Teaching time-series analysis. I. Finite Fourier analysis of ocean waves

American Journal of Physics

Volumn 69, Issue 4, 2001, Pages 490-496

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

^a UNITED STATES NAVAL ACADEMY   (United States)

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EID: 23044526129     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (22)

1. D. J. Whitford and G. A. Eisman, "Making science come alive - Bucking the declining science major enrollment trend at annapolis," J. Coll. Sci. Teach. XXVII (2), 109-113 (1997).
2. R. J. Higgins, "Fast Fourier transform: An introduction with some microcomputer experiments," Am. J. Phys. 44 (8), 766-773 (1976).
3. G. Whaite and J. Wolfe, "Harmonic or Fourier synthesis in the teaching labpratory," Am. J. Phys. 58 (5), 481-483 (1990).
4. B. Duchesne, C. W. Fisher, C. G. Gray, and K. R. Jeffrey, "Chaos in the motion of an inverted pendulum: An undergraduate laboratory experiment," Am. J. Phys. 59 (11), 987-992 (1991).
5. I. Bull and R. Lincke, "Teaching Fourier analysis in a microcomputer based laboratory," Am. J. Phys. 64 (7), 906-913 (1996).
6. D. J. Whitford, J. K. Waters, and M. E. C. Vieira, "Teaching time series analysis. II. Wave height and water surface elevation probability distributions," Am. J. Phys. 69, 497-504 (2001).
7. 0, (N/2) cosine terms, and ((N/2)-1) sine terms] is equal to the number of sampling points (N) in the data set. For more information on finite Fourier series, see G. M. Jenkins and D. G. Watts, Spectral Analysis and its Applications (Holden-Day, Oakland, CA, 1968), pp. 18-19.
8. J. S. Bendat and A. G. Piersol, Random Data: Analysis and Measurement Procedures (Wiley-Interscience, New York, 1971), pp. 299-300.
9. B. Kinsman, Ocean Waves and the Directional Spectrum: An Engineering Tool (Marine Sciences Research Center, State University of New York, Stony Brook, 1978), pp. 103-104.
10. The digitized data (in m) used for this figure and the subsequent exercise are: -0.01, -0.14, -0.09, 0.47, 1.3, 1.89, 2.16, 2.34, 2.37, 1.93, 1.09, 0.39, 0.1, -0.03, -0.17, -0.15, 0.15, 0.32, -0.01, -0.67, -1.24, -1.68, -2.18, -2.56, -2.39, -1.72, -1.03, -0.59, -0.2, 0.2, 0.31, and 0.00.
11. W. C. Elmore and M. A. Heald, Physics of Waves (McGraw-Hill, New York, 1969), p. 203. Note that in some physics texts this energy is considered an energy density per unit area, whereas in this paper we have used the oceanographic convention of reserving the term "density" to refer to the energy density spectrum as described in the next section. Also, the factor of "1/2" converts between peak and rms amplitudes in oscillatory phenomena such as elementary ac circuit formulas.
12. G. B. Airy, "On Tides and Waves," Encyclopaedia Metropolitan (B. Fellowes, London, 1845), pp. 241-396.
13. M. Sorensen, Basic Wave Mechanics for Coastal and Ocean Engineers (Wiley, New York, 1993), pp. 18-20.
14. Note that the wave amplitude is equal to one-half the wave height.
15. The Nyquist frequency is also called the cutoff or folding frequency. For more details on the Nyquist frequency, see C. Chatfield, The Analysis of Time Series, An Introduction (Chapman and Hall, London, 1984), 3rd ed., pp. 131-132,
16. J. S. Bendat and A. G. Piersol, Random Data: Analysis and Measurement Procedures (Wiley-Interscience, New York, 1971), pp. 228-231.
17. c=20 kHz and by solving Eq. (11) for Δt, we have Δt= 0.000025 s, i.e., the digitized data for this audio signal on the CD must be spaced no further apart than 1/40 of a millisecond.
18. R. G. Dean and R. A. Dalrymple, Water Wave Mechanics for Engineers and Scientists (World Scientific, Singapore, 1991), pp. 193-194.
19. M. J. Tucker, Waves in Ocean Engineering (Ellis Horwood, New York, 1991), pp. 41-43.
20. W. J. Emery and R. E. Thomson, Data Analysis Methods in Physical Oceanography (Pergamon, New York, 1998), pp. 380-387 and 422.
21. Although this, larger data set yields a smaller Δf, the output spectrum tends to have a greater random error, or "noise," associated with it. Fourier transforms of long records almost always require some form of data smoothing of the output. For a more thorough discussion of this topic, see Ref. 8, pp. 170-213.
22. R. J. W. Cooley and J. W. Tukey, "An algorithm for the machine calculation of complex Fourier series," Math. Comput. 19 (90), 297-301 (1965).