1. Introduction

Large ocean surface waves are a matter of great concern for mariners, designers of ships and offshore structures such as oil platforms, and many users of the seashore. In many situations, the overall roughness of the sea is of primary interest, but particular attention has been paid to the largest waves within a given sea state. A common criterion for designating a given wave as a “rogue” is that its trough to crest height H is at least 2.2 times the significant wave height H_s, defined as four times the standard deviation of the surface elevation. [See Dysthe et al. (2008) for a recent review.]

For the simple case of a narrowband spectrum of linear and independent waves, the surface elevation distribution is Gaussian and the height H has a probability density distribution (Longuet-Higgins 1952):

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The probability that a wave exceeds νH_s is exp(−2ν²), or (3.4 × 10⁻⁴, 6.3 × 10⁻⁵, 3.7 × 10⁻⁶) for ν = (2, 2.2, 2.5). The corresponding return periods for waves of period 10 s are 8.3 h, 2 days, 1 month. For narrowbanded Gaussian seas, similar results apply for the wave crest η_c = H/2.

These results need modification if the narrowbanded assumption is relaxed (Naess 1985) or if the probability of rogue waves is determined from data (Forristall 2005). The more rapid change in wave heights means that a large trough is less likely to be followed by a large crest so that extreme wave heights are now considerably less common (typically by a factor of 5 or more) than in the limit of a very narrow band. Allowance may also be made for nonresonant nonlinearity in the waves. In particular, allowing for phase-locked second and higher harmonics, crests become sharper and troughs flatter. This does not affect the distribution of wave heights, at least for a narrowband spectrum, but does mean that a crest height that occurs as frequently as a wave height of 2.2H_s is no longer just half this but, rather, 1.34H_s for typical situations (Dysthe et al. 2008).

What has been a matter of great concern is the possibility that extreme waves, whether recorded as wave height or crest height, occur more frequently than expected based on Gaussian theory, or on the modification of this theory to allow for shape changes associated with nonresonant nonlinear interactions. In particular, numerical simulations of a reduced equation set have suggested that, in seas with a narrow directional spread as well as a narrow peak in frequency, resonant nonlinear interactions (as in the Benjamin–Feir instability) will tend to transfer energy within a wave group, leading to amplification of one or more waves within that group and, hence, the generation of particularly large waves (Socquet-Juglard et al. 2005; Gramstad and Trulsen 2007). On the other hand, the observational evidence for this is limited, partly because of difficulties in measurement and partly because the statistical basis is not strong. [For reviews and discussion see Dysthe et al. (2008), the collection of papers in Müller and Henderson (2005), or Müller and Garrett (2007)].

What is clear from many cited examples of what observers describe as “freak” waves is that they tend to be much larger than the waves in the surrounding sea state, often appearing either singly or in small groups, without warning. In many situations, this unexpectedness is more dangerous than the wave height itself, for example, if mariners interpret an interval of several minutes of relatively small waves as an indication of a decreasing sea state. The question to be addressed in this note is whether unexpectedness is necessarily evidence for strongly nonlinear behavior or whether it could arise frequently from simple linear superposition (perhaps with some shape modification, as described above, to allow for nonresonant nonlinear interactions).

Here we report on Monte Carlo simulations of ocean surface waves, based on simple superposition of random phase sinusoids making up typical wave spectra, and pick out waves that are much larger than a significant number of previous waves, perhaps ignoring one or more waves immediately prior to the “unexpected” one. Our study complements the investigation of the average number of waves, in a group, with height above a prescribed threshold (Tucker and Pitt 2001, and references therein). In that situation, the prior waves need not be much smaller.

2. Simulations

We first choose the wave spectrum S( f ) as a function of frequency f (in hertz) [with ∫^∞₀S( f )df giving the variance of the surface elevation] to be one of the Joint North Sea Wave Project (JONSWAP) family (e.g., Holthuijsen 2007), with

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where A = 0.008 and σ = (0.07, 0.09) for ƒ(≤, >) f_p. Here γ is the peak enhancement factor, taken to be 1.0, 2.0, 3.3 for fully developed, developing, and young seas, respectively. We also take f_p = 0.1 Hz, though clearly all our results scale in time with this.

A random sea with a given spectrum may be simulated by adding contributions from a number of independent frequencies separated by a frequency interval Δƒ, which is then also the lowest frequency present. To avoid repetition, Δƒ must be taken to be no larger than the reciprocal of the record length (e.g., Tucker et al. 1984). We have chosen to generate time series that are each 10 h long so that Δƒ = 2.8 × 10⁻⁵ Hz. We then have 216 000 constituents between 0 Hz and our chosen cutoff frequency of 6 Hz.

We could take the phase of each frequency to be random and assign an amplitude of the square root of twice the required spectral amplitude. However, as discussed by Tucker et al. (1984), the spectral amplitudes should be random and come from a Rayleigh distribution, corresponding to separate sine and cosine terms having amplitudes that are independent and Gaussian, each with a mean square equal to the spectral amplitude. This scheme is implemented in the Matlab toolbox Wave Analysis for Fatigue and Oceanography (WAFO) (WAFO Group 2000), which employs inverse fast Fourier transform techniques to speed up the simulations. We use WAFO and, in order to generate reliable statistics, generate 60 000 independent time series for each situation.

A wave crest is defined as the maximum elevation η_c between a zero upcrossing and a zero downcrossing, thus ignoring possible intermediate crests. The wave height is taken as the elevation change from a similarly defined trough to the following crest. (We return later to consideration of including the change from crest to trough.)

The wave period is the time between a zero crossing and the second successor. The average wave period T_a for a JONSWAP spectrum is given approximately in terms of the peak period T_p = f ⁻¹_p by

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We define an unexpected wave as one having its height H or crest height η_c greater by a factor α than the wave or crest height of any of the preceding η_a waves (based on the average period T_a), excluding the m waves immediately prior to the unexpected one. In other words, the unexpected wave is preceded by na − m waves, none more than 1/α as big as the unexpected wave, and then a buildup of m intermediate waves.

a. Nonresonant nonlinearity

As remarked above, large waves tend to have sharper crests and flatter troughs than do sinusoids. This is a consequence of nonresonant nonlinearities; to second order in the wave steepness a periodic wave has an elevation (e.g., Holthuijsen 2007)

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where θ is the wave phase, kx − 2πƒt, for a wave with wavenumber k propagating in the x direction, and a is the amplitude of the first harmonic. The second-order correction is a forced wave, nonresonant since (2ƒ, 2k) do not satisfy the dispersion relation. The correction flattens the troughs and steepens the crests, but without any change in the trough to crest height for a periodic wave. Allowing for the correction has been found to better account for the shape of large waves (e.g., Walker et al. 2004). Some further improvement can be obtained by allowing for even higher harmonics as well, but we restrict our analysis to allowing for the second harmonic only so as to illustrate as simply as possible the consequence of nonresonant nonlinearity.

For the slowly varying wave trains considered here we implement the correction for crest heights by deriving the local frequency ƒ as the reciprocal of twice the time from upcrossing to downcrossing, determining k = (2πƒ)²/g from the dispersion relation and adding (1/2)kη²_c to the linear crest height η_c. This may slightly exaggerate the correction if the crest is not midway between the up- and downcrossings, but we ignore this. We could adopt a similar procedure for trough correction and would now get a slight change in the wave height as the linear crests and troughs will be of unequal magnitudes in a slowly varying wave train, but we ignore this detail and only concern ourselves with the correction to the crest height.

Two minor points need to be discussed. The first is that the mean square steepness per octave for the JONSWAP spectra is independent of ƒ_p, since the steepness spectral density is proportional to ƒ⁻¹ times the peak enhancement factor. Thus, even with the nonlinear crest height increase, our results are independent of our arbitrary choice of ƒ_p = 0.1 Hz and continue to scale in time with ƒ_p⁻¹.

Second, we note that by adding the second harmonic we are altering the spectrum of the waves. We have ignored this as the effect is small; the spectrum is increased by less than 1% up to 1.5ƒ_p, approximately 2% at 2ƒ_p, increasing to approximately 10% between 3ƒ_p and 4ƒ_p by which point the spectral contribution to the mean square wave height is very small.

3. Results

Figure 1 shows an example of an unexpected wave. In this case, the crest height at 374 s is 2.1 times as great as any in the preceding 310 s, which is 31 times the peak period of 10 s, or 43 times the average wave period of 7.1 s. In this example, there is no buildup, so the wave would qualify as unexpected, with n_a = 43 and α = 2.1 for any value of m including zero. The crest height of this particular unexpected wave is 1.4H_s—greater than the 1.1H_s that would qualify it as a rogue wave using linear theory. The trough to crest height, however, is only 1.9H_s, less than the 2.2H_s required for rogue designation (though the height change from the crest to the subsequent trough is 2.7H_s). We return later to a more general discussion of the extent to which unexpected waves meet a criterion for designation as rogue waves.

The occurrence of the large wave in Fig. 1 following a period of much smaller waves, purely as a consequence of random superposition, is not surprising qualitatively. What is not obvious is the frequency with which such events occur. Figure 2 shows the return period of an unexpected wave as a function of the amplitude multiplier α and number of prior waves no more than 1/α times as big, for m = 0, that is, allowing for no buildup. The rows are for different sea states and the columns for wave height based on linear waves, crest height η_lin for linear waves and crest height η_nl using the second-order correction described above.

For γ = 1, a wave with height H at least twice that of any of the preceding 30 waves (corresponding to 21 peak periods) occurs once every 7 × 10⁴ peak periods on average, giving a return period of 8 days if the peak period of the waves is 10 s. Also for γ = 1, unexpected crests are more probable than unexpected waves that are high from trough to crest; with α = 2 and n_a = 30 the return period is 4 days for the linear simulation and 2 days if the nonlinear enhancement is included.

Figure 2 shows that there is no major dependence of the probability on γ for a given α and n_a. For larger γ a given n_a corresponds to a larger n_p, but typically less real time as the peak period will be less for young and developing seas than for a fully developed sea.

Figure 3 is similar to Fig. 2, but now with m = 2, allowing for some buildup to the unexpected wave. This leads to an increase in the probability of an unexpected wave, more so for the narrowband spectrum characterized by γ = 3.3 than for a spectrum with γ = 1.

To illustrate the dependence of the probability on the group parameter m and the JONSWAP parameter γ, we focus on the particular choices α = 2 and n_a = 30, corresponding to the point at the center of the plots in Figs. 2 and 3. The figure confirms that, for all three values of m, unexpected crests are more probable than unexpected wave heights from trough to crest, particularly if the nonlinear enhancement is included. The circles in Fig. 4 show that for m = 0 the return period R in peak wave periods increases with γ, perhaps because increasing γ corresponds to a peakier spectrum, longer groups, and hence less likelihood of an isolated unexpected wave. Interestingly, the triangles and diamonds (m = 1 and 2) show the opposite trend to that for m = 0.

4. Discussion

The frequency with which unexpected waves occur even without the extra possible physics of resonant nonlinear interactions is remarkable and of scientific interest. It suggests that many reported freak waves may not be so freakish after all, but merely the simple consequence of linear superposition. Our predictions could be incorporated into maritime safety manuals or coastal warnings. To be sure, we have based our simulations on deep water scenarios and further analysis is required for nearshore situations, but it seems likely that similar results will emerge. Thus, mariners and tourists can be warned that even a few minutes of waves can be followed by a wave at least twice as high, with such an event likely every few days.

There are other possible interpretations and extensions of our analysis. We discuss these next.