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.

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.

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.

Existing theoretical results for attenuation of surface waves propagating on water of random fluctuating depth are shown to over-predict the rate of decay due to the way in which ensemble averaging is performed. A revised approach is presented which corrects this and is shown to conserve energy. New theoretical predictions are supported by numerical results which use averaging of simulations of wave scattering over finite sections of random bathymetry for which transfer matrix eigenvalues are used to accurately measure decay. The model of wave propagation used in this paper is derived from a linearised long-wavelength assumption whereby depth averaging leads to time harmonic waves being represented as solutions to a simple ordinary differential equation. In this paper it is shown how this can be adapted to incorporate a model of a continuous covering of the surface by fragmented floating ice. Attenuation of waves through broken ice of random thickness is then analysed in a similar manner as bed variations have been previously. General features of the predicted attenuation are discussed in relation to existing theoretical models for attenuation due to multiple scattering through random ice environments and to field data, particularly in the model's ability to capture a ‘rollover effect’ at higher frequencies.

Steep wave breaking on a vertical cylinder (a typical foundation supporting offshore wind turbines) will induce slam loads. Many questions on the important violent wave loading and the associated secondary load cycle remain unanswered. We use laboratory experiments with unidirectional waves to investigate the fluid loading on vertical cylinders. We use a novel three-phase decomposition approach that allows us to separate different types of nonlinearity. Our findings reveal the existence of an additional quasi-impulsive loading component that is associated with the secondary load cycle and occurs in the backwards direction against that of the incoming waves. This quasi-impulsive force occurs at the end of the secondary load cycle and close to the passage of the downward zero-crossing point of the undisturbed wave. Wavelet analysis showed that the impulsive force exhibits superficially similar behaviour to a typical wave-slamming event but in the reverse direction. To monitor the scattered wave field and extract run-up on the cylinder, we installed a four-camera synchronised video system and found a strong temporal correlation between the arrival time of the Type-II scattered wave onto the cylinder and the occurrence of this quasi-impulsive force. The temporal characteristics of this quasi-impulsive force can be approximated by the Goda wave impact model, taking the collision of the Type-II scattered waves at the rear stagnation point as the impact source.

This study investigates the potential use of an active device to efficiently absorb water waves propagating in a channel. The active device comprises a dipole source consisting of two sources in quasi-opposition of phase. We explore the feasibility of this approach to achieve perfect absorption of guided waves through interference phenomena. To accomplish this, we establish the law governing the waves emitted by the dipole source to optimize the absorption of specific incident waves. The validity of this law is demonstrated through numerical simulations and laboratory experiments, encompassing both the harmonic and transient regimes of the experimental set-up.

In this work, formal asymptotic solutions of a problem for linear water waves in a bounded basin are constructed. The solutions have the form of asymptotic quasimodes and are used for the description of standing water waves localised near the shoreline. Such short-wavelength quasimodes exist only for a discrete set of frequencies, which are determined by means of a quantisation-type condition. Some numerical results are also addressed.

In the interaction of water waves with marine structures, the interplay between wave diffraction and drag-induced dissipation is seldom, if ever, considered. In particular, linear hydrodynamic models, and extensions thereof through the addition of a quadratic force term, do not represent the change in amplitude of the waves diffracted and radiated to the far field, which should result from local energy dissipation in the vicinity of the structure. In this work, a series of wave flume experiments is carried out, whereby waves of increasing amplitude impinge upon a vertical barrier, extending partway through the flume width. As the wave amplitude increases, the effect of drag – which is known to increase quadratically with the flow velocity – is enhanced, thus allowing the examination of the far-field effect of drag-induced dissipation, in terms of wave reflection and transmission. A potential flow model is proposed, with a simple quadratic pressure drop condition through a virtual porous surface, located on the sides of the barrier (where dissipation occurs). Experimental results confirm that drag-induced dissipation has a marked effect on the diffracted flow, i.e. on wave reflection and transmission, which is appropriately captured in the proposed model. Conversely, when diffraction becomes dominant as the barrier width becomes comparable to the incoming wavelength, the diffracted flow must be accounted for in predicting drag-induced forces and dissipation.

In this paper, we study wave scattering and radiation by a surface-piercing vertical truncated metamaterial cylinder composed of a closely spaced array of thin vertical barriers, between which fluid can flow. A theoretical model is developed under full depth-dependent linearised water wave theory, where an effective medium equation and effective boundary conditions are employed, respectively, to describe the fluid motion inside the cylinder and match the flow between the fluid regions in and outside the metamaterial cylinder. A damping mechanism is introduced at the surface of the fluid occupied by the metamaterial cylinder to consider the wave power dissipation in narrow gaps between the thin vertical plates. The wave excitation forces acting on the cylinder and the hydrodynamic coefficients can be calculated straightforwardly in terms of the velocity potential inside the cylinder. An alternative way is by using the velocity potential outside the cylinder, the expression of which has the reduction of the integral and an infinite accumulation that are included in the straightforward expression. The results highlight the patterns of the radiated waves induced by the oscillation of the cylinder and the characteristics of the hydrodynamic coefficients. The metamaterial cylinder when fixed in place and with a damping mechanism included is found to capture more wave power than that of a traditional axisymmetric heaving wave energy converter over a wide range of wave frequencies.

We investigate and compare various types of acoustic trapped modes (TMs) in resonator–waveguide systems. The goal is to understand the commonality and difference between the mechanisms of common (symmetry protected, invisibility protected and symmetry–periodicity protected) and accidental TMs, occurring continuously and discretely in the resonator length–frequency two-parameter space. The latter type cannot yet be explained via an operator decomposition. Here, all TMs are explained in the same way by analysing why and how the propagating-wave loops in the eigenfunctions can satisfy the eigenmode condition (loop magnitude and phase constraints for closure) and the wave-trapping condition (loop zero-radiation condition) simultaneously. Firstly, the unified analysis reveals the commonality that one or multiple coupled propagating-wave loops satisfy TM conditions, and the difference. In common TMs, the loop zero radiation is independent of the single loop phase constraint that selects the TM frequency as a continuous function of resonator length. On the other hand, loop zero radiation in accidental TMs depends on the loop phase constraints and there are two phase constraints. Only at the crossing points of the two phase constraints can zero-radiation loops be ensured. Secondly, in contrast to previous studies, it suggests that modal degeneracy, avoided crossing and resonance-width bifurcation are not the mechanisms of accidental TMs.

Based on the multiple-scale expansion technique, a new set of extended nonlinear Schrödinger (ENLS) equations up to the third order is derived to account for the additional high-order bottom and dispersion effects as well as the nonlinear wave interaction on wave transformation over periodic sandbars of sinusoidal geometry. By employing the small-amplitude wave assumption, a closed-form analytical solution for Bragg scattering is obtained from the linearised ENLS equations, which demonstrates that a downshift of wave frequency of the maximum reflection is mainly due to the inclusion of the high-order bottom effect. The factors that affect the downshift of the resonant frequency are identified and a theoretical expression in parabolic form is derived to quantify the downshift magnitude. The fully ENLS equations are further analysed to reveal the additional wave nonlinear effects on Bragg scattering characteristics. Under the condition of infinitesimal sandbar amplitude, the ENLS equations render a theoretical expression of the critical value of $kh$ when the nonlinear wave self-modulation effect and the nonlinear wave cross-modulation effect are equal, whereas the former effect is responsible for wavenumber upshifting and the latter downshifting. When $kh$ is larger than the critical value, the increase of wave nonlinearity will enhance the downshift magnitude of the Bragg resonance, and vice versa. For finite amplitude of the bottom sandbar, the ENLS equations are solved numerically to examine the influence of both wave nonlinearity and sandbar amplitude on the characteristics of Bragg resonance. The results reveal that as the increase of sandbar amplitude, the critical $kh$ increases monotonically.

The refraction of surface gravity waves by currents leads to spatial modulations in the wave field and, in particular, in the significant wave height. We examine this phenomenon in the case of waves scattered by a localised current feature, assuming (i) the smallness of the ratio between current velocity and wave group speed, and (ii) a swell-like, highly directional wave spectrum. We apply matched asymptotics to the equation governing the conservation of wave action in the four-dimensional position–wavenumber space. The resulting explicit formulas show that the modulations in wave action and significant wave height past the localised current are controlled by the vorticity of the current integrated along the primary direction of the swell. We assess the asymptotic predictions against numerical simulations using WAVEWATCH III for a Gaussian vortex. We also consider vortex dipoles to demonstrate the possibility of ‘vortex cloaking’ whereby certain currents have (asymptotically) no impact on the significant wave height. We discuss the role of the ratio of the two small parameters characterising assumptions (i) and (ii) above, and show that caustics are significant only for unrealistically large values of this ratio, corresponding to unrealistically narrow directional spectra.

We report on an experimental study of a device comprising an array of submerged, inverted and periodic cylindrical pendula (resonators), whose objective is the attenuation of surface gravity waves. The idea is inspired by the concept of metamaterials, i.e. engineered structures designed to interact with waves and manipulate their propagation properties. The study is performed in a wave flume where waves are excited in a wide range of frequencies. We explore various configurations of the device, calculating the transmitted, reflected and dissipated energy of the waves. If the incoming wave frequencies are sufficiently close to the natural frequency of the pendula, we find a considerable wave attenuation effect. This behaviour is enhanced by the number of resonators in the array. Moreover, the device is also capable of reflecting the energy of selected frequencies of the incoming waves. These frequencies, predicted by a generalized Bragg scattering mechanism, depend on the spacing between the resonators. The presented results show promise for the development of an environmentally sustainable device for mitigating waves in coastal zones.

Wave interaction with graded metamaterials exhibits the phenomenon of rainbow reflection, in which broadband wave signals slow down and separate into their frequency components before being reflected. This phenomenon has been qualitatively understood by describing the wave field in the metamaterial using the local Bloch wave approximation (LBWA), which locally represents the wave field as a superposition of propagating wave solutions in the cognate infinite periodic media (so-called Bloch waves). We evaluate the performance of the LBWA quantitatively in the context of two-dimensional linear water-wave scattering by graded arrays of surface-piercing vertical barriers. To do this, we implement the LBWA numerically so that the Bloch waves in one region of the graded array are coupled to Bloch waves in adjacent regions. This coupling is computed by solving the scattering of Bloch waves across the interface between two semi-infinite arrays of vertical barriers, where the barriers in each semi-infinite array can have different submergence depths. Our results suggest that the LBWA accurately predicts the free surface amplitude across a wide range of frequencies, except those just above the cutoff frequencies associated with each of the vertical barriers in the array. This highlights the importance of decaying Bloch modes above the cutoff in rainbow reflection.

An analytical Green's function is developed to study the acoustic scattering by a flat plate with a serrated edge. The scattered pressure is solved using the Wiener–Hopf technique in conjunction with the adjoint technique. It is shown that the kernel decomposition proposed in recent literature appears only valid at high frequencies. We focus on this high-frequency regime and obtain the scattered pressure in the form of a contour integral. We show that such an integral, although complicated, can be evaluated exactly for any arbitrary piecewise linear serrations, for which closed-form analytical Green's functions are obtained. The derivation is validated by performing numerical integration of the contour integral showing excellent agreement. The Green's function is shown to agree well with the numerical results obtained using the finite element method at high frequencies. The noise directivity patterns are studied as a function of the frequency, serration amplitude, source position and Mach number. It is found that noise is often enhanced at low observer angles and may be slightly reduced at high observer angles, which may be understood from the perspective of an extended or removed rigid reflection surface. It is found that increasing the mean-flow Mach number leads to increasingly evident noise amplification at side angles, a seemingly strange Doppler behaviour exhibited in source-fixed coordinate frames. The analytical Green's function is applicable to both leading- and trailing-edge scatterings, and is particularly suitable for developing a three-dimensional trailing-edge noise model that is not only highly efficient but also capable of including non-frozen turbulence effects.

This paper presents and compares two different approaches to solving the problem of wave propagation across a large finite periodic array of surface-piercing vertical barriers. Both approaches are formulated in terms of a pair of integral equations, one exact and based on a spacing $\delta > 0$ between adjacent barriers and the other approximate and based on a continuum model formally developed by using homogenisation methods for small $\delta$. It is shown that the approximate method is simpler to evaluate than the exact method which requires eigenvalues and eigenmodes related to propagation in an equivalent infinite periodic array of barriers. In both methods, the numerical effort required to solve problems is independent of the size of the array. The comparison between the two methods allows us to draw important conclusions about the validity of homogenisation models of plate array metamaterial devices. The practical interest in this problem stems from the result that for an array of barriers there exists a critical value of radian frequency, $\omega _c$, dependent on $\delta$, below which waves propagate through the array and above which it results in wave decay. When $\delta \to 0$, the critical frequency is given by $\omega _c = \sqrt {g/d}$, where $d$ is the plate submergence and $g$ is the acceleration due to gravity, which relates to the resonance in narrow channels and is an example of local resonance, studied extensively in metamaterials. The results have implications for proposed schemes to harness energy from ocean waves and other problems related to rainbow trapping and rainbow reflection.

The trajectory of surface gravity waves in the potential flow regime is affected by the gravitational acceleration, water density and sea bed depth. Although the gravitational acceleration and water density are approximately constant, the effect of water depth on surface gravity waves exponentially decreases as the water depth increases. In shallow water, cloaking an object from surface waves by varying the sea bed topography is possible, however, as the water depth increases, cloaking becomes a challenge because there is no physical parameter to be engineered and subsequently affects the wave propagation. In order to create an omnidirectional cylindrical cloaking device for finite-depth/deep-water waves, we propose an elastic composite plate that floats on the surface around a to-be-cloaked cylinder. The composite plate is made of axisymmetric, homogeneous and isotropic annular thin rings which provide adjustable degrees of freedom to engineer and affect the wave propagation. We first develop a pseudo-spectral method to efficiently determine the wave solution for a floating composite plate. Next, we optimise the physical parameters of the plate (i.e. flexural rigidity and mass of every ring) using an evolutionary algorithm to minimise the energy of scattered waves from the object and therefore cloak the inner cylinder from incident waves. We show that the optimised cloak reduces the energy of scattered waves as high as 99 % for the target wave number. We quantify the effectiveness of our cloak with different parameters of the plate and show that varying the flexural rigidity is essential to control wave propagation and the cloaking structure needs to be at least made of four rings with a radius of at least three times of the cloaked region. We quantify the wave drift force exerted on the structures and show that the optimised plate reduces the exerted force by 99.9 %. The proposed cloak, due to its structural simplicity and effectiveness in reducing the wave drift force, may have potential applications in cloaking offshore structures from water waves.

Resonant filament-assisted mode conversion (FAMC) scattering of high harmonic fast waves (HHFW) by cylindrical field-aligned density inhomogeneities can efficiently redirect a fraction of the launched HHFW power flux into the parallel direction. Within a simplified analytic approach, this contribution compares the parallel propagation, reflection and dissipation of nearly resonant FAMC modes for three magnetic field line geometries in the scrape-off layer, in the presence of radio-frequency (RF) sheaths at field line extremities and phenomenological wave damping in the plasma volume. When a FAMC mode, excited at the HHFW antenna parallel location and guided along the open magnetic field lines, impinges onto a boundary at normal incidence, we show that it can excite sheath RF oscillations, even toroidally far away from the HHFW launcher. The RF sheaths then dissipate part of the power flux carried by the incident mode, while another part reflects into the FAMC mode with the opposite wave vector parallel to the magnetic field. The reflected FAMC mode in turn propagates and can possibly interact with the sheath at the opposite field line boundary. The two counter-propagating modes then form in the bounded magnetic flux tube a lossy cavity excited by the HHFW scattering. We investigate how the presence of field line boundaries affects the total HHFW power redirected into the filament, and its splitting between sheath and volume losses, as a function of relevant parameters in the model.

In this paper, a concept of a floating elastic wave energy converter consisting of a disk-shaped elastic plate is proposed. The floating plate is moored to the seabed through a series of power take-off (PTO) units. A theoretical model based on the linear potential flow theory and eigenfunction matching method is developed to study the hydroelastic characteristics and evaluate wave power absorption of the device. The PTO system is simulated as a discrete PTO, and moreover, it is also modelled as a continuum PTO to represent the case when the PTO system is composed of a large number of PTO units. The continuum PTO approximation is tested against the discrete PTO simulation for accuracy. Two methods are proposed to predict the wave power absorption of the device. After running convergence analysis and model validation, the present model is employed to do a multiparameter impact analysis. The device adopting a continuum PTO system is found to capture wave power efficiently in an extensive range of wave frequencies. For the continuum PTO system, it is theoretically possible to adopt optimised PTO damper and stiffness/mass to guarantee the absorption of 100 % of the energy flux available in one circular component of the plane incident wave.

Bragg scattering of nonlinear surface waves over a wavy bottom is studied using two-dimensional fully nonlinear numerical wave tanks (NWTs). In particular, we consider cases of high nonlinearity which lead to complex wave generation and transformations, hence possible multiple Bragg resonances. The performance of the NWTs is well verified by benchmarking experiments. Classic Bragg resonances associated with second-order triad interactions among two surface (linear incident and reflected waves) and one bottom wave components (class I), and third-order quartet interactions among three surface (linear incident and reflected waves, and second-order reflected/transmitted waves) and one bottom wave components (class III) are observed. In addition, class I Bragg resonance occurring for the second-order (rather than linear) transmitted waves, and Bragg resonance arising from quintet interactions among three surface and two bottom wave components, are newly captured. The latter is denoted class IV Bragg resonance which magnifies bottom nonlinearity. It is also found that wave reflection and transmission at class III Bragg resonance have a quadratic rather than a linear relation with the bottom slope if the bottom size increases to a certain level. The surface wave and bottom nonlinearities are found to play opposite roles in shifting the Bragg resonance conditions. Finally, the results indicate that Bragg resonances are responsible for the phenomena of beating and parasitic beating, leading to a significantly large local free surface motion in front of the depth transition.