Small-amplitude Waves Research Articles

Application of quadratic pressure drop condition in hydroelastic analysis of floating structures protected by perforated barrier

This paper presents the application of the quadratic pressure drop condition in hydroelastic analysis of large floating structures protected by perforated barrier. An iterative procedure is proposed to incorporate the quadratic pressure drop condition into the recently developed finite element – dual boundary element method. Numerical examples are carried out to compare the proposed quadratic model with the existing model involving the conventional linear pressure drop condition. It is found that the proposed quadratic model agrees rather well with the linear model for small wave amplitudes. However, for strong waves, there are significant differences between the quadratic model and linear model in terms of the resulting excitation force on the barrier and hydroelastic responses of floating structures. The excitation force from the quadratic model may be up to 100% higher than that from the linear model.

Hybrid linear potential flow - machine learning model for enhanced prediction of WEC performance

Numerical models based on the linear potential flow equations are of paramount importance in the design of wave energy converters (WECs). Over the years methods such as wave stretching, nonlinear Froude-Krylov and Morrison drag have been developed to overcome the short-comings of the underlying assumptions of small amplitude wave, small motion and inviscous flow. In this work we present a different approach to enhance the performance of the linear method: a hybrid linear potential flow – machine learning (LPF-ML) model. A hierarchy of high-fidelity models – Reynolds-Averaged Navier-Stokes, Euler and fully nonlinear potential flow – is used to create training data for correction factors targeting nonlinear hydrodynamics, pressure drag and skin friction, respectively. Long short-term memory (LSTM) networks are then trained and added to the LPF model. LSTM networks are heavy to train but fast to evaluate so the computational efficiency of the LPF model is kept high. Simple decay tests of generic bodies (sphere, box, etc) are used to validate the LPF-ML model. Finally, the LPF-ML is applied to a model-scale point-absorber WEC to assess the power production.

Theoretical study of magnetoacoustic mode in stellar atmospheres

Radiation magnetohydrodynamic (RMHD) theory is presented in the nonequilibrium diffusion grey radiation limit. Basic set of equations is perturbed to find the dispersion relation of small amplitude (linear) waves in RMHD plasma. It is found that magnetoacoustic waves have dispersion and damping in the RMHD plasma. It is also concluded that for very long wavelength magnetosonic mode, to create wave motion in the medium the restoring force is provide the radiation pressure and inertia is provided by matter. The effects of radiation energy, diffusivity and plasma beta on the dispersion and damping of magnetoacoustic mode in RMHD plasma are shown in the form of graphs. The model is applicable to the stellar atmosphere.

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.

Spectral instability of small-amplitude periodic waves of the electronic Euler–Poisson system

The present work shows that essentially all small-amplitude periodic traveling waves of the electronic Euler–Poisson system are spectrally unstable. This instability is neither modulational nor co-periodic, and thus requires an unusual spectral analysis and, beyond specific computations, newly devised arguments. The growth rate with respect to the amplitude of the background waves is also provided when the instability occurs.

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.

Solution Blow-Up and Global Solvability of the Cauchy Problem for the Equation of Moderately Long Longitudinal Waves in a Viscoelastic Rod

The Cauchy problem for a nonlinear Sobolev-type differential equation modeling moderately long small-amplitude longitudinal waves in a viscoelastic rod is investigated in a space of continuous functions defined on the entire number line that have limits at infinity. Conditions for the existence of a global solution and for finite time solution blow-up are examined.

Wheel-structured Triboelectric Nanogenerators with Hyperelastic Networking for High-Performance Wave Energy Harvesting.

Developing clean and renewable energy sources is an important strategy to reduce carbon emission and achieve carbon neutrality. As one of the most promising clean energy sources, large-scale, and efficient utilization of ocean blue energy remains a challenging problem to be solved. In this work, a hyperelastic network of wheel-structured triboelectric nanogenerators (WS-TENGs) is demonstrated to efficiently harvest low-frequency and small-amplitude wave energy. Different from traditional designs of smooth shell, the external blades on the TENG allow tighter interaction between the wave and the device, which can roll on the water surface like a wheel, continuously agitating internal TENGs. Moreover, the hyperelastic networking structure can stretch and shrink like a spring with stored wave energy, further intensifying the roll of the device, and connecting the WS-TENGs to form a large-scale network. Multiple driving modes with synergistic effects can be realized under wave and wind excitations. Self-powered systems are fabricated based on the WS-TENG network, showing the capability of the device in real wave environment. The work provides a new driving paradigm that can further enhance the energy harvesting capability toward large-scale blue energy utilization based on TENGs.

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.

Theoretical Analysis Method for Roll Motion of Popup Data Communication Beacons

The popup data communication beacon (PDCB) can send data to the shore and ships through the BeiDou navigation satellite system (BDS) when it surfaces. The data can be collected by a deep-sea landing vehicle (DSLV) and transmitted using a magnetic induction coil. PDCBs can reduce the cost of DSLV recovery and redeployment. Whether the data can be successfully sent mainly depends on the outlet height and roll angle of the PDCB. Thus, accurately assessing the effect of the roll angle on data transmission is crucial. In this study, first, the differential equation of roll motion was preliminarily established using the small-amplitude wave theory along with the shape characteristics of the PDCB. Next, the nonlinear term of the recovery moment was processed using the Linz Ted Poincaré method. Then, the wave current force was analyzed using the Morrison theoretical formula along with an additional inertia moment calculation formula that is suitable for slender cylindrical small buoys. Finally, the theoretical calculation results were verified using the computational fluid dynamics (CFD) method and pool test. The roll angle error of the theoretical calculation was within 5%. Thus, the heave and roll response of PDCBs can be evaluated using theoretical calculation methods. The proposed calculation formula of additional inertia moment has guiding significance for the further optimization of the structure.

Probing modified plasma waves in non-linear electrodynamics

Properties of modified plasma waves in non-linear electrodynamics are investigated. We consider a cold, uniform, collisionless, and magnetized plasma model. Initially, we also assume small amplitude waves and the non-relativistic approximation. For electrostatic waves, we obtain a modified Trivelpiece–Gould dispersion relation with a suitable change in the plasma frequency and analyze the stability of modes. Furthermore, electromagnetic waves related to the generalized Appleton–Hartree equation are established. In this case, we discuss modifications in circularly polarized waves and ordinary and extraordinary modes. After that, we apply our results to particular cases of low-energy quantum electrodynamics and a generalized Born–Infeld model. The correspondent dispersion relations and effects on the propagation regions are determined. Finally, we include the relativistic and large amplitude effects for circularly polarized waves. We obtain the dispersion relation within effective non-linear electrodynamics and examine the behavior of the refractive index when the frequency of the propagating wave converges to the plasma frequency.

Attenuation in hydroelastic response of floating-elastic-plate by porous membrane in two-layer fluid with bottom undulation

The prime intention of this article is to study the effects of bottom undulation and a porous membrane barrier on the hydroelastic response of an elastic plate floating in a two-layer fluid with changing bottom topography assuming the small amplitude waves and potential flow theory. Finite and semi-infinite plates coupled with porous membrane barriers are considered in this study. As a mathematical technique, the eigenfunction expansion method along with an approximation method mild-slope are adopted for the uniform bottom and variable bottom topographies, respectively. In the variable bottom topography, a system of differential equations is solved. By using matching and jump conditions, the solution is expressed as a linear algebraic system, from which the unknown constants of interest are computed. The effects of fluid density ratio, water depth ratio, bottom topography, and porous membrane on the bending moment, shear force, and deflection of the elastic plate are studied. It is found that when the density ratio becomes closer to one, the bending moments and shear forces acting on the elastic plates tend to diminish. The variations in the bending moments, shear forces, and plate deflection with respect to changing density ratios are found to be in opposite trends caused by surface and interfacial waves, respectively. With higher tension and a less porous membrane, the bending moment, shear forces, and pate deflection are reduced. Concave-up bottoms reflect more surface wave energy, whereas concave-down bottoms reflect more interfacial wave energy. The observations made in this article may help in analyzing the response of very-large floating structures in the presence of an undulating seabed.

Nonlinear Modulational Instabililty of the Stokes Waves in 2D Full Water Waves

The well-known Stokes waves refer to periodic traveling waves under the gravity at the free surface of a two dimensional full water wave system. In this paper, we prove that small-amplitude Stokes waves with infinite depth are nonlinearly unstable under long-wave perturbations. Our approach is based on the modulational approximation of the water wave system and the instability mechanism of the focusing cubic nonlinear Schrödinger equation.

Mechanical multi-level memory from multi-stable metamaterial

In this Letter, we consider the dynamics of a multi-stable metamaterial with an elastic substrate to realize a mechanical system within which the position of a transition wavefront can be precisely controlled and remotely determined. This ability is enabled, in part, by a (strain-)tunable potential energy landscape that conveys the wavefront from one stabilizing defect site to another. In separating two acoustically distinct domains, the wavefront reflects small-amplitude waves of appropriate frequency back toward the source whereupon the time interval between excitation and echo reveals the position of the front. In a numerical study, we exploit these mechanisms for mechanical multi-level memory, which may find applications, e.g., in soft robots as a flexible alternative to current rigid memory technologies. More generally, we anticipate that the concepts presented here toward a command of the transition wave position will enhance the development and applicability of multi-stable metamaterials.

Propagation of ion-acoustic wave and its fractal representations in spin polarized electron plasma

Propagation of small-amplitude quantum ion-acoustic waves and its fractal representations is investigated in an electron-ion quantum plasma with separated spin electrons in the framework of the KdV and EKdV equations derived using reductive perturbation technique. These two equations are transformed into planar dynamical systems by using suitable traveling wave transformation. Qualitatively different phase portraits of these two systems are drawn with their respective Sagdeev’s pseudopotential curves and surface plots. Distinct orbits in the phase portraits give rise to distinct wave solutions. Periodic and superperiodic wave solutions are investigated numerically and solitary wave solutions are derived analytically. The effect of parameters such as Mach number, direction cosine, spin polarization and frequency ratio on these wave solutions are presented. For instance, it is seen that the spin density polarization ratio has an impressive effect on the amplitude of the wave solution, while the frequency ratio has no effect on its amplitude. Also, we have provided a physical explanation for our finding. Further, the dynamical features of the original system and reconstructed system using fractal interpolation function are investigated under the external forcing term. Chaotic and quasiperiodic phenomena are observed for different initial conditions. The results may help to better understand the degenerate electron gas that exists in dense astrophysical bodies.

Effect of inconsistency of irradiation conditions of acoustic radiation force caused by tissue structure on shear wave velocity evaluation

Previous studies have shown that shear wave elastography of liver tissue can be unstable due to factors such as uncertainties in the acoustic radiation force (ARF) irradiation due to the influence of tissues near the surface and the complexity of the liver’s structure and its physical properties. This study aims to verify the influence of near-surface tissues on ARF and the effect of tissue structure on shear wave propagation and shear wave velocity (SWV) evaluation using wave propagation simulations by the elastic finite-difference time domain method. It is found that the ARF becomes weakly focused on multiple locations due to refraction of longitudinal waves by near-surface tissues, and multiple shear waves of small amplitude are propagated. However, a macroscopic SWV assessment, as in clinical practice, reduces the influence of near-surface tissues because the microscopic assessment results are averaged over the near-surface tissues.

Galilei-invariant and energy-preserving extensions of Benjamin–Bona–Mahony-type equations

The classical Benjamin–Bona–Mahony equation (BBM equation) models unidirectional propagation of long gravity surface waves of small amplitude. Unlike many other water wave models, it lacks the Galilean invariance, which is an essential property of physical systems. It is shown that by an addition of a higher asymptotic order nonlinear term, this deficiency can be corrected, giving rise to a new Galilei invariant Benjamin–Bona–Mahony equation (iBBM equation). Moreover, further additional higher-order terms can be chosen in a way that the augmented model preserves the energy conservation property along with Hamiltonian and Lagrangian structures. The resulting equation is referred to as energy-preserving Benjamin–Bona–Mahony equation (eBBM).It is shown that both the classical BBM equation and the energy-preserving eBBM equations belong to a one-parameter (α) family that shares essentially the same local and nonlocal symmetries, conservation laws, Hamiltonian, and Lagrangian structures, with the BBM and eBBM equations corresponding to parameter values α=0 and α=1, respectively. Symmetry and conservation law classifications reveal a special case α=1/3, which is shown to correspond to a rescaled version of the celebrated integrable Camassa–Holm (CH) equation. Local symmetries and conservation laws are computed, and numerical solution behaviour is compared for the three BBM-type modes and the CH-equivalent eBBM1/3 model.