lower and absorb them. This process increases the wave length in the upstream part of the trough, and the influence of the change is hi some way communicated to the downstream end, where the waves are first formed, with the result that larger and larger waves develop. Finally a master wave, with curling crest, rushes through the trough from end to end, and this has the effect of wiping out the irregularities and restoring the status of low intensity. Various phases of the cycle are illustrated by the profiles in figure 12 (p. 33). When combinations of velocity, resistance, and depth similar to those causing antidunes are made for a stream flowing through a rigid straight channel, without movable debris, the water develops surface waves, and these travel 2 Eger, Dix, and Seifert (Zeitschr. Bauwesen, vol. 56, pp. 325-328) found that under certain conditions dunes were developed in a sand of uniform grade but not in a finer sand composed of several grades. In our experiments the smooth phase of transportation had greater range with mixed grades than with single grades. 244 TRANSPOKTATION OF DEBKIS BY SUNNING WATEE. downstream faster than the current. They are initially rhythmic, but their period is unstable because their velocity of propagation varies with their size. A wave with slight advantage in size will overtake the one in front of it, and then the two will unite, making a wave with still higher velocity. Thus the system tends, for a time at least, toward reduction of the number of waves and increase of the wave interval. As the waves grow by composition, their fronts steepen and the culminating phase is that of a "roll wave," or bore.1 It may be noted, in passing, that the pulsations frequently observed in the overfall of a dam are probably rhythms of this type. It may be assumed, at least tentatively, that all the dominant rhythms observed in straight conduits are initiated at the intake. Those associated with low velocity and an immobile bed appear to be stationary, but with a mobile bed they develop dunes, and they then travel downstream with the dunes. Those associated with high velocity and an immobile bed develop waves of translation which travel downstream. With a mobile bed antidunes are developed, and surface waves travel with them upstream. It is possible that the antidune waves coexist with the waves of translation, and that the cycle of intensity in the antidune phenomena results from the interactions of the two systems. If there is warrant for the various correlations above indicated, the phenomena developed by our experiments in connection with modes of debris transportation afford a basis of classifi- cation for a considerable body of water rhythms. What we have called the smooth phase of traction marks a critical phase of water flow separating two types which are characterized by dominant rhythms, and the two dominant rhythms are in some way anti- thetic. The nature of their antithesis is not known to me, nor is the character of either rhythm; but it appears that the experimental and analytic study of the rhythms constitutes a field of research which is at the same time promising and important. I assume that the definite rhythms of water by which, when debris is present, the dunes and antidunes are caused are susceptible of analytic treatment, 1 See Cornish, Vaughan, Progressive wfcves in rivers: Geog. Jour. (London), vol. 29, pp. 23-31, 1907. and I believe that the experiment trough affords the means of preliminary delineation and ultimate verification. THE VERTICAL VELOCITY CURVE. No systematic study was made of the distri- bution of velocities within the streams of the laboratory, but incidentally, in connection with several different minor inquiries, the vertical distribution of velocities in the medial plane was recorded; and it happens that some of these records serve to illustrate the depend- ence of that distribution — the vertical velocity curve— on' certain conditions. The observa- tions were made with the Pitot-Darcy gage. The first to be reported were made ia con- nection with a comparison of the free outfall (fig. 1, p. 19) and the contracted outfall (fig. 3, FIGURE 73. — Modification of vertical velocity curve by approach to outfall. p. 25). The trough used with free outfall had smooth sides and bed, was 1 foot wide, and was set level. The stream of water, 0.734 ft.3/sec., carried no debris. Its surface slope gradually increased toward the outfall, and the mean velocity increased with the fall of the surface. Velocities were measured at three points, respectively, 5.5 feet, 2.3 feet, and 0.8 foot from the outfall. Their curves, given in figure 73, show that the acceleration as well as the velocity increased as the outfall was ap- proached, and that there was a coordinate modification of the shape of the curve. In the first curve it is evident, despite a discordance among the determined points, that the level of maximum velocity is somewhat above mid- depth; in the second the maximum is near mid- depth; and in the third it is below. The accel- eration, for Avhich an integrated expression appears in the horizontal space between curves, PROBLEMS ASSOCIATED WITH BHYTHM. 245 seems to have affected the lower part of the current more than the upper. The trough used with contracted outfall was 0.92 foot wide and was contracted, in the manner shown in figure 3 (p. 25) to 0.30 foot. The oblique walls producing the contraction .5 .4 J.S o.es \ 0.7S \ ! r w 0.2 1 : 1 \ \ \ \ .1 / i \ , ./ 1 J I 2 Velocity FIGURE 74. — Modification of vertical velocity curve by approach to contracted outfall. were 3 feet long. The first velocity station was 1.5 feet from the outfall, at a point where the width was 0.64 foot; the second and third were at 0.75 and 0.25 foot from the outfall, corresponding to widths of 0.47 and 0.35 foot. The Velocity curves, given in figure 74, show the descent of the plane of maximum velocity from a position slightly above middepth to FIGURE 75. — 1'lan of experiment trough with local contraction. The letters show stations at which velocities were observed, and are re- peated in figures 7G and 77. Scale, 1 inch«=2 feet. near the bottom. The acceleration, in this case connected with contraction of channel, as well as with the release at outfall from the channel resistance, is a function of depth. In the third arrangement a trough 0.92 foot wide, with a slope of 0.58 per cent, was con- tracted at one point in the manner indicated in figure 75. This gave to the stream such a FIGURE 76. — Profile of waU-r surface in trough shown in figure 75. Scale 1 inch=2feet. profile as is sketched in figure 76. Velocities were measured at four points, the distances from the point of extreme contraction being 2.0 feet, 1.0 foot, 0.5 foot, and 0, and the corre- sponding widths of current 0.92, 0.55, 0.37, and 0.19 foot. The velocity curves are plotted in figure 77. The level of maximum velocity is at or near the surface at the point of initial contraction, A, and then drops quickly, being below 0.9 depth in the narrowest strait, D. In this case also the acceleration appears to increase in regular manner with distance from the surface. To avoid the peculiarities observed near the outfall, as well as those appropriate to intake £ •1* c 1 .4- 1 f *, _ o 1 \ \ i I \ \ J 1 1 U J , Ji 3123 Velocity 4 FIGTOE 77. — Modification of vertical velocity curve by local contrac- tion of channel. The letters indicate positions of velocity stations in trough. (See fig. 75.) conditions, all other determinations of the curve were made near midlength of the trough, the ordinary distance from the outfall being 17 feet and the least distance 13 feet. In figure 78 are three curves illustrating the influence of slope of channel. The trough was 1.96 feet wide and its sides and bed were smooth; the discharge was 1.119ft.3/sec. Curve A shows velocities 17 feet from the out- FiGURE 78. — Modification of vertical velocity curve when mean velocity is increased by change of slope. fall, with the trough level; B corresponds to a slope of 0.26 per cent and C to 0.56 per cent. With the trough level, maximum velocity occurs at about 0.75 depth, and with both inclinations of trough its position is indicated at from 0.2 depth to 0.3 depth. As curve 0 differs in type from all others obtained, and as it may be made harmonious by rejecting a single observation, it is thought probable that the dotted line better expresses the fact. Its 246 TRANSPOKTATION OF DEBEIS BY KUNNING WATEE. substitution throws the maximum to the sur- face of the water and makes a consistent series of the three curves. In another series of observations the trough remained horizontal while the discharge was .923 Velocity FIGURE 79. — Modification of vertical velocity curve when mean velocity is increased by change of discharge. The corresponding discharges are indicated in ft.'/sec. varied. The width was 0.66 foot, and the dis- charges 0.182, 0.363, 0.545, 0.734, and 0.923 ft.3/sec. The resulting curves (fig. 79), with exception of that for the smallest discharge, form a consistent series, the level of maximum velocity rising slowly with increase of discharge. In this case also the discordant curve may be brought into harmony by the rejection of a 3=5J 01234 Velocity 1'iGUKE 80. — Modification of vertical velocity curve by changes in the roughness of the channel bed. Curve 1, paraffin; 2, roughness of debris grade (A); 3, roughness of grade (D); 4, roughness of grade (F). single observation, and a substitute curve is suggested in the diagram by a broken line.1 In another series the texture of the channel bed was made to vary. Constant features were the discharge, 0.734 ft.3/sec.; the width, 0.92 foot; the slope, 0.58 per cent; and the texture of the channel sides, which were planed and unpainted. The varieties of bed texture were (1) paraffin, coating a smooth board; (2) a ' In this instance, in the one before mentioned, and also in the case of three curves in figure 80 the aberrant point records a measurement which was made very near the surface of the water. In such positions the constant of the Pitot-Darcy gage has a special value, and it is on the whole probable that the apparent errors of observation are occasioned by an error of the rating formula. The matter is discussed in Appendix A. pavement of debris of grade (A); and (3 and 4) similar pavements with grades (D) and (F). The observed curves, given in figure 80, indi- cate that the resistance occasioned by a rough bed retards the whole current but retards the lower parts in greater degree than the upper, so that the level of maximum velocity is raised. The amount of retardation is greater as the texture of the bed is coarser. In figure 81 are two curves illustrating the influence of load on the vertical distribution of velocity. One curve was observed in a stream without load, flowing over a smooth and hori- zontal bed, the other in the same stream when carrying a load of 17 gm./sec. on a self-adjusted .3 !, Q Velocity FIGURE 81. — Modification of vertical velocity curve by addition of load to stream, with corresponding increase of slope. A, without load; B with load. slope of 0.64 per cent. To judge from the data shown in figure 78 (p. 245), the effect of slope alone would be to double or nearly double the mean velocity, but the actual increase was only 13 per cent. The acceleration due to slope was almost wholly neutralized by the retardation due to roughness of bed and to the work of traction. The retardation had also the effect of raising the level of maximum velocity. FIGUBE 82. — Modification of vertical velocity curve by addition and progressive increase of load. Similar contrasts are shown in figure 82, where the curve for an unloaded stream is com- pared with curves for the same stream when transporting three different loads. The con- stant factors in this case are: Width of channel, PROBLEMS ASSOCIATED WITH BHYTHM. 247 0.66 foot; discharge, 0.545 ft.3/sec.; grade of debris, (C). The variables are as follows: Load gm./sec 0 38 53 194 Slope of bed... per cent.. 0 0.38 0.5G 1.14 Depth at 17 feet from out- fall feet.. 0.353 0.313 0.303 0.226 Mean velocity... ft. /sec.. 2.57 2.64 2.73 3. 6G Level of maximum veloc- ity; measured from the surface as a fraction of depth 0.7 0.1 0 0 Letter indicating curve in figure82 IB C D With the slope of 0.38 per cent the sand moved in dunes; with 0.56 per cent the phase of trac- tion was transitional between the dune and the smooth; with 1.14 it was transitional between the smooth and the antidune. The conspicu- ous change associated with the addition of load is the raising of the level of maximum velocity, and this is correlated also with increase of slope, increase of mean velocity, decrease of depth, and modification of the mode of traction. The plotted points for velocities near the channel bed are irregular when the observations were made above a bed of loose debris, and little use has been made of them in drawing the curves. As previously mentioned, the presence of the gage caused a deflection of the lines of flow and the formation of a hollow in the bed. Not only was it impossible to observe with accuracy the relation of the instrument to the normal position of the bed, but the velocity observed was higher than that normally asso- ciated with the depth at which the instrument was placed. (See Appendix A.) In most of the groups of curves the variations of form are associated with simultaneous varia- tions of so many conditions that the nature of the control is not evident. For satisfactory interpretation a fuller series of observations seems to be required, but certain inferences may be drawn from those before us. Many of the peculiarities of form are con- nected with the position of the level of maximum velocity. The movements of the maximum in relation to depth of current are of two kinds. It rises with increase of depth when that increase is caused by increase of discharge (fig. 79). It falls with increase of depth when that increase is independent of discharge (figs. 78, 81, and 82). Apparently depth, considered by itself, is not a factor of control. The maximum rises with increase of mean velocity when that increase is due to increase of discharge (fig. 79), or of slope (fig. 78), but falls with increase of mean velocity when the increase is due to lessened resistance of the channel bed (figs. 73, 80, 81, and 82). Apparently mean velocity, consid- ered by itself, is not a factor of control. If we give attention to the three factors on which depth and mean velocity chiefly depend — namely, discharge, slope, and bed resistance — a more consistent relation is found. Variations of discharge affect only the group of curves in figure 79, and there the maximum rises with increase of discharge. Slope affects the groups in figures 78, 81, and 82, and in each case the maximum rises with increase of slope. Bed resistance affects the groups in figures 80, 81, and 82, and in each case the maximum rises with increase of resistance. The lowering of the level of maximum veloc- ity as the point of outfall is approached (fig. 73) is a harmonious feature, but in that case there is substituted for progressive reduction of bed resistance an abrupt cessation of all channel resistance. The resulting acceleration is propa- gated upstream, and its amount has a vertical distribution connected with pressure. In the case of contracted outfall (fig. 74), there is added an effect of convergence, which still fur^ ther illustrates the graduation of acceleration in relation to pressure. The influence of con- traction is important in other connections, but need not be further discussed in this place. The influence of outfall may extend to a considerable number of the curves here figured. In most of the experiments made without contraction at outfall there was progressive decrease of depth and increase of mean velocity, from some point near the head of the trough to its end. This was most marked when the bed of the trough was horizontal (figs. 73, 78-4, 79, 81-4, and 82-4). Reasoning from the observed fact that acceleration in- creases with depth, I think it probable that under such conditions the level of maximum velocity lies lower than it would with a uniform mean velocity. Returning to the consideration of discharge, slope, and resistance, we may note that the variable resistance with which variations of the curve have been definitely correlated is bed resistance. In all the experiments the channel sides had the same texture, so that the side resistance was approximately proportional to 248 TRANSPORTATION OF DEBRIS BY RUNNING WATER. the depth. The level of maximum velocity thus has the same relation to side resistance as to depth; it sometimes rises and sometimes falls when side resistance is increased. While a probability exists that side resistance influ- ences the position of the maximum, the nature of its influence is not shown by the observations under consideration. Discharge and slope, or the energy factors which they help to measure, urge the water forward, and their influence is applied to the whole stream. Bed resistance holds the stream back but is applied to its base only. The obvious tendency of these forces is to make the upper part of the stream move faster than the lower and produce a velocity curve with maxi- mum at the water surface. This tendency is opposed by some other factor, unknown, which tends to depress the level of the maximum. Whatever that other factor may be, it loses in relative importance when discharge, slope, or bed resistance is increased. A noteworthy feature of the curves is a tendency to change in character near the bed. The observations are not so precise nor so full as to afford a distinct characterization of the change, but there can be little question of its existence. It would appear that the peculiar conditions near the bed give great local impor- tance to some factor of velocity control which is elsewhere of minor importance. Attention may also be directed to the fact that none of the curves resembles an ordinary parabola with horizontal axis. Had these been the vertical velocity curves to which mathematical formulas were first fitted, the equation of the parabola would not have been used. The statement, above, that the factor tending to depress the level of maximum velocity below the water surface is unknown is perhaps rash, for several theories as to its nature have been advanced with confidence. To put the matter more cautiously — it has seemed to me that each theory of which I have read was effectually disposed of by the discus- sion which it aroused. However that may be, there is certainly room for another suggestion, and this I proceed to offer. Reynolds * arranged an experiment in which a liquid was made to flow over another denser liquid, the two being immiscible. Below a cer- i Roy. Soc. London Philos. Trans., vol. 174, pp. 943-944, 1883. tain velocity the surface of contact was smooth, but above the critical velocity the surface was occupied by a system of equal waves, which moved in the direction of flow but more slowly. With miscible liquids, or with two bodies of identical liquid, the waves are replaced by vor- tices. Under some conditions the vortices are as regularly spaced as the waves, but usually they are less regular, and various complications arise. The development of such vortices may readily be watched on a river surface wherever adjacent parts move with quite different veloci- ties or move in opposite directions. Such vor- tices have vertical axes, and their direction of rotation is determined by the differential mo- tion of the adjacent currents. If we conceive the water of a vortex as a body between parallel and opposed currents, then its direction of rota- tion is due to a mechanical couple contributed by the currents. Transferring attention to the longitudinal vertical section of R stream, we find ft couple of FIGURE S3.— Ideal longitudinal section of a stream, illustrating hypothe- sis to account for the subsurface position of the level of maximum velocity. which one element is the general forward move- ment of the current and the other is the bed resistance. These tend to produce and main- tain vortices with horizontal axes transverse to the channel and with forward rotation — that is, with the rotation of a wheel rolling forward in the direction of flow. To visualize these fea- t ures, figure 83 gives an ideal section of a stream, with flow from left to right, and the ovals A, B, C represent a system of forward-rolling vortices. The arrows within an oval show direction of rotation, and it is important to recognize that the motions they indicate are referred to the center of the vortex, or to the vortex as a whole, and not to the fixed bed of the channel. With reference to the bed all parts of the vortex are moving toward the right, the lower part merely moving slower than the upper. The tendency of vortices toward circular forms leaves certain tracts of the section unoc- cupied by the system of vortices. Consider the tract D, bounded below by the bed and above PROBLEMS ASSOCIATED WITH KHYTHM. 249 by parts of vortices A and B, and give attention to the motions by which the water in the tract is surrounded and influenced, taking care to re- fer each motion to the middle of the tract itself. Thus referred, the motion of the adjacent part of the rear vortex A is to the left and down- ward, that of the forward vortex B is to the left and upward, and that of the bed is to the left. The motions are indicated by arrows. The influences of the vortices tend to make the water about D rotate backward, while the influ- ence of the bed tends to make it rotate forward. The result is not a priori evident, but may be assumed to be something different from simple rotation. Its possibilities will again be re- ferred to. Now consider the tract E, bounded below by parts of A and B and above by the water surface, and give attention to the motions by which its water is affected. Motions being referred as before to the tract itself, that of vortex A is to the right and downward, and that of vortex B is to the right and upward. Above is the motion of the air, which, in the absence of wind, is to the left, as indicated by an arrow. The three influences to which the tract of water is subject all tend to give a backward rotation, as indicated in a similar position at F. The vortex F is secondary to the A, B, C system and rotates in the opposite direction. If its motions be referred to the fixed stream bed, it is evident that the water in its upper part moves in the direction of the current less rapidly than the water in its lower part. The existence of such a vortex therefore tends to reduce the average velocity at the surface of the current and increase it at some lower level. Abandoning now the specific and ideal case, we may state the hypothesis in general terms. Among the important causes of vortical motion in a river or other stream is the mechanical couple occasioned by the general forward motion of the water in conjunction with the resistance of the bed. This tends to form vortices with horizontal axes and forward roll; and the tendency is probably strongest in the lower part of the stream. In a space adjacent to two forward-rolling vortices there exists a tendency toward the development of a second- ary, backward-rolling vortex, but this tendency is apt to be nullified by other and adverse influences except in the upper part of the stream. The free surface of the water does not oppose the development of such reversed vortices. Wherever reversed vortices abound the velocity at the surface (averaged with respect to time) is less than at some level below the surface. The hypothesis as stated has no stationary element, but the phenomena of incipient dunes show that in certain cases repetitive motions are associated with stationary space divisions. A supplementary suggestion assigns these repetitive motions, or their initial phases, to the triangular space D in the ideal diagram, figure 83. The forces tending toward rotation in that space are antagonistic; and the sugges- tion is (1) that they produce some sort of alternating movement with regular periodicity, and (2) that the time interval of this move- ment, in combination with the forward move- ment of the major vortices A, B, C, yields a stationary space interval. An investigation based on this line of sugges- tion and designed to test it should lead also to an explanation of the observed changes in the method by which traction is accomplished. The dune method is associated with a depth which is large in relation to the mean velocity and with a moderate bed resistance. It is replaced by other methods in consequence of (1) a reduction of depth, or (2) an increase of velocity, or (3) an increase of resistance. Reduction of depth diminishes the space for development of the hypothetic vortices. In- crease of velocity or of resistance tends to enlarge their pattern. In either case the coercion of the water surface restricts the freedom of vortical movements and imposes conditions tending to modify their system. THE MOVING FIELD. The competent investigator is resourceful in the creation of new apparatus and methods as the need for them arises. While fully conscious of this fact, and of the further fact that no device is of assured value till it has been tried, I yet can not forbear to mention, for the benefit of others, a method which the Berkeley experi- ence leads me to think valuable for the study of the internal details of a current of water. In a current limited by the sides of a narrow, straight trough transverse movements are largely suppressed, so that most of the action can be learned from observation of what takes 250 TRANSPORTATION OF DEBRIS BY RUNNING WATER. place ill vertical planes parallel to the axis. The movements in a vertical plane may be exhibited by giving to the trough a glass side and by giving to the water in that plane ex- clusive illumination. When one of our labora- tory currents, being illuminated from above, was viewed from the side, small particles in suspen- sion were seen to be conspicuous, so conspicu- ous, in fact, as to resemble the motes in a sun- beam. Some of the particles were shreds of wood fiber worn from the trough, and these could be more easily followed by the eye because of their distinctive forms. Impressed by this phenomenon, I am confident that water movements in a vertical plane can be effec- tively revealed by giving to the water a suitable amount of suitable suspended material and by giving to the selected plane a brilliant and ex- clusive illumination. With the aid of such simple arrangements much may be seen, but measurement will be difficult; and vortical movements may not be easily discriminated from those which are merely sinuous. It is believed, however, that both these results may be achieved by aid of the moving field. In a very simple form this device was employed by us in the study of processes of traction (p. 27), and despite the crudity of the apparatus it was found to be highly efficient. An experiment trough having in its side a glass panel, A A in figure 4, bore a sliding screen, B, in which was an opening, C. Moving debris was watched through the open- ing at the same time that the screen and open- ing were moved in the direction of the current. To the field of view was thus given a hori- zontal motion, and that motion was, in effect, subtracted from the motions of the objects observed. The apparent motions were motions in relation to the moving field, and not to the fixed trough. No provision was made for de- termining the velocity of the field, nor for measuring the apparent motions of the objects viewed; but even without these devices for quantitative work, the possibilities of the method of observation were sufficiently evi- dent. With the necessary supplementary devices, the moving field promises to measure the horizontal and vertical components of motion hi any part of the vertical section, and it thus makes possible the complete delineation of velocities and directions of details of current, so far as those details may be exhibited in longitudinal vertical sections. By giving to the field a suitable velocity it should be possi- ble to see a traveling vortex which rotates in a vertical plane, just as vortices of horizontal rotation are seen on the surface of a stream. A notable defect in the Berkeley arrangement was the requirement that the observer move his head in unison with the moving field. This interfered with steadiness and also limited nar- rowly the space covered by an observation. It could be remedied by substituting for the slide a car which should carry both observer and peephole at a determined rate. Another suggested arrangement places the eye of the observer at a fixed telescope and A FIGURE 84.— Diagrammatic plan of suggested moving-field apparatus. moves the field by means of a rotating mirror. This is illustrated by figure 84, where the trough T is shown in plan. The telescope E views the trough through the mirror, which is pivoted on a vertical axis at M. As the plane of the mirror rotates from a to i the field com- manded by the telescope moves from A to B. It is evident that if the angular velocity of the mirror is constant the linear motion of the field along the trough will be relatively fast at A and B and relatively slow at 0. The error thus arising may be avoided by some device of the nature of linkage. It will be corrected with sufficient approximation if the axis of the mir- ror be controlled by a rigidly attached arm MD, which is in turn controlled by an arm about one-third as long, the two having sliding contact at D and the short arm revolving uni- formly about a vertical axis at F. APPENDIX A.— THE PITOT-DARCY GAGE. SCOPE OF APPENDIX. Many measurements of velocity were made with the Pitot-Darcy gage, but as only a few have been finally utilized in the preparation of the report, a discussion of the instrument seemed not appropriate to the main text. Certain phases of our experience, however, are thought worthy of record because they have practical bearing on the utility and the use of such gages, and these are the subject of the appendix. FORM OF INSTRUMENT. Darcy developed the Pitot tube by adding a second tube, differently related to the cur- rent, and by connecting the two above with a chamber from which the air was partly ex- hausted. The water columns in the two tubes were thus lifted from the vicinity of the water surface to a convenient position, where their difference in height could readily be measured. In the gages constructed for our use the aperture of one tube was directed upstream and that of the other downstream. The tubes were borings in a single piece of brass, which was shaped on the outside in smooth contours, designed to interfere the least possible with the movement of the water. The form first given was afterward modified, and figure 85 shows the third and last design, with which most of the work was done. The openings had a diameter of 0.1 inch. At the opposite or upper end of the brass piece were stopcocks, and above these connection was made with rubber tubes, which led to the complementary part of the apparatus, where the difference in height of the two water columns was observed. It is convenient to call tiie member exposed to the current the receiver, and the complemen- tary member the comparator. In the comparator were two glass tubes, straight and parallel, with internal diameters of about 0.8 inch. At the top they were connected by an arch, and at the summit of the arch was a branch tube, with a pet cock, used in regulating the amounts of air and water. At the bottom they communicated with the rubber tubes through a brass piece, in which were two stopcocks, connected by gearing so as to open and close together. These parts were mounted on a board which also carried a scale of inches and decimals. A sliding index was arranged so that it could be set by the meniscus of a water column and its position then read on the scale; and there was a fixed mirror behind the tubes to aid in avoiding error from parallax. The board was sup- ported in an inclined position, the slope given to the tubes and scale being that of 2£ hori- zontal to 1 vertical. This had the effect of 0.1 o.a root FIGURE 85.— Longitudinal section of lower end of receiver of Pitot-Darcy gage No. 3, with transverse sections at three points. making the movements of the columns 2.69 times as great for the same change of pressure as they would be if the tubes were vertical. In preparing for observation, the internal air pressure was so adjusted that the columns stood near the middle of the scale. The receiver was then held, by a suitable frame, in the selected part of the current, and the stop- cocks were opened. In the glass tube con- nected with the receiver opening facing up- stream the column rose; in the other tube it fell. When the full effect of the current had been realized, the stopcocks at the bottom of 251 252 TRANSPORTATION OF DEBRIS BY RUNNING WATER. the comparator were closed, and the heights of the columns were then read. The use of the gage to measure velocities close to the bed of debris proved impracticable because the presence of the receiver modified the movement of the water and thereby modi- fied the shape of the bed. (See pp. 26, 155.) This effect could have been reduced by using a different form of receiver. Darcy and Bazin * bent the tubes at the bottom in such a way that one or both openings met the water at some distance upstream from the vertical part of the tubes, and it is probable that the adop- tion of their design would have diminished the difficulty, although it could not have removed it. With such a design, however, the practi- cable forms for the second opening relate it to the piezometer, and the advantage of the down- stream opening is lost. That part of the design of the comparator which consists in the inclination of tubes and scale is not to be recommended. It refines by magnifying the reading, but it introduces pos- sibilities of error in other ways. If the tubes are not straight or are not equally inclined, an error is occasioned which does not enter if they and the scale are vertical. Evidence of such error was found in the fact that the still-water or zero-velocity readings of the two columns were not always identical, but no ready means of correction was discovered. Another source of error was detected in ine- qualities of sectional area of the glass tubes of the comparator. To show the nature of this error, let us assume that the pressure of the current at the upstream opening of the receiver is exactly equal to the negative pressure, or suction, at the downstream opening. If the glass tubes are of uniform and equal bore, one column moves upward just as much as the other moves downward, and the volume of air above the columns is unchanged. Now, assume that the tube containing the rising column has the greater diameter. It is evident that equal movement of the two columns will displace more air in the one tube than it will provide space for in the other, and the pressure of the confined air will be thereby increased. The effect of the increased pressure will be to lower i Bazin, F. A., Recherches experimentales sur 1'ecoulement de 1'eau dans les canaux decouverts: Acad. sci. Paris Mem. math, et phys., vol. 19, p. 49, PI. IV, 18fi5. This memoir was published also as part of "Recherches hydrauliques," by H. Darcy and F. A. Bazin. both columns. There is also a secondary effect of small amount connected with the fact that the pressure of the confined air plus the head of water between the tops of the columns and the surface of the stream, on the one hand, and the atmospheric pressure, on the other, are in equilibrium, but into this we need not here enter. As a means for the discussion of these errors the tubes weie calibrated, by Prof. J. N. Le Conte, in the following manner: The tubes being closed at the bottom, a weighed quantity of water was introduced into one and the height of its column was read. By lepeated additions of water and repeated readings the volumes of divisions of each tube were thus measured, the divisions being approximately 1.5 inches in length. In similar manner the volume was measured of the space above the straight tubes to the pet cock. The average sectional area of one tube was found to be 2.5 per cent greater than that of the other. The sectional area in the larger tube was found to vary through a range of at least 2.6 per cent, and the range for the smaller tube was 4.5 per cent. A table of corrections to readings was computed from the data of calibration, and this table was practically applied. In com- paring the rise of one column with the asso- ciated fall of the other the greatest correction applied amounted to a little less than 1 per cent. In the determination of velocity the largest correction applicable for this reason was about 0.3 per cent. These corrections, how- ever, pertain only to the discussions of the in- strument in the following pages. In the ordinary work of the gage they were not ap- plied, because it was found that the errors of this class were practically eliminated when the same methods were employed in the prepara- tion and in the use of rating formulas. The matter is mentioned here chiefly because errors from unequal tube caliber, which may some- times prove important, appear not to have been allowed for in the discussions of Pitot- Darcy gages. BATING METHODS. The first and second gages were rated by the method of floats; the third was twice rated by the running-water method and several times by the still-water method. The floats used weie THE PITOT-DARCY GAGE. 253 vertical cylinders of wood so adjusted that the submerged depth was twice the distance below the water surface of the apertures of the re- ceiver. Some of the still-water ratings were made at the Geological Survey's rating station at Los Angeles, where a car lunning on a track at the side of a reservoir drew the receiver through the water of the reservoir. The others were ma do la ter in the long trough of the Berkeley laboratory, the car in this case running above the water. In the application of the running- water method a measured discharge was passed through a rectangular trough and a survey of velocities throughout a cross section was made by means of the gage. By using in this survey the rating formula obtained by the still-water method, and then comparing the mean velocity thus computed with that computed from the discharge and sectional area, a correction is obtained which may be applied to the still- water rating formula. Only a single compari- son of this sort is practically available in con- nection with our instruments, and the terms of this are: Ft./sec. Mean velocity by discharge and area 1. 98 Mean velocity by gage (with still-water rating). 2. 08±0. 07 The resulting correction to the still-water ratings is — 5 per cent, but this determination has small value because of the large probable error of one of the compared determinations. The fact that the apparent correction is small is in accord with a property of the gage independently observed. By reason of the sinuosity of flow lines in a stream, the direc- tions of motion are not parallel to the axis of a straight channel. Therefore, a current meter wliich records the velocity in the direction of flow instead of the component of velocity par- allel to the channel axis yields an overestimate of mean velocity. Some Pitot-Darcy gages have been found to overestimate velocity when placed obliquely to the direction of flow, and for such the correction would be large. It was found, however, that the Berkeley gage when placed somewhat obliquely to the current gave a lower reading than when facing it squarely, and through this property it tended auto- matically to correct its readings for obliquity of current. If the correction were perfect, the still-water rating and running-water rating should be the same. RATING FORMULA. In the still-water ratings velocities inde- pendently determined were compared with resulting changes in the water columns of the comparator. Starting from the same level, one column rose with increase of speed and the other fell. Except for the influence of modi- fying conditions the changes should be equal, the positive velocity head being of the same amount as the negative velocity head. To test this point various sets of observations were plotted on section paper, the readings of the rising column being taken as abscissas and the readings of the f ailing column as ordinates. In most cases the plotted points fell well into line, and there was no question that the line repre- sented by them was straight. That is, the true ratio of the negative pressure at the down- stream opening of the receiver to the positive pressure at the upstream opening was constant, under the conditions of this particular series of trials. The value of the ratio was found to vary with conditions, and the several values found are so near unity as to confirm the theoretic belief that unity is the normal value. In considering the variations of value it is first to be noted that the gage with wliich the observations were made, No. 3, being sym- metric, could have either opening turned up- stream, and it was in fact used both ways, with record of its position. But its symmetry was only approximate and therefore the two posi- tions gave different results. Used in one way it will be called No. 3a, and in the other way No. 3b. It also happened that between the date of the Los Angeles ratings and that of the Berkeley ratings the receiver was accidently marred at one of its openings, and though its form was afterward restored as nearly . as possible, some difference remained which af- fected its constants. The values of the pressure ratio are accordingly arranged in four groups in Table 81. Two of these groups also are subdivided with reference to the position of the receiver in relation to the perimeter of the current. The ratio was notably larger after the acci- dent than before, and the change was greater for 3a than for 3b. The greatest value of the ratio was given by trials in which the receiver ran close to the side of the reservoir, which in 254 that case was a plank trough. The interpreta- tion of these results will be considered in another connection. TRANSPORTATION OF DEBRIS BY RUNNING WATER. Substituting in (118), we have iff- fl.- 82.28 fcF*. TABLE 81. — Ratio of the suction at one opening of the Pitot- Darcy gage to the pressure at the other. Place of rating. Conditions. Ratio, suction to pressure. Gage 3a. Gage 3b. Los Angeles .... Deep water 0.993±0.015 1.038± .007 1.005±0.005 1.010± .002 1.016± .003 1.014± .005 Do Do Do Shallow, near surface. . Shallow, near side 1.052± .004 Let A. represent the vertical space through which the column of water is raised by pressure from velocity V on the upstream opening of the receiver, and At the simultaneous depression of the column connected with the downstream opening. Then, each being assumed to equal the velocity head, V2 -(116) -(117) In practice the full velocity head is not realized in instruments of the Pitot-Darcy type, and the r) coefficient determined by rating is less than — . y It is a common experience also in practical application of hydraulic formulas to find that qualification is advisable in other respects. I therefore substituted tentatively for (117) the formula ^JcV* (118) and sought empiric values of u and &. The readings of the comparator corresponding to A, and A! may bo called H and 11^. The zero of the comparator scale being at its lower end, the difference between the readings corres- ponds to the sum of the spaces A. and \. The readings are in inches, while the unit used for A, A1; and g is the foot. Moreover, from the incli- nation of the comparator, the space between the two columns, as read on the scale, is 2.09 times the vertical space Ji + Ji^ Therefore, the product of 12 by 2.69 being 32.28, //-//, =32.28 (A + &,)_„. ..(119) or, making A'= 32.38 Jc, 11-11,= KVU (121) The observed quantities being //, 7/1; and V, it was possible to plot on logarithmic section paper any series of values of H— Ht in relation to the associated values of V, and thus compute graphically the corresponding values of u and K. The values were computed for all series of observations represented in Table 81, and they are given in Table 82. The mean of the seven values of u is 2.00, but their range is notable. The deviations from the normal may be ascribed ' in part to accidental errors. In the case of the third value, 1.90, and of the seventh, 1.94, the plotted positions are so scattered as to admit of considerable latitude in the drawing of the equation lines, but the control is much stronger for the values 2.09 and 2.12, and these could not be greatly reduced without violence to the facts of observation. It seems clear that the exponent is not whouy free from the influence of special conditions. TABLE 82.— Values of K and u in //-//I=A'T'«. Place of rating. Conditions. Gage. X u Los Angeles... Deep wator 3a 0 70 Do . .do.... 3b Berkeley. Do... do . 3b Do 3b 70 Do... 3b Do.. For the practical purpose of rating the in- strument, however, there is no advantage in departing from the normal exponent, and that was employed in the preparation of rating tables. The formula used for inferring veloci- ties from readings is .(122) in which The values of A} graphically computed, are given in Table 83, and these values were used in the computations of velocities. THE PITOT-DARCY GAGE. 255 TABLE 83. — -Values of A in V=A*^h — JJt and values of gage efficiency. ] 'lace of rating. Conditions. Values of A. Efficiency of gage. Gage 3a. (iageSb. 3a. 3b. Deep water 1.22±0.02 1.29± .04 1.2C±0.01 1.30± .02 1.24± .01 0.67 .60 0.63 .59 .66 .53 Do . . -. Shallow, near bed Do 1 37± .03 Do 1 26± 03 63 An inspection of the values of A with due attention to their probable errors serves to show that their differences are not to be re- garded as wholly accidental, but must be as- cribed in part to the variation of the instru- mental constant with conditions. That the nature of the variations may be appreciated, the conditions will be more fully described. As already stated, gages 3a and 3b are the same symmetric instrument, but with opposite faces turned toward the current, while the instru- ment was modified to some minute extent by an injury and repair occurring between the Los Angeles and Berkeley ratings. Practically there were four gages, but so far as the mech- anician could make them the four gages were identical in form. The reservoir at the Los Angeles rating station was broad and deep. The course followed by the receiver of the gage was several feet from the bank, at least 6 inches below the surface, and at least 1 foot above the bottom. The plank reservoir used in Berkeley was 1.96 feet wide, and the depth of water was 0.44 foot. In the ratings tabulated as at " mid- depth" the opening of the receiver was central in the cross section of the water. In the " near bottom" rating the center of the opening (its diameter being 0.01 foot) was 0.02 foot from the bottom; in the "near surface" rating the center was 0.03 foot below the surface of the water; and in the "near side" rating the cen- ter was 0.25 foot above the bottom and varied from 0.01 to 0.11 foot in its distance from the side, the course and the trough side not being quite parallel. The several relations of the opening to the water section are shown in fig- ure 86. There is nothing novel in the variation of the instrumental constant with minute differences in the shape of the receiver. This has been observed by all critical users of such instru- ments, and the custom is well established of giving a separate rating to each receiver. • So far as I am aware the variation of the constant with the relation of the receiver to the walls of the conduit and to the water surface has not previously been recorded, and this is a matter of considerable moment, for various elaborate FIGURE 8ti. — Cross section of prism of water in trough, showing positions given to gage opening in various ratings. discussions of the distribution of velocities within a cross section have assumed the con- stancy of the instrumental constant for all posi- tions. As clearly indicated by the tabulated results, the constant A increases when the free surface of the water is approached and dimin- ishes when a rigid wall of the conduit is approached. In the application of these results (and of cognate results with which it was thought best not to burden these pages) to our velocity de- terminations a graphic table was constructed to show the relation of the instrumental con- stant to the position of receiver in the conduit; and this table (fig. 87) supplied the constant to be used with each individual observation. This procedure had an important influence on the interpretation of the running-water rating, making the apparent correction for that rating less than it would be if the mid-depth constant were used exclusively. Its influence on the vertical velocity curves (figs. 74 to 82) was to give them less curvature near the water surface, greater curvature in approaching the channel bed, and (sometimes) a somewhat lower level of maximum velocity. 2o In equation (117) -^ V2 equals twice the theo- retic velocity head. If in equation (118) the exponent u be replaced by its mean value 2, 256 TRANSPORTATION OF DEBRIS BY RUNNING WATER. we have Jc V2 as an expression for the double head actually produced by the gage. The 2 ratio of Jc to — - measures the efficiency of the 9 instrument in realizing the theoretic head. the ure of efficiency is 2 32.16 1 0.996 32.28 A2 ' 2g 32.28 A2 ~~ A2 The values of the measure have been com- puted and are given in Table 83. The deep- water and mid-depth ratings being assumed to represent normal conditions, the mean of the corresponding efficiencies, namely, 0.62, may stand in a general way for the fraction of the theoretic head which is realized by this particu- lar type of the Pitot-Darcy gage. As the relation between velocity and observed head is that between agent and effect, it is evident that values of the efficiency ratio rather than those of the constant A should be compared in any attempt to explain the phe- nomena of variation. Restating from this viewpoint the results of comparative ratings, we have: The response of the head to changes of velocity is lessened when the receiver of the gage is brought near the free water surface and is increased when the receiver is brought near a rigid part of the stream's perimeter. It may be surmised that the differences in head are connected with the facility with which the flow lines of the water are diverted in passing around the instrument, regarded as an obstruc- tion. In midcurrent the diversion is resisted Water surface Bottom or side FIGURE 87.— Graphic table for interpolating values of A, in V= A -jH—Hi, for observations made with gage 3b in different parts of a stream. by the inertia of surrounding water. Near the surface the resistance of water is partly replaced by resistance of more mobile air. Near the conduit the resistance of water is partly replaced by the resistance of an immobile solid. APPENDIX B.— THE DISCHARGE-MEASURING GATE AND ITS RATING. THE GATE. The gate for measurement of discharge is mentioned at page 20, and its general relation to other apparatus is shown in figure 2. In a section on the measurement of discharge, page 22, it is briefly described, and the history of its use is outlined. Figure 88 is designed to ex- hibit its relations more fully and shows the arrangement of the general apparatus of the laboratory at the time of its calibration. The gate controlled an opening in the side of a vertical wooden shaft, of which the internal horizontal section measured 3 feet by 0.5 foot. At the top the shaft communicated freely with a long, narrow tank, to which water was con- tinuously delivered by a pump. The surface level of water in the tank was regulated by means of an overflow weir and a valve above the pump and was observed in a glass tube or gage, AB, in figure 88, outside of the tank. This tube was given a slope of 1 in 10, so that Overhead tank FIGURE 88. — Arrangement of apparatus connected with the rating of the discharge-measuring gate. the movements of its water column afforded a magnified indication of changes in the water level within the tank. Details of the gate are represented to scale in figure 89. A brass plate, PP, attached by screws to the outer face of the shaft, was pierced by an opening of 10.5 by 2 inches, the longer dimension being horizontal. About three sides the edges of the opening were beveled outward, leaving a 45° edge at the 20921°— No. 86— 14 IT inner face of the plate. A sliding plate, G, rested against the inner face of the fixed plate. This was the gate proper, its function being to close either the whole or a definite portion of the opening and thus regulate the width of the issuing water jet. It overlapped by half an inch the margins of the opening, above and below, and was guided by two brass pieces, which appear in sections 6 and c. To the end adjacent to the water jet was given a chisel 257 258 TRANSPORTATION OF DEBRIS BY RUNNING WATER. edge. The brass guides and the adjacent parts of the wooden shaft wall were shaped to a 45° bevel, which, if produced, would reach the edge of the opening in the fixed plate. Movements of the slide were controlled from the outside. A brass rod, 0, firmly attached to it and running parallel to its axis (fig. 89J), rested in a frame at the left of the opening (fig. 89«, e), where its motion was controlled by rack and pinion, the rack being cut on the upper side of the rod. At the left of the frame and pinion the rod slid along a brass scale graduated to inches and tenths; and the gate was set for any desired width of aperture by bringing an engraved index mark on the rod opposite the proper mark on the scale. The operation of the gate was found to be satisfactory. The head under which the jet issued, meas- ured from the middle of the opening to the water surface in the upper tank, was 6.0 feet, plus or minus a small fraction observed by means of the gage above mentioned. PLAN OF EATING. The method of rating was volumetric and empiric. The gate having been set at a par- ticular graduation, and the discharge having been, continued until the rate of flow through the stilling tank and experiment trough had become steady, the outflow of the trough was diverted for a measured time into a special p.- (a,) A O (d) Section through A-S p /Vj=d\Nx t ^ j4^^^Lup~^~^J^. — —s. — , Outer face of gate ^B' f ZS/////////X Section through C-D 0 5 10 Section through E-f 15 20 ZS Inches FIGURE 89.— Elevation and sections of gate for the measurement of discharge. reservoir where its volume was measured. The special reservoir is shown in figure 88 as " sump No. 2," and the diverting apparatus also is indicated. The "diverting trough" was piv- oted at the remote end and could be turned quickly by hand. In one position it received the discharge from the experiment trough and delivered it to sump No. 1, containing the supply for the pump; in another it permitted the water to fall into sump No. 2, arranged for measurement. The work of rating was per- formed by J. A. Burgess, and a full report upon it constituted his graduation thesis in engi- neering at the University of California. The details of apparatus and method were arranged by him, and the following account of the rating is essentially an abridgment of his report. CALIBRATION OF THE MEASURING RESERVOIR. Sump No. 2, constructed of concrete, was approximately rectangular but was not quite regular in form. No attempt was made to base computations of volume on its linear dimensions, but a scale of volumes was gradu- ated directly. A hook gage was first installed, the hook being attached to a vertical rod, to which slow motion could be given. An index borne by the rod near its upper end followed a smooth surface which had been prepared to receive a graduation but initially was un- THE DISCHARGE-MEASURING GATE AND ITS RATING. 259 marked. The index was so arranged that by pressure it could be made to indent the pre- pared surface, thereby producing a mark of graduation. Starting with just enough water in the sump to permit the use of the hook, a hue of graduation was marked. Then a cubic foot of water was added, the hook was read- justed, and a second line was marked. By repeating this process a scale was given to the gage, the unit of the scale being 1 cubic foot. The added units of water were measured in a wooden bottle made in the form of a cube with the opening at one corner ; and the volume of the bottle was so adjusted that it contained 62.3 pounds of water. The capacity of the reservoir was about 38 cubic feet. In the use of the scale thus provided, frac- tions of a cubic foot we/e read by means of a small free scale of equal parts applied obliquely to the space between two lines of graduation. THE OBSERVATIONS. The record of an observation included (1) the width of gate aperture, (2) the time inter- val, by stop watch, during which the water was delivered to sump No. 2, (3) two readings of the hook-gage scale, and their difference, giving the volume of water received by the sump, and (4) two gage readings of water level in the upper tank, one just before and one just after the period of volume measurement. The quotient of the volume of water by the time in seconds gave the discharge for the indicated width of gate opening. The readings of the high-tank water level gave data for a correction to the head, resulting in a small correction to the computed discharges. It was found that repeated observations with the narrower gate openings gave results nearly identical, while the results for wider openings showed more variation. The obser- vations with wide openings were accordingly multiplied to increase the precision of the averages. The variation was ascribed to pulsa- tions of the flowing water originating in the stilling tank. The general precision of the accepted values of discharge, listed in Table 2 (p. 23), is indicated by an average probable error of ±0.2 per cent. The largest computed probable error is ± 0.4 per cent, being that of the discharge for a gate opening of 6 inches. The general formula for discharge through a rectangular orifice when, as in the present case, the vertical dimension of the orifice is small in relation to the head, is in which h is the head, measured from the middle of the orifice, Id the area of the orifice, and c the constant of discharge. The observa- tions give the following values of c for different settings of the gate: Width of gate opening. Value of c. Inches. 1 2 3 4 5 • 6 Mean 0.704 .103 .667 .677 .677 .684 .686 For a "standard" orifice 2 inches square and a head of 6 feet Hamilton Smith's tables 1 give 0.604 as the value of c. The inner surface about the standard orifice is vertical and plane, whereas the surfaces about the orifice of our gate were oblique. The oblique guiding sur- faces served to increase the velocity at the orifice and thus enlarge the constant of dis- charge. The variation of the constant with width of opening is probably connected with the fact that the beveled surfaces were not symmetrically arranged about the opening. On three sides they made an angle of 45° with the vertical plane of the orifice, but the edge of the slide, constituting the fourth side, was beveled at a smaller angle. The constants of the gate were thus affected by the conditions of its setting and a new rating would be neces- sary with a different setting. i Hydraulics, p. 58. INDEX. •"•• Page. Accents, notation by 96 Acknowledgments 9 Adjustment of observations 55-95 Airy, Wilfred, on the law of stream traction 16,162 Alluvium, natural, capacity for 177-178 Antiduues, formation and movement of 31-34, 243 Apparatus, descriptions and figures of 1, 19, 257 Arntzen, "\VaIdemar, work of 9 B. Bazalgette, J. W., on flushing sewers 216 Bed , changes in roughness of, effect of, on velocity 246 characl er of, effect of, on flume traction 206 nature of, in streams and in flumes 15 of dcljris, diagrammatic longitudinal section of 57 profiles of 58 stream, composed of debris grains, ideal profile of 155 contoured plot of 198 Berlin, laboratory of river engineering at 16 Blackwell, T. E., observations by, on velocity competent for flume traction 216 Blasius, H., on deposits resembling dunes 232 on rjiythmic features of river beds foot note, 31 Blue, F. K., experiments by, on flume traction 217-218 Briggs, Lyman J., acknowledgments to 9 and Campbell, Arthur, work of, on viscosity as affected by suspended matter 228 Brigham, Eugene C., and Durham, T. C., work of, on viscosity as affected by suspended matter 228 Burgess, J. A., work of 9 C. Campbell, Arthur, and Briggs, L. J., work of, on viscosity as affected by suspended matter 228 Capacity, definition of 10, 35 for flume traction, in a semicylindric trough 214 in relation to discharge 209 in relation to fineness 210 in relation to slope 208 of mixed grades 212 table of adjusted values of 204-206 in relation to form ratio 213 lor stream traction, in relation to depth 164-168 in relation to discharge 137-149, 233-235 in relation to fineness 150-154, 235 in relation to form ratio 124-136, 236 in relation to slope 96-120, 233 in relation to velocity 155-163, 193-195 maximum 124, 130 of a natural alluvium 177, 180 of mixed grades 113-115,169-185 review of controls of, by conditions 186-193 table of values of, adjusted in relation to slope 75-87 readjusted in relation to discharge 137-138 readjusted in relation to fineness 151 for suspension 223-230 Channels, crooked, experiments with 196-198 curved, features of 198,220-221 form of 10 of fixed wjdth, relation of capacity to slope in 96-116 of similar section, relation of capacity to slope in 116-120 shaping of, by natural streams 221,222 widths of 22 See also Bed and Form ratio. Christy, S. B. , acknowledgments to 9 Competence, definition of 35 influm traction, data bearing on _ 215-216 simultaneous, for all controls of capacity 187 Competence constants, use of term 187 Computation sheet, logarithmic, figure showing 95 Contraction , local, eflect of, on velocity 245 Contractor, influence of 57 outfall, description of 25 Controls of capacity, review of. 186-195 Cornish, Vaughan, on antidunes footnote, 32 on progressive waves in rivers 244 Cunningham, Allan, on the velocities of streams 155 Current, nature of, at bends in channels. . . : 198, 220-221 See also Velocity. D. Darcy, H., and Bazin, F. A., modification of Pitot-Darcy gage by. 262 formulas of, for velocities in conduits and rivers 193-194 Deacon, G. F., experiments by 16 Di'-bris, collective movement of. 30-34 grades of, tables of 21, 199 mixed , causes of superior mobility of 178-179 mixtures of grades of, evidence from experiments with 113-1 15 experiments with 1 69-185 table of observations with 52-54 movement of particles of 155-156 natural, capacity for traction of. 177-178 table of observations with 54 source and sizes of 21-22, 152 table of observations with , 36-54 used in experiments, plate showing 22 See also Fineness and Sand. Depth of water, adjusted values of, table of. 89-93 adjustment of observations on 87-95 gage for measuring 21 in relation to slope, plot of observations on 87 in unloaded streams in flumes, table of 213 method of measuring 25-26 observations on, table of. 38-54 relation of capacity to 164-168 relation of velocity to, ideal curves showing 161 table of values of, for debris of grade (C) , when the width is 0.66 loot and the slope is 1.0 per cent 128 variation of, as related to variation of d ischarge 165 Dimensions of coefficients 64, 129, 139, 151, 186, 191 Discharge, change of, effect of, on velocity 246 competent, table of experimental data on 70 constant, relation of capacity to depth under 164 relation of capacity to velocity under 157 control of constants by 132-133 definition of. 35 diversity of, rn natural streams 221-222 gate for measuring, description and rating of 257-259 measurement of 22-23 mode of controlling 20 relation of , to « 66-67 relation of capacity to 137-149 values of, corresponding to gate readings 23 variation of, in relation to variation of depth 165 Discharge factor, applicability of, to natural streams 2)3-235 influence of 10 Dresden, laboratory of river engineering at 16 Dubuat-Nancay, L. G., experiments of, on competent velocity. 193,216 Dunes, definition of 31 formation and movement of 31, 231-232 longitudinal section illustrating 31 In relation to rhythm of current 242,244 interval between, length of 243 Dupuit, theory of, on suspension footnote, 224 Durham, T. C., and Brigham, Eugene C., work of, on viscosity as affected by suspended matter 228 Duty, constant for similar streams 239 definition of 36, 74 general formulation of 192 In experiments of F. K. Blue 217 in relation to discharge 144, 147 in relation to fineness 154 in relation to slope - 121 values of, corresponding to adjusted values of capacity, table of. 75-87 261 262 INDEX. E. Page. Efficiency, definition of 36 general formulation of 192 in relation to discharge 144, 148 In relation to fineness 154 in relation to slope 121-123 values of, corresponding to adjusted values of capacity, table of. 75-87 Efficiency and capacity, table comparing parameters in functions Of 123 Eger, Dix, and Seifert on scale of a model of Weser River 237 Energy, effect of load on 11, 225-227 relation of, to competent slope 64 Error, probable, for stream traction 56, 73-74,94, 113, 142-143, 151-153 probable, for flume traction 206 Eshleman, L. E., work of 9 Experiments, method of making 10, 22-26 nature and scope of 17-18 results of 240 F. Fargue, L., cited 197 on scale of model river footnote, 237 Feeding of debris, suggested apparatus for 241 influence of, on capacity 56 methods of 23 Fineness, bulk, definition of 21, 35, 183 bulk, relation of, to values of r< i u^ji i_ i u i tfM* * jeV ^, fl^ MAY M949J 1\^ MAY 3 1 1949 / ^LX^ / NOV 7 - 194^ 7 " / "-Y 1 3 1950 J ,,,3 OCT 16 1951 J 18 195 o ^/ LD 21-100m-9,'48(B399sl( >)476 YE 037" M517703 •<