Properties of sand waves formed by subcritical unidirectional water currents are investigated by statistical analyses of records of streambed profiles. Records of bed elevation y as a function of distance x along the channel, y = y(x), and time records at a fixed point of the channel, y = y(t), were collected in three laboratory flumes that were 8 inches, 2 ft and 8 ft wide and in a straight alluvial channel that was 55 ft wide. For all cases, the bed material was fine sand. The continuous analogue records were converted to discrete data points and were analyzed by digital computer.
The analyses show that both types of records, y(x) and y(t), can be approximately represented as stationary Gaussian processes. When the data are standardized and the length or distance are expressed as ratios of the mean duration between zero-crossings of y, the statistical properties of all the flume data are similar, with no distinguishing characteristics that can be attributed to size of flume or to whether the bed forms were ripples or dunes. The field data, however, reflect the influence of large alternate bars that were not present in the flumes.
The Gaussian assumption, together with the spectral properties of the records as expressed by a dimensionless parameter, 6, permit predicting the distributions of maximum and minimum values of y between successive zeros of y. These distributions represent the probability distributions of the depth of local scour and fill due to the formation and migration of sand waves, and the parameters that specify the distributions relate approximately to flow velocity and depth.
Observed values of the number of zero and h-level crossings, the mean duration between zero crossings, and the mean duration of upward excursions of the process y(t) above the fixed level h compared reasonably well with theoretical values for the Gaussian model. The distribution of the duration of upward excursions is the conditional probability distribution of the rest period of a particle, given that it is deposited on the downstream face of a ripple or dune at the level h. Observed distributions of these durations can be approximated by a gamma distribution with parameters that relate to h, where h is measured in units of standard deviation from the mean bed level. These distributions and other probability distributions that enter into stochastic models of sediment transport can be determined either from the theoretical model or empirically from the observed data. The results of the study ,-show that even though the bed elevation deviates somewhat from the postulated normal distribution, reasonable estimates of many properties of the bed profiles can be derived from fairly simple statistical models.
","language":"English","publisher":"U.S. Geological Survey","doi":"10.3133/ofr68201","usgsCitation":"Nordin, C., 1968, Statistical properties of dune profiles: U.S. Geological Survey Open-File Report 68-201, xiv, 137 p., https://doi.org/10.3133/ofr68201.","productDescription":"xiv, 137 p.","costCenters":[],"links":[{"id":491571,"rank":2,"type":{"id":11,"text":"Document"},"url":"https://pubs.usgs.gov/of/1968/0201/report.pdf","linkFileType":{"id":1,"text":"pdf"}},{"id":184022,"rank":1,"type":{"id":24,"text":"Thumbnail"},"url":"https://pubs.usgs.gov/of/1968/0201/report-thumb.jpg"}],"noUsgsAuthors":false,"publicationStatus":"PW","scienceBaseUri":"4f4e49dde4b07f02db5e1e3f","contributors":{"authors":[{"text":"Nordin, C.F. Jr.","contributorId":100852,"corporation":false,"usgs":true,"family":"Nordin","given":"C.F.","suffix":"Jr.","affiliations":[],"preferred":false,"id":254388,"contributorType":{"id":1,"text":"Authors"},"rank":1}]}} ,{"id":37240,"text":"rp51 - 1968 - Tiburon Marine Laboratory","interactions":[],"lastModifiedDate":"2016-11-02T15:04:41","indexId":"rp51","displayToPublicDate":"1994-01-01T00:00:00","publicationYear":"1968","noYear":false,"publicationType":{"id":18,"text":"Report"},"publicationSubtype":{"id":1,"text":"Federal Government Series"},"seriesTitle":{"id":79,"text":"Resource Publication","active":false,"publicationSubtype":{"id":1}},"seriesNumber":"51","title":"Tiburon Marine Laboratory","docAbstract":"No abstract available
","language":"ENGLISH","publisher":"U.S. Fish and Wildlife Service","publisherLocation":"Washington, D.C.","usgsCitation":"Bureau Of Sport Fisheries And Wildlife, 1968, Tiburon Marine Laboratory: Resource Publication 51, 2 p.","productDescription":"2 p.","costCenters":[],"links":[{"id":162353,"rank":0,"type":{"id":24,"text":"Thumbnail"},"url":"https://pubs.usgs.gov/thumbnails/usgs_thumb.jpg"}],"noUsgsAuthors":false,"publicationStatus":"PW","scienceBaseUri":"4f4e4b06e4b07f02db69a29f","contributors":{"authors":[{"text":"Bureau Of Sport Fisheries And Wildlife","contributorId":128243,"corporation":true,"usgs":false,"organization":"Bureau Of Sport Fisheries And Wildlife","id":529617,"contributorType":{"id":1,"text":"Authors"},"rank":1}]}} ,{"id":22416,"text":"ofr685 - 1968 - Some hydrologic effects of canal 111 near Homestead, Florida","interactions":[],"lastModifiedDate":"2012-02-02T00:08:07","indexId":"ofr685","displayToPublicDate":"1994-01-01T00:00:00","publicationYear":"1968","noYear":false,"publicationType":{"id":18,"text":"Report"},"publicationSubtype":{"id":5,"text":"USGS Numbered Series"},"seriesTitle":{"id":330,"text":"Open-File Report","code":"OFR","onlineIssn":"2331-1258","printIssn":"0196-1497","active":true,"publicationSubtype":{"id":5}},"seriesNumber":"68-5","title":"Some hydrologic effects of canal 111 near Homestead, Florida","language":"ENGLISH","publisher":"U.S. Dept. of the Interior, Geological Survey,","doi":"10.3133/ofr685","issn":"0094-9140","usgsCitation":"Barnes, H., Meyer, F., and Hartwell, J., 1968, Some hydrologic effects of canal 111 near Homestead, Florida: U.S. Geological Survey Open-File Report 68-5, 28 p. :ill., maps ;28 cm., https://doi.org/10.3133/ofr685.","productDescription":"28 p. :ill., maps ;28 cm.","costCenters":[],"links":[{"id":156632,"rank":0,"type":{"id":24,"text":"Thumbnail"},"url":"https://pubs.usgs.gov/thumbnails/usgs_thumb.jpg"}],"noUsgsAuthors":false,"publicationStatus":"PW","scienceBaseUri":"4f4e49e9e4b07f02db5e91ff","contributors":{"authors":[{"text":"Barnes, H.H. Jr.","contributorId":67916,"corporation":false,"usgs":true,"family":"Barnes","given":"H.H.","suffix":"Jr.","email":"","affiliations":[],"preferred":false,"id":188202,"contributorType":{"id":1,"text":"Authors"},"rank":1},{"text":"Meyer, F.W.","contributorId":25150,"corporation":false,"usgs":true,"family":"Meyer","given":"F.W.","email":"","affiliations":[],"preferred":false,"id":188201,"contributorType":{"id":1,"text":"Authors"},"rank":2},{"text":"Hartwell, J.H.","contributorId":98356,"corporation":false,"usgs":true,"family":"Hartwell","given":"J.H.","email":"","affiliations":[],"preferred":false,"id":188203,"contributorType":{"id":1,"text":"Authors"},"rank":3}]}} ,{"id":55852,"text":"ofr68162 - 1968 - Regional draft-storage relationships for the Grand River basin, Michigan","interactions":[],"lastModifiedDate":"2017-01-11T10:35:45","indexId":"ofr68162","displayToPublicDate":"1994-01-01T00:00:00","publicationYear":"1968","noYear":false,"publicationType":{"id":18,"text":"Report"},"publicationSubtype":{"id":5,"text":"USGS Numbered Series"},"seriesTitle":{"id":330,"text":"Open-File Report","code":"OFR","onlineIssn":"2331-1258","printIssn":"0196-1497","active":true,"publicationSubtype":{"id":5}},"seriesNumber":"68-162","title":"Regional draft-storage relationships for the Grand River basin, Michigan","docAbstract":"No abstract available.
","language":"English","publisher":"U.S. Geological Survey","publisherLocation":"Reston, VA","doi":"10.3133/ofr68162","usgsCitation":"Knutilla, R., 1968, Regional draft-storage relationships for the Grand River basin, Michigan: U.S. Geological Survey Open-File Report 68-162, 7 p., https://doi.org/10.3133/ofr68162.","productDescription":"7 p.","costCenters":[{"id":382,"text":"Michigan Water Science Center","active":true,"usgs":true}],"links":[{"id":183937,"rank":0,"type":{"id":24,"text":"Thumbnail"},"url":"https://pubs.usgs.gov/thumbnails/usgs_thumb.jpg"}],"country":"United States","state":"Michigan","otherGeospatial":"Grand River basin","noUsgsAuthors":false,"publicationStatus":"PW","scienceBaseUri":"4f4e4ac9e4b07f02db67c69a","contributors":{"authors":[{"text":"Knutilla, R. L.","contributorId":65451,"corporation":false,"usgs":true,"family":"Knutilla","given":"R. L.","affiliations":[],"preferred":false,"id":254367,"contributorType":{"id":1,"text":"Authors"},"rank":1}]}} ,{"id":38818,"text":"pp543J - 1968 - Effects of the Alaska earthquake of March 27, 1964, on shore processes and beach morphology","interactions":[{"subject":{"id":38818,"text":"pp543J - 1968 - Effects of the Alaska earthquake of March 27, 1964, on shore processes and beach morphology","indexId":"pp543J","publicationYear":"1968","noYear":false,"chapter":"J","title":"Effects of the Alaska earthquake of March 27, 1964, on shore processes and beach morphology"},"predicate":"IS_PART_OF","object":{"id":70048225,"text":"pp543 - 1966 - The Alaska earthquake, March 27, 1964: regional effects","indexId":"pp543","publicationYear":"1966","noYear":false,"title":"The Alaska earthquake, March 27, 1964: regional effects"},"id":1}],"isPartOf":{"id":70048225,"text":"pp543 - 1966 - The Alaska earthquake, March 27, 1964: regional effects","indexId":"pp543","publicationYear":"1966","noYear":false,"title":"The Alaska earthquake, March 27, 1964: regional effects"},"lastModifiedDate":"2022-02-15T20:19:06.712305","indexId":"pp543J","displayToPublicDate":"1994-01-01T00:00:00","publicationYear":"1968","noYear":false,"publicationType":{"id":18,"text":"Report"},"publicationSubtype":{"id":5,"text":"USGS Numbered Series"},"seriesTitle":{"id":331,"text":"Professional Paper","code":"PP","onlineIssn":"2330-7102","printIssn":"1044-9612","active":true,"publicationSubtype":{"id":5}},"seriesNumber":"543","chapter":"J","title":"Effects of the Alaska earthquake of March 27, 1964, on shore processes and beach morphology","docAbstract":"Some 10,000 miles of shoreline in south-central Alaska was affected by the subsidence or uplift associated with the great Alaska earthquake of March 27, 1964. The changes in shoreline processes and beach morphology that were suddenly initiated by the earthquake were similar to those ordinarily caused by gradual changes in sea level operating over hundreds of years, while other more readily visible changes were similar to some of the effects of great but short-lived storms. Phenomena became available for observation within a few hours which would otherwise not have been available for many years.\n\nIn the subsided areas—including the shorelines of the Kenai Peninsula, Kodiak Island, and Cook Inlet—beaches tended to flatten in gradient and to recede shoreward. Minor beach features were altered or destroyed on submergence but began to reappear and to stabilize in their normal shapes within a few months after the earthquake. Frontal beach ridges migrated shoreward and grew higher and wider than they were before. Along narrow beaches backed by bluffs, the relatively higher sea level led to vigorous erosion of the bluff toes. Stream mouths were drowned and some were altered by seismic sea waves, but they adjusted within a few months to the new conditions.\n\nIn the uplifted areas, generally around Prince William Sound, virtually all beaches were stranded out of reach of the sea. New beaches are gradually developing to fit new sea levels, but the processes are slow, in part because the material on the lower parts of the old beaches is predominantly fine grained. Streams were lengthened in the emergent areas, and down cutting and bank erosion have increased.\n\nExcept at Homer and a few small villages, where groins, bulkheads, and cobble-filled baskets were installed, there has been little attempt to protect the postearthquake shorelines. The few structures that were built have been only partially successful because there was too little time to study the habits of the new shore features and to design appropriate protection measures. Emergence of large areas that were once below water and permanent submergence of once-useful land areas have led to many problems of land use and ownership in addition to the destruction or relocation of wildfowl, shellfish, and salmon habitats.","largerWorkType":{"id":18,"text":"Report"},"largerWorkTitle":"The Alaska earthquake, March 27, 1964: Regional effects (Professional Paper 543)","largerWorkSubtype":{"id":5,"text":"USGS Numbered Series"},"language":"English","publisher":"U.S. Government Printing Office","publisherLocation":"Washington, D.C.","doi":"10.3133/pp543J","usgsCitation":"Stanley, K.W., 1968, Effects of the Alaska earthquake of March 27, 1964, on shore processes and beach morphology: U.S. Geological Survey Professional Paper 543, vi, 21 p., https://doi.org/10.3133/pp543J.","productDescription":"vi, 21 p.","numberOfPages":"34","costCenters":[{"id":380,"text":"Menlo ParkCalif. Office-Earthquake Science Center","active":false,"usgs":true}],"links":[{"id":277850,"type":{"id":15,"text":"Index Page"},"url":"https://pubs.usgs.gov/pp/0543j/index.html"},{"id":395994,"rank":4,"type":{"id":36,"text":"NGMDB Index Page"},"url":"https://ngmdb.usgs.gov/Prodesc/proddesc_99033.htm"},{"id":65742,"rank":300,"type":{"id":11,"text":"Document"},"url":"https://pubs.usgs.gov/pp/0543j/pp543j_text.pdf","linkFileType":{"id":1,"text":"pdf"}},{"id":119822,"rank":0,"type":{"id":24,"text":"Thumbnail"},"url":"https://pubs.usgs.gov/pp/0543j/report-thumb.jpg"}],"country":"United States","state":"Alaska","geographicExtents":"{\n \"type\": \"FeatureCollection\",\n \"features\": [\n {\n \"type\": \"Feature\",\n \"properties\": {},\n \"geometry\": {\n \"type\": \"Polygon\",\n \"coordinates\": [\n [\n [\n -154.8193359375,\n 56.68037378950137\n ],\n [\n -145.72265625,\n 56.68037378950137\n ],\n [\n -145.72265625,\n 63.93737246791484\n ],\n [\n -154.8193359375,\n 63.93737246791484\n ],\n [\n -154.8193359375,\n 56.68037378950137\n ]\n ]\n ]\n }\n }\n ]\n}","noUsgsAuthors":false,"publicationStatus":"PW","scienceBaseUri":"4f4e4a27e4b07f02db6106c9","contributors":{"authors":[{"text":"Stanley, Kirk W.","contributorId":8900,"corporation":false,"usgs":true,"family":"Stanley","given":"Kirk","email":"","middleInitial":"W.","affiliations":[],"preferred":false,"id":220494,"contributorType":{"id":1,"text":"Authors"},"rank":1}]}} ,{"id":15880,"text":"ofr68245 - 1968 - Availability of palynological material from Naval Petroleum Reserve No.4, IX: Gubik Test Well No. 1","interactions":[],"lastModifiedDate":"2025-06-27T18:36:11.982676","indexId":"ofr68245","displayToPublicDate":"1994-01-01T00:00:00","publicationYear":"1968","noYear":false,"publicationType":{"id":18,"text":"Report"},"publicationSubtype":{"id":5,"text":"USGS Numbered Series"},"seriesTitle":{"id":330,"text":"Open-File Report","code":"OFR","onlineIssn":"2331-1258","printIssn":"0196-1497","active":true,"publicationSubtype":{"id":5}},"seriesNumber":"68-245","title":"Availability of palynological material from Naval Petroleum Reserve No.4, IX: Gubik Test Well No. 1","docAbstract":"No abstract available.
","language":"English","publisher":"U.S. Geological Survey","doi":"10.3133/ofr68245","usgsCitation":"Scott, R., 1968, Availability of palynological material from Naval Petroleum Reserve No.4, IX: Gubik Test Well No. 1: U.S. Geological Survey Open-File Report 68-245, 3 p., https://doi.org/10.3133/ofr68245.","productDescription":"3 p.","costCenters":[],"links":[{"id":491552,"rank":2,"type":{"id":11,"text":"Document"},"url":"https://pubs.usgs.gov/of/1968/0245/report.pdf","linkFileType":{"id":1,"text":"pdf"}},{"id":148384,"rank":1,"type":{"id":24,"text":"Thumbnail"},"url":"https://pubs.usgs.gov/of/1968/0245/report-thumb.jpg"}],"noUsgsAuthors":false,"publicationStatus":"PW","scienceBaseUri":"4f4e4a9ae4b07f02db65d7f9","contributors":{"authors":[{"text":"Scott, Richard A.","contributorId":14414,"corporation":false,"usgs":true,"family":"Scott","given":"Richard A.","affiliations":[],"preferred":false,"id":171877,"contributorType":{"id":1,"text":"Authors"},"rank":1}]}} ,{"id":37248,"text":"rp59 - 1968 - 6 questions everybody asks at National Fish Hatcheries","interactions":[],"lastModifiedDate":"2016-11-02T15:16:31","indexId":"rp59","displayToPublicDate":"1994-01-01T00:00:00","publicationYear":"1968","noYear":false,"publicationType":{"id":18,"text":"Report"},"publicationSubtype":{"id":1,"text":"Federal Government Series"},"seriesTitle":{"id":79,"text":"Resource Publication","active":false,"publicationSubtype":{"id":1}},"seriesNumber":"59","title":"6 questions everybody asks at National Fish Hatcheries","docAbstract":"No abstract available
","language":"ENGLISH","publisher":"U.S. Fish and Wildlife Service","publisherLocation":"Washington, D.C.","usgsCitation":"Bureau Of Sport Fisheries And Wildlife, 1968, 6 questions everybody asks at National Fish Hatcheries: Resource Publication 59, 2 p.","productDescription":"2 p.","costCenters":[],"links":[{"id":162460,"rank":0,"type":{"id":24,"text":"Thumbnail"},"url":"https://pubs.usgs.gov/thumbnails/usgs_thumb.jpg"}],"noUsgsAuthors":false,"publicationStatus":"PW","scienceBaseUri":"53cd4935e4b0b290850eefac","contributors":{"authors":[{"text":"Bureau Of Sport Fisheries And Wildlife","contributorId":128243,"corporation":true,"usgs":false,"organization":"Bureau Of Sport Fisheries And Wildlife","id":529625,"contributorType":{"id":1,"text":"Authors"},"rank":1}]}} ,{"id":22459,"text":"ofr688 - 1968 - Turbulence measurement with a propeller flow meter","interactions":[],"lastModifiedDate":"2012-02-02T00:08:02","indexId":"ofr688","displayToPublicDate":"1994-01-01T00:00:00","publicationYear":"1968","noYear":false,"publicationType":{"id":18,"text":"Report"},"publicationSubtype":{"id":5,"text":"USGS Numbered Series"},"seriesTitle":{"id":330,"text":"Open-File Report","code":"OFR","onlineIssn":"2331-1258","printIssn":"0196-1497","active":true,"publicationSubtype":{"id":5}},"seriesNumber":"68-8","title":"Turbulence measurement with a propeller flow meter","language":"ENGLISH","publisher":"U.S. Geological Survey,","doi":"10.3133/ofr688","issn":"0094-9140","usgsCitation":"Bennett, J.P., 1968, Turbulence measurement with a propeller flow meter: U.S. Geological Survey Open-File Report 68-8, xv, 171 p. :ill. ;28 cm., https://doi.org/10.3133/ofr688.","productDescription":"xv, 171 p. :ill. ;28 cm.","costCenters":[],"links":[{"id":156022,"rank":0,"type":{"id":24,"text":"Thumbnail"},"url":"https://pubs.usgs.gov/thumbnails/usgs_thumb.jpg"}],"noUsgsAuthors":false,"publicationStatus":"PW","scienceBaseUri":"4f4e4afee4b07f02db6976c6","contributors":{"authors":[{"text":"Bennett, J. P.","contributorId":52103,"corporation":false,"usgs":true,"family":"Bennett","given":"J.","email":"","middleInitial":"P.","affiliations":[],"preferred":false,"id":188294,"contributorType":{"id":1,"text":"Authors"},"rank":1}]}} ,{"id":32794,"text":"pp352I - 1968 - Rates of slope degradation as determined from botanical evidence, White Mountains, California","interactions":[],"lastModifiedDate":"2012-02-02T00:09:20","indexId":"pp352I","displayToPublicDate":"1994-01-01T00:00:00","publicationYear":"1968","noYear":false,"publicationType":{"id":18,"text":"Report"},"publicationSubtype":{"id":5,"text":"USGS Numbered Series"},"seriesTitle":{"id":331,"text":"Professional Paper","code":"PP","onlineIssn":"2330-7102","printIssn":"1044-9612","active":true,"publicationSubtype":{"id":5}},"seriesNumber":"352","chapter":"I","title":"Rates of slope degradation as determined from botanical evidence, White Mountains, California","largerWorkType":{"id":18,"text":"Report"},"largerWorkTitle":"Erosion and sedimentation in a semiarid environment","largerWorkSubtype":{"id":5,"text":"USGS Numbered Series"},"language":"ENGLISH","doi":"10.3133/pp352I","usgsCitation":"LaMarche, V.C., 1968, Rates of slope degradation as determined from botanical evidence, White Mountains, California: U.S. Geological Survey Professional Paper 352, p. 341-377, https://doi.org/10.3133/pp352I.","productDescription":"p. 341-377","costCenters":[],"links":[{"id":119887,"rank":0,"type":{"id":24,"text":"Thumbnail"},"url":"https://pubs.usgs.gov/pp/0352i/report-thumb.jpg"},{"id":60774,"rank":400,"type":{"id":17,"text":"Plate"},"url":"https://pubs.usgs.gov/pp/0352i/plate-1.pdf","linkFileType":{"id":1,"text":"pdf"}},{"id":60775,"rank":401,"type":{"id":17,"text":"Plate"},"url":"https://pubs.usgs.gov/pp/0352i/plate-2.pdf","linkFileType":{"id":1,"text":"pdf"}},{"id":60776,"rank":300,"type":{"id":11,"text":"Document"},"url":"https://pubs.usgs.gov/pp/0352i/report.pdf","linkFileType":{"id":1,"text":"pdf"}}],"noUsgsAuthors":false,"publicationStatus":"PW","scienceBaseUri":"4f4e4a7fe4b07f02db64893b","contributors":{"authors":[{"text":"LaMarche, Valmore C. 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