Comprehensive refractive manipulation of water waves using electrostriction.

The control of wave propagation based on refraction principles offers unparalleled possibilities as shown by the striking example of optics. This approach is unfortunately limited for water waves as it relies mainly on variations of the liquid depth which, while controlling the wave velocity, also trigger nonlinearities and damping. In this article, we show experimentally that electrostriction allows to implement extensive refraction-based control of water waves in a precise and contactless manner. The setup consists of an electrode under high voltage placed above the grounded conductive water. The waves propagating under the electrode can be slowed down up to approximately half their speed compared to free propagation. We characterize the Snell-Descartes laws of refraction and the total internal reflection for the water waves. We implement emblematic refraction-based devices such as electrically tunable focusing lenses, waveguides without obstacles, and beam splitters based on frustrated internal reflection to perform interference experiments.

Modelling attenuation of irregular wave fields by artificial ice floes in the laboratory.

A summary is given on the utility of laboratory experiments for gaining understanding of wave attenuation in the marginal ice zone, as a complement to field observations, theory and numerical models. It is noted that most results to date are for regular incident waves, which, combined with the highly nonlinear wave-floe interaction phenomena observed and measured during experimental tests, implies that the attenuation of regular waves cannot necessarily be used to infer the attenuation of irregular waves. Two experiments are revisited in which irregular wave tests were conducted but not previously reported, one involving a single floe and the other a large number of floes, and the transmission coefficients for the irregular and regular wave tests are compared. The transmission spectra derived from the irregular wave tests agree with the regular wave data but are overpredicted by linear models due to nonlinear dissipative processes, regardless of floe configuration. This article is part of the theme issue 'Theory, modelling and observations of marginal ice zone dynamics: multidisciplinary perspectives and outlooks'.

Storm Waves May Be the Source of Some "Tsunami" Coastal Boulder Deposits.

Coastal boulder deposits (CBD) provide what are sometimes the only remaining signatures of wave inundation on rocky coastlines; in recent decades, CBD combined with initiation of motion (IoM) analyses have repeatedly been used as primary evidence to infer the existence of ancient tsunamis. However, IoM storm wave heights inferred by these studies have been shown to be highly inaccurate, bringing some inferences into question. This work develops a dimensionless framework to relate CBD properties with storm-wave hindcasts and measurements, producing data-driven relations between wave climate and boulder properties. We present an elevation-density-size-inland distance-wave height analysis for individual storm-transported boulders which delineates the dynamic space where storm-wave CBD occur. Testing these new relations against presumed tsunami CBD demonstrates that some fall well within the capabilities of storm events, suggesting that some previous studies might be fruitfully reexamined within the context of this new framework.

Prediction of the free-surface elevation for rotational water waves using the recovery of pressure at the bed.

This paper considers the pressure-streamfunction relationship for a train of regular water waves propagating on a steady current, which may possess an arbitrary distribution of vorticity, in two dimensions. The application of such work is to both near shore and offshore environments, and in particular, for linear waves we provide a description of the role which the pressure function on the seabed plays in determining the free-surface profile elevation. Our approach is shown to provide a good approximation for a range of current conditions.This article is part of the theme issue 'Nonlinear water waves'.

On three-dimensional Gerstner-like equatorial water waves.

This paper reviews some recent mathematical research activity in the field of nonlinear geophysical water waves. In particular, we survey a number of exact Gerstner-like solutions which have been derived to model various geophysical oceanic waves, and wave-current interactions, in the equatorial region. These solutions are nonlinear, three-dimensional and explicit in terms of Lagrangian variables.This article is part of the theme issue 'Nonlinear water waves'.

Recent advances on the global regularity for irrotational water waves.

We review recent progress on the long-time regularity of solutions of the Cauchy problem for the water waves equations, in two and three dimensions. We begin by introducing the free boundary Euler equations and discussing the local existence of solutions using the paradifferential approach. We then describe in a unified framework, using the Eulerian formulation, global existence results for three- and two-dimensional gravity waves, and our joint result (with Deng and Pausader) on global regularity for the gravity-capillary model in three dimensions. We conclude this review with a short discussion about the formation of singularities and give a few additional references to other interesting topics in the theory.This article is part of the theme issue 'Nonlinear water waves'.

Hamiltonian models for the propagation of irrotational surface gravity waves over a variable bottom.

A single incompressible, inviscid, irrotational fluid medium bounded by a free surface and varying bottom is considered. The Hamiltonian of the system is expressed in terms of the so-called Dirichlet-Neumann operators. The equations for the surface waves are presented in Hamiltonian form. Specific scaling of the variables is selected which leads to approximations of Boussinesq and Korteweg-de Vries (KdV) types, taking into account the effect of the slowly varying bottom. The arising KdV equation with variable coefficients is studied numerically when the initial condition is in the form of the one-soliton solution for the initial depth.This article is part of the theme issue 'Nonlinear water waves'.

Analytical approximation and numerical simulations for periodic travelling water waves.

We present recent analytical and numerical results for two-dimensional periodic travelling water waves with constant vorticity. The analytical approach is based on novel asymptotic expansions. We obtain numerical results in two different ways: the first is based on the solution of a constrained optimization problem, and the second is realized as a numerical continuation algorithm. Both methods are applied on some examples of non-constant vorticity.This article is part of the theme issue 'Nonlinear water waves'.

Existence and amplitude bounds for irrotational water waves in finite depth.

We prove the existence of solutions to the irrotational water-wave problem in finite depth and derive an explicit upper bound on the amplitude of the nonlinear solutions in terms of the wavenumber, the total hydraulic head, the wave speed and the relative mass flux. Our approach relies upon a reformulation of the water-wave problem as a one-dimensional pseudo-differential equation and the Newton-Kantorovich iteration for Banach spaces.This article is part of the theme issue 'Nonlinear water waves'.

Capturing the flow beneath water waves.

Recently, the authors presented two numerical studies for capturing the flow structure beneath water waves (Nachbin and Ribeiro-Junior 2014 Disc. Cont. Dyn. Syst. A34, 3135-3153 (doi:10.3934/dcds.2014.34.3135); Ribeiro-Junior et al. 2017 J. Fluid Mech.812, 792-814 (doi:10.1017/jfm.2016.820)). Closed orbits for irrotational waves with an opposing current and stagnation points for rotational waves were some of the issues addressed. This paper summarizes the numerical strategies adopted for capturing the flow beneath irrotational and rotational water waves. It also presents new preliminary results for particle trajectories, due to irrotational waves, in the presence of a bottom topography.This article is part of the theme issue 'Nonlinear water waves'.

Evolution of statistically inhomogeneous degenerate water wave quartets.

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'.

Laboratory investigations of wave-induced transport of plastic debris over a rippled bottom.

Laboratory experiments are conducted in a wave flume to investigate the effect of water waves on the transport of plastic pellets over a rippled bottom. The horizontal velocities of plastic debris are analyzed over the rippled bottom for different wave conditions and plastic elements with different properties. Laboratory investigations determined the characteristic transport patterns of wave-induced plastic debris with a density of ∼2.0g/cm3 moving along the rippled bottom. In the first, swing-type motion, the grains move only in the ripple trough with velocities lower than 0.10 m/s. For sliding-type movement, the grains move along the entire rippled surface with velocities in the range of 0.10-0.13 m/s. For higher velocities in the range of 0.15-0.20 m/s, a saltation-type motion becomes dominant. The results show that plastic grains may move up to 2-3 cm above the ripple crest depending on hydrodynamic conditions. The analysis shows that for velocity-skewed flows, sliding-type motion and onshore transport dominate. For acceleration-skewed flows, saltation-type motion and offshore transport dominate, which is attributed to higher boundary layer thickness and phase lag effects. The analysis of the relationship between the particle Reynolds number and the thickness of the turbulent boundary layer reveals that for values of Rep≥1000 and a boundary layer thickness mm saltation-type motion becomes dominant. The direction of transport is affected not only by the density of the sediment and the wave skewness coefficients but also by the dimensions of the bottom ripples. The laboratory investigations also provide insight into the hydrodynamic conditions affecting the transport of plastic debris along the bottom covered with ripples in oscillating nonlinear water flows.

Cellulose-based superhydrophobic wrinkled paper and electrospinning film as green tribolayer for water wave energy harvesting.

Water waves are viable low-carbon and renewable sources of power that can be optionally combined with triboelectric nanogeneration (TENG). Herein, we report on the synthesis of a TENG device based on green wrinkled paper tribolayers (W-TENG) assembled in grids (G-TENG) with channels that enable contact-separation modes involving metal balls that roll in phase with the waves. The paper's wrinkle wavelength and amplitude were adjusted by using a crepe blade at a given angle with respect to a drying cylinder, as well as the speed and torque. Polar hierarchical superhydrophobic cellulose micro/nanostructures, proposed as positive tribolayers with enhanced contact area and triboelectric density. The negative (biodegradable) tribolayers were prepared by electrospinning aqueous suspensions of polyvinyl alcohol and poly (ethylene oxide) reinforced with cellulose nanofibers. The charge transfer by the W-TENG reached up to 40 nC in air and retained 27 nC under 85 % relative humidity, ~5 and 7 times higher than those measured in planar TENG counterparts. A G-TENG array charging time (100-µF capacitor) of ~188 s was measured when the voltage of the capacitor raised to ~1.5 V. Overall, we introduce a new, scalable TENG system that is demonstrated for its remarkable ability to harvest blue energy.

Computational and numerical wave solutions of the Caudrey-Dodd-Gibbon equation.

The Caudrey-Dodd-Gibbon ( CDG ) model, a variation of the fifth-order KdV equation (fKdV) with significant practical consequences, is solved in this study using a precise and numerical technique. This model shows how gravity-capillary waves, shallow-water waves driven by surface tension, and magneto-acoustic waves move through a plasma medium. With a focus on accuracy, new computational and approximation methods have been made possible by recent improvements in analytical and numerical methods. Numeric information is represented visually in the tables. All simulation results are shown in two and three dimensions to show both the numerical and fundamental behavior of the single soliton. Recent research shows that this method is the best way to solve nonlinear equations that are common in mathematical physics.

The modified extended tanh technique ruled to exploration of soliton solutions and fractional effects to the time fractional couple Drinfel'd-Sokolov-Wilson equation.

The modified extended tanh technique is used to investigate the conformable time fractional Drinfel'd-Sokolov-Wilson (DSW) equation and integrate some precise and explicit solutions in this survey. The DSW equation was invented in fluid dynamics. The modified extended tanh technique executes to integrate the nonlinear DSW equation for achieve diverse solitonic and traveling wave envelops. Because of this, trigonometric, hyperbolic and rational solutions have been found with a few acceptable parameters. The dynamical behaviors of the obtained solutions in the pattern of the kink, bell, multi-wave, kinky lump, periodic lump, interaction lump, and kink wave types have been illustrated with 3D and density plots for arbitrary chose of the permitted parameters. By characterizing the particular benefits of the exemplified boundaries by the portrayal of sketches and by deciphering the actual events, we have laid out acceptable soliton plans and managed the actual significance of the acquired courses of action. New precise voyaging wave arrangements are unambiguously gained with the aid of symbolic computation using the procedures that have been announced. Therefore, the obtained outcomes expose that the projected schemes are very operative, easier and efficient on realizing natures of waves and also introducing new wave strategies to a diversity of NLEEs that occur within the engineering sector.

Bifurcation analysis for axisymmetric capillary water waves with vorticity and swirl.

We study steady axisymmetric water waves with general vorticity and swirl, subject to the influence of surface tension. This can be formulated as an elliptic free boundary problem in terms of Stokes' stream function. A change of variables allows us to overcome the generic coordinate-induced singularities and to cast the problem in the form "identity plus compact," which is amenable to Rabinowitz's global bifurcation theorem, whereas no restrictions regarding the absence of stagnation points in the flow have to be made. Within the scope of this new formulation, local curves and global families of solutions, bifurcating from laminar flows with a flat surface, are constructed.

Scattering of surface waves by a vertical truncated structured cylinder.

This paper describes the solution to the problem of scattering of plane incident waves on water of constant depth by a bottom mounted circular cylinder, extending partially through the depth, which has an internal structure comprised of closely spaced thin vertical barriers between which fluid is allowed to flow. The problem is solved under full depth-dependent linearized water wave theory using an effective medium equation to describe the fluid motion in cylinder and effective boundary conditions to match that flow to the fluid region outside the cylinder. The interest in this problem lies in the development of novel solution methods for fully three-dimensional water wave interaction with bathymetric plate arrays. Results computed using this theory are compared with a shallow water approximation based on the recent work of Marangos & Porter (2021 Shallow water theory for structured bathymetry. Proc. R. Soc. A 477, 20210421.) and with accurate computations of an exact representation of the geometry using a discrete set of plates. Other results highlight the resonant directional lensing effects of this type of cylindrical plate array device.

On the energy of nonlinear water waves.

This article presents results concerning the excess kinetic and potential energies for exact nonlinear water waves. In particular, it is proven, for periodic travelling irrotational water waves, that the excess kinetic energy density is always negative, whereas the excess potential energy density is always positive, in the steady reference frame. A characterization of the total excess energy density as a weighted mean of the kinetic energy along the wave surface profile is also presented.

On the accuracy and applicability of a new implicit Taylor method and the high-order spectral method on steady nonlinear waves.

This paper presents an investigation and discussion of the accuracy and applicability of an implicit Taylor (IT) method versus the classical higher-order spectral (HOS) method when used to simulate two-dimensional regular waves. This comparison is relevant, because the HOS method is in fact an explicit perturbation solution of the IT formulation. First, we consider the Dirichlet-Neumann problem of determining the vertical velocity at the free surface given the surface elevation and the surface potential. For this problem, we conclude that the IT method is significantly more accurate than the HOS method when using the same truncation order, M, and spatial resolution, N, and is capable of dealing with steeper waves than the HOS method. Second, we focus on the problem of integrating the two methods in time. In this connection, it turns out that the IT method is less robust than the HOS method for similar truncation orders. We conclude that the IT method should be restricted to M = 4, while the HOS method can be used with M ≤ 8. We systematically compare these two options and finally establish the best achievable accuracy of the two methods as a function of the wave steepness and the water depth.

The mixed Rossby-gravity wave on the spherical Earth.

This work revisits the theory of the mixed Rossby-gravity (MRG) wave on a sphere. Three analytic methods are employed in this study: (a) derivation of a simple ad hoc solution corresponding to the MRG wave that reproduces the solutions of Longuet-Higgins and Matsuno in the limits of zero and infinite Lamb's parameter, respectively, while remaining accurate for moderate values of Lamb's parameter, (b) demonstration that westward-propagating waves with phase speed equalling the negative of the gravity-wave speed exist, unlike the equatorial ß-plane, where the zonal velocity associated with such waves is infinite, and (c) approximation of the governing second-order system by Schrödinger eigenvalue equations, which show that the MRG wave corresponds to the branch of the ground-state solutions that connects Rossby waves with zonally symmetric waves. The analytic conclusions are confirmed by comparing them with numerical solutions of the associated second-order equation for zonally propagating waves of the shallow-water equations. We find that the asymptotic solutions obtained by Longuet-Higgins in the limit of infinite Lamb's parameter are not suitable for describing the MRG wave even when Lamb's parameter equals 104. On the other hand, the dispersion relation obtained by Matsuno for the MRG wave on the equatorial ß-plane is accurate for values of Lamb's parameter as small as 16, even though the equatorial ß-plane formally provides an asymptotic limit of the equations on the sphere only in the limit of infinite Lamb's parameter.