We revise the solutions of the forced Korteweg–de Vries equation describing a resonant interaction of a solitary wave with external pulse-type perturbations. In contrast to previous work where only the limiting cases of a very narrow forcing in comparison with the initial soliton or a very narrow soliton in comparison with the width of external perturbation were studied, we consider here an arbitrary relationship between the widths of soliton and external perturbation of a relatively small amplitude. In many particular cases, exact solutions of the forced Korteweg–de Vries equation can be obtained for the specific forcings of arbitrary amplitude. We use the earlier developed asymptotic method to derive an approximate set of equations up to the second-order on a small parameter characterising the amplitude of external force. The analysis of exact solutions of the derived equations is presented and illustrated graphically. It is shown that the theoretical outcomes obtained by the asymptotic method are in a good agreement with the results of direct numerical modeling within the framework of forced Korteweg–de Vries equation.

This review studies some of the existing ideas which create a fundamentally new direction for wave energy converter (WEC) development, obtaining energy from the elliptical motion of particles in water waves using a rotor. The review focuses on three main aspects of rotor‐based WECs: experimental study of the developed prototypes, derivation and development of the mathematical models, and control effectors and philosophies for the proposed devices. The range of developed small scale prototypes and their experimental studies are presented. The shortcomings of the current mathematical and hydrodynamic models are identified, while an overview of the proposed and new possible control effectors and strategies is conducted. This allows us to see the state of the development of the different concepts and problems to be solved in bringing wave energy rotors to operational reality.

The commercial development and deployment of wave energy converters (WECs) will require arranging these devices in groups known as ‘arrays’, similar to the deployment other large-scale renewable energy systems, such as wind farms, or tidal arrays. This study explores a novel control co-design (CCD) strategy for heterogeneous arrays of point absorber-type WECs, focusing on the simultaneous optimisation of buoy hull geometry and array layout to harness multi-directional wind and swell wave energy. The WEC array operates under a newly developed global centralised control algorithm, which supports displacement constraints, but allows for the assessment of array performance in the frequency domain. This approach has the potential to significantly speed up the numerical solution of the control co-design optimisation problem, compared to more traditional time-domain-based methods. The array optimisation problem is solved using a global optimisation method. The performance function aims to optimise the positive network effect of interactions between devices in the array, while simultaneously considering cost issues, quantified by device sizes. The investigation identifies optimal device geometry and array layouts for clusters of three, four, and five WECs, in two different wave climates: Irish and Portuguese coasts, allowing the sensitivity of optimal solutions to different wave climates to be studied.

The development of optimal control strategies for cyclorotor wave energy converters (WECs) is at an early stage. In this paper, we present new methods and solutions for optimal pitch and/or rotational velocity control strategies for different configurations of cyclorotor-based WECs, in both monochromatic and panchromatic waves. The cyclorotor is modelled with the use of an approximate two-dimensional mathematical model, where hydrofoils are approximated as point source vortices in potential waves. The goal of the developed control strategy is to determine the optimal velocity profile, and/or pitch angle variations, for maximum energy conversion, in terms of generated mechanical shaft power. The solutions, obtained with the use of spectral methods, show a clear benefit in using a variable velocity for cyclorotors with either one or two hydrofoils, in monochromatic waves, with a typical increase in energy capture of 30-40%, while the optimal pitching did not significantly increase the value of the absorbed wave energy. We also present control results for panchromatic waves, assuming all the properties of incoming wave packages, within a 10 s time interval, are known. The obtained solutions have shown significant benefit in joint optimal pitch and velocity control, especially in the case of panchromatic waves. It has been also shown that successful implementation of cyclorotor control strategies requires optimal configuration of the static characteristics of the cyclorotor. In conclusion, we discuss the optimal control benefits and highlight problems which must be solved for further successful development of these control strategies.

We analytically study a scattering of long linear surface waves on stationary currents in a duct (canal) of constant depth and variable width. It is assumed that the background velocity linearly increases or decreases with the longitudinal coordinate due to the gradual variation of duct width. Such a model admits an analytical solution of the problem in hand, and we calculate the scattering coefficients as functions of incident wave frequency for all possible cases of sub-, super-, and transcritical currents. For completeness we study both cocurrent and countercurrent wave propagation in accelerating and decelerating currents. The results obtained are analyzed in application to recent analog gravity experiments and shed light on the problem of hydrodynamic modeling of Hawking radiation.

Recently conducted analytical assessment of the potential performance of cyclorotor wave energy converters (WECs) have shown that such devices offer the best wave absorption behaviour, if energy capture can be optimised through suitable control. Such claims require additional investigation. This article is dedicated to validation of the control-oriented point vortex model of cyclorotor WECs against numerical and experimental assessments conducted by various research groups. The validation is conducted in terms of the traditional metrics for cyclorotor WECs: (a) cancellation of incoming waves; (b) generation of lift and drag forces (c) mechanical power generation. It is shown that the point vortex model generally confirms the previously conducted analytical assessment of device performance. However, accounting for the influence of the hydrofoil induced wakes decreases performance estimates to some extent. It is also shown that, overall, wave cancellation metrics are more optimistic than actual shaft power generation. Analysis of the lift and drag coefficients, which were derived from experimental data, reveal a range of hydrodynamic and mechanic effects which could influence actual device performance. It has been shown that, due to the complexity of hydrodynamic effects, lift and drag coefficients for the control-oriented model should be considered not only as functions of the Reynolds number and angle of attack, but also related to submergence of the foils and direction of their rotation with respect to the free surface. This method allows us to achieve the best validation against experimental results in terms of generation of tangential and radial forces.