Small Wave Research Articles

Equations for small amplitude shallow water waves over small bathymetric variations

A generalized version of the a b c d -Boussinesq class of systems is derived to accommodate variable bottom topography in two-dimensional space. This extension allows for the conservation of suitable energy functionals in some cases and enables the description of water waves in closed basins with well-justified slip–wall boundary conditions. The derived systems possess a form that ensures their solutions adhere to important principles of physics and mathematics. By demonstrating their consistency with the Euler equations and estimating their approximation error, we establish the validity of these new systems. Their derivation is based on the assumption of small bathymetric variations. With practical applications in mind, we assess the effectiveness of some of these new systems through comparisons with standard benchmarks. The results indicate that the assumptions made during the derivation are not overly restrictive. The applications of the new systems encompass a wide range of scenarios, including the study of tsunamis, tidal waves and waves in ports and lakes.

DAMPING OF FINITE AMPLITUDE SOLITARY WAVES IN A FLUME

Solitary wave is a permanent wave when the dissipation is ignored. Many analytical solutions have been developed for finite amplitude solitary waves. In additional to the perturbation solutions, closed form solutions are also available (e.g., McCowan 1891, Clamond and Fructus 2003, being denoted as CF from hereon), which are more accuracy, especially for larger amplitude solitary waves which were discussed in Wang and Liu (2022). In Wang and Liu (2022) ’s laboratory experiments, solitary waves are slowly damped along the wave flume, which can be attributed to the energy dissipation inside the boundary layers on the bottom and sidewalls. Keulegan (1948) first derived an approximate solution to describe the wave damping in the wave flume for small amplitude solitary wave. The basic idea of calculating the damping effect is that the rate of energy dissipation inside the boundary layers must be the same as the rate of wave energy loss. In this work, the same principle of energy balance is adopted for calculating the wave damping for finite amplitude solitary waves in a wave flume. The closed form solutions provided in CF are employed here for analyses.

On the development of a generalized atmospheric boundary layer velocity profile for offshore engineering applications considering wind–wave interaction

Within industrial standards, effects due to wind–wave interaction on the marine atmospheric boundary layer (ABL) may be included via the Charnock sea-surface roughness parameter for open-sea and near-coastal waters. This roughness parameter does not accommodate the wide variety of wave states possible, nor does it modify the ABL to include speed-up effects resulting from an undulating wave profile. In an attempt to provide an updated definition of the general ABL profile for offshore wind engineering applications, an experimental and numerical study is performed to assess the interaction of the ABL on a fixed wave geometry. By extracting the mean velocity field during the wind–wave interaction, bespoke values of the velocity profile power exponent and wind risk factor can be obtained. A range of wave heights and wave lengths are considered from which a Kriging surrogate model is trained to supply the relevant wind profile parameters depending on the wave state. These newly garnered parameters enable the practitioner to define a reference wind velocity, and then adjust the velocity profile characteristics to contain the influence of the wave. This approach provides a new inlet velocity condition that can be used within computational wind engineering investigations without the need to explicitly model the wave surface, as well as flexibility in specifying the underlying wave conditions. Application of the re-calibrated velocity profile shows that wind forces are significantly greater throughout an offshore wind turbine (OWT) swept blade area when large (H= 15 m) and small (H= 1.5 m) wave heights are compared.

Hybrid Model Predictive Control of a Two-Body Wave Energy Converter with Mechanically Driven Power Take-Off

In this paper, a variable damper is proposed to regulate the efficiency of a two-body wave energy converter (WEC) with mechanically driven power take-off (PTO). The variable damper introduces logic constraints into the WEC system, which can be translated into a mixed logical dynamical form with the dynamics of real-valued variables, the dynamics of logic variables, and their interactions. A hybrid model predictive control (MPC) method is used to determine the control inputs, which has the capacity to handle various constraints. The performance is assessed through simulations to evaluate the effectiveness of the proposed method. The achievable performance improvements of the proposed hybrid MPC are shown by means of comparative analysis with uncontrolled WEC devices. The results show that the proposed hybrid MPC has a high requirement on the lower bound of the variable damper and the maximum damping is used only at low relative velocities to achieve the optimum phase, like latching control. The hybrid MPC performs exceptionally well under wave conditions with a small significant wave height and long wave period, improving the power generation of the uncontrolled system up to 22.5%. And, the prediction error has a significant effect on hybrid MPC performance, especially for long prediction horizon.

Are small waves fondle and big waves overturn? Market reaction and corporate governance during four COVID-19 waves

PurposeThis study examines the effects of corporate governance on market returns during the first four waves of the COVID-19 crisis.Design/methodology/approachEvent study and linear regression methods were applied on a sample of 293 US firms.FindingsThe results show that differences in abnormal returns are more significant during the second wave of COVID-19 and the two following waves. Moreover, estimations show that good corporate governance alleviated the effect of COVID-19 during the second wave and the two following waves. However, corporate governance did not affect abnormal returns during the first wave. Furthermore, evidence highlights that the effect of corporate governance is more observed in the industries most affected by COVID-19 than in the least affected industries.Originality/valueMany studies have attempted to investigate the effect of corporate governance on stock returns during the first wave of the pandemic. However, to the authors' knowledge, this is the first study that focuses on different waves that occurred during 2020 and 2021.

Asymptotic stability manifolds for solitons in the generalized Good Boussinesq equation

We consider the generalized Good-Boussinesq model in one dimension, with power nonlinearity and data in the energy space H1×L2. This model has solitary waves with speeds −1<c<1. When |c| approaches 1, Bona and Sachs showed orbital stability of such waves. It is well-known from a work of Liu that for small speeds solitary waves are unstable. In this paper we consider in more detail the long time behavior of zero speed solitary waves, or standing waves. By using virial identities, in the spirit of Kowalczyk, Martel and Muñoz, we construct and characterize a manifold of even-odd initial data around the standing wave for which there is asymptotic stability in the energy space.

Numerical Evaluation of Hydrodynamic Coefficients in Roll Motion

The determination of ship motions in a seaway heavily depends on hydrodynamic coefficients. However, the coefficients related to roll-motion exhibit nonlinear behavior due to viscous effects and relatively small wave damping, which pose significant challenges. To overcome these challenges, several studies have employed computational fluid dynamics (CFD) to analyze hydrodynamic performance, leveraging the advancements in computational performance. Nevertheless, limited research has investigated the error and uncertainty of these numerical solutions. In this study, we conducted a forced oscillation test on a ship-like section that induces lateral vortex shedding using the OpenFOAM with overset grid method. We validated the numerical results of the roll hydrodynamic coefficients with experiments and the results of potential theory. We used the procedure proposed by the ITTC to perform uncertainty analysis on the grid size and time step size, which presented the overall uncertainty of the numerical analysis results. Our findings indicate that the grid uncertainty of the added moment of inertia was significant in the low frequency region, while the time step uncertainty was significant in the high frequency region.

The dominant instability of near-extreme Stokes waves

The instability of Stokes waves, steady propagating waves on the surface of an ideal fluid of infinite depth, is a fundamental problem in the field of nonlinear science. The dominant instability of these waves depends on their steepness. For small amplitude waves, it is well known that the Benjamin-Feir or modulational instability dominates the dynamics of a wave train. We demonstrate that for steeper waves, an instability caused by disturbances localized at the wave crest vastly surpasses the growth rate of the modulational instability. These dominant localized disturbances are either coperiodic with the Stokes wave or have twice its period. In either case, the nonlinear evolution of the instability leads to the formation of plunging breakers. This phenomenon explains why long propagating ocean swell consists of small-amplitude waves.

Study on U hull modifications with concave design to improve the tourist ship stability

Stability is an important aspect that was studied in ship design as it relates to the safety of the ship when facing ocean waves. The tourist ship needed to have a good image in order to attract tourists to use its services. Good stability of the tourist ship was crucial in enhancing the image of the ship and attracting tourists to utilize its services. This research aimed to investigate the effect of concave on the U hull to improve ship stability. The study compared the stability test results of two hull designs: U hull with concave and without concave. The stability tests of both ship models were conducted through simulation methods. The simulations were performed under two conditions: when the tourist ship was empty and when it was fully loaded with passengers. The simulation results indicated that the design of the U hull with concave had a positive influence on ship stability, leading to improved stability. The concave design enhanced the ship's ability to return to its original position when encountering waves. Even in the presence of small waves, the tourist ship did not experience instability.

COVID’s future: Viral multi-lineage evolution and the dynamics of small epidemic waves without seasonality in COVID-19

Since its emergence, SARS-CoV-2 virus, the causative agent of COVID-19, has undergone over three years of evolutionary changes. Throughout this evolutionary process, multiple variants, such as Alpha, Beta, Gamma, Delta, and Omicron, have emerged, leading to widespread outbreaks and pandemics. A comprehensive analysis of the evolution of SARS-CoV-2 over the past three years reveals its progression through three distinct stages.

Particle paths beneath forced small amplitude periodic waves in a shallow channel with constant vorticity

Particle paths beneath small amplitude periodic forced waves in a shallow water channel are investigated. The problem is formulated in the forced Korteweg-de Vries equation framework which allows to approximate the velocity field in the bulk fluid. We show that the flow can have zero, one or three stagnation points. Besides, differently from the unforced problem, stagnation points can arise for small values of the vorticity as long as the moving disturbance travels sufficiently fast.

Spectral Analysis of Ionospheric Density Variations Measured With the Large Radio Telescope in the Low‐Latitude Region

AbstractThe low‐latitude ionosphere is a dynamic region with a wide range of disturbances in temporal and spatial scales. The Giant Metrewave Radio Telescope (GMRT) situated in the low‐latitude region has demonstrated its ability to detect various ionospheric phenomena. It can detect total electron content (TEC) variation with precision of 10−3 TECU and also can measure TEC gradient with an accuracy of about 7 × 10−4 TECU km−1. This paper describes the spectral analysis of previously calculated TEC gradient measurements and validates them by comparing their properties using two bands. The analysis tracked individual waves associated with medium‐scale traveling ionospheric disturbances (MSTIDs) and smaller waves down to wavelengths of ∼10 km. The ionosphere is found to have unanticipated changes during sunrise hours, with waves changed propagation direction as the sun approached the zenith. Equatorial spread F disturbances during sunrise hours is observed, along with smaller structures moving in the same direction.

Correlations of tensor field components in isotropic systems with an application to stress correlations in elastic bodies.

Correlation functions of components of second-order tensor fields in isotropic systems can be reduced to an isotropic fourth-order tensor field characterized by a few invariant correlation functions (ICFs). It is emphasized that components of this field depend in general on the coordinates of the field vector variable and thus on the orientation of the coordinate system. These angular dependencies are distinct from those of ordinary anisotropic systems. As a simple example of the procedure to obtain the ICFs we discuss correlations of time-averaged stresses in isotropic glasses where only one ICF in reciprocal space becomes a finite constant e for large sampling times and small wave vectors. It is shown that e is set by the typical size of the frozen-in stress components normal to the wave vectors, i.e., it is caused by the symmetry breaking of the stress for each independent configuration. Using the presented general mathematical formalism for isotropic tensor fields this finding explains in turn the observed long-range stress correlations in real space. Under additional but rather general assumptions e is shown to be given by a thermodynamic quantity, the equilibrium Young modulus E. We thus relate for certain isotropic amorphous bodies the existence of finite Young or shear moduli to the symmetry breaking of a stress component in reciprocal space.

An improved analytical approach to establish the soliton solutions to the time‐fractional nonlinear evolution models

The time‐fractional Schrödinger (higher dimensional) and the Zakharov–Kuznetsov (ZK) models are the noteworthy modeling equations to interpret the propagation of signal in nonlinear optical fibers, small‐amplitude gravity waves on the surface water, high frequency acoustic waves, the interactions of small amplitude waves, the ion acoustic waves, nonlinear optics, Bose‐Einstein condensates, dynamics of plasma particles, etc. In this article, in order to develop advanced and broad‐spectrum closed‐form soliton solutions of the previously described nonlinear models, the advanced auxiliary equation approach has been put in used with reference to the beta‐derivative concept. The established soliton solutions are further comprehensive and typical and found in the form fractional, exponential, hyperbolic, and trigonometric functions and their variation. We depict 3D shape of the obtained solitons and explain the physical significance of the solitons. The accomplished solutions affirm that the method is effective and powerful and yields compatible solutions to fractional nonlinear evolution equations.

Hydrodynamic performance of wave energy converter integrated with pile restrained floating structure near a partially reflecting seawall

The integration of a Wave Energy Converter (WEC) with a Pile-Restrained Rectangular Floating Breakwater (PRFB) in the presence of a partially reflecting vertical seawall is analysed to enhance the hydrodynamic performance and WEC efficiency of the integrated breakwater-WEC device based on small amplitude wave theory using the Boundary Element Method (BEM). The rectangular floating breakwater is designed to have heave motion with a pile-restrained floating structure placed in a position to attenuate the incoming wave in the transmitted region and the linear power take-off (PTO) damping is employed to calculate the absorbed power. The study is performed to understand the effectiveness of wave energy conversion and its hydrodynamic performance due to changes in the seawall's porosity, relative structural width, relative structural draft, wave energy conversion power take-off damping coefficients, and the relative gap of the WEC integrated with PRFB from the seawall. The study demonstrated that in the presence of a fully reflecting seawall, the wave energy extraction is enhanced for the integrated WEC system without compromising the defined threshold wave reflection coefficient but at the expense of a constrained range of wavenumbers that correspond near the system's fundamental natural frequency. Moreover, the capture width ratio is noted to be higher for relatively smaller structural drafts, while the wave reflection coefficient shows precisely the reverse trend. However, under such circumstances, the integrated WEC system operates as a motion-trapping structure, especially when the reflection coefficient of the seawall, CR≥0.75. Thus, the present study will assist the designer in determining the appropriate degrees of efficiency of the WEC device without sacrificing hydrodynamic performance by fine-tuning the hybrid floating breakwater system's geometrical parameters.

Barrier and bottom topography effects on hydroelastic response of floating elastic plate in a two-layer fluid

The prime intention of this article is to investigate the effect of variable bottom topography and a bottom-sitting porous barrier on the hydroelastic response of an elastic plate floating on a two-layer fluid using small amplitude wave theory. Galerkin's single-mode approximation in each layer for variable bottom topography and the method of eigenfunction expansion for the fluid region of uniform bottom topography are used as mathematical tools to detail the phenomena. In the variable bottom topography, a system of differential equations is solved. By applying matching conditions, jump conditions, and the appropriate boundary conditions, the solution is expressed as an algebraic linear system from which all the unknown constants are evaluated. The effects of different parameters related to the fluid, bottom topography, and porous barrier on the bending moment, shear force, and deflection of an elastic plate are explored. The variations in the bending moments, shear forces, and plate deflection with respect to fluid density are found to be in opposite trends, caused by surface and interfacial waves, respectively. Further, as the density ratio becomes closer to one, the bending moments, shear forces, and plate deflection tend to diminish for interfacial waves. The bottom effect on bending moments, shear forces, and plate deflection is minimal due to surface waves but significant due to interface waves with maximum amplitude in a concave up bottom. The plate deformation can be further reduced using a suitable barrier, as investigated in this article. The findings hold good potential for furthering our understanding of installing an elastic plate in a stratified fluid with fluctuating water depth.

Forces on a Vertical Dam due to Solitary Impulse Wave Run-Up and Overtopping

Impulse waves feature tsunami-like characteristics and are generated by very rapid mass wasting, including landslides and avalanches, into water bodies. In engineered reservoirs, these waves may run up and overtop the dam, thereby exerting hydrostatic and dynamic forces on its structure. For this work, solitary waves were applied as a proxy for impulse waves and the forces acting on vertical structures during wave run-up and overtopping were investigated with hydraulic laboratory experiments. The solitary waves were generated in a wave channel with a piston-type wave generator and the horizontal pressure forces at a vertical wall as well as a vertical dam-like structure were measured with multiple pressure sensors. The discrete pressure data were used to interpolate pressure distributions and the resulting horizontal forces. Empirical equations were derived, approximating the measured maximum forces and their respective centers of pressure within ±7% for both cases with and without wave overtopping. For small freeboards, already small relative wave amplitudes may exert forces similar to those induced by an earthquake as estimated with a pseudo-static approach.

Identifiying the PID-N method performance for speed control of BLDC motor propeller on catamaran ships model

Some archipelagic countries must have wide sea. This condition means that there is transportation of goods and people problem. Therefore, an effort has been made to explore renewable energy especially solar energy. In this research, a BLDC (Brush Less Direct Current) motor propeller is applied, and PID-N (Proportional, Integral, and Derivative – Filter Coefficient) method for Catamaran ship speed control. Where PID-N has additional filters such as P, I, and D; however, the advanced performance of this control uses RLS (Recursive Least Square) actuator modeling is the most systematical and flexible design. The experiment includes the speed response of ships with payload and without payload, as well as changes in speed settings. The results show that the ship's speed control using the PID-N has smooth and definite speed control. The PID-N design is systematical, easy and quick for the control designer to change the actuator or sensor performance, if there will be any actuator or sensor replacement with different specifications, which is required in commercial. As it is known, in commercial design, time and method are crucial and has to be effective. The PID-N control method shows its superiority. In detail, the Catamaran speed setting is incremented from 0.4 m/s to 0.6 m/s without payload, the average settling time is 6 seconds with the average error speed 0.071 m/s in the set speed 0.6 m/s. For setting speed increased by 0.4 m/s to 0.6 m/s with payload, settling time gets longer, 7 to 13 seconds with the average error speed 0.01 m/s in the set speed 0.06 m/s. It performs small errors and fasts settling time. The ship model has experimented on an artificial pond, it is related to the proportional small sea wave. The research also shown the ratio of electric power consume toward speed setting is in exponential formula

Identification of Subsurface Mesoscale Crack in Full Ceramic Ball Bearings Based on Strain Energy Theory

Subsurface mesoscale cracks exist widely in the outer ring of full ceramic ball bearings (FCBBs), which is a potential threat for the stable operation of related devices such as aero engines, food processing machinery, and artificial replacement hip joints. This paper establishes a dynamic model of subsurface mesoscale cracks in the outer ring of FCBBs based on strain energy theory, and the influence of different crack lengths on the running state is analyzed. The existence of mesoscale cracks is regarded as weakening on the stiffness coefficient, and the deterioration degree of outer ring stiffness of subsurface cracks is thereby quantified. It is found that a small wave peak appears in the vibration time-domain signal when there is a mesoscale crack on the outer ring subsurface, and the crack evolution is evaluated by the amplitude of the corresponding feature frequency. Finally, the accuracy of the model is verified by experiments. The model realizes the identification and degree evaluation of subsurface mesoscale cracks in FCBBs, and provides theoretical references for the diagnosis and status monitoring for FCBB rotor systems.

Manipulation of nonautonomous nonlinear wave solutions of the generalized coupled Gross–Pitaevskii equations with spin–orbit interaction and weak Raman couplings

Using unitary and similarity transformations, explicit solutions of the coupled Gross–Pitaevskii models with spin–orbit coupling and varying nonlinearities and harmonic plus linear potentials are constructed by a mapping to the integrable one-dimensional Manakov system under the constraints of weak Raman coupling and small amplitude waves. An intriguing example for controllability of the dynamics is obtained for the case of a linear potential that induces complex spatial curvatures onto the density profiles. Peregrine solitons are also found demonstrating that the system may sustain rogue waves. It is shown that adiabatic modulations of external potentials and/or two-body atomic interactions may be used to efficiently control the dynamics of the constructed nonlinear waves. Numerical simulations demonstrate the stability of the obtained solutions.