Interaction Of Water Waves Research Articles

Interaction of surface water waves with a vertical elastic plate: a hypersingular integral equation approach

In this paper, we present an alternative method to investigate scattering of water waves by a submerged thin vertical elastic plate in the context of linear theory. The plate is submerged either in deep water or in the water of uniform finite depth. Using the condition on the plate, together with the end conditions, the derivative of the velocity potential in the direction of normal to the plate is expressed in terms of a Green’s function. This expression is compared with that obtained by employing Green’s integral theorem to the scattered velocity potential and the Green’s function for the fluid region. This produces a hypersingular integral equation of the first kind in the difference in potential across the plate. The reflection coefficients are computed using the solution of the hypersingular integral equation. We find good agreement when the results for these quantities are compared with those for a vertical elastic plate and submerged and partially immersed rigid plates. New results for the hydrodynamic force on the plate, the shear stress and the shear strain of the vertical elastic plate are also evaluated and represented graphically.

Modeling the Interaction of Water Waves with Porous Coastal Structures

AbstractThe mathematical modeling of the interaction of water waves with porous coastal structures has continuously been among the most relevant challenges in coastal engineering research and practice. Finding a tool to better predict essential processes, relevant to the functionality and stability of breakwaters and jetties, and how they are affected by permeability, has been hampered by computational limitations that are being overcome. Over the last 60 years, the Journal of Waterway, Port, Coast, and Ocean Engineering has witnessed gradual developments leading from linearized solutions based on wave theories and constant friction coefficients to very sophisticated Eulerian or Lagrangian solvers of the Navier-Stokes (NS) equations, including turbulence within porous media. Today, although not without difficulty, the first steps are being made toward addressing the simulation of a fully three-dimensional interaction of complete sea states with porous structures at prototype scale. In this paper, after po...

Water wave interaction with two symmetric inclined permeable plates

We consider the reflection and the transmission of the surface water waves by two inclined identical permeable plates that are symmetric with respect to a line which is perpendicular to the horizontal line connecting the middle points of the plates. The problem is reduced to finding the solutions of two hypersingular integral equations of the second kind for the discontinuity in the velocity potential across one of the plates. The unknowns of the integral equations are expanded in terms of a finite series of Chebyshev polynomials multiplied by an appropriate weight function and then solved by collocating at suitable discrete points. Using these solutions, the numerical estimates for the reflection and the transmission coefficients, amplitudes of the hydrodynamic forces and moments and the wave dissipation coefficient are computed. Numerical results for these quantities are presented graphically in a number of figures. These results are compared with the published results for a single vertical as well as horizontal permeable plates and a submerged horizontal impermeable plate.

Numerical studies of the KP line-solitons

The Kadomtsev–Petviashvili (KP) equation admits a class of solitary wave solutions localized along distinct rays in the xy-plane, called the line-solitons, which describe the interaction of shallow water waves on a flat surface. These wave interactions have been observed on long, flat beaches, as well as have been recreated in laboratory experiments. In this paper, the line-solitons are investigated via direct numerical simulations of the KP equation, and the interactions of the evolved solitary wave patterns are studied. The objective is to obtain greater insight into solitary wave interactions in shallow water and to determine the extent the KP equation is a good model in describing these nonlinear interactions.

Hydrodynamic interactions of water waves with a group of independently oscillating truncated circular cylinders

In this study, we examine the water wave radiation by arrays of truncated circular cylinders. Each cylinder can oscillate independently in any rigid oscillation mode with a prescribed amplitude, including translational and rotational modes such as surge, sway, heave, pitch, roll, and their combinations. Based on the eigenfunction expansion and Graf’s addition theorem for Bessel functions, we developed an analytical method that includes the effects of evanescent modes in order to analyze such arrays of cylinders. To investigate the effects of several influential factors on convergence, our objective is to dramatically reduce the number of tests required and determine the influencing relationships between truncation number and convergence behavior for different factor combinations. We use the orthogonal test method to fulfill the objective. Lastly, we present our results regarding the effects of evanescent modes on hydrodynamic coefficients.

Interaction of modulated gravity water waves of finite depth

We consider the capillary-gravity water wave problem of finite depth with a flat bottom of one or two horizontal dimensions. We derive the modulation equations of leading and next-to-leading order in the hyperbolic scaling for three weakly amplitude-modulated plane wave solutions of the linearized problem in the absence of quadratic and cubic resonances. We justify the derived system of macroscopic equations in the case of gravity waves using the stability of the finite depth water wave problem on the time scale O(1/ϵ).

Surface wave interaction with rigid plates lying on water

In this paper, a variety of problems concerned with the interaction of water waves with fixed horizontal plates lying on the surface of a fluid are investigated.Firstly, solutions are presented to the problem of the scattering of incident waves by: (i) infinitely-long plates of constant finite width (often referred to as the two-dimensional ‘finite dock problem’) and (ii) finite plates which are either rectangular or parallelogram-shaped. Secondly, hydrodynamic coefficients due to forced motions of plates are also considered. Finally, eigenvalue problems associated with free oscillations of the surface in long channels of uniform width and finite rectangular holes in an otherwise infinite rigid plate covering the surface are considered.A common method of solution is applied to all problems which involves using Fourier transforms to derive integral equations for unknown potentials over finite regions of space occupied by either plates or the free surface. Integral equations are converted, using the Galerkin method, into second-kind infinite systems of algebraic equations. In each problem numerical approximations to the solutions are found to converge rapidly with increasing truncation size of the infinite system making this approach both numerically efficient and accurate. Some comparisons with existing results are made, and new results for finite plates are demonstrated.

The interaction of oblique flexural gravity waves with a small bottom deformation on a porous ocean-bed: Green’s function approach

The interaction of oblique incident water waves with a small bottom deformation on a porous ocean-bed is examined analytically here within the framework of linear water wave theory. The upper surface of the ocean is assumed to be covered by an infinitely extended thin uniform elastic plate, while the lower surface is bounded by a porous bottom surface having a small deformation. By employing a simplified perturbation analysis, involving a small parameter δ(=1), which measures the smallness of the deformation, the governing Boundary Value Problem (BVP) is reduced to a simpler BVP for the first-order correction of the potential function. This BVP is solved using a method based on Green’s integral theorem with the introduction of suitable Green’s function to obtain the first-order potential, and this potential function is then utilized to calculate the first-order reflection and transmission coefficients in terms of integrals involving the shape function c(x) representing the bottom deformation. Consideration of a patch of sinusoidal ripples shows that when the quotient of twice the component of the incident field wave number propagating just below the elastic plate and the ripple wave number approaches one, the theory predicts a resonant interaction between the bed and the surface below the elastic plate. Again, for small angles of incidence, the reflected wave energy is more as compared to the other angles of incidence. It is also observed that the reflected wave energy is somewhat sensitive to the changes in the flexural rigidity of the elastic plate, the porosity of the bed and the ripple wave numbers. The main advantage of the present study is that the results for the values of reflection and transmission coefficients obtained are found to satisfy the energy-balance relation almost accurately.

Bäcklund transformation, infinite conservation laws and periodic wave solutions to a generalized (2+1)-dimensional Boussinesq equation

Under investigation in this paper is a generalized (2+1)-dimensional Boussinesq equation, which can be used to describe the water wave interaction. By using Bell polynomials, a lucid and systematic approach is proposed to systematically study the integrability of the equation, including its bilinear representation, soliton solutions, periodic wave solutions, Bäcklund transformation and Lax pairs, respectively. Furthermore, by virtue of its Lax equations, the infinite conservation laws of the equation are also derived with the recursion formulas. Finally, the asymptotic behavior of periodic wave solutions is shown with a limiting procedure.

Singular boundary method for water wave problems

This article presents the singular boundary method (SBM) for water wave-structure interaction analysis. The SBM is a novel recent boundary meshless collocation method, which uses the singular fundamental solutions of governing equation as basis function in the interpolation formulation and introduces the concept of source intensity factors to avoid the singularities of the fundamental solutions at origin. It is a strong-form meshless boundary collocation technique with the merits being easy-to-implement and integration-free. In this study, two types of water wave-structure interactions, water wave interaction of multiple-cylinder-array structure and semicircular harbor, are considered. Numerical results show the efficiency and accuracy of the present SBM scheme in comparison with the analytical solutions, the method of fundamental solutions and the null field boundary integral element method. Then the near trapped mode phenomenon of ten-cylinder structure and the focusing effect of the semicircular harbor are revisited by using the present SBM.

Mathematical Modeling of Partial-Porous Circular Cylinders with Water Waves

The interaction of water waves with partially porous-surfaced circular cylinders was investigated. A three-dimensional numerical modeling was developed based on the complete mathematical formulation of the eigenfunction expansion method in the potential flow. Darcy’s law was applied to describe the porous boundary. The partial-porous cylinder is composed of a porous-surfaced body near the free surface, and an impermeable-surfaced body with an end-capped rigid bottom below the porous region. The optimal ratio of the porous portion to the impermeable portion can be adopted to design an effective ocean structure with minimal hydrodynamic impact. To scrutinize the hydrodynamic interactions inNpartial-porous circular cylinders, the computational fluid domain is divided into three regions: an exterior region,Ninner porous body regions, andNregions beneath the body. Wave excitation forces and wave run-up on multibodied partial-porous cylinders are calculated and compared for various porous-portion ratios and wave conditions, all of which significantly influence the hydrodynamic property.

Numerical analysis of a hybrid substructure for offshore wind turbines

For the reliable design of substructure supporting offshore wind turbines it is very important to reduce the effects of wave forces. Since the substructure is strongly influenced by the effects of wave forces as the size of substructure increases. In the present study, the hybrid substructure with multi-cylinder is newly suggested to reduce the effects of wave forces. Using diffraction theory the scattering waves in a fluid region are expressed by an Eigenfunction expansion method with three dimensional potential theory to calculate the wave force acting on the hybrid substructure. The wave force and wave run-up acting on the hybrid substructure is presented to examine the water wave interaction according to the variation of cylindrical size and the distance among cylinders. It is found that the suggested hybrid substructure with multi-cylinder is very useful to reduce the effects of wave forces acting on the substructure for offshore wind turbines.

Wave reflection and vortex evolution in Bragg scattering in real fluids

The vortex generation and dissipation accompanying Bragg scattering of water waves propagating over a series of submerged rectangular breakwaters are investigated both numerically and experimentally. The present model, which takes effects of non-linearity, viscosity and turbulence into consideration, is applied to simulate the entire vortex evolution process as water waves pass over a series of artificial rectangular bars. Particle image velocimetry is used to measure the velocity field in the vicinity of the bars. The numerical model is validated via a comparison with measured water surface elevations and velocity fields; the results show good agreement. The mechanism of vortex evolution and its influence on the interaction of water waves with submerged structures for cases with and without resonance are studied. The corresponding wave reflection coefficients for both cases are calculated and compared with experimental data and solutions based on linear wave theory. Examination of the turbulence properties shows that the turbulent intensity decreases on the weather side but increases on the lee side under the Bragg scattering conditions.

Hydroelastic analysis of VLFS based on a consistent coupled-mode system and FEM

Three models for the interaction of water waves with large floating elastic structures are analysed. The first model, based on the Euler–Bernoulli beam theory, has already been extensively studied. The second is based on the Rayleigh beam equation. The third approach utilises the Timoshenko approximation and is thus capable of incorporating shear deformation and rotary inertia effects. A novelty of the proposed hydroelastic systems is the consistent local mode expansion of the underlying hydrodynamic field interacting with the floating structure, which leads to coupled-mode systems with respect to the modal amplitudes of the wave potential and the surface elevation. This representation is rapidly convergent to the solution of the full hydroelastic problem. The dispersion relations of these models are derived and analysed, supporting at a next stage the efficient development of finite element method solvers of the coupled system.

Interaction of dispersive water waves with weakly sheared currents of arbitrary profile

A set of depth-integrated equations describing combined wave–current flows is derived and validated. To account for the effect of turbulence induced by interactions between waves and currents with arbitrary horizontal vorticity, new additional stress terms are introduced. These stresses are functions of a parameter b that relates the relative importance of wave radiation stress and bottom friction stress to the wave–current interaction. To solve the equations, a fourth-order MUSCL-TVD scheme with an approximate Riemann solver is adopted. As a first-order check of the model, the Doppler shift effect and wave dispersion over linearly sheared currents are analytically shown to be retained appropriately in the equation set. The model results are then validated through comparisons with three experimental data sets. First, based on the experiments of Kemp and Simons (1982, 1983), a reasonable functional form of b is estimated. Second, simulations examining the propagation of a weakly dispersive wave over a depth-uniform or linearly sheared current are performed. Finally, the model is applied to a more complex configuration where bichromatic waves interact with spatially varying currents. Simulated results indicate that the model is capable of predicting nearshore interactions of waves with currents of arbitrary vertical structure. One of the unique properties of the developed model is its ability to assimilate an external current field from any source, be it from a circulation model or an observation, and predict the interaction of a nonlinear and dispersive wave field with that current.

Linearised theory for surface and interfacial waves interacting with freely floating bodies in a two-layer fluid

We derive a linear system of equations governing the interaction of water waves with partially or totally submerged freely floating structures in a two-layer fluid. We establish conditions for the stability of equilibrium and, by considering time-harmonic motions, rewrite the problem as a spectral boundary-value problem consisting of a differential equation and an algebraic system, coupled through boundary conditions. We give also a suitable variational formulation for the problem and provide examples of configurations where the problem admits only the trivial solution.

Theory and experiment of the resonance sound wave interactions in water

Equations governing nonlinear interactions among acoustic waves in water which satisfy the resonance conditions are derived from the Burgers equation. Taking into account of the second-order nonlinear, the three-wave interaction is the fundamental process of the interaction equations. Then, taking the three-wave interaction equation as an example, the energy transfer mechanism among the three acoustic waves is quantitatively analyzed. And the three-wave sound energy propagation is studied through numerical calculation. An interesting phenomenon is found that, when the three acoustic waves meet the relationship of a weak low-frequency sound and two strong high-frequency sounds, the energy of low-frequency sound will be amplified or reduced in some regions during the three-wave propagation. And the location and size of the regions are affected by the acoustic wave amplitude, frequency, and phase. The variation severities of low-frequency acoustic wave energy are mainly determined by other two high-frequency acoustic wave energy. The frequencies of two high-frequency sound waves have little effects on the energy of low-frequency sound wave. The region whether is amplification or reduction is determined by the phase difference of three waves. The variation laws of low-frequency sound energy are also verified by the experimental results in river.

Darboux Transformations and N-soliton Solutions of Two (2+1)-Dimensional Nonlinear Equations

Two Darboux transformations of the (2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawaka (CDGKS) equation and (2+1)-dimensional modified Korteweg-de Vries (mKdV) equation are constructed through the Darboux matrix method, respectively. N-soliton solutions of these two equations are presented by applying the Darboux transformations N times. The right-going bright single-soliton solution and interactions of two and three-soliton overtaking collisions of the (2+1)-dimensional CDGKS equation are studied. By choosing different seed solutions, the right-going bright and left-going dark single-soliton solutions, the interactions of two and three-soliton overtaking collisions, and kink soliton solutions of the (2+1)-dimensional mKdV equation are investigated. The results can be used to illustrate the interactions of water waves in shallow water.

WKB analysis of the Schrödinger–KdV system

We consider the behavior of solutions to the water wave interaction equations in the limit ε→0+. To justify the semiclassical approximation, we reduce the water wave interaction equation into some hyperbolic-dispersive system by using a modified Madelung transform. The reduced system causes loss of derivatives which prevents us to apply the classical energy method to prove the existence of solution. To overcome this difficulty we introduce a modified energy method and construct the solution to the reduced system.

Steady-state resonance of multiple wave interactions in deep water

AbstractThe steady-state resonance of multiple surface gravity waves in deep water was investigated in detail to extend the existing results due to Liao (Commun. Nonlinear Sci. Numer. Simul., vol. 16, 2011, pp. 1274–1303) and Xu et al. (J. Fluid Mech., vol. 710, 2012, pp. 379–418) on steady-state resonance from a quartet to more general and coupled resonant quartets, together with higher-order resonant interactions. The exact nonlinear wave equations are solved without assumptions on the existence of small physical parameters. Multiple steady-state resonant waves are obtained for all the considered cases, and it is found that the number of multiple solutions tends to increase when more wave components are involved in the resonance sets. The topology of wave energy distribution in the parameter space is analysed, and it is found that the steady-state resonant waves indeed form a continuum in the parameter space. The significant roles of the near-resonance and nonlinearity were also revealed. It is found that all of the near-resonant components as a whole contain more and more wave energy, as the wave patterns tend from two dimensions to one dimension, or as the nonlinearity of the steady-state resonant wave system increases. In addition, the linear stability of the steady-state resonant waves is analysed. It is found that the steady-state resonant waves are stable, as long as the disturbance does not resonate with any components of the basic wave. All of these findings are helpful to enrich and deepen our understanding about resonant gravity waves.