Teaching time-series analysis. II. Wave height and water surface elevation probability distributions

American Journal of Physics

Volumn 69, Issue 4, 2001, Pages 497-504

  Whitford, Dennis J ^a   Waters, Jennifer K ^a   Vieira, Mario E C ^a  

^a UNITED STATES NAVAL ACADEMY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 23044528202     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (36)

1. Note that random waves as observed in the ocean are not typical "white noise" signals with a "flat line" frequency distribution. They possess distinguishable, narrow-band (observable surface gravity wind waves fall between 0.03 and 1.0 Hz) frequency distribution features which are Gaussian.
2. The first part is contained in D. J. Whitford, M. E. C. Vieira, and J. K. Waters, "Teaching Time-Series Analysis. I. Finite Fourier Analysis of Ocean Waves," Am. J. Phys. 69, 490-496 (2001).
3. Since this exercise's procedures are equally applicable to lakes, bays, and rivers, we have adopted the term "water surface elevations" for this article rather than the more traditional term of "sea surface elevations."
4. M. J. Tucker, Waves in Ocean Engineering: Measurement, Analysis, Interpretation (Ellis Horwood, New York, 1991), 41 p.
5. Further information on Gaussian, or normal, distributions can be found in any introductory probability text such as M. R. Spiegel, Schaum's Outline of Theory and Problems of Probability and Statistics (McGraw-Hill, New York, 1975), pp. 109-111.
6. For additional information, see R. M. Sorensen, Basic Coastal Engineering, 1st ed. (Wiley, New York, 1978), p. 121.
7. Note that elevation (η) is a signed zero-to-value amplitude, that is, it measures from the still water level or baseline, and can be positive or negative. In contrast, wave height (H), as used in this paper, is a peak-to-peak measure of water surface elevation, and is a positive value (i.e., never less than zero).
8. M. S. Longuet-Higgins, "On the statistical distribution of the heights of sea waves," J. Mar. Res. 11 (5), 245-266 (1952):
9. Alternatively, see R. G. Dean and R. A. Dalrymple, Water Wave Mechanics for Engineers and Scientists (World Scientific, Singapore, 1991), pp. 187-193,
10. or B. Kinsman, Wind Waves (Dover, New York, 1984), pp. 342-347.
11. Sample water surface elevation data from the U.S. Naval Academy's wave tank can be found and downloaded at http://www.usna.edu/NAOE/ courses/en475.htm under "Additional Course Materials." Other web sites, such as the National Oceanic and Atmospheric Administration's National Data Buoy Center (http://seaboard.ndbc.noaa.gov/) have actual historical and real-time oceanic data available for downloading in a wide variety of formats.
12. Note that although the signal generated is not truly random (i.e., the signal is known and can be repeated), the wave field it creates simulates an irregular or "random" sea. Thus it is referred to as a "random wave signal" and the generated waves are called "random waves." Students are cautioned to appreciate the difference between what is truly random and what is a programmed simulation of a random process. For additional information, see J. Zseleczky, "Wave Generating Software at the U.S. Naval Academy," 24th American Towing Tank Conference, Texas A&M University, College Station, Texas, November 1995 (unpublished).
13. Further information on more advanced topics in time series analysis (e.g., filtering and windowing) can be found in C. Chatfield, The Analysis of Time Series - An Introduction (Chapman and Hall, New York, 1984), 3rd ed.
14. or G. Jenkins and D. Watts, Spectral Analysis and Its Applications (Holden-Day, Oakland, CA, 1968), among others.
15. Reference 5, p. 163.
16. For steep deep water waves, as well as waves in intermediate and shallow water depths, significant wave height determined by the wave-by-wave method will be increasingly larger than significant wave height determined by the standard deviation of the demeaned 77 data. See E. F. Thompson and C. L. Vincent, "Significant Wave Height for Shallow Water Design," Journal, Waterway Port Coastal and Ocean Engineering Division, American Society of Civil Engineers, September, 828-842 (1985).
17. Reference 5, p. 153.
18. Your spreadsheet software will dictate the tabular format of the histogram output. Thus your resulting format may not be identical to Tables I and II.
19. This renormalization by Δη is similar, but different, from that carried out in Ref. 2 using Δf.
20. M. R. Spiegel, Schaum's Outline of Theory and Problems of Probability and Statistics (McGraw-Hill, New York, 1975), pp. 40-41.
21. W. J. Pierson, "An Interpretation of the Observable Properties of Sea Waves in Terms of the Energy Spectrum of the Gaussian Record," Trans. Am. Geophys. Union 35, 747-757 (1954).
22. Several authors have compared the Rayleigh distribution with measured wave heights and found that the distribution yields acceptable values for most storms. See M. D. Earle, "Extreme Wave Conditions During Hurricane Camille," J. Geophys. Res. 80 (3), 377-379 (1975)
23. or S. K. Chakrabarti and R. P. Cooley, "Statistical Distribution of Periods and Heights of Ocean Waves," J. Geophys. Res. ibid. 82 (9), 1363-1368 (1971).
24. For more explanatory information on the Rayleigh distribution beyond Longuet-Higgins' (1952) original work, one may peruse R. G. Dean and R. A. Dalrymple, Water Wave Mechanics for Engineers and Scientists (World Scientific, Singapore, 1991), pp. 187-193
25. R. M. Sorensen, Basic Coastal Engineering (Wiley, New York, 1997), 2nd ed., pp. 155-159.
26. In time series analysis, "ergodicity" means that "... for most stationary processes which are likely to be met in practice, the sample moments of an observed record length T converge to the corresponding population moments as T→∞. In other words, time averages for a single realization converge to ensemble averages." (Chatfield, 1985, p. 65). For further information on ergodicity, see C. Chatfield, The Analysis of Time Series - An Introduction (Chapman and Hall, New York, 1984), 3rd ed, p. 65,
27. or G. Jenkins and D. Watts, Spectral Analysis and Its Applications (Holden-Day, Oakland, CA, 1968), p. 179, among others.
28. Although other math software programs such as MATLAB are capable of performing a FFT on this entire data set, it is still useful to use a spread-sheet program here, despite a possible 1024-point limitation. In addition to introducing the concept of ensemble averaging to the student, performing, this task tends to bring about a general discussion of the limitations that exist with all software. Virtually all data analysis programs and computer hardware reach some limitation in real data analysis. Discovering methods to work within these limitations is many times part of conducting research.
29. This spectrum was based on wave records acquired in the North Atlantic Ocean for 20-40 knots. It applied to deep water, fully arisen seas, i.e., seas in equilibrium with the wind speed which had blown over unlimited fetch and for unlimited duration. See W. J. Pierson and L. Moskowitz, "A Proposed Spectral Form for Fully Developed Wind Seas Based on Similarity Theory of S. A. Kitaigorodskii," J. Geophys. Res. 69, 5181-5190 (1964).
30. 4].
31. M. K. Ochi, Ocean Waves, the Stochastic Approach (Cambridge U.P., Cambridge, UK, 1991), p. 39.
32. Reference 5, pp. 119-121.
33. This spectrum was based on Eq. (2), was empirically derived, and assumed no correlation between individual wave periods and heights. See C. L. Bretschneider, "Wave Variability and Wave Spectra for Wind-Generated Gravity Waves," Technical Memorandum No. 118, U.S. Army Beach Erosion Board, Washington, DC, 1959.
34. This spectrum was a modification of the PM spectrum to account for fetch-limited seas. See K. Hasselmann, T. P. Barnett, E. Bouws, H. Carlson, D. E. Cartwright, K. Enke, J. A. Ewing, H. Gienapp, D. E. Hasselmann, P. Kruseman, A. Meerburg, P. Muller, D. J. Gibers, K. Richter, W. Snell, and H. Walden, "Measurements of Wind-Wave Growth and Swell Decay During the Joint North Sea Wave Project (JONSWAP)," Report to the German Hydrographic Institute, Hamburg, Germany, 1973.
35. This spectrum accounted for waves that were moving into shallower water. See E. Bouws, H. Gunther, W. Rosenthal, and C. L. Vincent, "Similarity of the Wind Wave Spectrum in Finite Depth Water. I. Spectral Form," J. Geophys. Res. 90, 975-986 (1985).
36. c = 1/[2Δt]) are discussed in detail in the first exercise described in Ref. 2.