Efficient non-hydrostatic modelling of surface waves interacting with structures

An efficient three-dimensional non-hydrostatic model is applied to simulate free-surface waves interacting with structures. The model employs an implicit Crank–Nicholson scheme to discretize the Navier–Stokes equations under a Cartesian staggered grid framework. An integration method is introduced to account for the full effects of non-hydrostatic pressure at the free-surface layer. A domain decomposition method is proposed to effectively solve the resulting matrix system. The model is first validated by simulating three-dimensional sloshing waves in a container. The model is then applied to simulate waves propagating over two-dimensional and three-dimensional submerged structures, in which the effects of non-linearity and dispersion are important. The model results show that the model using only two vertical layers are in all favorable agreements with experimental data, demonstrating the efficiency and accuracy of the model on simulating surface waves interacting with structures.     

Finite element modelling of surface waves in a channel

A variational formulation of the vertically-integrated differential equations for free surface wave motion is presented. A finite element model is derived for solving this nonlinear system of hydrodynamic equations. The time integration scheme employed is discussed and the results obtained demonstrate its good stability and accuracy.Several applications of the model are considered: the first problem is an open channel of uniform depth and the second an open channel of linearly varying depth. The ‘inflow’ boundary condition is prescribed in terms of the velocity which represents a wavemaker and/or a flow source, while the ‘outflow’ boundary condition is specified in terms of the water elevation. The outflow condition is adjusted for two cases, a reflecting boundary (finite channel) and a non-reflecting boundary (open-ended channel). The latter boundary condition is examined in some detail and the results obtained show that the numerical model can produce the non-reflecting boundary that is similar to the analytical radiation condition for waves. Computational results for a third problem, involving wave reflection from a submerged cylinder, are also presented and compared with both experimental data and analytical predictions.The simplicity and the performance of the computational model suggest that free surface waves can be simulated without excessively complicated numerical schemes. The ability of the model to simulate outflow boundary conditions properly is of special importance since these conditions present serious problems for many numerical algorithms.     

Multi-bump standing waves with a critical frequency for nonlinear Schrödinger equations

In this paper we study the existence and qualitative property of standing wave solutions for the nonlinear Schrödinger equation with E being a critical frequency in the sense that . We show that if the zero set of W−E has several isolated connected components Z_i(i=1,…,m) such that the interior of Z_i is not empty and ∂Z_i is smooth, then for ?>0 small there exists, for any integer k,1?k?m, a standing wave solution which is trapped in a neighborhood of , where is any given subset of . Moreover the amplitude of the standing wave is of the level . This extends the result of Byeon and Wang (Arch. Rational Mech. Anal. 165 (2002) 295) and is in striking contrast with the non-critical frequency case , which has been studied extensively in the past 20 years.     

Two-phase flow modelling of spilling and plunging breaking waves

A two-phase flow model, which solves the flow in the air and water simultaneously, has been employed to investigate both spilling and plunging breakers in the surf zone with a focus during wave breaking. The model is based on the Reynolds-averaged Navier–Stokes equations with the k–?$k''–''?$ turbulence model. The governing equations are solved using the finite volume method, with the partial cell treatment being implemented in a staggered Cartesian grid to deal with complex geometries. The PISO algorithm is utilised for the pressure–velocity coupling and the air–water interface is modelled by the interface capturing method via a high-resolution volume of fluid scheme. Numerical results are compared with experimental measurements and other numerical studies in terms of water surface elevations, mean flow and turbulence intensity, in which satisfactory agreement is obtained. In addition, water surface profiles, velocity and vorticity fields during wave breaking are also presented and discussed. It is shown that the present model is capable of simulating the wave overturning, air entrainment and splash-up processes.     

Hamiltonian model for coupled surface and internal waves in the presence of currents

We examine a two dimensional fluid system consisting of a lower medium bounded underneath by a flatbed and an upper medium with a free surface. The two media are separated by a free common interface. The gravity driven surface and internal water waves (at the common interface between the media) in the presence of a depth-dependent current are studied under certain physical assumptions. Both media are considered incompressible and with prescribed vorticities. Using the Hamiltonian approach the Hamiltonian of the system is constructed in terms of ‘wave’ variables and the equations of motion are calculated. The resultant equations of motion are then analysed to show that wave–current interaction is influenced only by the current profile in the ‘strips’ adjacent to the surface and the interface. Small amplitude and long-wave approximations are also presented.     

A new formulation of the water wave problem for Stokes waves of constant vorticity

We consider steady symmetric gravity water waves on finite depth with constant vorticity and a monotone surface profile between crests and troughs. The problem is transformed into one concerning the vertical velocity. A representation formula for the stream function in terms of the surface and the vorticity is presented, and we show that the surface can be determined from the vertical velocity.     

Remark on peakon solution to the nonlinear surface wind waves equation

We prove that the existence of peakon as weak traveling wave solution and as global weak solution for the nonlinear surface wind waves equation, so as to correct the assertion that there exists no peakon solution for such an equation in the literature. Copyright © 2017 John Wiley & Sons, Ltd.     

Solitary waves of a generalized Ostrovsky equation

We consider the existence and stability of traveling waves of a generalized Ostrovsky equation ${(u_t-\beta u_{xxx}-f{\left(u\right)}_x)}_x=\gamma u$, where the nonlinearity $f\left(u\right)$ satisfies a power-like scaling condition. We prove that there exist ground state solutions which minimize the action among all nontrivial solutions and use this variational characterization to study their stability. We also introduce a numerical method for computing ground states based on their variational properties. The class of nonlinearities considered includes sums and differences of distinct powers.     

Stability of solitary waves for the Ostrovsky equation

Considered herein is the Ostrovsky equation which is widely used to describe the effect of rotation on the surface and internal solitary waves in shallow water or the capillary waves in a plasma. It is shown that the solitary-wave solutions are orbitally stable for certain wave speeds.

     

Bifurcations and exact solutions for a nonlinear surface wind waves equation

By using the method of dynamical systems, for the nonlinear surface wind waves equation, which is given by Manna, we study its dynamical behavior to determine all exact explicit traveling wave solutions. To guarantee the existence of the aforementioned solutions, all parameter conditions are determined. Our procedure shows that the nonlinear surface wind waves equation has no peakon solution. Copyright © 2015 John Wiley & Sons, Ltd.     

Solitary wave solutions of the improved KdV equation by VIM

The variational iteration method (VIM) is applied to solve numerically the improved Korteweg-de Vries equation (IKdV). A correction function is constructed with a general Lagrange multiplier that can be identified optimally via the variational theory. This technique provides a sequence of functions with easily computable components that converge rapidly to the exact solution of the IKdV equation. Propagation of single, interaction of two, and three solitary waves, and also birth of solitons have been discussed. Three invariants of motion have been evaluated to determine the conservation properties of the problem. This procedure is promising for solving other nonlinear equations.     

On surface waves in diffraction gratings

We discuss the existence criterion of surface waves based on the augmented scattering matrices. Such matrices arise if one takes into account not only oscillating waves but also those which grow (attenuate) in amplitude far from the grating. A family of planar dielectric gratings with periodic modulation of the refraction index is considered. Asymptotic and numerical analysis of the model are given. We represent various examples of gratings which support (or do not support) surface waves. Copyright © 2000 John Wiley & Sons, Ltd.     

Instability of the standing waves for nonlinear Klein-Gordon equations with damping term

This paper is concerned with the nonlinear Klein-Gordon equations with damping term. In terms of the variational argument, the sharp conditions for blowing up and global existence are derived out by applying the potential well argument and using the concavity method. Further, the instability of the standing waves is shown.     

Long distance wave computation using nonlinear solitary waves

A recently developed method is described to propagate short wave equation pulses over indefinite distances and through regions of varying indices of refraction, including multiple reflections. The method, “Wave Confinement”, utilizes a newly developed nonlinear partial differential equation (pde) that propagates basis functions according to the wave equation. These basis functions are generated as stable solitary waves where the discretized equation can be solved without any numerical dissipation. The method can also be used to solve for harmonic waves in the high frequency (Eikonal) limit, including multiple arrivals. The solution involves discretizing the wave equation on a uniform Eulerian grid and adding a simple nonlinear “Confinement” term. This term does not change the amplitude (integrated through each point on the pulse surface) or the propagation velocity, or arrival time, and yet results in capturing the waves as thin surfaces that propagate as thin nonlinear solitary waves and remain ∼2-3 grid cells in thickness indefinitely with no numerical spreading. A new feature described in this paper involves computing scattering of short pulses from complex objects such as complete aircraft. A simple “immersed surface” approach is used, that utilizes the same uniform grid as the propagation and avoids complex, body fitted or adaptive grid schemes.The new method should be useful in areas of wave propagation, from radar scattering and long distance communications to cell phone transmission.     

Generalized solitary waves in the gravity-capillary Whitham equation

We study the existence of traveling wave solutions to a unidirectional shallow water model, which incorporates the full linear dispersion relation for both gravitational and capillary restoring forces. Using functional analytic techniques, we show that for small surface tension (corresponding to Bond numbers between 0 and 1/3) there exists small amplitude solitary waves that decay to asymptotically small periodic waves at spatial infinity. The size of the oscillations in the far field are shown to be small beyond all algebraic orders in the amplitude of the wave.     

Propagation of torsional surface waves in a homogeneous layer of finite thickness over an initially stressed heterogeneous half-space

In the present paper, the dispersion equation which determines the velocity of torsional surface waves in a homogeneous layer of finite thickness over an initially stressed heterogeneous half-space has been obtained. The dispersion equation obtained is in agreement with the classical result of Love wave when the initial stresses and inhomogeneity parameters are neglected. Numerical results analyzing the dispersion equation are discussed and presented graphically. The result shows that the initial stresses have a pronounced influence on the propagation of torsional surface waves. It has also been shown that the effect of density, directional rigidities and non-homogeneity parameter on the propagation of torsional surface waves is prominent.     

Stability of periodic traveling waves for complex modified Korteweg-de Vries equation

We study the existence and stability of periodic traveling-wave solutions for complex modified Korteweg-de Vries equation. We also discuss the problem of uniform continuity of the data-solution mapping.     

Surface waves supported by thin-film/substrate interactions

A systematic approximation to the linear equations for small-amplitudesurface waves in an elastic half space, interacting with a residuallystressed thin film of different material bonded to its planeboundary, is developed in powers of the film thickness, assumingthe latter to be small compared to the wavelength of the disturbance.The theory is illustrated by calculating asymptotic expansionsof the wave speeds for Love and Rayleigh waves valid to secondorder in the dimensionless film thickness for a transverselyisotropic film bonded to an isotropic substrate.