• Hasselmann, K., and Coauthors, 1973: Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP). Ergänzungsheft zur Deutchen Hydrograph Zeitschrift, Reiche A (8°), Nr. 12, 95 pp.

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    • Export Citation
  • Hwang, P. A., 2004: Influence of wavelength on the parameterization of drag coefficient and surface roughness. J. Oceanogr., 60 , 835841.

    • Search Google Scholar
    • Export Citation
  • Plant, W. J., , W. C. Keller, , and K. Hayes, 2005: Simultaneous measurement of ocean winds and waves with an airborne coherent real aperture radar. J. Atmos. Oceanic Technol., 22 , 832846.

    • Search Google Scholar
    • Export Citation
  • Walsh, E. J., , D. W. Hancock III, , D. E. Hines, , R. N. Swift, , and J. F. Scott, 1989: An observation of the directional wave spectrum evolution from shoreline to fully developed. J. Phys. Oceanogr., 19 , 670690.

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    • Export Citation
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    Plots of the relationship between F(k) and F( f ) for two different values of spectral widths specified by γ. They are related by cg/(2πk), also shown.

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    The deep-water ocean wave dispersion relation and the true relation between kp and fp for two different spectral widths.

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The Ocean Wave Height Variance Spectrum: Wavenumber Peak versus Frequency Peak

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  • 1 Applied Physics Laboratory, University of Washington, Seattle, Washington
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Abstract

Many authors assume that the frequency peak and the wavenumber peak of an ocean wave height variance spectrum are related by the ocean wave dispersion relationship. This note shows that this is not true and that the true relationship depends on the shape of the spectrum, thereby introducing an element of randomness into the relationship.

Corresponding author address: William J. Plant, Applied Physics Laboratory, University of Washington, 1013 NE 40th St., Seattle, WA 98105-6698. Email: plant@apl.washington.edu

Abstract

Many authors assume that the frequency peak and the wavenumber peak of an ocean wave height variance spectrum are related by the ocean wave dispersion relationship. This note shows that this is not true and that the true relationship depends on the shape of the spectrum, thereby introducing an element of randomness into the relationship.

Corresponding author address: William J. Plant, Applied Physics Laboratory, University of Washington, 1013 NE 40th St., Seattle, WA 98105-6698. Email: plant@apl.washington.edu

1. Spectral conversions

Ocean wave height variance spectra can be functions of either wavenumber or frequency. We refer to the first as the wavenumber spectrum and the second as the frequency spectrum. In a recent paper, I briefly mentioned that the peak of the frequency spectrum ( fp) cannot in general be related to the peak of the wavenumber spectrum (kp) by the ocean wave dispersion relationship (Plant et al. 2005). Unfortunately, this fact does not seem to be widely recognized despite a general understanding of how to relate one spectrum to another. This has caused inaccuracies, even errors, in some papers relating the two types of spectral peaks (Walsh et al. 1989; Hwang 2004). In this brief note, I show that, except in the case in which the ocean wave spectrum is a delta function, the peak of the wavenumber spectrum is not related to the peak of the frequency spectrum by the ocean wave dispersion relation. In general, their relationship depends on the shape of the spectrum.

The relationship between the omnidirectional (i.e., integrated over all azimuth angles) frequency spectrum F( f ) and the omnidirectional wavenumber spectrum F(k) is
i1520-0485-39-9-2382-e1
as dictated by the requirement that the total wave height variance must not depend on which independent variable is used. This equation immediately implies that
i1520-0485-39-9-2382-e2

2. The relationship between fp and kp

To illustrate how this equation affects the fpkp relationship, I will use the Joint North Sea Wave Project (JONSWAP) spectrum (Hasselmann et al. 1973):
i1520-0485-39-9-2382-e3
where
i1520-0485-39-9-2382-eq1
and will take σ to be 0.2. Since I am not interested in spectral magnitudes here, I will normalize both F(k) and F( f ) to their peak values in the plots to follow. Furthermore, to keep cg/(2πk) on scale in these plots, I divide it by 2 times its value at fp in the plots. Then the relationships among the three quantities are shown in Fig. 1, where F(k) is plotted versus f and not versus k. The dispersion relation does relate these two independent variables, and therefore the abscissa could be easily converted to a more standard wavenumber axis. In these illustrations, I assume deep water.

Note that the peaks of F(k) and F( f ) are not at the same frequency in either panel of Fig. 1 but that they are much closer together for the narrower spectrum. They are different because of the weighting applied to F( f ) by cg/(2πk) to obtain F(k).

A value for kp can be obtained from the plots in Fig. 1 by converting the frequency at which F(k) maximizes to wavenumber via the deep-water dispersion relation k = (2πf )2/g.

By letting fp take on a range of values, the relationship between kp and fp shown in Fig. 2 is obtained for two values of γ. Also shown in Fig. 2 is the deep-water dispersion relation. This figure shows that 1) the relationship between kp and fp is not generally the dispersion relationship but depends on spectral shape and 2) kp is closer to the value obtained from fp via the dispersion relation for narrower spectra. In the case of a delta-function spectrum, and only then, fp and kp are related by the dispersion relation.

The dependence of the kpfp relationship on spectral shape introduces a source of randomness into the relationship, causing the two variables to be at least somewhat decorrelated. This is in addition to the effects of currents and statistical uncertainty, which have not been considered here.

REFERENCES

  • Hasselmann, K., and Coauthors, 1973: Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP). Ergänzungsheft zur Deutchen Hydrograph Zeitschrift, Reiche A (8°), Nr. 12, 95 pp.

    • Search Google Scholar
    • Export Citation
  • Hwang, P. A., 2004: Influence of wavelength on the parameterization of drag coefficient and surface roughness. J. Oceanogr., 60 , 835841.

    • Search Google Scholar
    • Export Citation
  • Plant, W. J., , W. C. Keller, , and K. Hayes, 2005: Simultaneous measurement of ocean winds and waves with an airborne coherent real aperture radar. J. Atmos. Oceanic Technol., 22 , 832846.

    • Search Google Scholar
    • Export Citation
  • Walsh, E. J., , D. W. Hancock III, , D. E. Hines, , R. N. Swift, , and J. F. Scott, 1989: An observation of the directional wave spectrum evolution from shoreline to fully developed. J. Phys. Oceanogr., 19 , 670690.

    • Search Google Scholar
    • Export Citation
Fig. 1.
Fig. 1.

Plots of the relationship between F(k) and F( f ) for two different values of spectral widths specified by γ. They are related by cg/(2πk), also shown.

Citation: Journal of Physical Oceanography 39, 9; 10.1175/2009JPO4268.1

Fig. 2.
Fig. 2.

The deep-water ocean wave dispersion relation and the true relation between kp and fp for two different spectral widths.

Citation: Journal of Physical Oceanography 39, 9; 10.1175/2009JPO4268.1

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