Wave measurements from fixed gauges in a flume produce time-series of surface elevation. To characterize the waves and enable a comparison with theory, additional spatial information is needed. This is commonly obtained by simultaneous measurements at pairs of gauges with spacing much smaller than a typical wavelength, assuming the surface profile to be frozen between the gauges. Although a “wave” can be defined as the signal between successive positive or negative zero-crossings, between successive crests, or between successive troughs, experimental measurements are not agnostic to the choice of metric. We show that wave-by-wave measurements of phase velocity or wavelength are most stable and consistent over repeated experiments when using wave-by-wave cross-correlation, as opposed to the passage of zero-crossings, crests, or troughs. The high variability of these other metrics may contribute to some of the difficulty in measuring nonlinear wave properties in flume experiments.
}
The goal of this work is to investigate particle motions beneath unidirectional, deep-water waves up to the third-order in nonlinearity. A particular focus is on the approximation known as Stokes drift and how it relates to the particle kinematics as computed directly from the particle trajectory mapping. The reduced Hamiltonian formulation of Zakharov and Krasitskii serves as a convenient tool to separate the effects of weak nonlinearity, in particular, the appearance of bound harmonics and the mutual corrections to the wave frequencies. By numerical integration of the particle trajectory mappings, we are able to compute motions and resulting drift for sea-states with one, two and several harmonics. We find that the classical Stokes drift formulation provides a slight underestimate of the drift at the surface and a slight overestimate at depth. Incorporating difference harmonic terms into the formulation yields an improved agreement with the drift obtained from nonlinear wave theories, particularly at greater depth. The consequences of this are explored for regular and irregular waves, as well as parametric wave spectra.
}
With the growing abundance of man-made cylindrical structures located on or close to the seabed, it is important to be able to assess their potential environmental impact. Herein, a model is presented of the viscous-thermal boundary layer in the vicinity of a circular cylinder resting on, or partially buried in, an otherwise flat seabed. To model the influence of wave-induced motions near such a cylinder, we assume oscillatory flow in which the water particle displacements are small with respect to the cylinder radius. A perturbation expansion is utilised to derive solutions of the boundary layer equations, leading to analytical solutions at multiple orders. The unsteady temperature field for various burial depths is then determined numerically using a Crank–Nicolson scheme, and quantitative results, such as the Nusselt number at the cylinder surface, are deduced. Both diffusion and steady convection are responsible for the unsteady transport of temperature. The dynamics of the convective field enhance overall heat transfer from the cylinder and lead to the temperature being transported radially outward near to the seabed.
}
The classical water-wave theory often neglects water compressibility effects, assuming acoustic and gravity waves propagate independently due to their disparate spatial and temporal scales. However, nonlinear interactions can couple these wave modes, enabling energy transfer between them. This study adopts a dynamical systems approach to investigate acoustic-gravity wave triads in compressible water flow, employing phase-plane analysis to reveal complex bifurcation structures and identify steady-state resonant configurations. Through this framework, we identify specific parameter conditions that enable complete energy exchange between surface and acoustic modes, with the triad phase (also known as the dynamical phase) playing a crucial role in modulating energy transfer. Further, incorporating spatial dependencies into the triad system reveals additional dynamical effects that depend on the wave velocity and resonance conditions: we observe that travelling-wave solutions emerge, and their stability is governed by the Hamiltonian structure of the system. The phase-plane analysis shows that, for certain velocity regimes, the resonance dynamics remains similar to the spatially independent case, while in other regimes, bifurcations modify the structure of resonant interactions, influencing the efficiency of energy exchange. Additionally, modulated periodic solutions appear, exhibiting changes in wave amplitudes over time and space, with implications for wave-packet stability and energy localisation. These findings enhance the theoretical understanding of acoustic-gravity wave interactions, offering potential applications in geophysical phenomena such as oceanic microseisms.
}
In this manuscript we investigate the Benjamin–Feir (or modulation) instability for the spatial evolution of water waves from the perspective of the discrete, spatial Zakharov equation, which captures cubically nonlinear and resonant wave interactions in deep water without restrictions on spectral bandwidth. Spatial evolution, with measurements at discrete locations, is pertinent for laboratory hydrodynamic experiments, such as in wave flumes, which rely on time-series measurements at fixed gauges installed along the facility. This setting is likewise appropriate for experiments in electromagnetic and plasma waves. Through a reformulation of the problem for a degenerate quartet, we bring to bear techniques of phase-plane analysis which elucidate the full dynamics without recourse to linear stability analysis. In particular we find hitherto unexplored breather solutions and discuss the optimal transfer of energy from carrier to sidebands. We show that the maximal energy transfer consistently occurs for smaller side-band separation than the fastest linear growth rate. Finally, we discuss the observability of such discrete solutions in light of numerical simulations.
}
This paper sets out to explore the modulational (or Benjamin-Feir) instability of a monochromatic wave propagating in the presence of damping such as that induced by sea ice on the ocean surface. The fundamental wave motion is modelled using the spatial Zakharov equation, to which either uniform or nonuniform (frequency-dependent) damping is added. By means of mode truncation the spatial analog of the classical Benjamin-Feir instability can be studied analytically using dynamical systems techniques. The formulation readily yields the free surface and its envelope, giving insight into the physical implications of damping on the modulational instability. The evolution of an initially unstable mode is also studied numerically by integrating the damped, spatial Zakharov equation, in order to complement the analytical theory. This sheds light on the effects of damping on spectral broadening arising from this instability.
}
}
The stability of waves in deep water has classically been approached via linear stability analysis, with various model equations, such as the nonlinear Schrödinger equation, serving as points of departure. Some of the most well-studied instabilities involve the interaction of four waves – so called Type I instabilities – or five waves – Type II instabilities. A unified description of four and five wave interaction can be provided by the reduced Hamiltonian derived by Krasitskii (1994). Exploiting additional conservation laws, the discretised Hamiltonian may be used to shed light on these four and five wave instabilities without restrictions on spectral bandwidth. We derive equivalent autonomous, planar dynamical systems which allow for straightforward insight into the emergence of instability and the long time dynamics. They also yield new steady-state solutions, as well as discrete breathers associated with heteroclinic orbits in the phase space.
}
We develop a new methodology for the deterministic forecasting of directional ocean surface waves based on nonlinear frequency corrections. These frequency corrections can be pre-computed based on measured energy density spectra and, therefore, come at no additional computational cost compared to linear theory. The nonlinear forecasting methodology is tested on highly nonlinear synthetically generated seas with a variety of values of average steepness and directional spreading and is shown to consistently outperform a linear forecast.
}
We develop a general framework to describe the cubically nonlinear interaction of a degenerate quartet of deep-water gravity waves in one or two spatial dimensions. Starting from the discretised Zakharov equation, and thus without restriction on spectral bandwidth, we derive a planar Hamiltonian system in terms of the dynamic phase and a modal amplitude. This is characterised by two free parameters: the wave action and the mode separation between the carrier and the sidebands. For unidirectional waves, the mode separation serves as a bifurcation parameter, which allows us to fully classify the dynamics. Centres of our system correspond to non-trivial, steady-state nearly resonant degenerate quartets. The existence of saddle-points is connected to the instability of uniform and bichromatic wave trains, generalising the classical picture of the Benjamin-Feir instability. Moreover, heteroclinic orbits are found to correspond to discrete, three-mode breather solutions, including an analogue of the famed Akhmediev breather solution of the nonlinear Schrödinger equation.
}
We study a simple model of three resonantly-interacting nonlinear waves. Linear stability analysis allows for an easy identification of stable and unstable initial conditions. Subsequently, a reformulation of the problem allows us to establish the onset of phase coherence, demonstrating its critical role in the growth of instabilities. Furthermore, we are able to provide explicit expressions for the time-varying moments of the modal amplitudes, as well as for the bispectrum which measures phase coherence. We provide direct insight into the long-time statistics of the system without Monte-Carlo simulation. This includes long-time asymptotics, which show that the system desynchronises, leading to the convergence of the second order moments and decay in the bispectrum.
}
We provide an in-depth exploration of the mass-transport properties of Pollard's exact solution for a zonally propagating surface water-wave in infinite depth. Without resorting to approximations we discuss the Eulerian mass transport of this fully nonlinear, Lagrangian solution. We show that it has many commonalities with the linear, Eulerian wave-theory, and also find Pollard-like solutions in the first and second order Lagrangian theory.
}
We present a theoretical model of the temperature distribution in the boundary layer region close to the seabed. Using a perturbation expansion, multiple scales and similarity variables, we show how free-surface waves enhance heat transfer between seawater and a seabed with a solid, horizontal, smooth surface. Maximum heat exchange occurs at a fixed frequency depending on ocean depth, and does not increase monotonically with the length and phase speed of propagating free-surface waves. Close agreement is found between predictions by the analytical model and a finite-difference scheme. It is found that free-surface waves can substantially affect the spatial evolution of temperature in the seabed boundary layer. This suggests a need to extend existing models that neglect the effects of a wave field, especially in view of practical applications in engineering and oceanography.
}
We study the temporal distribution of wave energy in a wave field that is generated by the reflection of a wave spectrum from a vertical wall. Weakly nonlinear wave fields over finite, constant depth are considered, and the reflection induces large correlations between different wave components of the wave field. The nonlinear time evolution of such an inhomogeneous random wave field is studied by means of an equation developed by Crawford, Saffman and Yuen in 1980. We show that, depending on the spectrum and the water depth, there is a significant increase in the probability of freak waves, whose height is more than twice the significant wave height, created by the reflection off the wall.
}
We derive a simple algebraic form of the nonlinear wavenumber correction of unidirectional surface gravity waves in deep water, based on temporal measurements of the water surface and the spatial Zakharov equation. This allows us to formulate an improvement over linear deterministic wave forecasting with no additional computational cost. Our new formulation is used to forecast both synthetically generated as well as experimentally measured seas and shows marked improvements over the linear theory.
}
This study investigates deterministic wave forecasting from the perspective of the Zakharov equation. Forecasts based on linear dispersion, weakly nonlinear amplitude dispersion and the Zakharov equation are compared for reference ocean surfaces generated from the Joint North Sea Wave Observation Project and Pierson-Moskowitz spectra. This approach allows for the role of nonlinearity to be isolated, and demonstrates the success of simple frequency corrections in forecasting wave fields up to moderate steepness. The role of second-order bound waves is investigated by means of analytical formulae, and their impact on forecast accuracy is illustrated for a range of forecast times.
}
This article is concerned with the non-linear interaction of homogeneous random ocean surface waves. Under this umbrella, numerous kinetic equations have been derived to study the evolution of the spectral action density, each employing slightly different assumptions. Using analytical and numerical tools, and providing exact formulas, we demonstrate that the recently derived generalized kinetic equation exhibits blow up in finite time for certain degenerate quartets of waves. This is discussed in light of the assumptions made in the derivation, and this equation is contrasted with other kinetic equations for the spectral action density.
}
We review the theory of wave interaction in finite and infinite depth. Both of these strands of water-wave research begin with the deterministic governing equations for water waves, from which simplified equations can be derived to model situations of interest, such as the mild slope and modified mild slope equations, the Zakharov equation, or the nonlinear Schrödinger equation. These deterministic equations yield accompanying stochastic equations for averaged quantities of the sea-state, like the spectrum or bispectrum. We discuss several of these in depth, touching on recent results about the stability of open ocean spectra to inhomogeneous disturbances, as well as new stochastic equations for the nearshore.
}
Two expressions for the nonlinear dispersion relation for gravity waves on water of constant depth are derived, one for wave fields with discrete amplitude spectra, the other for wave fields with continuous wavenumber energy spectra. Numerical examples for wave quartets and for two-dimensional Pierson-Moskowitz spectra are given, and an important possible application is discussed.
}
Using the IsraeliMediterranean as an example, we address the impact of resource variability and device survivability on the design of floating-body wave-energy converters (WECs). Employing a simplified heaving cylinder as a prototypical WEC, several device sizes, corresponding to the most frequently encountered and most energetic sea states in the Israeli Mediterranean, are investigated. The mean annual energy production is calculated based on the scatter-diagram/power-matrix approach. Subsequently, a measure for significant device motions under irregular sea-states akin to the spectral significant wave-height is developed, and cutoffs to regular operation are explored fromthe perspective of these significant displacements. The impact of thisWEC downtime is captured in a refinement of mean annual energy production, which consists of supplementing the scatter-diagram/power-matrix calculations by a Boolean displacement matrix. In the Israeli Mediterranean, where most of the annual incident wave power comes in infrequent winter storms, larger WECs outperform smaller WECs by a greater margin when downtime is taken into account. Analogous displacement cutoffs for refining calculations of mean annual energy production may inform WEC design for other sites.
}
We discuss the hydrodynamics of a wave energy converter consisting of two vertically floating, coaxial cylinders connected by dampers and allowed to heave, surge and pitch. This design, viable in deep water and able to extract energy independent of the incident wave direction, is examined for monochromatic waves as well as broad-banded seas described by a Pierson Moskowitz spectrum. Several possible device sizes are considered, and their performance is investigated for a design spectrum, as well as for more severe sea states, with a view towards survivability of the converters. In terms of device motions and captured power, a quantitative assessment of converter design as it relates to survival and operation is provided. Most results are given in dimensionless form to allow for a wide range of applications.
}
A discretized equation for the evolution of random surface wave fields on deep water is derived from Zakharov’s equation, allowing for a general treatment of the stability and long-time behaviour of broad-banded sea states. It is investigated for the simple case of degenerate four-wave interaction, and the instability of statistically homogeneous states to small inhomogeneous disturbances is demonstrated. Furthermore, the long-time evolution is studied for several cases and shown to lead to a complex spatio-temporal energy distribution. The possible impact of this evolution on the statistics of freak wave occurrence is explored. This article is part of the theme issue ‘Nonlinear water waves’.
}
Recently, Stiassnie et al. (J Ocean Eng Mar Energy 2(1):47–57, 2016) studied the potential for capturing wave energy over a large ocean basin via a toy model of a wave farm attacked by unidirectional wave fields. In the present work, we develop an approach to model macroscopically the behaviour of sparse arrays consisting of infinite rows of floating, axisymmetric wave energy converters in deep water. This approximate framework allows for such arrays to be characterized by frequency- and direction-dependent transfer functions. The example of a self-reacting converter consisting of vertically floating, coaxial cylinders moving in three modes of motion is discussed in detail, and the performance of large arrays of such devices, attacked by directional JONSWAP spectra and taking into account wave growth by the wind is investigated. This allows for fast and flexible estimates of power absorption by arrays as well as of their effects on the wave field.
}
We investigate the different wave-modes generated by a wave-maker in compressible flow. In addition to the propagating and evanescent waves found in the incompressible case, new radially propagating acoustic-gravity modes appear. We discuss the asymptotic behaviour of these waves, and give an example for a simple line wave-maker configuration.
}
We discuss some of the latent potential for harnessing wave power in open seas. Large farms of wave-energy converters in the open sea may extract energy several times over the course of an ocean basin, allowing the waves to grow under the influence of the wind, and capturing energy otherwise dissipated. Our calculations show that such an approach results in increasing the wave power potential by an order of magnitude compared to coastal capture alone. To carry out such calculations one needs the captured, reflected, and transmitted energy transfer functions of the farms. Here we simulate the functioning of the farms by one simplified two-dimensional converter consisting of two vertical floating plates, for which explicit transfer functions are calculated. Our main goals are to increase the awareness of the scientific community to the importance of harvesting wave power in open seas, and to provide a preliminary picture for the geometry and size of wave energy farms in open seas.
}
We give a review of developments concerning the Gerstner wave solution to the incompressible water wave equations, including many recent contributions that have successfully extended the Gerstner wave theory to geophysical and stratified fluids. We also highlight aspects of the mass transport of Gerstner waves, which serves to contrast the Gerstner solution with linear and nonlinear irrotational theories.
}
We treat the particle motion in Stokes linear edge wave along a uniformly sloping beach. By a rotation of the coordinate frame, we show that there is no particle motion in the direction orthogonal to the sloping beach, and conclude that particles have a longshore drift in the direction of wave propagation which decreases with depth and distance from the shoreline. We discuss the application of this rotated coordinate frame to higher mode (Ursell) and weakly nonlinear (Whitham) edge waves, and show that the weakly nonlinear case is identical to that for two-dimensional deep-water Stokes waves.
}
We provide an explicit solution to the full, nonlinear governing equations for gravity water waves describing internal edge waves along a sloping bed. This solution is based on the Gerstner edge wave. We discuss the relation of this internal, trochoidal edge wave to the analogous wave found in the linear theory, compare it with the classical Gerstner wave, as well as discuss the inclusion of Coriolis forces in the f-plane approximation.
}
We present a new exact solution describing progressive waves on a blunt interface based on Gerstner's trochoidal wave. The second-order irrota-tional theory is developed for a sharp interface, and subsequently for three fluid layers, the upper and lower of which may approach one another to form the so-called blunt interface. This situation is captured analogously by our exact rotational solution. We establish remarkable agreement between the exact and second-order theories, and present applications to surface water waves.
}
We give an explicit solution describing internal waves with a still-water surface, a situation akin to the well-known dead-water phenomenon, on the basis of the Gerstner wave solution to the Euler equations.
}
We demonstrate that, for a two-dimensional, steady, solitary wave profile, a flow of constant vorticity beneath the wave must likewise be steady and two-dimensional, and the vorticity will point in the direction orthogonal to that of wave propagation. Constant vorticity is the hallmark of a harmonic velocity field, and the simplified vorticity equation is used along with maximum principles to derive the results.
}
Building upon recent work in the applicability of soliton theory to tsunami propagation, we discuss the effects of shear flow on the KdV balance. This leads in the shallow-water limit to the Burns condition, and we see that for shear which does not yield critical layer solutions, the speeds determined by the Burns condition arise again in the KdV balance. In the event of waves propagating counter to the shear, KdV dynamics arise earlier, while their appearance is delayed in the case of waves propagating with the shear, the magnitude of this effect depending on the surface shear velocity.
}
Building on previous investigations, we show that Gerstner's famous deep water wave and the related edge wave propagating along a sloping beach, found within the context of water of constant density, can both be adapted to provide explicit free surface flows in incompressible fluids with arbitrary density stratification.
}
We investigate the Chilean tsunami of 1960 to determine the role of KdV dynamics. On the basis of the scales involved, and making use of recent advances, we put on a rigorous foundation the fact that KdV dynamics were not influential in this catastrophic event.
}