1. Introduction
Rogue waves, often associated with severe disasters and casualties, were once dismissed as maritime folklore until numerous encounters with vessels and offshore platforms were documented (Draper Reference Draper1964; Didenkulova & Pelinovsky Reference Didenkulova and Pelinovsky2020; Didenkulova, Didenkulova & Medvedev Reference Didenkulova, Didenkulova and Medvedev2023). These anomalous ocean waves (with wave height reaching 2 or 2.2 times the significant wave height, by definition) occur with unexpectedly high probability (Dysthe, Krogstad & Müller Reference Dysthe, Krogstad and Müller2008; Mori et al. Reference Mori, Waseda and Chabchoub2023; Bitner-Gregersen et al. Reference Bitner-Gregersen, Gramstad, Trulsen, Magnusson, Støle-Hentschel, Aarnes and Breivik2024).
Rogue wave formation essentially results from wave energy focusing through linear or nonlinear mechanisms. Numerous hypotheses regarding these focusing processes have been proposed (Kharif, Pelinovsky & Slunyaev Reference Kharif, Pelinovsky and Slunyaev2009; Onorato et al. Reference Onorato, Residori, Bortolozzo, Montina and Arecchi2013; Adcock & Taylor Reference Adcock and Taylor2014; Dematteis et al. Reference Dematteis, Grafke, Onorato and Vanden-Eijnden2019), which can be systematically categorised based on the role of nonlinearity. Linear focusing mechanisms include dispersive focusing and spatial focusing due to bathymetric or current-induced reflection and refraction (Chien, Kao & Chuang Reference Chien, Kao and Chuang2002). Nonlinear focusing mechanisms encompass bound wave nonlinearity (Fedele et al. Reference Fedele, Brennan, Ponce de León, Dudley and Dias2016; Knobler et al. Reference Knobler, Malila, Tayfun, Liberzon and Fedele2025), modulation (Benjamin–Feir) instability (Benjamin & Feir Reference Benjamin and Feir1967; Onorato et al. Reference Onorato2009; He et al. Reference He, Slunyaev, Mori and Chabchoub2022; Zhai et al. Reference Zhai, Klahn, Onorato and Fuhrman2025) and non-equilibrium dynamics (NED) due to significant and rapid environmental changes (Trulsen Reference Trulsen2018).
Recently, NED has attracted considerable interest as it provides a universal explanation of rogue wave formation under rapidly varying wind, current or bathymetric conditions (Trulsen, Zeng & Gramstad Reference Trulsen, Zeng and Gramstad2012; Annenkov & Shrira Reference Annenkov and Shrira2015; Toffoli et al. Reference Toffoli, Proment, Salman, Monbaliu, Frascoli, Dafilis, Stramignoni, Forza, Manfrin and Onorato2017; Zhang et al. Reference Zhang, Ma, Tan, Dong and Benoit2023; Mendes, Teutsch & Kasparian Reference Mendes, Teutsch and Kasparian2025), or under unrealistic initial conditions (Shemer, Sergeeva & Liberzon Reference Shemer, Sergeeva and Liberzon2010; Tang et al. Reference Tang, Barratt, Bingham, van den Bremer and Adcock2022). In particular, being relevant in coastal areas, NED induced by drastic depth reduction has been intensively investigated in both numerical (Zeng & Trulsen Reference Zeng and Trulsen2012; Gramstad et al. Reference Gramstad, Zeng, Trulsen and Pedersen2013; Kashima, Hirayama & Mori Reference Kashima, Hirayama and Mori2014; Viotti & Dias Reference Viotti and Dias2014; Zheng et al. Reference Zheng, Lin, Li, Adcock, Li and van den Bremer2020; Lawrence et al. Reference Lawrence, Gramstad and Trulsen2021a ; Zhang, Ma & Benoit Reference Zhang, Ma and Benoit2024a , Reference Zhang, Ma and Benoitb , Reference Zhang, Mendes, Benoit and Kasparianc ) and experimental flumes (Trulsen et al. Reference Trulsen, Zeng and Gramstad2012, Reference Trulsen, Raustøl, Jorde and Rye2020; Bolles, Speer & Moore Reference Bolles, Speer and Moore2019; Zhang et al. Reference Zhang, Benoit, Kimmoun, Chabchoub and Hsu2019; Wang et al. Reference Wang, Ludu, Zong, Zou and Pei2020; Lawrence et al. Reference Lawrence, Trulsen and Gramstad2021b ; Li et al. Reference Li, Draycott, Adcock and van den Bremer2021a ; Samseth & Trulsen Reference Samseth and Trulsen2025), focusing on statistical distributions of free surface elevation and kinematics, evolution of statistical moments and wave forces on structures. Theoretical investigations on NED have been conducted from both stochastic (Majda, Moore & Qi Reference Majda, Moore and Qi2019; Majda & Qi Reference Majda and Qi2020; Mendes et al. Reference Mendes, Scotti, Brunetti and Kasparian2022; Mendes & Kasparian Reference Mendes and Kasparian2023) and deterministic perspectives (Li et al. Reference Li, Draycott, Zheng, Lin, Adcock and van den Bremer2021b ; Moss et al. Reference Moss, Schultz, Parkes, Li and Draycott2025).
Research on NED in three-dimensional (3-D) cases is gaining significant momentum. Using fully nonlinear potential flow models, Lawrence, Trulsen & Gramstad (Reference Lawrence, Trulsen and Gramstad2022) and Draycott & Li (Reference Draycott and Li2025) investigated the interplay between long-crested waves and 2-D bathymetry (circular shoals), both of which show a significant kurtosis enhancement due to NED. With the high-order spectral method, Ducrozet & Gouin (Reference Ducrozet and Gouin2017) showed that directional spreading strongly suppresses rogue wave formation due to depth variation, while Tang et al. (Reference Tang, Moss, Draycott, Bingham, van den Bremer, Li and Adcock2023) observed a weaker directional effect. Conversely, Lyu, Mori & Kashima (Reference Lyu, Mori and Kashima2023) showed with a 2-D depth-modified nonlinear Schrödinger equation that short-crested wave fields with a larger spreading angle may increase the occurrence probability of extreme waves. Remarkably, Mei et al. (Reference Mei, Chen, Yang and Gui2023) found through a fully nonlinear Boussinesq model that directional spread can decrease the excess kurtosis in intermediate water but also increase it in shallow water. Evidently, previous studies on NED of 3-D waves are either based on numerical simulations or have been constrained by experimental facility limitations (Tang et al. Reference Tang, Moss, Draycott, Bingham, van den Bremer, Li and Adcock2023). Contradictory conclusions indicate that the effect of the wave directionality on rogue wave formation remains unclear. To address this knowledge gap, we conduct a systematic, large-scale experimental study of NED in wave fields with varying directional spreading and incident direction. We explore how the non-equilibrium parameters, skewness and kurtosis, evolve after a rapid depth variation.
2. Experimental set-up
The experimental campaign was conducted at the Multi-functional Test Basin of the National Marine Environmental Monitoring Centre (NMEMC) in Dalian, China. The wave tank is 49 m long, 47 m wide and 1.2 m deep. Waves are generated by a snake-type (consisting of 80
$\times$
0.5 m paddles) multi-directional wavemaker installed on one shorter side of the tank. Porous medium damping zones along the opposing end and lateral boundaries are set to minimise wave reflection.
In the wave tank, an isosceles trapezoidal prism parallel to the wavemaker was installed on an otherwise flat bottom. It consists of a central flat section (
$8$
m in width and
$0.36$
m in height) flanked by two transitional slopes (with a horizontal extent of
$1.75$
m, thus a gradient of
$1/4.86$
). The upslope of the trapezoidal prism starts
$15$
m away from the wavemaker. During the experimental campaign, the water depth near the wavemaker was set to
$h_1=0.61$
m, thus
$h_2=0.25$
m over the submerged structure. Twenty-six resistance-type wave gauges were deployed during the tests, with a sampling frequency
$f_s=50$
Hz. Two arrays of five probes with the same configuration as introduced in Nwogu (Reference Nwogu1989) were used to estimate the directional spectra before and above the submerged bar. The remaining probes are adopted to capture the wave evolution. The schematic of the experimental basin, as well as the locations of the wave gauges, are shown in figure 1.
Figure 1. The NMEMC wave tank (a); the layout of wave gauge array following Nwogu (Reference Nwogu1989) for the estimation of directional spectrum. (b); the experimental wave tank and locations of the wave gauges (c).
The wave fields are described by their directional spectrum
$S(f,\theta )=S_J(f)D(\theta |f)$
, with
$S_J(f)$
denoting the JOint North Sea WAve Project spectrum and
$D(\theta |f)$
the directional spreading function
\begin{equation} S_J(f)=\frac {\alpha _J{g}^2}{\left (2\pi \right )^4}\frac {1}{f^5}\exp {\left [-\frac {5}{4}\left (\frac {f_{\!p}}{f}\right )^4\right ]}\gamma ^{\exp {\left [-\frac {\left (f-f_{\!p}\right )^2}{2\left (\sigma _{\!J} f_{\!p}\right )^2}\right ]}}, \end{equation}
where
$g$
is the acceleration due to gravity,
$f_{\!p}=1/T_p$
denotes the spectral peak frequency,
$\alpha _J$
the spectral energy parameter,
$\gamma$
the peakedness parameter and
$\sigma _{\!J}$
width parameter,
$\sigma _{\!J}=0.07$
for
$f\lt f_{\!p}$
and
$\sigma _{\!J}=0.09$
for
$f\geq f_{\!p}$
. Here, the classical Mitsuyasu-type directional spreading function (Goda Reference Goda2010) is adopted
\begin{equation} D(\theta |f) = \frac {2^{2s-1}}{\pi }\frac {\varGamma ^2(s+1)}{\varGamma (2s+1)}\cos ^{2s}\left (\frac {\theta -\theta _{\textit{inc}}}{2}\right )\!, \end{equation}
with
$\varGamma$
denoting the gamma function,
$\theta _{\textit{inc}}$
the dominant wave direction and
$s(f)$
the frequency-related angular spreading parameter,
$s=s_{\textit{max}} (f/f_{\!p})^{5}$
for
$f \leq f_{\!p}$
and
$s=s_{\textit{max}} (f/f_{\!p})^{-2.5}$
for
$f\geq f_{\!p}$
. The peak value of
$s$
,
$s_{\textit{max}}$
, governs the width of the directional spreading. To avoid phase locking and ensure ergodicity of the generated wave field, we adopt the modified double summation approach for wave generation, which associates a random phase
$\phi _{\textit{ij}}\in [0,\, 2\pi ]$
with each frequency-direction bin
$(f_{\textit{ij}},\theta _j)$
(Luo et al. Reference Luo, Liu, Li and Jia2020)
\begin{equation} \eta (y,t) = \sum _{i=2}^{N_{\!f}} \sum _{j=1}^{N_{\theta }} a_{\textit{ij}}\cos \left (yk_{\textit{ij}}\sin \theta _j - 2\pi f_{\textit{ij}}t + \phi _{\textit{ij}}\right )\!, \end{equation}
where
\begin{equation} \begin{cases} f_{\textit{ij}} = \hat f_i - \dfrac {1}{2}\Delta f + {\left (j-1+{R}_{\textit{ij}}\right )}\Delta f/N_{\theta }, \\[3pt] \hat f_i=( f_{i-1}+ f_i)/2, \end{cases} \end{equation}
with
$N_{\!f}$
,
$N_{\theta }$
denoting the discretisation points in frequency and direction, respectively, and
$\Delta f$
,
$\Delta \theta$
being the uniform intervals. The cutoff range for the directional spectrum
$S(f,\theta )$
is
$f_i\in [0.5f_{\!p},\, 3.5f_{\!p}]$
and
$\theta _j\in [-\pi ,\,\pi ]$
. Here,
$a_{\textit{ij}}$
,
$k_{\textit{ij}}$
and
$\phi _{\textit{ij}}$
denote the amplitude, wavenumber and random phase of the component harmonics, respectively, and
$R_{\textit{ij}}$
denotes a random number in
$[0, 1]$
with a uniform probability distribution.
Table 1. Incident wave conditions and key non-dimensional parameters.
The configurations of incident wave fields are listed in table 1, together with non-dimensional parameters, wave steepness
$\varepsilon \equiv \sqrt {2}k_p\sigma$
and relative water depth
$\mu \equiv k_p h$
, where
$\sigma$
denotes the standard deviation of the measured free surface elevation (FSE) and
$k_p$
the spectral peak wavenumber. The subscripts
$0$
and
$f$
denote deeper-water and shallower-water quantities, respectively. In all cases, the waves generated offshore are of relatively mild nonlinearity. Three peak periods
$T_p=\{1.2,\, 1.6,\, 2.0\}$
s were examined, corresponding to weak, intermediate and strong non-equilibrium scenarios, respectively. Here, the cases with
$T_p=1.6$
and 2.0 s are shown. The peakedness parameter
$\gamma =3.3$
was maintained constant across all cases. Three directional spreading conditions
$s_{\textit{max}}=\{10, \, 35, \, 85\}$
were tested, in which 95 % of the integrated spectral energy of the total energy is concentrated within the ranges
$[-0.39, 0.39]\pi$
,
$[-0.22, 0.22]\pi$
and
$[-0.14, 0.14]\pi$
, respectively. The B2 and B4 cases were tested with three oblique incidence conditions,
$\theta _{\textit{inc}}=\pi /12$
,
$\theta _{\textit{inc}}=\pi /6$
and
$\theta _{\textit{inc}}=\pi /4$
. Note that the wave gauges are shifted from
$y=20$
to
$y=24$
m in the cases with
$\theta _{\textit{inc}}=\pi /4$
to capture wave evolution. Each experimental run lasts 6 minutes for wave generation and data acquisition, in order to limit energy accumulation due to long-wave reflection. To ensure statistical stability, several runs with different random phase seeds were performed such that the total sample duration exceeded 5000
$T_p$
for each case.
3. Results and discussion
To verify the quality of directional wave field generation in the offshore area and to show the spectral evolution as waves propagate over the bar, two wave gauge arrays were deployed in the wave tank. Figure 2(a.i–d.i) shows the target spectra of the normal incidence B1–B4 cases with
$\theta _{\textit{inc}}=0$
and four values of
$s_{\textit{max}}$
, the corresponding spectra measured offshore (a.ii–d.ii), and on top of the bar (a.iii–d.iii). The directional estimation is achieved by using a classical iterative maximum likelihood approach. In figure 2, the measured and target spectra are in good agreement offshore, indicating that the directional wave fields were generated properly. At the top of the bar, the second-order harmonics are excited significantly after shoaling, consistent with the observations in long-crested wave scenarios. In figure 2, no evident transverse waves (propagating with
$\theta \approx \pm \pi /2$
) induced by sidewall reflection are present in the current colour scale, indicating the transverse waves are of amplitudes lower than those of the primary harmonics by at least two orders of magnitude. The wave reflection by the end wall is also small and can thus be ignored. Moreover, it is noticed that the enhancement of the second-order harmonics slightly decreases as
$s_{\textit{max}}$
decreases. The same good wavemaking quality also applies to other cases with normal/oblique incidence.
Figure 2. Target directional spectra for the normal incident cases with
$s_{\textit{max}}=10,\, 35,\, 85$
and
$\infty$
in panels (a.i–b.i); the corresponding experimental spectra measured offshore (a.ii–d.ii) and on top of the bar (a.iii–d.iii).
Following previous works (see Onorato, Osborne & Serio Reference Onorato, Osborne and Serio2005; Trulsen et al. Reference Trulsen, Raustøl, Jorde and Rye2020; Li et al. Reference Li, Draycott, Zheng, Lin, Adcock and van den Bremer2021b
, for example), we focus on the evolution of third- and fourth-order statistical moments, skewness and kurtosis, of the FSE, which characterise the non-Gaussianity of sea states. Skewness,
$\lambda _3(\eta )\equiv \langle (\eta -\langle \eta \rangle )^3\rangle /\sigma ^3$
, with
$\langle \boldsymbol{\cdot }\rangle$
being an averaging operator, measures the asymmetry of the wave profile in the vertical direction. Kurtosis,
$\lambda _4(\eta )\equiv \langle (\eta -\langle \eta \rangle )^4\rangle /\sigma ^4$
, serves as a proxy of rogue wave intensity. In addition, the asymmetry parameter,
$\lambda _3[\mathcal{H}(\eta )]$
, measures the horizontal asymmetry of the wave profile, with
$\mathcal{H}(\boldsymbol{\cdot })$
denoting Hilbert transform. In the following, the spatial evolutions of the statistical parameters are discussed, including the normalised significant wave height, asymmetry parameter, skewness and the net change of kurtosis
$\Delta \lambda _4$
, defined by subtracting the mean value of
$\lambda _4$
prior to the shoal
\begin{equation} \Delta \lambda _4(x) = \lambda _4(x) \; - \langle \lambda _4\rangle |_{0 \leq x \leq 15\,\text{m}}. \end{equation}
Using
$\Delta \lambda _4$
, we can isolate depth-induced effects from small influences arising from different incident sea-state conditions, thereby focusing on the relative influence of shoaling across different wave conditions.
Figure 3 shows the spatial evolution of the statistical parameters in cases A1–A4 with different directional spreadings. Figure 3(a) shows that the normalised significant wave height is slightly increased over the bar due to shoaling. Figures 3(b) and 3(c) indicate that the NED effects result in increased asymmetry of the mean wave profile in both horizontal and vertical directions. In figure 3(d), kurtosis is moderately enhanced over the bar, implying higher rogue wave probability. These observations are in line with those reported in unidirectional wave studies (see Lawrence et al. Reference Lawrence, Trulsen and Gramstad2021b
; Zhang et al. Reference Zhang, Benoit, Kimmoun, Chabchoub and Hsu2019; Zheng et al. Reference Zheng, Lin, Li, Adcock, Li and van den Bremer2020, for instance). Figure 4 shows the statistical parameter evolution in the normal incident B1–B4 cases with varying
$s_{\textit{max}}$
. The evolution trend is very similar to that of cases A1–A4, except for a more pronounced NED response, i.e. higher peaks of skewness and kurtosis due to higher steepness atop the shoal
$\varepsilon _{\!f}$
and smaller relative water depth
$\mu _{\!f}$
(see table 1).
Figure 3. Spatial evolution of normalised significant wave height (a), asymmetry parameter (b), skewness (c) and net change of kurtosis (d) of cases A1–A4 (all with normal incidence,
$\theta _{\textit{inc}}=0$
).
Figure 4. Same as figure 3, but for normal incident directional wave cases (
$\theta _{\textit{inc}}=0$
) B1–B4.
Combining figures 3 and 4, we conjecture that directional spreading induces only minor changes in the statistical parameters, especially on the top of the bar, where waves are out of equilibrium. The normalised significant wave height (
$H_s / H_{s,0}$
) is slightly disturbed by the directional spread in the shoaling zone, while it is strongly disturbed in the de-shoaling zone, which is presumably due to wave reflection and refraction. The skewness and the asymmetry parameter are almost unchanged for different
$s_{\textit{max}}$
because of the similar level of wave nonlinearity (steepness) over the bar. Only minor differences are seen over the downslope area, and in the deeper flat region after it. Meanwhile, the net change of kurtosis is mildly reduced for broader directional spreading. In the B4 case with
$s_{\textit{max}}=\infty$
and
$\theta _{\textit{inc}}=0$
, the total kurtosis maximum value achieves 4.61 over the bar, while in the B1 case with
$s_{\textit{max}}=10$
and
$\theta _{\textit{inc}}=0$
, a comparable value, 4.44 is achieved. Both indicate a strongly non-Gaussian sea state resulting from non-equilibrium wave evolution, and a much higher probability of rogue waves than that expected in a Gaussian sea state.
Figure 5(a–d) shows the evolution of statistical parameters for the oblique incident unidirectional wave B4 cases, with
$s_{\textit{max}}=\infty$
and varying
$\theta _{\textit{inc}}$
. It is worth mentioning that in panel (a), the normalised significant wave height decreases dramatically after the downslope for the case with
$\theta _{\textit{inc}}=\pi /4$
. This is due to a limitation of the experimental facility: the wavemaking paddles are installed on one side of the wave tank, thus leaving the last four wave gauges outside the effective experimental zone. Despite the decrease in significant wave height, this case is included for discussion anyway, as the non-Gaussian behaviour on top of the bar is of more interest and is not affected by the facility’s limitation. From figure 5(c–d), we notice that, not only the maximum values, but also the locations where the maxima of skewness and kurtosis are achieved, vary considerably with
$\theta _{\textit{inc}}$
. We attribute this to the effective slope gradient, which is crucial for the wave non-equilibrium evolution. The effective slope, defined as the ratio of the bar height and horizontal upslope length in the direction of wave propagation, reads
$1/4.86$
,
$1/5.03$
,
$1/5.61$
and
$1/6.87$
for
$\theta _{\textit{inc}}=0$
,
$\pi /12$
,
$\pi /6$
and
$\pi /4$
, respectively.
Figure 5. Same as figure 3, but for oblique incident unidirectional cases (
$s_{\textit{max}}=\infty$
).
Figure 6(a)–(d) shows the evolution of statistical parameters in cases that comprise both directionality and oblique incidence. In this case, as the wave fields are of a relatively broad band in directional spreading, wave energy can be transmitted to a broader range in the wave tank; the significant wave height in the case with
$\theta _{\textit{inc}}=\pi /4$
does not reduce as significantly as in figure 5(a). The evolution trends of the asymmetry, skewness and kurtosis parameters in figure 6(b)–(d), are very similar to those observed in figure 5(b)–(d); their maximum values increase with increasing effective slopes, and the locations where the maxima are achieved shift towards downstream. Inversely, the out-of-equilibrium sea states adapt faster to the new water depth when the effective slope is milder.
Figure 6. Same as figure 3, but for oblique incident directional cases (
$s_{\textit{max}}=35$
).
To further illustrate the role played by the directionality and the oblique incidence on the non-equilibrium wave dynamics, in figure 7, the maximum values of skewness and kurtosis in the B1–B4 cases are extracted and plotted as functions of
$s_{\textit{max}}$
and
$\theta _{\textit{inc}}$
. It is clearly shown in figures 7(a.i) and 7(a.ii) that increasing the directional spreading
$s_{\textit{max}}$
results in only approximately 10 % lower skewness and kurtosis than in the unidirectional case. Our result is in line with the observations of Tang et al. (Reference Tang, Moss, Draycott, Bingham, van den Bremer, Li and Adcock2023), and in contrast to those of Lyu et al. (Reference Lyu, Mori and Kashima2023). The main reason is that, in the latter work, the relative water depth of the shallower region is around
$\mu _{\!f} = 1.1$
. Although it falls within the typical NED range
$0.5 \leq \mu _{\!f} \leq 1.3$
(Trulsen et al. Reference Trulsen, Raustøl, Jorde and Rye2020; Zhang et al. Reference Zhang, Ma, Tan, Dong and Benoit2023), it corresponds to weakly non-equilibrium wave evolution. Thus, the effect of directionality on the non-equilibrium response (e.g. the skewness and net change in kurtosis) is weak and comparable to the statistical error atop the bar, and stronger downslope. In figures 7(b) and 7(c), the effects of oblique incidence, which have been rarely discussed, are shown. We find that the incident angle plays an important role in the magnitude of the non-equilibrium wave response. This is attributed to the effective bottom slope. However, it should be noted that the wave gauge locations were chosen to capture the maximum skewness and kurtosis for the normal incident case, but are likely not optimal for cases with oblique incidence. Therefore, the maximum values of skewness and kurtosis could be underestimated to some extent. Whereas, from the relatively smooth evolution of skewness in figures 5(c) and 6(c), we believe that the conclusion will not be overturned when the ‘actual’ maximum values of skewness and kurtosis are considered. Further investigations of oblique incident cases with more wave gauges over the bar, using numerical simulations and statistical distributions, will be discussed in a subsequent paper.
Figure 7. Maximum values of
$\lambda _3$
and
$\Delta \lambda _4$
as functions of
$\theta _{\textit{inc}}$
and
$s_{\textit{max}}$
, in normal incident directional wave cases (a), oblique incident unidirectional wave cases (b) and oblique incident directional wave cases with
$s_{\textit{max}}=35$
(c).
4. Conclusions
The present study provides a comprehensive experimental investigation into the effect of wave directionality on extreme wave formation during nonlinear shoaling, focusing on the NED induced by rapid depth changes. Unlike previous research, which predominantly explored unidirectional irregular waves or directional waves with small directional spread, this work systematically examines multidirectional wave fields using a large-scale wave tank over realistic steep smooth slopes. An oblique incident angle
$\theta _{\textit{inc}}$
was chosen up to
$\pi /4$
(the limit of our experimental set-up), and a wide range of directional spreadings from
$s_{\textit{max}}=10$
to a unidirectional condition
$s_{\textit{max}} = \infty$
have been tested. Our results indicate that directional spreading has a minor impact on reducing statistical moments such as skewness and kurtosis for a relatively steep slope, contrasting with previous numerical studies that suggest evident suppression of rogue wave formation due to energy dispersion with directionality.
In contrast to the literature’s limited exploration of oblique incidence, the present study highlights the significant role of the incidence direction in NED, driven by the effective bottom slope. This finding extends previous works, which focused on normal incidence, by showing that obliqueness significantly suppresses non-Gaussian behaviour and rogue wave formation. These experimental results, using a Mitsuyasu-type directional spreading function and varying incidence angles, offer a more nuanced understanding than the contradictory numerical simulations, suggesting that future research should prioritise experimental validation and finer resolution studies to clarify this dynamics.
Acknowledgements
The authors wish to acknowledge Dr Z. Guanting from NMEMC and Dr T. Ting from DUT for their technical help during the experimental campaign.
Declaration of interests
The authors report no conflicts of interest.
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