## 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 ( *f _{p}*) cannot in general be related to the peak of the wavenumber spectrum (

*k*) 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.

_{p}*F*(

*f*) and the omnidirectional wavenumber spectrum

*F*(

*k*) isas dictated by the requirement that the total wave height variance must not depend on which independent variable is used. This equation immediately implies that

## 2. The relationship between *f*_{p} and *k*_{p}

_{p}

_{p}

*f*–

_{p}*k*relationship, I will use the Joint North Sea Wave Project (JONSWAP) spectrum (Hasselmann et al. 1973):whereand will take

_{p}*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

*c*/(2

_{g}*πk*) on scale in these plots, I divide it by 2 times its value at

*f*in the plots. Then the relationships among the three quantities are shown in Fig. 1, where

_{p}*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 *c _{g}*/(2

*πk*) to obtain

*F*(

*k*).

A value for *k _{p}* 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*= (2

*πf*)

^{2}/

*g*.

By letting *f _{p}* take on a range of values, the relationship between

*k*and

_{p}*f*shown in Fig. 2 is obtained for two values of

_{p}*γ*. Also shown in Fig. 2 is the deep-water dispersion relation. This figure shows that 1) the relationship between

*k*and

_{p}*f*is not generally the dispersion relationship but depends on spectral shape and 2)

_{p}*k*is closer to the value obtained from

_{p}*f*via the dispersion relation for narrower spectra. In the case of a delta-function spectrum, and only then,

_{p}*f*and

_{p}*k*are related by the dispersion relation.

_{p}The dependence of the *k _{p}*–

*f*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.

_{p}## 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.

Hwang, P. A., 2004: Influence of wavelength on the parameterization of drag coefficient and surface roughness.

,*J. Oceanogr.***60****,**835–841.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****,**832–846.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****,**670–690.