We consider the generation of extreme waves of flexible features in deep-water surface gravity waves, in the context of the fully nonlinear wave model, i.e. Euler equations of the free surface. This is achieved through a recently developed GenWave approach, coupled with the high-order spectral method to solve the Euler equations. Through an iterative refinement procedure, GenWave progressively refines the initial phases associated with a given wave spectrum, which leads to extreme-wave generation at specified design time and location, with their probability of exceedance quantified. In this study, we consider two types of extreme waves, enabled by the flexibility of GenWave. The first type is a spatially and temporally localised rogue wave, and second is a spatially localised extreme-wave group propagating at a constant speed. The two generated extreme-wave profiles are compared with their prototypes – the Peregrine soliton and regular soliton (or bright soliton) in the context of the nonlinear Schrödinger equation (NLS), with reasonable agreement. For the persistent group, comparison with linear evolution and the NLS identifies nonlinear persistence, faster propagation and bound-wave content beyond the leading-order NLS description.

We formulate a model of the two-way interactions between surface gravity waves and ocean currents. The model couples the transport of wave action in the four-dimensional (horizontal) position–wavevector phase space with the Craik–Leibovich system for the currents. Coupling is via the Doppler shift in the dispersion relation governing action transport, and wave pseudomomentum in the Craik–Leibovich system. The velocity in the Doppler shift is a vertical integral of the Lagrangian mean velocity of the currents, with a weight that is consistent with the vertical structure of the pseudomomentum. This consistency ensures conservation of momentum and energy in the coupled wave–current system. The conservation properties of the wave–current model stem from an underlying variational structure. We derive this structure from that of the rotating Euler equations for an incompressible fluid with free surface by introducing a Lagrangian wave–mean decomposition, making simplifying approximations, and Whitham averaging. The wave–mean decomposition is constructed such that the Lagrangian mean velocity is non-divergent and satisfies a rigid-lid boundary condition. We apply the wave–current model to the problem of generation of inertial oscillations by surface waves originally considered by Hasselmann.

We investigate the evolution of Peregrine breather (PB) rogue waves in narrow-banded random background wave fields using a fully nonlinear wave model. The influence of the peak wavenumber, the significant wave height, and the spectral width of the background field on the maximum crest height, the timing of rogue wave occurrence, and the spatiotemporal distribution of rogue waves is analysed. The results show that when the background peak wavenumber matches that of the PB carrier, nonlinear focusing is enhanced and the phase distribution is modified. The maximum crest height increases by 4.4 % on average, the mean time of first occurrence decreases by 15.9 %, and rogue wave activity extends over a larger spatiotemporal domain. These effects weaken as the wavenumber detuning increases. Increasing the significant wave height of the background field enhances nonlinear interactions and leads to more pronounced changes in the relative phases among wave components, resulting in broader distributions of rogue waves. Over the range considered, the ensemble-averaged maximum crest height increases by up to 5.4 %, while the mean time of first occurrence decreases by up to 40.4 %. The influence of spectral width depends on wavenumber detuning: spectral broadening has little effect when the two peak wavenumbers coincide, but gradually reduces the maximum crest height when they are separated. These findings demonstrate that narrow-banded random background wave fields can significantly modulate the evolution of PB rogue waves through combined effects on energy distribution and phase relationships, providing new insights into rogue wave dynamics in realistic ocean environments.

A pressure disturbance in steady rectilinear translation applied at a capillary–gravity interface radiates waves only above a minimum speed $c_{\textit{min}}$, where it experiences a wave resistance; below this threshold it generates no waves, and the response is merely a localised deformation of the interface around the source, free of wave resistance. Modulating the source amplitude at a frequency $\varOmega$ shifts the forcing to the sideband pair $kV\pm \varOmega$, where $k$ is the wavenumber and $V$ is the translation speed, opening resonance conditions absent from the steady problem; radiation then sets in below the steady threshold $c_{\textit{min}}$. Here, we derive the mean wave resistance $\langle R\rangle$ of a line source in closed form, and show it to be strictly positive at every non-zero speed $V$: the drag-free below-threshold regime disappears for any $\varOmega \neq 0$. Crossing the radiation transitions – where the group velocity matches the source speed – produces integrable resonant peaks in $\langle R\rangle$, whose exponents we derive, including at the cusp where two transition branches merge. Below threshold, where a steadily moving source feels no drag, $\langle R\rangle$ rises from zero as $V\varOmega ^{5}$. Throughout, surface tension is essential to $\langle R\rangle$: because it controls the short, dominant capillary-scale waves, discarding it misrepresents $\langle R\rangle$ over almost the entire speed–frequency plane.

This paper investigates the temporal, spatiotemporal and nonlinear instability of a weakly viscoelastic liquid film flowing down a vertical cylindrical fibre. The fluid is described by the Walters’ liquid B model, and the analysis combines long-wave asymptotic modelling, a long-wave-weighted Orr–Sommerfeld stability analysis and nonlinear numerical simulations. Linear stability analysis reveals that while viscoelasticity is temporally destabilising, it significantly expands the domain of absolute instability in parameter space, facilitating the upstream propagation of disturbances. In the nonlinear regime, viscoelasticity exhibits a distinct dual effect: it monotonically suppresses the saturation amplitude $r_{\textit{max}}$ with increasing elasticity $M$, whilst accelerating the onset of finite-time singularity during rupture dominated by absolute instability at larger elasticity parameters. The rupture dynamics obeys a self-similar scaling $h_{\textit{min}} \sim (t_c - t)^m$. Driven by gravity-induced axial stretching, the system deviates from classical pure-capillary pinch-off, shifting the Newtonian baseline to a friction-dominated drainage regime with $m \approx 0.50$ and an asymmetric spatial collapse. Based on this gravitationally modified baseline, increased elasticity further reduces the temporal exponent $m$, indicating that elastic effects accelerate the local thinning dynamics while preserving the main upstream similarity structure. Travelling-wave solutions further reveal elasticity-induced changes in the phase-space topology of the saturated states: in the solitary-wave regime, increasing elasticity promotes a transition from near-homoclinic trajectories to stable limit cycles, whereas in the parasitic-wave regime it produces more complex homoclinic-like trajectories associated with satellite droplets in physical space.

The present study examines the evolution of the phase velocity of surface gravity waves propagating against an opposing current in the vicinity of wave-blocking conditions, defined as flow regimes in which the wave-energy flux can no longer propagate upstream relative to an adverse background current. We adopted a novel experimental technique by deploying passive devices (‘flow conditioners’), which offer robust and repeatable control over the current profile to represent several shear structures. The classical evaluation of phase velocity through cross-correlation of wave-gauge signals is used as a reference for testing a novel experimental method based on high-resolution velocity fields obtained by particle image velocimetry. The two experimental estimates show generally good agreement. The experimental data are compared with theoretical predictions obtained from eigenvalue-based approximation schemes for the governing problem and with the full numerical solution of the classical Rayleigh eigenvalue problem. Second-order approximation models give the best theoretical estimates, while results from the full numerical solution are slightly more dispersed. The hypothesis of a uniform current largely underestimates the experimental values and breaks down systematically. Our results indicate that bulk current strength alone is insufficient to characterise the wave kinematics and that the vertical distribution of the adverse current within the wave-active layer exerts a significant additional influence. This helps explain why shear-weighted dispersion approximations perform well, contrary to the uniform-current assumption. This work is particularly relevant for the full-scale characterisation and prediction of compound flooding events arising from the combination of river flood discharges with intense marine storm conditions, offering quantitative considerations on the best predictive models that can be used in practical situations.

The nonlinear hydroelastic response of a floating ice sheet to a moving load is investigated in two dimensions through a systematic reduction, spanning from the fully nonlinear Euler equations to a low-dimensional dynamical system. By incorporating linear damping to model dissipation, we demonstrate that the steady-state response exhibits an S-shaped bifurcation structure, giving rise to bistability and hysteretic transitions. By projecting the forced–dissipative Whitham equation onto a two-dimensional manifold, we establish a formal analogy between this hydroelastic wave problem and the forced–damped Duffing oscillator – the prototypical model for such folding bifurcations. This reduction reveals that the spatial lag between the load and the peak ice deflection manifests as a temporal phase shift, which the system adjusts spontaneously to satisfy a global power-balance condition. A critical ratio of damping to load magnitude is identified as the governing threshold for the fold bifurcation, furnishing a predictive criterion for the onset of nonlinear jump phenomena in the ice-sheet response.

We investigate how negatively buoyant inertial spheres settle in progressive surface gravity waves, focusing on the phase-dependent modulation of the particle–fluid relative motion. Starting from a Maxey–Riley framework with a quasi-steady nonlinear-drag closure, we formulate a phase-resolved equation of motion in terms of the slip velocity and obtain a perturbation solution in the limit of small wave amplitude. The leading state corresponds to the quiescent-water terminal slip, the first-order response yields a dominant harmonic modulation induced by the wave kinematics and second-order terms introduce higher harmonics arising from higher-order interactions. A key element is that, under nonlinear drag with finite particle Reynolds number, the particle relaxation time and hence the Stokes number depend on the instantaneous slip. This dependence is retained consistently in the perturbation expansion, allowing the vertical-slip phase at finite Stokes number to be predicted without the additional ordering adopted in previous work. We test predictions using wave flume experiments with simultaneous particle tracking velocimetry/particle image velocimetry, which provide slip estimates along particle trajectories and phase-binned ensemble statistics. The analysis first assesses whether particles attain a terminal settling regime within the observation window, then quantifies the phase-dependent oscillation of the horizontal and vertical slip. For cases that attain the terminal regime, the perturbation solution reproduces the amplitude and the phase of the dominant slip oscillation. However, cases that remain in a wave-acceleration-dominated relaxation regime show settling hindrance near the water surface; the wave-averaged vertical slip approaches the quiescent terminal velocity with depth, and no clear settling enhancement is observed.

The generation of renewable power from ocean waves at a commercial scale requires arrays of wave energy converters (WECs) to be designed for efficient power capture over broad frequency bands and a wide range of incident wave directions. Here, array designs are developed in a three-dimensional model based on linear potential-flow theory, by arranging heaving-buoy-type WECs into rows, and spatially grading the resonant properties of successive rows via linear spring–damper power take-off (PTO) mechanisms. The grading leverages theories for rainbow reflection and rainbow absorption, and involves analyses of Bloch waves and pole–zero pairs in complex frequency space to inform the array layout and choice of PTO parameters. The spacing within an array alters the resonant properties of the rows and the behaviour of higher-order passbands, which has implications for the efficiency of an array. Strategies are developed to grade arrays composed of rows with infinitely many WECs, for efficient, broadband absorption over a broad target frequency range, and average absorptions exceeding $90\,\%$ are achieved over a wide range of wave directions. The strategies are applied to finite arrays, which demonstrate broadband absorption in directionally spread irregular sea states, and are shown to outperform the absorption of uniform arrays across a wide range of peak periods.

Scalars in the upper ocean are transported by surface gravity waves. Their net transport is often attributed to the advective Stokes drift, however, this assumption neglects potential effects of waves on diffusion. In this study, we conduct a fully Lagrangian analysis of the advection–diffusion equation to show that non-breaking waves influence diffusion in two ways: (i) the unsteady strain from the wave orbital motion causes diffusion to vary as a function of wave phase, maximising horizontal diffusion under the crests and minimising it under the troughs, and conversely for vertical diffusion, (ii) the Lagrangian mean water level induces a mean vertical strain which yields a net enhancement of the horizontal diffusion. These effects are driven by the normal strain of the wave and therefore represent a mechanism distinct from the drift-induced shear dispersion, which we refer to as ‘distortion’. This means that the net transport of diffusive scalars under surface waves differs from that under only the wave-averaged Stokes drift, and similarly that the net transport under a standing wave differs from diffusion in a quiescent fluid. Asymptotic expressions for the diffusivity distortion are derived for monochromatic progressive waves, standing waves and a wave spectrum, which are then validated with numerical simulations. Finally, we also investigate the influence of the Lagrangian mean water level on buoyant particle distributions, showing how it can skew the interpretation of vertical transport models commonly used for studying ocean microplastics.

Flexure of Antarctic ice shelves under excitation from long ocean waves induces mechanical ice-shelf stresses that amplify fractures and, hence, contribute to calving events. Here, a solution method is developed for a hydroelastic mathematical model of wave-induced ice-shelf flexure, based on the conventional theory of a Kirchoff–Love plate floating on shallow water under linearised conditions, but allowing wave forcing of ice shelves with variations in both horizontal dimensions, and where the ice shelves are of arbitrary shape, including non-uniform thickness. The method uses finite elements specifically designed for the high-order hydroelastic system, and a Dirichlet-to-Neumann map to bound the computational domain in the open ocean. Following verification, the method is used to conduct novel studies on how the ice-shelf deflection is affected by the ice-shelf shape, the incident wave direction and the proportion of the shelf that is grounded. The efficiency of the method allows the studies to be conducted over a broad frequency range, such that resonant responses are identified. The method is also used to model the response of the Drygalski Ice Tongue to long-wave forcing.

We study large-amplitude travelling gravity waves on the free surface of an infinitely deep fluid with a shear layer of constant vorticity extending to a finite depth. A boundary-integral formulation is developed by using conformal mapping to describe the two free boundaries of the problem and is solved numerically using a pseudospectral method. The linear wavespeed $c$ satisfies a cubic dispersion relation, which essentially arises from the combination of two gravity modes and a shear mode. For certain ranges of parameters, the collision of these modes leads to linear instability. The nonlinear solutions corresponding to one of the three values of $c$ were studied by Milinazzo & Saffman (1990 J. Fluid Mech. 216, 93–101). These solutions are dominated by the surface gravity mode; however, the solutions associated with the remaining two values of $c$ were overlooked. We study these new solutions in detail and find distinct properties from those calculated by Milinazzo & Saffman, such as the emergence of large-amplitude waves on the vorticity interface and changes in the relative phase between the two free boundaries. Moreover, bifurcations between shear and gravity modes are shown to emerge between solution branches of disparate wavenumbers under resonance conditions, which results in significant qualitative changes in the solution profiles across very small regions of parameter space.

Surface-swimming animals generate capillary-gravity waves whose patterns reflect the unsteady, sinuous nature of their trajectories. For uniform circular motion, cusp formation and wavefront folding have been observed at supercritical speeds. Here, we show that slaloming trajectories at supercritical speeds produce wavefront singularities of higher codimension than previously reported, namely a swallowtail catastrophe. Combining observations of surface-swimming snakes with numerical simulations of unsteady wakes, we identify a swallowtail catastrophe embedded in the wave envelope. This is explained by an integral formulation with a delay term accounting for the timing of wave emission along the trajectory, enabling a description of singularity formation and degeneracy within catastrophe theory. These results provide a general framework for predicting the emergence of caustics and cusps in wavefronts generated by unsteady motions.

An improved model for two-dimensional wind-driven water waves is developed by incorporating both wind pressure and tangential viscous stress. These aerodynamic forces are determined via numerical simulations of turbulent flow over a slowly moving, wavy boundary, and are validated against established experimental and numerical data. The model accounts for wind input and viscous dissipation simultaneously, bypassing the oversimplifications of irrotational and inviscid flow assumptions. Utilising a multi-scale asymptotic expansion, we derive a theoretical solution for the wave profile and introduce a complex, time-dependent parameter that quantifies the effects of wind on wave evolution. Its real part governs the wave growth or decay rate, while the imaginary part modulates the phase shift. This formulation explicitly captures the often-overlooked role of tangential viscous stress. Furthermore, unsteady numerical simulations using a conformal mapping technique confirm the theoretical growth rates. Finally, the theoretical solution enables parametrising wave skewness and asymmetry. Incorporating an empirical relationship between wave slope and inverse wave age, the model yields predictions that closely align with laboratory and field observations and provides an improved description of nonlinear wave profiles.

Axisymmetric waves occur across a wide range of scales. This study analyses large-scale (with a jet height of up to 6 m) gravity-dominated axisymmetric waves (Bond number $ \textit{Bo}=\textit {O}(10^{5})$ and Weber number $\textit{We}=\textit {O}(10^{4}{-}10^{6})$), for which surface-tension effects are negligible. Our aim is to clarify the dynamics of highly nonlinear axisymmetric jet formation, cavity collapse and the consequent generation of secondary jets. The newly developed three-dimensional framework OceanSPHysics3D, combining unsteady potential flow with SPH, provides an integrated platform capable of handling multidirectional and axisymmetric waves, thereby enabling the full simulation of axisymmetric jet initiation and collapse. The computed free-surface elevations and jet evolution agree well with the experiments of McAllister et al. (2022 J. Fluid Mech. 935, A5) and an analytical jet-tip-angle formulation of Longuet-Higgins (1983 J. Fluid Mech. 127, 103–121). The simulations elucidate how the falling primary jet induces a secondary jet. The mechanisms forming the pre-jet trough and post-jet cavity are fundamentally different. The pre-jet trough arises geometrically from directional focusing of the constituent waves, yielding a self-similar shape when appropriately scaled. In contrast, the post-jet cavity is formed inertially by the falling continuous jet, and lacks both spatial and temporal self-similarity. Its collapse also differs: the cavity pinches off at the neck to generate upward and downward secondary jets, with local accelerations reaching $150g$. The primary-jet scale governs the ensuing secondary-jet dynamics, including vortex-ring formation and strong vertical mixing. These findings illustrate the complexity of axisymmetric jet dynamics, and demonstrate the ability of the present framework to reproduce the key coupled processes in such extreme free-surface events.

Nonlinear hydrodynamic effects influence the power capture of resonant wave energy converters (WECs), particularly within the frequency band where these devices are designed to operate. In this study, wave flume experiments are conducted on a semi-immersed spherical model with prescribed heave motions, emulating the response to an incident broad-banded wave group. Several body motion amplitudes (up to a maximum displacement of greater than half the radius) are tested at different peak frequencies, revealing progressive reductions in both the first-harmonic radiation forces and radiated wave elevations, per unit motion amplitude, as the motion amplitude is increased. A simple partially nonlinear potential flow model incorporating instantaneous ‘volume flow’ corrections shows very good agreement with the measured radiated waves, capturing the dominant nonlinear trends. An experimental decomposition of the radiated field into channel modes is also performed, based on linear theory while accounting for linear viscous dissipation. This is then used to calculate the power in the radiated field compared to the input power from the moving body over a frequency band of interest. The model and experiments show a consistent, monotonic decrease in normalised radiated power as the amplitude of body motion increases. Overall, the findings indicate that linear theory is remarkably accurate even for quite large body motions. The most important nonlinear corrections in the first-harmonic frequency range arise from third-order potential flow interactions, fundamentally caused by the spherical geometry, and demonstrate the potential utility of volume flow models for assessing WEC radiation.

This paper considers the theoretical operation of a novel wall-mounted oscillating water column wave energy device designed to be resonant over a broad range of frequencies. A curved duct with a submerged opening in a vertical wall is partitioned into a number of separate narrow channels of uniform width using thin annular baffles. Each channel connects the submerged opening to its own internal free surface whose rise and fall drives the air enclosed above through a Wells-type turbine through which energy is harvested. The different channel lengths within the duct encourages, when subjected to forcing from incident waves, resonance across a range of frequencies. The two-dimensional problem described in this work is analysed using classical linearised water wave theory and the geometric complexity of the partitioned duct is simplified by homogenisation. This allows the solution to the water wave problem to be reduced to a scalar integral equation whose solution is approximated using a standard numerical method. It is shown that it is possible, even with the most basic choice of turbine power take-off strategies, to achieve efficiencies close to 100 % across much of the range of frequencies defined by the resonance associated with the longest and shortest channels of the device. Viscous damping within the channels is shown to have a negligible effect for practical configurations.

The shape of depth-limited overturning waves strongly influences turbulence and sediment transport in the surf zone. Bathymetric slope $s$ and initial wave height $H_0/h_0$ are key parameters that govern wave overturning shape, yet their effect has not been quantified. We use a two-dimensional fully nonlinear potential flow model to study shoaling and overturning solitary waves across a range of $s$ and $H_0/h_0$ representing spilling and plunging waves. Near-constant domain volume and energy demonstrate the model’s accuracy. Gentler slopes yield smaller, more inclined overturns with thinner jets, whereas steeper slopes produce larger, more horizontally inclined overturns with thicker jets. Wave breaking and overturning parameters such as breaker depth index, overturn area, jet area, aspect ratio and overturn angle are examined. Non-dimensional overturn and jet areas increase with steeper slopes and smaller $H_0/h_0$. Aspect ratios increase similarly but more slowly, while overturn angles shift from inclined to horizontal. A scaling parameter $s/(H_0/h_0)^{1/4}$ scales these wave breaking and overturning parameters well. During shoaling, front-face wave steepening occurs more rapidly for larger slopes and smaller $H_0/h_0$. Wave overturn shape is linked to wave steepening rate, which also scales with slope and $H_0/h_0$. Empirical expressions for overturning parameters and wave steepening rate are found. The jet potential energy, related to the wave dissipation at jet impact, is a strong function of jet area, linking overturning wave shape to the turbulence that drives sediment suspension and transport.

This study investigates a novel class of flexible plate metamaterials (FPMMs), consisting of a periodic array of bottom-mounted flexible structures, designed for the sub-wavelength manipulation of surface gravity waves. A theoretical framework is developed by coupling the matched eigenfunction expansion method with a modal expansion technique, incorporating Bloch’s theorem to characterise the band structures and complex dispersion relations. Numerical analysis reveals that the structural flexibility facilitates the formation of low-frequency, broadband bandgaps, primarily driven by a local resonance (LR) mechanism. It is shown that wave propagation can be effectively suppressed within these bandgap regimes through the synergistic interaction between the incident wave field and the structural vibration modes. The theoretical predictions are validated through laboratory experiments conducted in a wave flume, where substantial wave attenuation was observed under resonant conditions. Notably, the LR-induced bandgaps occur at significantly lower frequencies and exhibit broader bandwidths compared to those arising from traditional Bragg resonance. These findings demonstrate that the proposed FPMMs provide a tunable and effective mechanism for sub-wavelength wave attenuation, offering fundamental insights into localised hydrodynamic wave control.

This study investigates the oblique incident wave and the wave crossing in nonlinear wave evolution under variable directional spreading and bathymetry. Building on the depth-modified nonlinear Schrödinger (dNLS) model for a two-dimensional random wavefield, we incorporate wave refraction and symmetric crossing wave systems to numerically simulate realistic ocean conditions more accurately. Monte Carlo simulations are conducted to analyse wave spectrum, surface morphology and statistical moments (skewness and kurtosis). Results show that small incident angles have a very limited effect on extreme events and nonlinear interactions in a single system, but even small-angle wave crossing leads to an increase in the occurrence probability of freak waves from single system to multi-system. As the crossing angle increases, the enhanced dispersion suppresses transverse wave groupiness and reduces the likelihood of extreme events.