DOI: 10.1098/rsta.1957.0002. The statistical analysis of a random, moving surface

The statistical analysis of a random, moving surface

10.1098/rsta.1957.0002

Crossref journal-article

The Royal Society

Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences (175)

Abstract

The following statistical properties are derived for a random, moving, Gaussian surface: (1) the probability distribution of the surface elevation and of the magnitude and orientation of the gradient; (2) the average number of zero-crossings per unit distance along a line in an arbitrary direction; (3) the average length of the contours per unit area, and the distribution of their direction; (4) the average density of maxima and minima per unit area of the surface, and the average density of specular points (i.e, points where the two components of gradient take given values); (5) the probability distribution of the velocities of zero-crossings along a given line; (6) the probability distribution of the velocities of contours and of specular points; (7) the probability distribution of the envelope and phase angle, and hence (8) when the spectrum is narrow, the probability distribution of the heights of maxima and minima and the distribution of the intervals between successive zero-crossings along an arbitrary line. All the results are expressed in terms of the two-dimensional energy spectrum of the surface, and are found to involve the moments of the spectrum up to a finite order only. (1), (3), (4), (5) and (6) are discussed in detail for the special case of a narrow spectrum. The converse problem is also studied and solved: given certain statistical properties of the surface, to find a convergent sequence of approximations to the energy spectrum. The problems arise in connexion with the statistical analysis of the sea surface. (More detailed summaries are given at the beginning of each part of the paper.)

Bibliography

The statistical analysis of a random, moving surface. (1957). Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 249(966), 321â387.

Authors 0

None

References 38 Referenced 610

Barber N. F. Ursell F. D arbyshire J . & Tucker M. J . 1946 A frequency analyser used in the study of ocean waves. Nature Lond. 158 329-332. (10.1038/158329a0)
10.1098/rsta.1948.0005
Barber N. F. 1950 Ocean waves and swell. Lecture Institution of Civil Engineers London 2 2 pp.
10.1098/rspa.1956.0041
C ourant R. & H ilbert D. 1953 Methods o f mathematical physics. 1st English edition. New York: Interscience Publishers Inc.
10.1364/JOSA.44.000838
{'key': 'p_7', 'first-page': '198', 'article-title': "£ Statistics of the sea surface derived from the sun's glitter. J", 'volume': '13', 'author': 'Cox C.', 'year': '1954', 'journal-title': 'Mar. Res.'} / Mar. Res. / £ Statistics of the sea surface derived from the sun's glitter. J by Cox C. (1954)
Cram er H. 1937 Random variables and probability distributions. C am bridge M athem atical T ract no. 36. C am bridge University Press.
Duntley S. Q . 1950 T he visibility of submerged objects. P art I. O ptical effects of w ater waves. Report Visibility Lab. Mass. Inst. Techn.
Eckart C. 1946 The sea surface and its effect on the reflection o f sound and light. U niversity of California Division of W ar Research R eport M 407 (unpublished).
10.1121/1.1907123
10.1063/1.1721204
10.1103/PhysRev.91.784
Euler L. 1 7 5 2 -3 Elem enta doctrinae solidorum. Comment. Acad. Sci. Imp. Petropol. 4 109-160.
K endall M . G. 1952 The advanced theory o f statistics vol. 1. London: Griffin. 457 pp.
10.1007/BF01449156
Legendre A. M. 1811 Exercises de calcul integrale t. 1 . Paris: Courcier. 386 pp.
Longuet-Higgins M . S. & Barber N. F. 1946 Four theoretical notes on the estimation of sea conditions. A dm iralty Research L aboratory T eddington R eport N 1 -N 4 103.30/W (unpublished).
10.1098/rsta.1950.0012
{'key': 'p_20', 'first-page': '245', 'article-title': 'O n the statistical distribution of the heights of sea waves. J', 'volume': '9', 'author': 'Longuet-Higgins M .', 'year': '1952', 'journal-title': 'Mar. Res.'} / Mar. Res. / O n the statistical distribution of the heights of sea waves. J by Longuet-Higgins M . (1952)
10.1029/TR035i005p00758
Pierson W. J. 1952 A unified mathematical theory fo r the analysis propagation and refraction o f storm generated ocean surface waves Part I. R eport Dept of Meteorology N.Y. U niv. 336 pp. (10.5962/bhl.title.39077)
10.1080/14786448008626893
Rayleigh Lord 1945 The theory o f sound 1st A m erican edition. New York: Dover Publications.
10.1002/j.1538-7305.1944.tb00874.x
10.1002/j.1538-7305.1945.tb00453.x
Rudnick P. 1950 Correlograms for Pacific O cean waves. Proc. 2nd Berkeley Symposium on Mathematical Statistics and Probability pp. 627-638 University of California Press. (10.1525/9780520411586-046)
{'key': 'p_28', 'first-page': '1', 'article-title': 'O n the motions of ships in confused seas', 'volume': '61', 'author': 'St Denis M .', 'year': '1953', 'journal-title': 'Trans. Soc. Nav. Archit., N .Y .'} / Trans. Soc. Nav. Archit., N .Y . / O n the motions of ships in confused seas by St Denis M . (1953)
Schumacher A. 1952 Results of exact wave measurements (by stereophotogramm etry) with special reference to more recent theoretical investigations. Symposium on G ravity Waves W ashington Ju n e 1951. Nat.Bur. Stand. Circ. no. 521 U .S. Govt. Printing Office.
10.1364/JOSA.44.000037
Sommerville D. M. Y. 1929 An introduction to the geometry o f n dimensions. London: M ethuen. 196 pp.
10.1112/plms/s2-20.1.196
Tukey J . W. 1949 The sampling theory o fpower spectrum estimates. Symposium on application of autocorrelation analysis to physical problems Woods Hole Mass. Ju n e 1949. Office of Naval Research W ashington D.G.
Tukey J . W. & H am m ing R. W. 1949 Measuring noise color. Bell Telephone Laboratories Internal M em orandum M M 49.
{'key': 'p_35', 'first-page': '409', 'article-title': 'D istribution of height in ocean waves. N .Z', 'volume': '34', 'year': '1953', 'journal-title': 'J. Sci. Tech. B'} / J. Sci. Tech. B / D istribution of height in ocean waves. N .Z (1953)
{'key': 'p_36', 'first-page': '789', 'article-title': 'U ber die analytische Darstellbarkeit sogenannter willkiirlicher Funktionen reeller Argumente', 'volume': '633', 'author': 'Weierstrass K. T. W.', 'year': '1885', 'journal-title': 'S. B. Akad. Wiss. Berl.'} / S. B. Akad. Wiss. Berl. / U ber die analytische Darstellbarkeit sogenannter willkiirlicher Funktionen reeller Argumente by Weierstrass K. T. W. (1885)
Weierstrass K. T. W. 1903 W erke vol. 3 1-37. Berlin.
W hittaker E. T. & W atson G. N. 1952 Modern analysis. 4th edition. Cam bridge University Press.

Dates

Type	When
Created	18 years, 8 months ago (Dec. 18, 2006, 6:42 p.m.)
Deposited	7 months, 1 week ago (Jan. 12, 2025, 11:34 a.m.)
Indexed	2 weeks, 3 days ago (Aug. 6, 2025, 8:50 a.m.)
Issued	68 years, 6 months ago (Feb. 21, 1957)
Published	68 years, 6 months ago (Feb. 21, 1957)
Published Online	28 years, 7 months ago (Jan. 1, 1997)
Published Print	68 years, 6 months ago (Feb. 21, 1957)

Funders 0

None

BibTeX

@article{1957, moving surface_1957, volume={249}, ISSN={2054-0272}, url={http://dx.doi.org/10.1098/rsta.1957.0002}, DOI={10.1098/rsta.1957.0002}, number={966}, journal={Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences}, publisher={The Royal Society}, year={1957}, month=feb, pages={321–387} }

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