Waves in the ocean at a specific position are described by a time signal. An irregular signal will have phases that are commonly understood to be uniformly distributed in (-π,π]. Previous study defined maximal waves and pseudo-maximal waves. A maximal wave is a wave with all phases zero and has maximal amplitude equal to the integration of its two-sided absolute spectrum. Thus, we call a signal with all phases zero at some time (say t=0) a maximal signal, as MS(t)=∫|ηˇ0(ω)|cos⁡ωtdω. At t=0, all wave components contribute to a constructive interference, hence MS(0)=∫|ηˇ0(ω)|dω. This is the highest amplitude that is possible for given spectrum, ηˇ0(ω). In view of the assumption of uniform distribution of the phases, the chance for such a maximal wave vanishes.

A pseudo-maximal (pm) wave is a partly coherent wave, that is in between a completely irregular wave and a fully coherent maximal wave. For a given signal with random phase θ(ω)∈(-π,π] as a function of wave frequencies with θ(ω)=-θ(-ω), we consider a pm signal as the signal for which the phases are restricted for certain α∈(0,1) to the phases θα(ω)=αθ(ω), as ηpm(t)α=∫|ηˇ0(ω)|cos⁡θα(ω)-ωtdω. By taking a fraction α of the random phase, the maximal amplitude decreases and the background increases for increasing α. For α= 0, it is a maximal wave with coherent phases while for α= 1 it is an irregular wave and the freak wave may disappear completely.

The phases of all frequencies in a pm signal are constrained as |θ(ω)|<απ. We now define a weak pseudo-maximal (wpm) signal, ηr(t), by restricting the phases of only the frequencies of large energy-carrying modes (see Fig.  for an illustration). We also illustrate the importance of such restrictions for coherence by plotting the maximal, pm and wpm signals of a given Jonswap spectrum in Fig. .

The restriction of wpm signal is typically for frequencies within one (or a half) standard deviation (SD) around the mean frequency, |ω-ωm|<σω or less. Then we consider a signal for α=[0,1] and define θα as θα(ω)=αθ(ω),for|ω-ωm|<σωθ(ω),elsewhere and get a signal that has maximal amplitude less than the maximal amplitude of the pm signal: ηr(t)α=∫-∞∞|ηˇ(ω)|cos⁡(θα(ω)-ωt)dω≤ηpm(0)α.

In general, the mean frequency is not necessarily equal to the peak frequency because the spectrum of waves that is usually of Jonswap shape is not symmetric around the peak frequency.

Figure  illustrates that the value of α significantly affects the maximal crest height and the wave evolution along the x axis. The smaller the value of α, the higher the value of the generated crest. On the other hand, variations in σω influence much less the maximum elevation of the influx signal. In any case, the wave evolution is tremendously affected and the maximum amplitude during the evolution can be much higher for larger σω. In Fig.  it is shown that at an influx position (x≈ -3600 m), the maximum amplitudes are quite the same for various fractions of SD, but near the focusing position a larger fraction of SD produces a higher maximum amplitude. This is the consequence of the fact that the larger fraction of SD gives more wave components with coherent phases.

Although the signal coherence can describe and measure the appearance of freak waves, the concepts use the whole interval of the time signal. However, not the whole interval will contribute in generating a freak wave since the waves propagate with their own group and phase velocity. The freak wave will be generated from local waves' interaction. Therefore, we will investigate the local energy propagation using wavelet transformation. This is expected to give a more refined measure of the appearance of the freak wave.

In Fourier analysis we transform a function that depends on time into a function that depends on the frequency as a single variable. Given a time signal η(t), Fourier transformation gives the relations ηˇ(ω)=12π∫η(t)eiωtdtη(t)=∫ηˇ(ω)e-iωtdω=2∫0∞|ηˇ(ω)|cos⁡(ωt+θ(ω))dω. The Fourier transform of η(t) is the complex valued function ηˇ(ω)=|ηˇ(ω)|eiθ(ω), in which |ηˇ(ω)| is the amplitude spectrum and θ(ω) is the phase of the signal. The spectral energy density of the signal is defined by |ηˇ(ω)|2 that describes how the energy of the signal is distributed with frequency. Any local (time) information is not directly contained in Fourier transform, but is hidden in the spectrum and phase. At a certain local time, t=t0, we have η(t0)=2∫0∞|ηˇ(ω)|cos⁡(ωt0+θ(ω))dω. The term inside the integral represents the amplitude spectrum and phase distribution with the frequency at a single time. Then we may define a local energy spectrum, E(t0,ω), as E(t0,ω)=|ηˇ(ω)|cos⁡(ωt0+θ(ω))2, presenting the local information of the signal directly. More generally, we will not only consider the energy at a single instant but will also analyze the energy in the neighborhood. Therefore, we will use wavelet transformation for the local energy analysis since it will show the distribution of the local energy spectrum better because it includes energy contributions from neighboring times instead of only one local time. Figures  and  illustrate the local energy distribution computed by Fourier and wavelet transform for a dispersive focusing wave and an irregular wave that will be used as study cases in Sects.  and . The plots show that the wavelet transform gives a more refined description of the local energy distribution as a function of time and frequency.

The wavelet transform is an extension of Fourier transformation. The basis function in Fourier transform is a sinusoidal of a specific frequency, and the L2 inner product with the signal leads to the Fourier coefficient of that frequency only. A wavelet is composed of a mixture of frequencies (which is indicated by its own Fourier transform). As a consequence, the wavelet coefficients refer to this mixture of frequencies, not a single frequency. We will now provide a summary of the main notions needed in the following sections.

Definition 2.1. A mother wavelet is a zero average function, ψ, as ψ∈L2R:∫ψ(t)dt=0.

Definition 2.2. A wavelet family is family of functions generated from any type of mother wavelet, ψ, through dilatation (s>0) and translation (u∈R): ψu,s(t)=1sψt-us.

There are many types of mother wavelets: Morlet, Haar, Daubechies, Meyer, etc. (see ). In this paper, we use the Morlet wavelet consisting of a plane wave modulated by a Gaussian, ψ(t)=e-t2/2e-iω0t, which is given in the Fourier domain by ψ^(ω)=2πe-(ω-ω0)2/2 with the central frequency ω0.

Definition 2.3. The continuous wavelet transform of a signal η(t) at the scale s and at the time u is calculated by correlating η with the wavelet family, ψu,s: Wη(u,s)=〈η,ψu,s〉=∫η(t)1sψ*t-usdt, where ψ* is the complex conjugate of ψ.

From Definition 2.3, the wavelet transform of a time signal η(t) gives a complex valued function Wη(u,s). For the Morlet wavelet, we obtain Wη(u,s)=∫η(t)1se-(t-u)2/2s2eiω0(t-u)/sdt. By substituting s=ω0/ω and writing F(t;u,ω)=1s(ω)e-(t-u)2/2s(ω)2, the equation above gives Wη(u,ω)=∫η(t)Ft;u,ωeiω(t-u)dt. The function F(t;u,ω) applies as a Gaussian window function to the signal η(t). This shows that the wavelet transform can be interpreted as the Fourier transform of a windowed signal in the neighborhood of t=u. The magnitude of the wavelet transform, |Wη(u,ω)|, represents the energy distribution of the signal over frequency and time and its angle, arg⁡Wη(u,ω), represents the local phase of the signal.

Similar to Fourier transform, it is possible to rebuild the signal from the wavelet transform, the so-called inverse wavelet transform. It is given by η(t)=1Cψ∫0∞∫Wη(u,s)1sψt-usdudss2=∫0∞1ω0Cψ∫Wη(u,ω)F(t;u,ω)e-iω(t-u)dudω, with Cψ=∫0∞|ψ^(ω)|2ωdω. As an example, for ω0=6 the Morlet wavelet above produces Cψ≈ 1.883. Different from Eq. () that gives the local energy spectrum computed at one time, Eq. () shows that the local energy spectrum from the wavelet transform is not only computed at the local time but it also includes the contribution of the signal surrounding that time.

The choice of the central frequency ω0 should be such that the Morlet wavelet satisfies the admissibility condition, Cψ<∞, which is equivalent to ψ^(0)= 0. Then the (real) wavelet transform is complete and preserves the quantity of energy: ∫|η(t)|2dt≈1ω0Cψ∫0∞∫|Wη(u,ω)|2dudω. Actually, the Morlet wavelet satisfies the condition only approximately because ψ^(0)=2πe-ω02/2 does not vanish exactly. A proper choice of ω0 can make the wavelet at least practically admissible and allows one to apply it widely to the signal decomposition . The defined Morlet wavelet is sufficiently admissible if we choose ω0> 5 (see ), hence ω0=6 is taken to be sufficient since then ψ^(ω)≤3.8×10-8 for ω≤0. Parseval's identity gives a relation between the signal and its Fourier transform as ∫|η(t)|2dt=2π∫|ηˇ(ω)|2dω≈∫1ω0Cψ∫|Wη(u,ω)|2dudω. Therefore, the spectral energy density of a signal can be computed through the wavelet transform, i.e, |ηˇ(ω)|2≈12πω0Cψ∫|Wη(u,ω)|2du. This equation shows that the energy distribution from the wavelet transform behaves locally, and its integration over the time shift u is approximately the spectral energy density obtained by Fourier transform.

defines a wave group as an uninterrupted sequence of waves with wave heights higher than an arbitrarily chosen, but usually high, threshold value. Instead of a wave group, we define a group event based on a chosen local energy level as threshold, which is determined by the contour level of the spectral energy determined by the wavelet transform. A group event of a time signal η(t) is part of the time signal with |Wη(u,ω)| higher than a threshold value. We denote the set of group events with respect to the threshold value ϵ, by WGϵ: WGϵ=ηi(t):i=1,2,…,Ngηi(t)=η(t):t=[t1,t2]⊂[0,T]||Wηi(u,ω)|≥ϵ. WGϵ is the assembly of Ng group events; each group is determined by the time interval during which the wavelet transform is larger than a specified value ϵ. The selection of the group events depends on the chosen threshold value ϵ. In practice, we normalize the value of |Wη| with its maximum, so that the value of ϵ is chosen in (0,1). The choice depends on the background waves since it aims to ignore the waves that do not contribute to the evolution of the group under consideration. When the background waves are high, we should choose a large ϵ, but when the background waves are small, we can choose a small value of ϵ. In this paper, we choose ϵ≈ 0.65 for the random signals and ϵ≈ 0.2 for the maximal signal.

From all the group events determined in this way, we characterize the groups that may lead to a freak wave. For a given time signal, η(t), t∈[0,T], we define a total energy signal, ET, as ET=∫0T|η(t)|2dt≈1ω0Cψ∫0T∫|Wη(u,ω)|2dωdu. For each value of the total energy signal, there can be a maximal wave with a coherent state.

Next, we define the total energy threshold to eliminate group events which unlikely generate a freak wave. The remaining groups are so-called critical group events. WGcrit=WGϵ∩ηi(t)|∫t1t2ηi(t)2dt≥ρ2ET in which ρ=1.25Hs∫|ηˇ(ω)|dω is a freak wave threshold normalized by the amplitude of a maximal signal. Based on their local energy, these critical group events could generate a freak wave forward or backward, but the probability depends on the phases.

We start from the complex value of the wavelet transform of η(t), Wη(u,ω)=|Wη(u,ω)|eiΘ(u,ω). It gives the spectral energy distribution |Wη(u,ω)| and the phase information Θ(u,ω) as a function of time and frequency. From this we may look at the frequencies that carry most energy as a function of time denoted by ωm(u) from the critical group event: ωm(u)=ω||Wη(u,ω)|=max⁡ω|Wη(u,ω)|. Convergence of waves will occur when long waves catch up with shorter waves. Hence, when the local wave length increases, i.e., when the wave frequency decreases, the waves will converge at a later time and vice versa. Therefore, the distinction is determined by the frequency in the time interval: when decreasing in forward time, this leads to a focusing energy, and an increase leads to defocusing energy. Since continuity of the local wave frequency in the random waves cannot be guaranteed, we approximate the local wave frequency by a linear interpolation, ωm(u)≈ω̃(u), so that we

can distinguish the two cases: ω̃(u)=Ãu+B̃Ã>0,defocusing/diverging energyÃ≤0,focusing/converging energy.

Moreover, we can also look at the most energetic waves as a function of wave frequency. This leads to a local time of each wave contribution. In the case of a dispersive focusing wave, focusing of the energy occurs when all wave contributions are in phase at one local time.

Motivated by this, for each critical group event in a local time interval [t1,t2], we define a function τm(ω) representing the local time of the maximal energy, τm(ω)∈t1,t2, as τm(ω)=u||Wη(u,ω)|=max⁡u|Wη(u,ω)|. Hence, if the critical group event gives a constant τm(ω), all frequencies contribute at the same time, which leads to local coherence at that time. If the frequencies are decreasing over the local time interval, it may indicate a local focusing at a later time.

The observations of the most energetic waves in either time or frequency can be used to see whether a freak wave may appear in forward or backward time, but the generation of a freak wave is still not assured, since the amplitude is not determined yet. The local information of the energy and phase gives a method to investigate locally the relation between the local coherence and freak wave occurrence. In this subsection, we measure the local coherence of the group event along its evolution and we will show that the highest amplitude occurs when the local coherence is maximum in the restricted frequency interval. As the wavelet transformation gives a function of frequency and time, we define a time spreading of the most energetic waves [φ(ω)]τ∈[-π,π] for each time τ∈(t1,t2) as follows: φ(ω)τ=τm(ω)-τmod2π that is taken at the time at which the absolute mean is minimal. φ(ω)=φ(ω)τ||φ‾(ω)|τ=min⁡τ|φ‾(ω)|τ. The time spreading is exactly zero at a certain frequency interval when τm(ω) is constant at that interval. To investigate the local coherence, we determine the maximum (M), the mean (μ) and the standard deviation (σ) of the absolute value of the time spreading normalized by π. Accordingly, we define three quantities depending on position that can represent local coherence, ΓM,μ,σ(x)∈[0,1], depending on the choice for the parameters, M, μ or σ: ΓM(x)=1-MΓμ(x)=1-2μΓσ(x)=1-3σ. These values represent a somewhat different measure of local coherence. Note that the extreme case (ΓM=Γμ=Γσ=1) occurs for the maximal signal, when all the phases are zero. Note also that this measure is different from the degree of phase coherence defined in , as it corresponds to the local time spreading of the most energetic waves of a group event. To investigate the dependence between the local coherence and the occurrence of freak waves, we compute the correlation between the local coherence and the maximum amplitude normalized by its time-averaged local energy, Corr(ΓM,μ,σ,Am). For N number of time signals at the positions (x1,x2,…,xN), the correlation is computed by CorrΓ,Am=∑i=1NΓ(xi)-μΓAm(xi)-μA(N-1)σΓσA,withAm(xi)=max⁡t∈t1,t2|η(xi,t)|1t2-t1∫t1t2η(xi,t)2dt, and Γ(xi) is the local coherence of the time signal at xi.

The case is a focusing wave that will lead to a maximal wave. We consider a dispersive focusing wave with significant wave height 0.013 m, for which measurements at several positions are available from an experiment at MARIN (Case 202002). The experiment was executed at a water depth of 1 m. Here, we use the elevation at the first measurement position after the wave flap as the influx signal for the numerical simulation by both the linear and nonlinear AB equation. The spectral shape of the influx signal with peak frequency of approximately 5 rads-1 is shown in Fig. . The result of the numerical simulation of a dispersive focusing wave using both the linear and nonlinear AB equations have been previously verified with the measurements .

Referring to Fig. , the influx signal only consists of one group event with almost zero background, which is therefore the only critical group event. This is an idealized case as the freak wave turns out to be a maximal wave that is generated from all wave components in the initial signal. This can be observed from the evolution of the influx signal; the shorter (slower) waves are followed by longer (faster) waves such that at the focusing point at 50.2 m all waves have vanishing phase. See Fig.  for various plots of snapshots of the dynamics at successive measurement positions.

During the evolution, the changes of the distribution of the local energy in the time–frequency frame are described well by the filled contour plot of the local energy. The local energy distribution from one group event is squeezed into a maximal wave. This is also shown by the decreasing width of the time intervals towards the focusing point in Fig. . We can see at x= 20 m that the energy is distributed in 20 s, at x= 40 m it is distributed approximately in 10 s and at the focusing point the energy is only distributed in 3 s. Moreover, the pure maximal wave is shown by the zeroes of the time spreading at x= 50.2 m in Fig. b. The profile of the maximal wave can be seen in Fig. .

In order to show that the occurrence of the freak wave is related to a local coherence, and to illustrate the three different measures of coherence introduced above, we show the evolution of these coherence measures for the linear and nonlinear evolution in Table . It can be observed from this table that the correlation of each of the three measures of coherence and the occurrence of the maximum amplitude at x= 50.2 m is very strong (≈ 0.95), although outside the focusing position the values of the three Γ's can be rather different. ΓM and Γμ seem to be much better indicators for the focusing than Γσ.

The second and third case are synthetic signals of irregular waves that are generated from a Jonswap spectrum with normal and thunderstorm sea conditions at a water depth of 480 m (deep water). The wave evolutions are computed linearly by the AB equation as the nonlinear effect for these cases is not significant. However, a freak wave is still found in both cases.

The fourth case is an irregular wave, for which measurements at several positions are available from MARIN experiment with a water depth of 0.6 m (Case 103001). It has 1.697 s peak period and significant wave height of approximately 0.06 m. We use the time signal from the first measurement position after the wave flap as the influx signal. The spectral shape of the influx signal is shown in Fig. . The local energy distribution of the signal is presented in Fig. . There are six critical group events from the influx signal as shown in Fig. . The largest local energy signal of the wave groups is found around t≈ 240 s and it develops a freak wave.

The evolution of the time signal around the critical group event and its energy distribution at several positions are shown in Fig. a. Even though the energy spectral distribution does not show clearly the development of the critical group event into a freak wave, the change of the time spreading shows the development of its local coherence (see Fig. b). A freak wave occurs at x= 103.7 m when its time spreading is near coherent for a short carrying wave mode. The freak wave is shown in Fig. . From Fig. , we can also see that there is unclear increasing or decreasing wave frequencies of the most energetic wave. The local coherences are measured and presented in Table . The three values of Γ's present quite high correlation (≥ 0.75) between the local coherences and the maximum amplitude in both the linear and nonlinear evolution. In this case, Γμ performs as the best indicator for the freak wave appearance.