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tional spread of the wave spectrum is available at the same location for this event.

The field measurements presented in figures 1 and 2 can be used to get a rough quantitative idea of the magnitudes of some relevant parameters characterizing these waves. We will describe the wavetrain as a slow modulation around a central wave (with wave vector k0) of the energy spectrum. The central frequency can be estimated from figure 2 to be f0 ≈ 0.074 Hz. The depth at 16/11-E is h ≈ 70 m, and the bottom is virtually horizontal in a large neighborhood of this site. The dispersion relation for gravity waves on finite depth is

ω2=gk tanh kh, (1)

where ω = 2πf and g = 9.8 m/s2. The central wavenumber can be estimated as k0 ≈ 0.024 m–1, and the normalized inverse depth is (k0h)–1 ≈ 0.60. The wave amplitude is seen to be about a ≈ 5 m, and the central wave steepness is therefore of the order k0a ≈ 0.12.

The deviation in frequency (half the width of the “top” of the spectrum) associated with the slow modulation can be estimated to be Δf ≈ 0.018 Hz. As an estimate for frequency bandwidth we therefore have Δf/f0 ≈ 0.24. An estimate of the wavenumber bandwidth in the direction of the central wave vector can then be obtained by substituting

ω = ω0 ± Δω and k = k0 ± Δk|| (2)

into the dispersion relation (1). To linear order in the bandwidths one has


The factor multiplying the frequency bandwidth is 2 for deep water and decreases to unity for decreasing depth. In our case (3) provides the estimate for the wavenumber bandwidth along the direction of the central wave vector Δk||/k0 ≈ 0.40.

The time series in figure 1 cannot give information on the directional spread in the two horizontal dimensions. However, these wind-driven wave systems typically have angular deviations between 20° and 30°. This gives .

We seek to explain the occurrence of freak waves in the absence of ocean currents or non-uniform bottom topography by nonlinear self modulation of a wave train. Even though the wind is essential to produce the wave spectrum, it can be neglected over the relatively short scales characteristic for a freak wave event. The directional spread of the spectrum is assumed to be important and a three-dimensional model must therefore be employed.

Figure 1: Ten minute wave elevation time series measured by a downward pointing radar at 16/11-E in the Norwegian sector of the North Sea. First axis is time in minutes, second axis is elevation n in meters. Data courtesy of J.I.Dalane and O.T.Gudmestad of Statoil.

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