We introduce a new model equation for Stokes gravity waves based on conformal transformations of Euler's equations. The local version of the model equation is relevant for the dynamics of shallow water waves. It allows us to characterize the travelling periodic waves both in the case of smooth and peaked waves and to solve the existence problem exactly, albeit not in elementary functions. Spectral stability of smooth waves with respect to co-periodic perturbations is proven analytically based on the exact count of eigenvalues in a constrained spectral problem.

We develop a time-dependent conformal method to study the effect of viscosity on steep surface waves. When the effect of surface tension is included, numerical solutions are found that contain highly oscillatory parasitic capillary ripples. These small-amplitude ripples are associated with the high curvature at the crest of the underlying viscous-gravity wave, and display asymmetry about the wave crest. Previous inviscid studies of steep surface waves have calculated intricate bifurcation structures that appear for small surface tension. We show numerically that viscosity suppresses these. While the discrete solution branches still appear, they collapse to form a single smooth branch in the limit of small surface tension. These solutions are shown to be temporally stable, both to small superharmonic perturbations in a linear stability analysis, and to some larger amplitude perturbations in different initial-value problems. Our work provides a convenient method for the numerical computation and analysis of water waves with viscosity, without evaluating the free-boundary problem for the full Navier–Stokes equations, which becomes increasingly challenging at larger Reynolds numbers.

Gravito–capillary waves at free surfaces are ubiquitous in several natural and industrial processes involving quiescent liquid pools bounded by cylindrical walls. These waves emanate from the relaxation of initial interface distortions, which often take the form of a cavity (depression) centred on the symmetry axis of the container. The surface waves reflect from the container walls leading to a radially inward propagating wavetrain converging (focussing) onto the symmetry axis. Under the inviscid approximation and for sufficiently shallow cavities, the relaxation is well-described by the linearised potential-flow equations. Naturally, adding viscosity to such a system introduces viscous dissipation that enervates energy and dampens the oscillations at the symmetry axis. However, for viscous liquids and deeper cavities, these equations are qualitatively inaccurate. In this study, we decompose the initial localised interface distortion into several Bessel functions and study their time evolution governing the propagation of concentric gravito–capillary waves on a free surface. This is carried out for inviscid as well as viscous liquids. For a sufficiently deep cavity, the inward focussing of waves results in large interfacial oscillations at the axis, necessitating a second-order nonlinear theory. We demonstrate that this theory effectively models the interfacial behaviour and highlights the crucial role of nonlinearity near the symmetry axis. This is rationalised via demonstration of the contribution of bound wave components to the interface displacement at the symmetry axis Contrary to expectations, the addition of slight viscosity further intensifies the oscillations at the symmetry axis although the mechanism of wavetrain generation here is quite different compared with bubble bursting where such behaviour is well known (Duchemin et al., Phys. Fluids, vol. 14, issue 9, 2002, pp. 3000–3008). This finding underscores the limitations of the potential flow model and suggests avenues for more accurate modelling of such complex free-surface flows.

Nonlinear hydroelastic waves along a compressed ice sheet lying on top of a two-dimensional fluid of infinite depth are investigated. Based on a Hamiltonian formulation of this problem and by applying techniques from Hamiltonian perturbation theory, a Hamiltonian Dysthe equation is derived for the slowly varying envelope of modulated wavetrains. This derivation is further complicated here by the presence of cubic resonances for which a detailed analysis is given. A Birkhoff normal form transformation is introduced to eliminate non-resonant triads while accommodating resonant ones. It also provides a non-perturbative scheme to reconstruct the ice-sheet deformation from the wave envelope. Linear predictions on the modulational instability of Stokes waves in sea ice are established, and implications for the existence of solitary wave packets are discussed for a range of values of ice compression relative to ice bending. This Dysthe equation is solved numerically to test these predictions. Its numerical solutions are compared with direct simulations of the full Euler system, and very good agreement is observed.

Floating fluid-filled membrane breakwater (FFMB) is a temporary structure that can attenuate waves in the deep sea. In this paper the hydrodynamic performance of the FFMB is analysed by using the eigenfunction expansion boundary element method (EEBEM) and physical model experiments. A general motion equation is derived that considers both the dynamic tension and curvature of the membrane. Moreover, an integral expression for the dynamic tension is provided. On this basis, a linear model for solving wave–membrane interaction is established through the EEBEM. Newly designed physical experiments are performed to verify the model and elucidate the nonlinear characteristics of the FFMB. Following verification of the model, this paper investigates the effects of various structural parameters of the FFMB on the wave transmission coefficient, reflection coefficient, horizontal wave force, vertical wave force and dynamic tension. Furthermore, the interrelationship between the structural resonant response and the hydrodynamic performance is elucidated, and the optimal density and filling ratio of the FFMB for engineering applications are proposed. The results demonstrate that the numerical and experimental results are in good agreement, indicating that the model and the motion equation are both practical and highly accurate. By optimizing the structural parameters, the FFMB is capable of effectively attenuating waves within a specific frequency band, while minimizing the wave force.

Metastructures composed of a closely spaced plate array have been widely used in bespoke manipulation of waves in contexts of acoustics, electromagnetics, elasticity and water waves. This paper focuses on wave scattering by discrete plate array metastructures of arbitrary cross-sections, including isolated vertical metacylinders, periodic arrays and horizontal surface-piercing metacylinders. A suitable transform-based method has been applied to each problem to reduce the influence of barriers in a two-dimensional problem to a set of points in a one-dimensional wave equation wherein the solution is constructed using a corresponding Green's function. A key difference from the existing work is the use of an exact description of the plate array rather than an effective medium approximation, enabling the exploration of wave frequencies above resonance where homogenisation models fail but where the most intriguing physical findings are unravelled. The new findings are particularly notable for graded plate array metastructures that produce a dense spectrum of resonant frequencies, leading to broadband ‘rainbow reflection’ effects. This study provides new ideas for the design of structures for the bespoke control of waves with the potential for innovative solutions to coastal protection schemes or wave energy converters.

Depth-limited overturning wave shape affects water turbulence and sediment suspension. Experiments have shown that wind affects shoaling and overturning wave shape, with uncertain mechanism. Here, we study wind effects (given by the wind Reynolds number) on solitary wave shoaling and overturning with the two-phase direct numerical simulations model Basilisk run in two dimensions on steep bathymetry for fixed wave Reynolds number and Bond number. For all wind, the propagating solitary wave sheds a two-dimensional turbulent air wake and has nearly uniform speed with minimal wave energy changes over the rapidly varying bathymetry. Wave-face slope is influenced by wind, and shoaling wave shape changes are consistent with previous studies. As overturning jet impacts, wind-dependent differences in overturn shape are quantified. The non-dimensional breakpoint location and overturn area have similar wind dependence as previous experience, whereas the overturn aspect ratio has opposite wind dependence. During shoaling, the surface viscous stresses are negligible relative to pressure. Surface tension effects are also small but grow rapidly near overturning. In a wave frame of reference, surface pressure is low in the lee and contributes 2–5 % to the velocity potential rate of change in the surface dynamic boundary condition, which, integrated over time, changes the wave shape. Reasons why the overturn aspect ratio is different than in experiment and why a stronger simulated wind is required are explored. The dramatic wind effects on overturning jet area, and thus to the available overturn potential energy, make concrete the implications of wind-induced changes to wave shape.

This study investigates the influence of surface wave characteristics, specifically wave steepness and directional spreading, on intermittency in deep-water gravity wave turbulence through long-term numerical simulations of three-dimensional potential fully nonlinear periodic gravity waves. We conducted this investigation by estimating the scaling exponent of the surface elevation under different sea state conditions. With our numerical methods, we were able to evaluate the scaling exponents of the structure-function up to 12th order. The observed increased intermittency in directionally narrower sea states and in higher steepness conditions aligns with known effects of quasi-resonant wave–wave interactions and wave breaking. Comparative analyses reveal that both the conventional She–Leveque model and the multifractal models, also used to represent intermittency in wave turbulence of a different nature, exhibit a strong correlation in this study. This observation underscores the universality of intermittency phenomena within wave turbulence.

Even without breaking or wind influence, ocean surface waves are observed to produce turbulence in the water, possibly influencing ocean surface dynamics and air–sea interactions. Based on the water-side free-surface simulations, recent studies suggest that such turbulence is produced through the interaction between the waves and the near-surface Eulerian current associated with the viscous attenuation of waves. To clarify the dynamical role of the air–water interface in the turbulence production, the attenuating interfacial gravity waves were simulated directly using a newly developed two-phase wave-resolving numerical model. The air–water coupling enhanced the wave energy dissipation through the formation of a strong shear at the air-side viscous boundary layer. This led to an enhancement of the wave-to-current momentum transfer and the formation of the down-wave Eulerian mean sheared current, which is favourable for the CL2 instability responsible for the production of Langmuir circulations. As a result, the water-side turbulence grew stronger compared with the corresponding free surface (water-only) wave-resolving simulation. The evolution of the wave-averaged field was well reproduced with the Craik–Leibovich equation with the upper boundary condition provided with the virtual wave stress based on linear theory. The wave energy dissipation by air–water coupling plays a significant role in the quantitative understanding of the wave-induced turbulence at the laboratory and field scales.

The depth-integrated horizontal momentum equations and continuity equation are employed to develop a new model. The vertical velocity and pressure can be expressed exactly in terms of horizontal velocities and free-surface elevation, which are the only unknowns in the model. Dividing the water column into elements and approximating horizontal velocities using linear shape function in each element, a set of model equations for horizontal velocities at element nodes is derived by adopting the weighted residual method. These model equations can be applied for transient or steady free-surface flows by prescribing appropriate lateral boundary conditions and initial conditions. Here, only the wave–current–bathymetry interaction problems are investigated. Theoretical analyses are conducted to examine various linear wave properties of the new models, which outperform the Green–Naghdi-type models for the range of water depth to wavelength ratios and the Boussinesq-type models as they are capable of simulating vertically sheared currents. One-dimensional horizontal numerical models, using a finite-difference method, are applied to a wide range of wave–current–bathymetry problems. Numerical validations are performed for nonlinear Stokes wave and bichromatic wave group propagation in deep water, sideband instability, regular wave transformation over a submerged shoal and focusing wave group interacting with linearly sheared currents in deep water. Very good agreements are observed between numerical results and laboratory data. Lastly, numerical experiments of wave shoaling from deep to shallow water are conducted to further demonstrate the capability of the new model.

Arrays of heaving buoy type wave energy converters (WECs) are a promising contender to harness the renewable power of ocean waves on a commercial scale but require strategies to achieve efficient capture of wave energy over broad frequency bands for economic viability. A WEC-array design is proposed for absorption over a target frequency range in the two-dimensional water wave context by spatially grading the resonant properties of WECs via linear spring–damper power take-off mechanisms. The design is based on theories for rainbow reflection and rainbow absorption, which incorporate analyses based on Bloch wave modes and pole–zero pairs in complex frequency space. In contrast to previous applications of these theories, the influence of a higher-order passband and associated pole–zero pairs are shown to influence absorption at the high-frequency end of the target interval. The theories are used to inform initialisations for optimisation algorithms, and an optimised array of only five WECs is shown to give near-perfect absorption ($\geq$99 %) over the target interval. Broadband absorption is demonstrated when surge and pitch motions are released, for irregular sea states, and for incident wave packets in the time domain, where the time-domain responses are decomposed into Bloch modes to connect with the underlying theory.

The impact of shoaling on linear water waves is well known, but it has only been recently found to significantly amplify both the intensity and frequency of rogue waves in nonlinear irregular wave trains atop coastal shoals. At least qualitatively, this effect has been partially attributed to the ‘rapid’ nature of the shoaling process, i.e. shoaling occurs over a distance far shorter than that required for waves to modulate themselves and adapt to the reduced water depth. Through a theoretical model and highly accurate nonlinear simulations, we disentangle the respective effects of the length and angle of a shoal's slope. We investigate the effects of the shoaling process rapidness on the evolution of key statistical and spectral sea-state parameters. We let the wave field evolve over a slope with constant angle in all cases while we vary the slope length. Our results indicate that the non-equilibrium dynamics is not affected by the slope length, because further extending the slope length does not influence the magnitude of the statistical and spectral measures as long as the non-equilibrium dynamics dominates the wave evolution. Thus, the shoaling effect on rogue waves is deduced to be mainly driven by the slope magnitude rather than the slope length.

To study two-dimensional dispersive waves propagating through turbulent flows, a new and less restrictive fast waves approximation is proposed using a multiscale setting. In this ansatz, large and small scales of the turbulence are treated differently. Correlation lengths of the random small-scale turbulence components can be considered negligible in the wave packet propagating frame. Nevertheless, the large-scale flow can be relatively strong, to significantly impact wavenumbers along the propagating rays. New theoretical results, numerical tools and proxies are derived to describe ray and wave action distributions. All model parameters can be calibrated robustly from the large-scale flow component only. We illustrate our purpose with ocean surface gravity waves propagating in different types of surface currents. The multiscale solution is demonstrated to efficiently document wave trapping effects by intense jets.

Trapped surface waves have been observed in a swimming pool trapped by, and rotating around, the cores of vortices. To investigate this effect, we have numerically studied the free-surface response of a Lamb–Oseen vortex to small perturbations. The fluid has finite depth but is laterally unbounded. The numerical method used is spectrally accurate, and uses a novel non-reflecting buffer region to simulate a laterally unbounded fluid. While a variety of linear waves can arise in this flow, we focus here on surface gravity waves. We investigate the linear modes of the vortex as a function of the perturbation azimuthal mode number and the vortex rotation rate. We find that at low rotation rates, linear modes decay by radiating energy to the far field, while at higher rotation rates modes become nearly neutrally stable and trapped in the vicinity of the vortex. While trapped modes have previously been seen in shallow water surface waves due to small perturbations of a bathtub vortex, the situation considered here is qualitatively different owing to the lack of an inward flow and the dispersive nature of non-shallow-water waves. We also find that for slow vortex rotation rates, trapped waves propagate in the opposite direction to the vortex rotation, whereas, above a threshold rotation rate, waves corotate with the flow.

The particle trajectories in irrotational, incompressible and inviscid deep-water surface gravity waves are open, leading to a net drift in the direction of wave propagation commonly referred to as the Stokes drift, which is responsible for catalysing surface wave-induced mixing in the ocean and transporting marine debris. A balance between phase-averaged momentum density, kinetic energy density and vorticity for irrotational, monochromatic and spatially periodic two-dimensional water waves is derived by working directly within the Lagrangian reference frame, which tracks particle trajectories as a function of their labels and time. This balance should be expected as all three of these quantities are conserved following particles in this system. Vorticity in particular is always conserved along particles in two-dimensional inviscid flow, and as such even in its absence it is the value of the vorticity that fundamentally sets the drift, which in the Lagrangian frame is identified as the phase-averaged momentum density of the system. A relationship between the drift and the geometric mean water level of particles is found at the surface, which highlights connections between the geometry and dynamics. Finally, an example of an initially quiescent fluid driven by a wavelike pressure disturbance is considered, showing how the net momentum and energy from the surface pressure disturbance transfer to the wave field, and recognizing the source of the mean Lagrangian drift as the net momentum required to generate an irrotational surface wave by any conservative force.

When studying two-dimensional fluid–body interactions in the low-Froude limit, traditional asymptotic theory predicts a waveless free surface at every order. The waves are, in fact, exponentially small and it has been well-established that such waves ‘switch on’ seemingly instantaneously across so-called Stokes lines, partitioning the fluid domain into wave-free regions and regions with waves. In three dimensions, the Stokes-line concept extends to higher-dimensional Stokes surfaces. This article is concerned with the archetypal problem of uniform flow over a point source, reminiscent of, but separate to, the famous Kelvin wave problem. In prior research, the intersection of the Stokes surface with the free surface was found, in implicit form, for this case of a point source. However, on account of the algebraic manipulations required, it is not clear how this approach can be extended to more challenging settings. Here we develop a numerical-based procedure that allows the Stokes surface to be computed. The intersections of the Stokes surfaces with both the free surface and the deeper fluid are discussed for the case of the point source. Crucially, this procedure provides an important tool for generalising exponential asymptotics to the case of nonlinear (non-point-source) wave-generating bodies.

The shoreline hazard posed by ocean long waves such as tsunamis and meteotsunamis critically depends on the fraction of energy transmitted across the shallow near-shore shelf. In linear setting, bathymetric inhomogeneities of length comparable to the incident wavelength act as a protective high-pass filter, reflecting long waves and allowing only shorter waves to pass through. Here, we show that, for weakly nonlinear waves, the transmitted energy flux fraction can significantly depend on the amplitude of the incoming wave. The basis of this mechanism is the formation of dispersive shock waves (DSWs), a salient feature of nonlinear evolution of long water waves, often observed in tidal bores and tsunami/meteotsunami evolution. Within the framework of the Boussinesq equations, we show that the DSWs efficiently transfer wave energy into the high wavenumber band, where reflection is negligible. This is phenomenologically similar to self-induced transparency in nonlinear optics: small amplitude long waves are reflected by the bathymetric inhomogeneity, while larger amplitude waves that develop DSWs blueshift into the transparency regime and pass through. We investigate this mechanism in a simplified setting that retains only the key processes of DSW disintegration and reflection, while the effects such as bottom dissipation and breaking are ignored. The results suggests that the phenomenon is a robust, order-one effect. In contrast, the increased transmission due to the growth of bound harmonics associated with the steepening of the wave is weak. The results of the simplified modelling are validated by simulations with the FUNWAVE-TVD Boussinesq model.

The spatial evolution of various statistical parameters of fetch-limited waves generated by steadily blowing wind over mean water flow in a wind-wave flume is investigated experimentally. Measurements are performed in both along- and against-wind current conditions, and compared with measurements in the absence of current. A rake of capacitance-type wave gauges is used to measure surface elevation for a wide range of wind and water current velocities; additionally, an optical wave gauge is used to measure the directional properties of the wind-wave field in the presence of a mean water current at multiple locations. The variation with fetch of essential wave parameters such as characteristic wave energy, dominant frequency, power spectra and temporal coherence, as well as higher-order statistical moments that characterize wave shape, is presented for co- and counter-wind water currents, and compared with the no-current condition. The findings in the presence of mean water flow are interpreted in the framework of the viscous shear flow instability model of Geva & Shemer (Phys. Rev. Lett., vol. 128, 2022, 124501).

The Whitham equation is a non-local, nonlinear partial differential equation that models the temporal evolution of spatial profiles of surface displacement of water waves. However, many laboratory and field measurements record time series at fixed spatial locations. To directly model data of this type, it is desirable to have equations that model the spatial evolution of time series. The spatial Whitham (sWhitham) equation, proposed as the spatial generalization of the Whitham equation, fills this need. In this paper, we study this equation and apply it to water-wave experiments on shallow and deep water. We compute periodic travelling-wave solutions to the sWhitham equation and examine their properties, including their stability. Results for small-amplitude solutions align with known results for the Whitham equation. This suggests that the systems are consistent in the weakly nonlinear regime. At larger amplitudes, there are some discrepancies. Notably, the sWhitham equation does not appear to admit cusped solutions of maximal wave height. In the second part, we compare predictions from the temporal and spatial Korteweg–de Vries and Whitham equations with measurements from laboratory experiments. We show that the sWhitham equation accurately models measurements of tsunami-like waves of depression, waves of elevation, and solitary waves on shallow water. Its predictions also compare favourably with experimental measurements of waves of depression and elevation on deep water. Accuracy is increased by adding a phenomenological damping term. Finally, we show that neither the sWhitham nor the temporal Whitham equation accurately models the evolution of wave packets on deep water.

We consider steady surface waves in an infinitely deep two-dimensional ideal fluid with potential flow, focusing on high-amplitude waves near the steepest wave with a 120$^{\circ }$ corner at the crest. The stability of these solutions with respect to coperiodic and subharmonic perturbations is studied, using new matrix-free numerical methods. We provide evidence for a plethora of conjectures on the nature of the instabilities as the steepest wave is approached, especially with regards to the self-similar recurrence of the stability spectrum near the origin of the spectral plane.