Hans Albert Einstein: Innovation and Compromise in Formulating Sediment Transport by Rivers

Robert Ettema1 and Cornelia F. Mutel2

Abstract: This paper is written to mark the hundredth anniversary of the birth of Hans Albert Einstein ͑1904–1973͒. It casts his career as that of the archetypal researcher protagonist determined to master intellectually the way water flows and conveys alluvial sediment in rivers. In that effort, Einstein personified the mix of success and frustration experienced by many researchers who have attempted to formulate the complicated behavior of alluvial rivers in terms of mechanically based equations. His formulation of the relationship between rates of bed-sediment transport ͑especially bedload transport͒ and water flow comprised an innovative departure from the largely empirical approach that prevailed at the time. He introduced into that relationship the emerging fluid-mechanic concepts of turbulence and boundary layers, and concepts of probability theory. Inevitably the numerous complexities attending sediment transport mire formulation and prompt his use of several approximating compromises in order to make estimating bed-sediment transport practicable. His formulation nonetheless is a milestone in river engineering. DOI: 10.1061/͑ASCE͒0733-9429͑2004͒130:6͑477͒ CE Database subject headings: Sediment transport; Rivers; Alluvial streams; Fluid mechanics; Turbulence; Boundary layer.

Introduction ally earned an undergraduate degree in civil engineering from the Swiss Federal Institute of Technology ͑ETH͒. Albert then encour- Hans Albert Einstein, born in May 1904, might have remained aged his son to come to Germany ͑where Albert was a professor one of countless civil engineers whose work, although locally at the University of Berlin͒. Albert facilitated this move by help- important, had little impact on the world as a whole. However, his ing him locate a job at the steel construction firm of August trenchant independence of spirit and famous father, Albert Ein- Klonne, in Dortmund, where Einstein worked as a structural en- stein, launched him into a productive career as a researcher and gineer focusing on bridge construction. However, by 1931 Albert educator fascinated with the mechanics of bed-sediment transport was becoming increasingly apprehensive about the growing Nazi and water flow in alluvial rivers. By virtue of the times in which power in Germany. Understanding well the threat posed to Jews, he lived ͑1904–1973͒, the trans-Atlantic span of his life, and his and concerned about his son’s safety, Albert encouraged a return name, Hans Albert Einstein’s ͑hereinafter called Einstein͒ career to Switzerland. Seven years later, Albert would again feel the forms a convenient course along which to view the advance of pressure to ensure his son’s safety, and would facilitate a second alluvial-river mechanics as an engineering science. This paper move ͑this time to the United States͒ and job change. Thus Ein- follows part of his career, viewing his efforts to understand and stein’s career was also marked by historic movements; each shift formulate two central issues in alluvial-river behavior: the rela- in its course was induced by a change in the political climate tionship between bed-sediment transport and water flow, and that linked with such movements. between flow depth and flow rate. This paper discusses how despite the politically encouraged Although Einstein lived most of his youth with his mother moves, or perhaps because of them, Einstein emerged as a leading Mileva, who had separated from Albert when Einstein was 10 expert in alluvial-river mechanics, his expertise being sought years old, his career was strongly marked by his father’s influ- around the world. The paper does so with scant inclusion of equa- ence. Family correspondence reveals that, though Albert first dis- tions. Practically every major textbook on alluvial-river mechan- suaded his son from entering civil engineering, he later fostered ics and sediment transport ͓e.g., the books by Einstein’s doctoral and partly directed that career. Until 1927, Einstein and his students Graf and Chien ͑Graf 1971; Chien and Wan 1999͔͒ mother resided in Zurich, where he attended school and eventu- present the main equations comprising Einstein’s formulations. For a broad technical assessment of Einstein’s contributions to 1IIHR-Hydroscience and Engineering, Dept. of Civil and alluvial-river mechanics, the writers defer to the useful synopsis Environmental Engineering, College of Engineering, The Univ. of Iowa, by Shen ͑1975͒, another of his doctoral students. The Proceedings Iowa City, IA. E-mail: [email protected] of a symposium, to honor Einstein on the occasion of his retire- 2IIHR-Hydroscience and Engineering, Dept. of Civil and ment, lists his publications and the graduate students with whom Environmental Engineering, College of Engineering, The Univ. of Iowa, he worked ͑Shen 1972͒. Iowa City, IA. E-mail: [email protected] A theme running through this paper is innovation and compro- Note. Discussion open until November 1, 2004. Separate discussions mise. Though springing innovatively from emerging concepts of must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing turbulent flow and probability theory, concepts that were becom- Editor. The manuscript for this paper was submitted for review and posing well established in engineering only during the early decades sible publication on January 8, 2004; approved on February 6, 2004. This of the twentieth century, Einstein’s formulation of sediment trans- paper is part of the Journal of Hydraulic Engineering, Vol. 130, No. 6, port becomes beleaguered by confounding physical details and June 1, 2004. ©ASCE, ISSN 0733-9429/2004/6-477–487/$18.00. the natural variability of sediment and flow conditions in rivers.

JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JUNE 2004 / 477 and supervising the construction of ETH’s impressive hydraulics laboratory with which to undertake it. A French engineer, Du Boys ͑1879͒, had done some simple flume experiments and proposed the first mechanistic formula for estimating bed sediment transport as bedload, the portion of bed sediment transport whereby bed particles move on or near the bed. A difficulty with his formula, though, was its basis on a misconceived notion of bed-particle movement. Du Boys had assumed that bed sediment moves as a series of superimposed shearing layers, and had arrived at a formula relating rate of bedload transport as per unit width of channel to a critical flow condition beyond which flow mobilized bed sediment, and an excess of average shear stress exerted on the bed. Subsequent flume experiments showed the sliding layer view of bed-sediment move- Fig. 1. Alpine Rhine constrained to a single, straightened channel ment to be fallacious ͑e.g., Schoklitsch 1914; Gilbert 1914͒. Nev- just upstream of Lake Constance, Switzerland ertheless, the notion of a critical shear stress ͑or flow rate, flow depth͒ associated with bed-sediment transport was conceptually appealing. Consequently formulas similar to Du Boys’ were con- sidered best suited for estimating not only bedload transport but Inevitably, simplifying assumptions, empiricism, and other judi- also the total rate of bed-sediment transport; prior to the 1930s, it cious compromises are needed to prop formulation so that it is of was moot whether engineers actually distinguished between the practical engineering use. It is a theme common to many efforts in two transport terms. formulating sediment and water movement in alluvial rivers. Perhaps the most advanced at sizing alluvial channels were the British, who had sought an improved design method for irrigation canals dug through sandy terrain in parts of the Indian subconti- Beginnings: Meyer-Peter’s Flume nent and Egypt. The method, termed the Regime Method ͑e.g., Lacey 1929͒, relied almost entirely on empirical relationships to Professor Eugene Meyer-Peter of the Swiss Federal Institute of characterize channels under long-term equilibrium or ‘‘regime.’’ Technology ͑ETH͒ in Zurich needed to know how much sediment The Regime Method was still in development, and its applicabil- moved with water flowing along the Alpine Rhine, especially the ity to the Alpine Rhine with its gravel bed was uncertain. amount of coarser sediment, gravels and sands, that moved along The problems with the Alpine Rhine clearly showed that the the river’s bed. This need was to bring the young Einstein from few existing formulas were far from being dependable or useful. Germany, where he then worked, back to Switzerland, the country More understanding of fundamental processes was needed. Ac- of his birth. cordingly, Meyer-Peter implemented a comprehensive plan entail- In the late 1920s, the Swiss federal government and the local ing field measurements in the Alpine Rhine, as well as hydraulic cantonal government of St. Gallen, responding to concerns about modeling and flume experiments to be conducted in ETH’s new an alarming increase in the frequency with which the river hydraulics lab. flooded, had contracted Meyer-Peter to recommend an effective To recruit research assistants, Meyer-Peter placed an advertise- modification to the Alpine Rhine over a 20-km reach extending ment in a Zurich newspaper. The ad caught the attention of Mil- from the Alps to the head of Lake Constance. The river, which eva Einstein, and she contacted Albert. He had briefly thought and ͓͑ ͔ ͒ wends through the Swiss Alps to Lake Constance ͑Fig. 1͒, was written about aspects of river mechanics Albert Einstein 1926 , aggrading, and likely would break out of its leveed banks and appreciated the importance of Meyer-Peter’s work, and saw a disastrously flood its valley. Political considerations gave Meyer- promising, safer career opportunity for his and Mileva’s elder son. Peter’s work urgency, as the Alpine Rhine above Lake Constance In 1931 Einstein joined Meyer-Peter’s research effort and started formed an international border between Switzerland, Austria, and working toward the doctoral degree. Liechtenstein. The key question weighing on Meyer-Peter’s mind At first a rather lackadaisical and playful research assistant ͑ ͒ was by how much to narrow the channel so that it deepened Fig. 2 , not that well regarded by Meyer-Peter, Einstein eventu- sufficiently to convey its loads of water and sediment. ally became intrigued by gravel-particle movement along Meyer- Though many efforts at channel modification had been at- Peter’s flume. After a few years, Einstein and colleagues wrote a tempted elsewhere in Europe prior to 1930, they had relied on series of papers presenting research findings stemming from little more than rules of thumb aided by cut-and-fill adjustments Meyer-Peter’s plan. The papers were on bedload transport, hy- to arrive eventually at suitably sized, nominally stable channels. draulic radius and flow resistance, measurement of sediment ͑ The few formulas purporting to relate flow rate and depth for transport, and hydraulic modeling Einstein 1934; Meyer-Peter ͒ water in alluvial channels were so empirically tied to local chan- et al. 1934; Einstein 1935; Einstein and Mu¨ller 1939 . nel conditions that they could not reliably help Meyer-Peter. Also, the few formulas for estimating bed-sediment transport were sketchy and, at best, loosely related bed-sediment transport New Insight: Railway Schedules and Galton’s Board to water discharge or depth through a particular reach of river. Schoklitsch ͑1930͒, a leading European authority on river- While observing particle movement along the flume ͑Fig. 3͒, Ein- engineering at the time, wrote in his highly regarded textbook Der stein realized that a distribution describing the rates of travel of Wasserbau that ‘‘at the present stage of research, a ‘‘calculation’’ identical particles could be used to determine an ‘‘average travel of sediment load was out of the question.’’ Meyer-Peter realized velocity’’ for a group of particles. He borrowed this notion from he faced a complicated task, and in 1927 had set about designing railway-schedule terminology, implying the total distance traveled

478 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JUNE 2004 Fig. 4. Galton’s board

Fig. 2. A playful Einstein amidst a pile of fellow Swiss Federal would be the case for low rates of gravel transport, for which Institute of Technology students water flow dislodges particles from the bed and moves them downstream until lodging in some momentarily secure seating. At intense transport rates, however, a blizzard of particles would divided by the total time of travel including stops. With such a bounce along the bed, each particle pausing for the barest of velocity determined, it might be possible to formulate the compo- moments, if pausing at all. The assumption simplified the proba- nent of bed-sediment transport called bedload, the transport of bilistic analysis and lent it symmetry. bed particles in successive contact with the bed of a channel. Polya suggested that Galton’s board would be a convenient Einstein also realized he needed a course in probability theory statistical device for describing and tracking the movement of a if he were to define distributions and average values of particle bed particle. Galton, a cousin of Charles Darwin, was a statisti- velocity and travel distance. Fortunately, an excellent mathemati- cian who developed a board comprising two perpendicular axes, cian at ETH, Professor George Polya, offered the course Einstein of which one represents distance traveled, and the other repre- needed. Polya took keen interest in the problem Einstein wished sents duration of pause ͑Fig. 4͒. to formulate, though he initially was daunted by its complexity. To play bed-particle movement on Galton’s board, Einstein He guided Einstein through some of the probabilistic aspects of first had to define the board’s properties in terms of movement on bed-particle movement. The two men drew closer as Einstein the bed. Since water flow conditions along the flume were con- sought Polya’s advice on the development of the theoretical as- stant, he assumed that the likelihood of particle motion was the pects of his doctoral thesis ͑George Polya, letter to Albert Ein- same at any point and at any time on the bed. Using Galton’s stein, 1935͒. board Einstein arrived at a formula giving the travel distribution Einstein viewed gravel particle movement as a succession of of a set of particles along the bed ͑or board͒. The formula is alternating forward leaps and rest pauses. Einstein assumed the expressed in terms of a probability distribution that describes the forward leaps to be relatively brief compared to the rest pauses. In number of particles located a distance increment downstream analogy with the movement of a commuter train, the particle is from the particle source ͑the origin of the board͒ since a given taken to move over a distance that is long compared to the indi- period had elapsed. From the known characteristic distribution vidual distances between stops, and that travel periods were neg- ͑and the distribution moments͒, Einstein could determine the av- ligibly brief compared to stop periods. To simplify the formula- erage distance traveled and average resting period of bed particles tion he assumed that the forward leaps take no time. Certainly this moving across the board, and thereby potentially estimate the rate of sediment transport. In correspondence with his father, Einstein described his research, explaining its objective, the difficulties he faced, and the approach he was taking. Albert responded with interest and en- couragement, offering suggestions intended to clarify the process Einstein was attempting to formulate. Albert gave considerable thought to his son’s research subject, and took pleasure in sug- gesting ways to formulate particle motion. For example in one letter from Princeton in 1936, Albert proposed a way to eliminate the approximating assumption whereby the periods of particle motion were taken as negligibly short compared to the periods that the particles were at rest on the bed. That assumption becomes weak at high intensities of bedload transport for which almost the entire bed is mobilized. Usually Albert’s suggestions, though probing Einstein’s formulation, were not fruitful. They led to complicated mathematical equations whose solution then en- tailed dubious simplifications. Fig. 3. Observation, by Einstein, of 22-mm-diameter gravel from the Einstein approached the probabilistic formulation of the bed- Alpine Rhine moving in Meyer-Peter’s new flume at Swiss Federal load transport of bed particles from two standpoints. One stand- Institute of Technology point aimed at determining the distribution of particle travel dis-

JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JUNE 2004 / 479 Fig. 6. U.S. Soil Conservation Service’s Enoree-River Flume, South Carolina, 1939. The flume was designed for measuring sediment Fig. 5. Sample of data from Einstein’s work with Meyer-Peter’s loads and flow in an actual river flume at Swiss Federal Institute of Technology cated in a region of South Carolina that had experienced severe soil erosion problems incurred with intensive farming. Einstein tances in the flume, whence Einstein could attempt to relate worked with colleagues Joe Johnson and Alvin Anderson on ways average particle velocity to the water flow parameters. The second to measure sediment transport. They quickly realized the need to one sought to simulate the capture of bedload particles by a bed- distinguish two distinct populations of sediment conveyed by load basket, such as he had tested in the ETH flume. His experi- water in the river. In one paper ͑Einstein et al. 1940͒ they coined ments with the flume led to data curves, like those in Fig. 5, the term ‘‘washload’’ to describe the river’s load of suspended relating bedload transport rate per unit width of channel versus an fine silt and clay-size particles derived from soil erosion and usu- average speed of particle travel, with particle shape as a third ally not comprising the river’s bed, the source of bed-sediment variable. load. A major step in his formulation required relating the average Einstein continued trying to translate the findings from his characteristics of particle travel to turbulent flow behavior. This thesis research into a practical method for describing and predict- step also required that Einstein deal with the experimental diffi- ing bedload transport of sediment in rivers and streams. He was culty of particles departing the end of the flume during an experi- unconvinced by the critical-shear-stress approach used by several ment. As big as it was, the ETH flume was too short. Without prior formulations ͑e.g., Du Boys 1879; Shields 1936͒.Inhis information on the distances traveled by those lost particles it was opinion, bedload movement was better related to flow turbulence difficult to define the average characteristics of particle move- near the bed. Accordingly he took the principal conclusions from ment. Einstein earned Polya’s commendation by working around his thesis and used them as a basis for a new approach that this conundrum statistically. equated the volumetric rate of bedload transport to the total num- Einstein attained the doctorate degree from ETH in 1937 ͑Ein- ber and volume of particles likely to be in motion. In turn, the stein 1937͒, submitting a thesis in which he novelly applied prob- number and volume of moving particles depended on the probability theory to describe bed-particle movement in turbulent ability that water flow would lift or eject an individual particle flow. Though the scope of his thesis research did not include from its seating on the bed and move it downstream in a given formulation of a method for estimating rates of bed sediment period. Einstein viewed that probability as reflecting the stochas- transport under given water flow and channel conditions, Ein- tic nature of water velocities close to the bed. stein’s insights into individual motion of bed particles formed the The difficulty lay in determining the probability that the hy- basis for his new approach to formulating bedload transport. In a drodynamic lift on any particle on the bed is about to exceed the letter to Einstein, Meyer-Peter described Einstein’s doctoral study particle’s weight within a given period of time. From a different as producing ‘‘some intriguing ideas, but not exactly useful for perspective, the probability could be viewed as the part of the bed my Alpine Rhine study.’’ for which hydrodynamic lift force exceeds particle weight. The probability problem is comprised of two parts. One part concerned the need for an equation for hydrodynamic lift; particle Formulation: Enoree-River Flume weight is relatively easy to formulate. The other part concerned finding a meaningful expression of time; transport rate implies Albert, who had moved to the United States in 1934 because of movement per unit time. Einstein adapted a well-known and stan- his concern about political movements in Germany, persuaded dard formula for hydrodynamic lift, writing it in terms of a local Einstein to come to the United States in 1938. His arrival coin- velocity of water flow at a level near the bed. Here, though, as- cided with the recent establishment of the U.S. Soil Conservation sumptions were needed regarding estimation of the velocity and Service ͑SCS͒, reflecting a great national concern about soil ero- lift coefficient. sion and the condition of many rivers. Albert assisted his son in The trickier problem concerned the inclusion of a time period. securing a position as a cooperative agent with SCS’s newly es- The most reasonable period to use was the average time required tablished field laboratory on the Enoree River ͑Fig. 6͒, near for the water to remove one particle from the bed. Unfortunately Greenville, S.C. The lab was established for measuring sediment there is no way to express the time required for hydrodynamic lift loads in the Enoree River in order to better understand the rela- to pick up a particle. Einstein assumed that lift involves some tionships between sediment transport and water flow. It was lo- characteristic dynamic feature of the flow field around a particle

480 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JUNE 2004 falling in still water. Particle diameter divided by particle fall velocity expresses a characteristic time. Up to this stage, his formulation was reasonably rigorous, once the under-girding assumptions about average particle step length were accepted. But the subjective use of fall velocity for particles in a description of particles rolling and bouncing along the bed was unsettling. By combining formulas for the weight rate of bed particles moving as bedload, hydrodynamic lift on a bed particle, bed particle weight, and characteristic time based on bed particle fall velocity, Einstein arrived at a seemingly simple relationship between intensity of sediment discharge and the probability of particle entrainment from the channel bed. In terms of a more detailed formulation ͑repeated in most textbooks͒, Einstein expressed this relationship as

A⌽ϭ f ͑B⌿͒ (1) Fig. 7. Relationship between ␾ and ␺; modified from Einstein ͑ ͒ The two parameters, ⌽ and ⌿, were central to Einstein’s charac- 1942 terization of bedload transport; ⌽ϭdimensionless expression for intensity of sediment transport; and ⌿ϭdimensionless expression for flow intensity or gross shear force exerted on the bed. A and bed channels. He queried his own assumption that all bedload Bϭconstants incorporating awkward details about particle shape particles moved in steps of constant length proportional to particle and step length, as well as water velocity distribution. diameter, unaffected by flow conditions. His work at ETH had Einstein could not derive the exact form of the relationship suggested this to be the case for the gravel beds at fairly low between ⌽ and ⌿. Too many variables were unknown. He instead intensities of transport for which the probability of particle en- had to find the relationship from plots of bedload data interpreted trainment was moderate or low. He conjectured that, with increas- as ⌽ versus ⌿. If his formulations were correct conceptually, the ing intensity of transport, the probability of entrainment is high data would lie systematically along a single curve signifying a and the step lengths increase from the constant length at low single general relationship, or ‘‘law,’’ for bedload transport. intensities. As step length increases, the area and number of particles starting movement together increases, and consequently so does the rate of bedload transport. This refinement of his theory Validation Test: Gilbert’s Data modified the relationship between ⌽ and ⌿, and led to a second curve with a common stem as the original curve, but which Obtaining reliable data from which to determine the relationship, veered away in almost the same manner as the cluster of Gilbert’s however, was not straightforward. Einstein used the only two sand-bed data. The new curve, though, still did not run through comprehensive sets of lab flume data readily available to him at those data. Einstein wondered if Gilbert’s data were tainted with the time: his from ETH ͑Meyer-Peter et al. 1934͒, and those pub- measurement error. lished by Karl Grove Gilbert about 20 years earlier ͑Gilbert By 1941, Einstein had sufficiently ordered his thoughts on a 1914͒. Gilbert, a prote´ge´ of John Wesley Powell, had conducted method for describing and predicting bedload transport that he novel and comprehensive flume experiments at the University of was able to get them published as an ASCE Proceedings paper California-Berkeley. The great river surveys of the 1800s ͓notably ͑Einstein 1942͒. As was the practice of the ASCE Transactions Humphreys and Abbot ͑1861͒, Powell ͑1875͔͒ were accompanied Journal, which subsequently published his paper, his paper was by engineering and scientific desire to know more about the me- accompanied by discussions by researchers interested in alluvial chanics of rivers in the United States Gilbert’s data encompassed sediment transport. It drew praise for its attempt to relate sedi- a greater range of sediment and flow conditions than did Ein- ment movement and flow mechanics, but it raised questions about stein’s ETH data. Incidentally, Gilbert too had used a railway the main assumptions spanning the gap between formulation con- analogy to characterize water and flow and sediment transport in cepts and presentation of a practical predictive method. In par- rivers; the term ‘‘grade,’’ meaning channel slope, was borrowed ticular, it was criticized for purporting to be on greater rational from the grade of railway tracks, which commonly were laid basis than were the current formulations based on the concept that along the relatively level ground of flood plains alongside rivers a critical value of bed shear stress or rate of water flow be ex- and streams ͑Pyne 1980͒. ceeded before bed particles could move. Difficult questions in- By and large the two sets of data fell along a curve in accor- cluded: Why base the formulation on lift force alone? Why should dance with his formulation, except for a range of conditions re- settling velocity be included in a formulation of bedload trans- flecting high intensities of sand transport. Those data veered sub- port? One discusser, Anton Kalinske, remarked that Einstein evi- stantially away from Einstein’s postulated curve, and clustered dently had ‘‘stepped over into the realm of abstract dimensional along their own curve ͑Fig. 7͒. The deviant data suggested that analysis’’ when he used particle settling velocity as a convenient bedload transport, or rather bed-sediment transport, could not be parameter to put the probability of particle motion in a time con- fully described using his method. What disconcerted him was the text. Kalinske had attempted to include turbulence in formulating realization that the deviant data were not merely a batch of results the movement of bed particles ͑Kalinske 1942͒, and provided from a set of extreme hydraulic conditions, but in fact were rep- insightful comments that Einstein eventually would have to con- resentative of flow and sediment transport in the sand-bed chan- sider in advancing his formulation. nels representative of most rivers in the United States. Another discusser, Samuel Shulits, who in the early 1930s had The deviation caused Einstein to review the formulation of his prepared an English translation of Schoklitsch’s book Der Wass- method, and to question the accuracy of Gilbert’s data for sand erbau, wrote that Einstein’s ‘‘scholarly probe into the universal

JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JUNE 2004 / 481 creek. He and technicians were out at the enlivened creek imme- diately the next morning, recording its discharges of water and sediment. The equipment worked well and the measurements proved, at least to Einstein’s satisfaction, that his concepts were valid for a little river like Mountain Creek ͑Einstein 1944͒.He subsequently obtained further data from another little river, West Goose Creek in Mississippi.

Method Extended: Caltech Flume

With the entry of the United States into World War II, and after the modest yield of results from the Enoree River, the SCS wound down its work at Enoree Field Station and reassigned the station’s personnel. In 1943 Einstein was transferred to SCS’s laboratory at Fig. 8. Mountain Creek, Miss., 1941. The creek was fitted with a the California Institute of Technology, Pasadena. Albert was en- size-reduced version of the sediment-measurement apparatus used for thused about the move and encouraged his son to contact The- the Enoree River ͑Fig. 6͒. odore von Ka´rman, a renowned Caltech fluid mechanician. Von Ka´rman, however, was busy with war-related matters, and he never developed Albert’s hoped-for relationship with Einstein. law for the transportation of bed load is inspiring.’’ Shulits then As his part of the war effort, Einstein was seconded to quickly tempered his praise by comparing Einstein’s formula, Eq. Caltech’s Hydrodynamics Laboratory to work on shock waves ͑1͒, with bedload formulas based on the notion of a critical flow produced by explosives and projectiles breaking the sound barrier. condition beyond which bed particles of a characteristic size However, he still had opportunities to continue developing his began moving. He thought the latter formulas more directly ex- bedload method and to investigate several pressing problems pressed the relationship between flow and sediment transport, emerging in the wake of dam building and other engineering ac- whereas Eq. ͑1͒, notwithstanding all the interesting background to tivities along rivers, in particular along the Rio Grande River. As its formulation, essentially devises a functional relationship be- Einstein saw things, accurate estimation of bed-sediment load, not tween two dimensionless parameters. In his closure to the discus- just bedload, was the most important problem in alluvial-bed river sions, Einstein ͑1942͒ stoutly defended his approach and dispar- engineering. The ability to predict bed-sediment load in a river aged the notion of a critical flow condition, calling it ‘‘a condition would enable engineers to predict the river’s response to changes that does not exist in nature.’’ The significance of his approach, he in its water and sediment loads, thereby reducing the uncertainty argued, was not so much its immediate outcome, Eq. ͑1͒ and Fig. associated with utilizing the river as a resource for water and 7, but rather its grappling with the problem of formulating bed- hydropower. sediment transport in terms of actual turbulent-flow behavior. He Convinced of the essential correctness of his bedload method, acknowledged that ‘‘the problem is far from solved.’’ Einstein set about extending it by addressing several complicated aspects of bed-sediment transport: bedform development, transport of nonuniform bed sediment, and combined bedload and Mountain Creek: A Little River suspended-load transport of bed sediment ͑i.e., total bed-sediment load͒. In contrast with the Alpine Rhine and the Enoree, Mountain SCS researchers at Caltech, Arthur Ippen and notably Hunter Creek in South Carolina was a mere ditch ͑Fig. 8͒. Yet, to Ein- Rouse, had formulated an equation for the vertical distribution of stein, Mountain Creek was an ideal little river. The creek pos- suspended bed sediment over the depth of flow. The equation, and sessed most of the characteristics of alluvial rivers that Einstein lab data supporting it, were written up by Rouse ͑1939͒, and sought to understand and formulate. Moreover, it was conve- subsequently elaborated by another SCS researcher, Vito Vanoni niently small so that Einstein could measure its water and sedi- ͑1946͒. Commonly called the Rouse equation, it is one of the ment loads. The Enoree River field station had proven disappoint- more successful formulations of sediment transport. However, it ing for obtaining field data on bedload because of insufficiently gives only the distribution of suspended-sediment concentration frequent large flows. relative to some reference elevation near the bed, showing that the Mountain Creek could help in calibrating or linking his concentration decreases rapidly with higher elevation in a flow. A laboratory-flume insights and equations to the behavior of a sand- practical difficulty was that the equation does not give the abso- bed river. He had learned from his Alpine-Rhine work at ETH lute suspended-sediment load. To get that, the relative distribution that laboratory results, and formulations based only on the results has to be tied to a known, or estimated, sediment load concentra- of laboratory idealizations of rivers, usually are regarded skepti- tion at some level near the bed. Here, Einstein saw an opportunity cally by practical engineers dealing with real rivers. If he could to link the Rouse equation with his formulation of bedload and, in show that his ideas worked for sediment movement in Mountain his words, to produce ‘‘a unified method for calculating the part Creek as well as in his ETH flume, then showing that they worked of the sediment load in an alluvial river that is responsible for for a river would be a matter of simple geometry. maintaining the channel in equilibrium’’ ͑Einstein 1950͒. As perverse luck would have it, the summer and autumn of Einstein surmised that the suspended-load distribution as de- 1941 were relatively dry in South Carolina. Flows in the creek scribed by Rouse’s equation could be spliced to the top of the barely moved any sediment. Only a few inches of rain fell, though bedload layer as described by the bedload formulation. The con- a single storm did drop an inch-and-a-half of rain during the centration of particles moving at the top of the bedload layer evening of almost the last day Einstein intended to monitor the would serve as a convenient and reasonable reference concentra-

482 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JUNE 2004 tion with which to set the maximum concentration at the bottom Prominent among the 10 was Einstein, whose new approach to of the suspended load distribution. However, the splicing of bed- bedload transport estimation Vanoni described as ‘‘a radical de- load and suspended load is easier said than accurately done. It parture from all previous bed-load formulas.’’ meant also that Einstein needed to strengthen the rigor of his Einstein addressed the participants on two issues of keen in- bedload formulation. His statistical observations on particle mo- terest with regard to the sediment troubles along the Rio Grande: tion would have to be modified, including his earlier assumption; measuring and predicting the rate at which rivers move sediment the average step of a certain particle is the same even if the along their bed ͑Einstein 1948͒. He likened the middle Rio hydraulic conditions or the composition of the bed changes. Gil- Grande to Mountain Creek, asserting that it behaved essentially bert’s data and the data from Mountain Creek showed that this like the creek. Unlike the other speakers, Einstein could draw on assumption did not hold at the intense rates of sediment transport and describe European as well as U.S. experience. Moreover, for occurring for sandy rivers. many participants the name Einstein held beguiling promise of To see how particles move under conditions of intense rates of major breakthroughs in understanding and formulating the me- sediment transport, and to get adequately detailed data on bed- chanical laws of sediment transport by rivers. Within several sediment transport at high intensity rates for which bed sediment months of the Interagency Sedimentation Conference, former would be transported in suspension as well as along the bed, SCS colleague Joe Johnson facilitated Einstein joining the engi- Einstein needed more flume experiments. During the period neering faculty of the University of California at Berkeley. 1944–1946, while others at Caltech were largely occupied by Over the following 2 years, Einstein completed a detailed defense-related research, Einstein used his spare time to churn write-up of his bed-sediment transport method and published it as water and sediment through a small recirculating flume in the U.S. Department of Agriculture Report 1026 ͑Einstein 1950͒, now SCS lab. widely recognized as a milestone in alluvial-river mechanics. His method became widely known thereafter as the Einstein method, and was used extensively by the Bureau, the U.S. Corps of Engi- Recognition: Rio Grande River neers ͑USACE͒, the U.S. Geological Survey, and many others. Report 1026 elaborated and better explained Einstein’s probabil- While concerns about soil conservation and sediment transport ity approach to bed-sediment transport. Moreover, it presented an had been set aside during World War II, these concerns returned elegant splicing of the bedload and suspended-load components urgently right after the war, at which time Einstein was well po- of bed-sediment transport, and it introduced new concepts aimed ⌽ ⌿ sitioned to play a leading role in addressing them. In May 1947 at reducing some of the empiricism in the versus relationship ͑ ͒ the various federal agencies concerned with rivers and their wa- introduced by Einstein 1942 . tersheds convened at the Denver headquarters of the U.S. Bureau The concepts included modifying the flow intensity parameter ⌿ of Reclamation to hold the nation’s first meeting focused on the so that it could be used for estimating the transport rates of sedimentation troubles facing engineers and soil conservationists particle-size fractions comprising a bed of nonuniform sediment. in the United States. All the federal agencies sent representatives. Further, it involved estimation of the flow energy expended on Also in attendance were engineers and scientists from diverse bed-particle roughness, not on the entire bed; introduction of two state agencies, universities, and a number of overseas organiza- adjustments to account for the velocity of flow locally around a tions. The conference placed Einstein center-stage as one of the particle; and pressure distributions at a bed surface of nonuniform nation’s leading authorities on sediment transport at a time when sized sediment. The modified parameter is the full implications of the nation’s sediment troubles were be- ⌿ ϭ␧Y͑␤2/␤2͒⌿Ј (2) coming pressing. * x The need for the conference had arisen from a growing na- in which ␧ϭfactor intended to account for the sheltering of tional recognition of the widespread, adverse consequences that smaller particles amidst larger particles in a bed of nonuniform sediment troubles were posing for river-basin development and sediment; Yϭpressure-correction factor, which together with for the conservation of land and water resources. The national ␤2 ␤2 / x , is intended to account for the influence of particle size scope of the troubles had become increasingly worrisome during nonuniformity on hydrodynamic lift; and ⌿Јϭ⌿ modified in an the mid-1930s, shortly after the federal government had initiated effort to account for bed sediment development of bedforms on numerous programs to enhance irrigation, hydropower, naviga- channel beds. Though quite readily conceived, near-bed com- tion, flood control, and soil conservation. Severely eroded water- plexities in flow and particle disposition meant that ␧ and Y have sheds, river-channel aggradation or degradation, reservoir sedi- to be determined empirically from flume data. Textbooks on sedimentation, and the adverse environmental effects of muddied ment transport ͑e.g., Chien and Wan 1999͒ explain the details waters all indicated that much more needed to be learned about associated with the terms in Eq. ͑2͒, and elaborate on the subse- watershed and river behavior. Prominent among the sediment quent work examining ␧ and Y. Using Gilbert’s data and his own troubles discussed were those along the Rio Grande River. ETH data, Einstein arrived at the following relationship between SCS colleague Vito Vanoni ͑1948͒ outlined for conference par- probability for motion, p, the parameter ⌿ , and the intensity of ticipants a history of the development of predictive relationships * ⌽ bedload transport for particles in the particle-size fraction, i for sediment-transport and water flow in alluvial rivers. He laid ϭ ⌽ ϭ * (iB /ib) , where ib and iB fractions are the fractions of a out the big questions to be addressed in order to better understand given particle size in the bed and the bedload, respectively: how rivers move sediment, and ended his presentation by lament- (1/7)⌿ Ϫ2 ing the lack of scientific and engineering attention given in the 1 i 2 43.5⌽ ϭ Ϫ ͵ * Ϫt ϭ *i United States to sediment-transport problems, problems whose p 1 ␲1.2 e ϩ ⌽ (3) Ϫ(1/7)⌿ Ϫ2 1 43.5 i national importance he ranked with the more popular contempo- *i * rary problems of atomic energy and rocket propulsion. Fewer than This equation is commonly referred to as the Einstein bedload 10 professionals in the United States, he estimated, were devoting function ͑e.g., Chien and Wan 1999͒. Fig. 9 shows a data curve the major part of their time to the study of sediment transport. relating ⌿ and ⌽ . Though the Einstein method in Report * *i JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JUNE 2004 / 483 questions in the Missouri itself and in its tributaries. This is the only way to find the relative importance of the various influ- ences’’ ͑Einstein, unpublished report to Missouri River Division, U.S. Army Corps of Engineers, Omaha, 1948͒. He realized that the efficacy of the predictive methods would have to be checked by the simultaneous measurement of sediment transport and the flow variables of the river itself. One immediate matter was a little delicate. The method selected for estimating the rates of bed-sediment transport through the river was the bedload equation proposed by Professor Lorenz Straub, a prominent hydraulics engineer who had been a USACE engineer in the 1930s. Straub ͑1935͒ had proposed the method while working on House Document 238 ͑Missouri River Report͒, a detailed assessment of the flow and sediment problems posed to ⌿ ⌽ ͑ ͒ Fig. 9. Relationship between IL and from Einstein 1950 engineering use of the Missouri River. As Straub was the senior, * and initially the most vocal, board member, and since his method had been developed expressly with the Missouri River in mind, 1026 was comprehensive, it was somewhat cumbersome to apply, his was the method that the division had decided to adopt. Ein- and some of its components were found to need adjustment. stein was uncomfortable with the method. In a long letter report to the division, he outlined the steps that needed to be taken to gauge the sediment load conveyed by the river, and he went Application: Missouri River through the shortcomings of Straub’s method. Besides being essentially an extension of the shear-stress ͑or discharge͒ excess For 2 months during the summer of 1948, about 4 months after approach proposed earlier by Du Boys and others ͑e.g., Schokl- his first meeting as a board member for the Missouri River Divi- itsch 1934͒, Straub’s method assumed that the river kept its cross- sion’s Sedimentation Studies program, Einstein was in North Da- sectional shape and its roughness for the full range of water flow kota and Montana, at the headquarters of the USACE’s Garrison and while the river’s bed degraded or aggraded. These assump- District. Over the remainder of his career he maintained a produc- tions seemed unreasonable to Einstein, and they were not sup- tive relationship with USACE, assisting it with sediment concerns ported by measurements of flow depth and flow rate for selected along the Missouri, the Arkansas, and other rivers, and working to reaches of the river. better formulate sediment transport and water flow in rivers. Einstein together with the division’s engineers had examined USACE’s Missouri River Division was charged to oversee a data on the Missouri River and several of its tributaries in an vast watershed that covered about one-fifth of the continental U.S. effort to better understand the relationship between flow depth In 1948, the Division’s efforts were concentrated largely on and flow rate for these rivers. An explanation ventured in terms of implementing the Pick-Sloan Act, which prescribed a plan to con- changing channel shape failed because channel shapes were trol and regulate the river’s flow for the purposes of flood control, found not to change appreciably as flow varied. A more promising hydropower generation, and navigation. However, the Division explanation related flow energy loss to the intensity of bed- found implementing Pick-Sloan to be fraught with more difficul- sediment transport ͑and thereby Einstein’s modified flow-intensity ties than the plan had envisioned. All of the dam projects called parameter ␺Ј), and variations in bedforms and thereby bed for were facing difficulties and setbacks attributable to the river’s roughness. sediment ͑Fig. 10͒. Both Straub’s and Einstein’s methods were used for estimating By September 1948, the technical problems facing the division bed-sediment transport, though Straub’s was soon abandoned. were clear to Einstein. He summarized them in a brief report to Einstein, though, encountered an unexpected complication: down- the division, stating ‘‘every attempt must be made to study the stream of Fort Peck Dam, the river’s degrading bed became ar- mored with coarser bed sediment. No sediment-transport method had taken armoring into account.

Further Reﬁnement: Berkeley’s Flumes

The University of California-Berkeley’s ability to attract talented graduate students, combined with Einstein’s link to USACE, enabled him to undertake at Berkeley a sustained research effort aimed at better understanding and formulating sediment-transport processes. It was an effort largely undertaken by graduate students and USACE engineers working under Einstein’s guidance. They embarked on a comprehensive series of flume investigations aimed at illuminating key aspects of sediment and flow behavior. Moreover, Einstein’s Berkeley appointment enabled him to teach, something he enjoyed ͑Fig. 11͒. ͑ ͒ Fig. 10. Board members Einstein, Straub, Vanoni, Lane and Corps Briefly mentioned here are two examples illustrative of that engineers ponder bed degradation of the Missouri River downstream effort. An especially important issue, and one that has challenged of Fort Peck Dam, Mont., 1948 formulation of sediment transport in rivers, concerns what hap-

484 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JUNE 2004 Confronting Complexity The complex mix of processes at play in natural alluvial rivers has defied ͑so far at least͒ reliable prediction of bed-sediment transport and flow depth; uncertainties of 100% or more are common for predicting rates of bed-sediment transport. Engineers and scientists have long recognized that rivers are complex, and accordingly have used largely empirical as well as analytical ap- proaches to characterize alluvial-river behavior. Commonly, the practical design engineer and the scientist in the field have found the empirical approach more practicable and have been skeptical of sophisticated, predictive methods based on advanced fluid mechanics and data from laboratory flumes. Proponents of the more empirical approach and those of the largely mechanistic approach are quick to debate each other’s methods, especially when one claims to be the superior. The following exchange is an example of the debate, and illustrates Einstein’s conviction about the ulti- mate truth of the concepts supporting his method for estimating bed-sediment load. The exchange follows a paper published by Ning Chien, Einstein’s student. Shortly before he returned to China where he was to play a leading role addressing that country’s river problems, Chien in 1954 published two ASCE Proceedings papers that drew a salvo Fig. 11. Einstein at a Berkeley flume explaining flow processes to of criticism from a leading exponent of the ͑empirical͒ Regime students method approach to river behavior. One paper ͓‘‘The present status of research on sediment transport’’ ͑see Chien 1956͔͒ addressed the relationship between water discharge and bed- pens if the bed sediment comprises a wide range of particle sizes. sediment load. In it, Chien described the reliance of sediment- This situation, of course, is the norm for most riverbeds. A basic transport formulation on accurate formulation of water flow. assumption underpinning the Einstein method needed further Appended to Chien’s paper was a stern discussion criticizing work; i.e., that all particle sizes in a river may be equally avail- Chien’s neglect of the body of understanding collectively termed able at the bed surface and within the bed. Over about 1 year, the Regime Method. The discusser, Thomas Blench, a very ca- 1950 to 1951, graduate student Ning Chien and Einstein carried pable hydraulic engineer and a leading proponent of that method, out a series of flume experiments and were in a position to pro- argued that Chien’s paper presented knowledge limited only to vide detailed descriptions of the processes whereby different sized findings from ‘‘laboratory flumes with trifling flows.’’ He further bed particles segregate in the upper layer of a river bed, how argued that Chien had neglected ‘‘the vast amount of observations armoring occurs, and how a riverbed acts much like a reservoir on canals in the field, the dynamical aspect of the formulas for sediment, storing it during periods of reduced water flow, and evolved there-from, and the fact that these formulas provide a releasing it during periods of greater water flow. simple and adequate means of practical design that has been used With doctoral student Robert Banks, Einstein began investigat- widely for many years.’’ The Regime Method’s formulas, Blench ing how several factors contribute to flow resistance in rivers. claimed, ‘‘represent what real channels actually do.’’ This effort would provide more insight for his sediment-transport Einstein, though not a coauthor of Chien’s paper, wrote an method, and it would help address a crucial companion issue additional closure discussion to that by Chien. He took issue with concerning the relationship between water discharge and depth in the Blench’s claim about the sufficiency of the ‘‘superiority of the alluvial rivers. The total resistance opposing the flow consists of ‘simple and adequate’ ’’ Regime formulas. Those formulas, he the combined effect of resistance attributable to surface rough- pointed out, were developed by curve-fitting of data from ‘‘a very ness, bedforms ͑or bar resistance as he expressed it͒, and vegeta- narrow range of bedload conditions.’’ He went on to express, tion. Einstein wondered if the total resistance could be expressed among other things, his doubt that the Regime formulas would as the sum of these components. This thought was not new. It had work for rivers in the United States. In his closure following been used successfully in determining flow resistance in flow Blench’s cutting discussion of another ASCE Proceedings paper around bodies. ͑see Einstein and Chien 1956͒, Einstein presented figures showing The notion of dividing flow resistance into two parts, particle the inadequate performance of the Regime formulas. roughness drag and bedform drag, was new for alluvial-river me- Since Einstein ͑1950͒ numerous methods have been developed chanics. Its first practical implementation is the Einstein- for estimating the relationships between water discharge and bed- Barbarossa method for estimating the relationship between flow sediment transport in alluvial rivers. Some methods have built on depth and flow rate in alluvial channels. Einstein and Nicholas the Einstein method laid out in Report 1026, or modified the Barbarossa, an engineer with the USACE’s Omaha District, used method for better accuracy and more convenient use ͑e.g., Colby data from the Missouri, several of its tributaries, and two Califor- and Hembree 1955; Bishop et al. 1965͒. Others have developed nia rivers to find a relationship between Einstein’s parameter ⌿ from improved insights, and still others have remained resignedly * and that part of flow-energy loss attributable to bedforms. Addi- empirical ͑e.g., Brownlie 1981͒. Meyer-Peter’s research plan for tionally, they used the Manning-Strickler equation to estimate en- the Alpine Rhine led to another quite widely used method for ergy loss attributable to surface roughness. Publication of their estimating bedload transport ͑Meyer-Peter and Mu¨ller 1948͒. method ͑Einstein and Barbarossa 1952͒ was a further milestone in Ironically, Einstein through his early work at ETH ͑Meyer-Peter formulating alluvial-river mechanics. et al. 1934͒ had a significant role in developing that method.

JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JUNE 2004 / 485 he and friend Don Bondurant, a retired USACE engineer, had outlined a book on alluvial rivers. It was not to be the usual format of textbook, but rather an approach that introduces typical engineering problems arising between people and alluvial rivers, then explains the knowledge and methods needed to solve the problems. The book, like Einstein’s work to formulate sediment transport, was a task that unfortunately remained unﬁnished. In 1988, the American Society of Civil Engineers established the Hans Albert Einstein Award ‘‘to honor Hans Albert Einstein for his outstanding contributions to the engineering profession and his advancement in the areas of erosion control, sedimentation and alluvial waterways.’’

Acknowledgments Fig. 12. Einstein maintained a life-long interest in sediment The writers thank Professor Daniel Vischer of ETH-Zurich for his movement assistance with background material used in preparing this paper. They also thank the paper’s reviewers and Pierre Julien, JHE Editor. Latter Years

Much of Einstein’s career can be cast as the archetypal story of References the researcher protagonist determined to master intellectually the way water flows and conveys alluvial sediment in a river. In that Bishop, A. A., Simons, D. B., and Richardson, E. V. ͑1965͒. ‘‘Total bed- effort, he personifies the mixed success and frustrations experi- material transport.’’ J. Hydraul. Eng., 91͑2͒, 175–191. ͑ ͒ enced by many researchers who have attempted to describe the Brownlie, W. R. 1981 . ‘‘Prediction of flow depth and sediment dis- complicated behavior of alluvial rivers in terms of rationally charge in open channels.’’ Rep. No. KH-R-43A, W. M. Keck Labora- tory of Hydraulics and Water Resources, California Institute of Tech- based equations. The effort begins keenly with apparent good nology, Pasadena, Calif. promise of success, based on innovative new insights into com- Chien, N. ͑1956͒. ‘‘The present status of research on sediment transport.’’ ponent processes. Formulation seems within reach and progress is Trans. Am. Soc. Civ. Eng., 121, 833–868. made, but then judicious assumptions and curve-fitting empiri- Chien, N., and Wan, Z. ͑1999͒. Mechanics of sediment transport, Ameri- cism have to be invoked as approximating compromises to ac- can Society of Civil Engineers, Reston, Va. commodate the many confounding complexities inevitably faced. Colby, B. R., and Hembree, C. H. ͑1955͒. ‘‘Computations of total sedi- Einstein retained his long fascination with alluvial rivers ͑Fig. 12͒ ment discharge, Niobrara River near Cody, Nebraska.’’ Water Supply and continued his efforts to understand and formulate how they Paper 1357, U.S. Geological Survey. ͑ ͒ ´ convey sediment. During his latter years, his fascination broad- Du Boys, M. P. 1879 .‘‘Etudes du re´gime du Rhoˆne et de l’action ´ ` ´ ened to include sediment transport in coastal waters. exercee par les eaux sur un lit a fond de graviers indefiniment affouil- lable.’’ Ann. Ponts Chausseés, Se´rie 5, Paris, 8, 141–195 ͑in French͒. Einstein retired from his active professorship at the University Einstein, A. ͑1926͒.‘‘U¨ ber die Ursachen der Maänderbildung der Flu¨sse of California on July 1971, at the age of 67. His retirement earned und Baersschen Gesetzes.’’ Naturwissenschaften, 14, 223–225 ͑in him the Berkeley Citation, an award ‘‘for distinguished achieve- German͒. ment and notable service to the university,’’ and a Certificate of Einstein, H. A. ͑1934͒. ‘‘Der Hydraulische oder Profil-Radius.’’ Sch- Merit from the U.S. Department of Agriculture, ‘‘for pioneering weizer Bauzeitung, Band 103͑8͒, 89–91 ͑in German͒. research in developing the bed-load function of sediment trans- Einstein, H. A. ͑1935͒. ‘‘Die Eichung des im Rhein verwendeten Ge- port by streams, and leadership in developing application of fluid scheibefangers.’’ Schweizer Bauzeitung, Band 110͑12–15͒, 29–32 ͑in dynamics theory in solving engineering problems in the field of German͒. ͑ ͒ soil and water conservation.’’ Eight months later the American Einstein, H. A. 1937 . Der Geschiebetrieb als Wahrscheinlichkeitsprob- Society of Mechanical Engineers presented him a certificate of lem. Mitt. Versuchsanst. fur Wasserbau, an der Eidgenossische Tech- nische Hochschule in Zurich, Zurich, Switzerland. recognition for his 20 years of ‘‘devoted and distinguished ser- Einstein, H. A. ͑1942͒. ‘‘Formulas for the transport of bed sediment.’’ vices to applied mechanics reviews.’’ Trans. Am. Soc. Civ. Eng., 107, 561–574. However, none of these accolades seem to have meant as Einstein, H. A. ͑1944͒. ‘‘Bed-load transport in Mountain Creek.’’ Tech. much to him as the sedimentation symposium that had been held Paper SCS-TP-55, U.S Soil Conservation Service. in his honor a few weeks earlier in June 1971. About 80 profes- Einstein, H. A. ͑1948͒. ‘‘Determination of rates of bed-load movement.’’ sors, researchers, and former students of Einstein’s had attended, Proc., Federal Interagency Sedimentation Conference, Denver, 75– many of the students flying in with their spouses from distant 90. locations to honor their former professor. The symposium report- Einstein, H. A. ͑1950͒. ‘‘The bed-load function for sediment transporta- edly was very moving for Einstein, who greatly enjoyed the oc- tion in open channel flows.’’ Tech. Bulletin No. 1026, U.S. Dept of casion. Perhaps foremost among his contributions were the 20 Agriculture, Soil Conservation Service, Washington, D.C. Einstein, H. A., Anderson, A., and Johnson, J. ͑1940͒. ‘‘A distinction doctoral graduates he guided while at Berkeley. Many of them between bed load and suspended load.’’ Trans. Am. Geophys. Union, became leading figures in the study of alluvial rivers and hydrau- 21, 628–633. lic engineering. Einstein, H. A., and Barbarossa, N. L. ͑1952͒. ‘‘River channel rough- Late June 1973, while a Visiting Scholar at Woods Hole ness.’’ Trans. Am. Soc. Civ. Eng., 117, 1121–1146. Oceanographic Institute in Massachusetts, Einstein suffered a Einstein, H. A., and Chien, N. ͑1956͒. ‘‘Similarity of distorted models.’’ heart attack and shortly thereafter died. At the time of his death, Trans. Am. Soc. Civ. Eng., 121, 440–457.

486 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JUNE 2004 Einstein, H. A., and Mu¨ller, R. ͑1939͒.‘‘U¨ ber die A¨ hnlichkeit bei Rouse, H. ͑1939͒. ‘‘An analysis of sediment transport in the light of Flussbaulichen Modellversuchen.’’ Schweizer Archive fur Angewandte turbulence.’’ Tech. Paper SCS-TP-25, U.S. Soil Conservation Service, Wissenschaft und Technik, Heft 8, Zurich, Switzerland. Washington D.C. ¨ Gilbert, G. K. ͑1914͒. ‘‘Transportation of debris by running water.’’ Pro- Schoklitsch, A. ͑1914͒.‘‘Uber schleppkraft und geschiebebewegung,’’ fessional Paper No. 86, U.S. Geological Survey, Washington, D.C. Englemann, Leipzig, Germany. ͑ ͒ ͓ Graf, W. H. ͑1971͒. Hydraulics of sediment transport, McGraw-Hill, New Schoklitsch, A. 1930 . Der Wasserbau. Springer, Vienna. English trans- ͑ ͔͒ York. lation by Shulits, S. 1937 . Hydraulic structures, American Society Humphreys A. A., and Abbot, H. L. ͑1861͒. Upon the physics and hy- of Mechanical Engineers, New York. ͑ ͒ draulics of the Mississippi River, J. B. Lippincott & Co., Philadelphia. Schoklitsch, A. 1934 . Geschiebetrieb und die Geschiebefracht, Wasserkraft & Wasserwirtschaft, Jgg. 39, Heft 4 ͑in German͒. Kalinske, A. ͑1942͒. ‘‘Criteria for determining sand transportation by Shen, H. W., ed. ͑1972͒. Sedimentation. A Symposium to Honor Professor surface creep and saltation.’’ Trans. Am. Geophys. Union, Part II, 28, H. A. Einstein, Colorado State Univ., Ft. Collins, Colo. 266–279. Shen, H. W. ͑1975͒. ‘‘Hans A. Einstein’s contributions in sedimentation.’’ Lacey, G. ͑1929͒. ‘‘Stable channels in alluvium.’’ Proc., Inst. Civ. Eng., J. Hydraul. Eng., 101͑5͒, 469–488. 27, 259–384. Shields, A. ͑1936͒. Anwendung der A¨ hnlichkeitsmechanik und der Tur- ͑ ͒ Meyer-Peter, E., Favre, H., and Einstein, H. A. 1934 . ‘‘Neuere Ver- bulenzforschung auf die Geschiebebewegung. Mitt. der Preussischen suchresultate u¨ber den Geschiebetrieb.’’ Schweizer Bauzeitung, Band Versuchanstalt fur Wasserbau und Schiffbau, Heft 26, Berlin ͑in Ger- ͑ ͒ ͑ ͒ 103 4 , 89–91 in German . man͒. ͑ ͒ Meyer-Peter, E., and Mu¨ller, R. 1948 . ‘‘Formulas for bed-load trans- Straub, L. G. ͑1935͒. Missouri River Report, House Document 238, Ap- port.’’ Proc., Int. Association for Hydraulic Research, 2nd Meeting, pendix XV, Corps of Engineers, U.S. Army to 73rd U.S. Congress. Stockholm, Sweden. Vanoni, V. ͑1946͒. ‘‘Transportation of suspended sediment by water.’’ Powell, J. W. ͑1875͒. Exploration of the Colorado River of the West and Trans. Am. Soc. Civ. Eng., 111, 67–133. its tributaries, U.S. Government Printing Ofﬁce, Washington, D.C. Vanoni, V. ͑1948͒. ‘‘Development of the mechanics of sediment transpor- Pyne, S. J. ͑1980͒. Grove Karl Gilbert: A great engine of research, Univ. tation.’’ Federal Interagency Sedimentation Conference, Denver, of Texas Press, Austin, Tex. 209–221.