Surface Wave Problem Research Articles

A model problem of vortical surface waves

A model problem of vortical surface waves

Surface waves on an inviscid shear flow in a channel

The three-dimensional problem of surface waves propagating on a horizontal velocity profile is considered. The waves are assumed to propagate with a fixed form and a constant speed, but the transverse shape may vary. The flow is allowed to have vorticity, but is assumed to be inviscid and incompressible. A simple linear velocity profile is studied, and the flow is confined to a channel. An exact solution to the linear problem is given, but the wavespeed must be determined numerically. The results show that there are numerous wavespeeds that exist for a given wavenumber. A comparison with potential flow waves and other rotational waves shows that the different wavespeeds correspond to sloshing modes. The effect of shear in the primary flow is also examined.

Note on the reflexion of water waves at a nearly vertical cliff in the presence of surface tension

In this note the two-dimensional problem of incoming surface waves against an approximately vertical wall or cliff in water of infinite depth is examined, using velocity potential formulation and linearized boundary-value problem theory for time-harmonic motion. The cliff has arbitrary profile but for simplicity is taken to be vertical at the free surface. The approximate first-order solution is determined subject to a dynamical edge condition apposite to the presence of surface tension, and contains partially reflected outgoing waves. The solution is obtained by perturbation theory in a form involving known unperturbed and first-order correction potentials that is applicable also to water of finite constant depth. The motivation for the note is to point out a correction in principle to the results of a recent investigation for a specific profile, in which reflexion is ignored and another error made in obtaining a first-order solution by a method that is restricted to infinite depth.

A numerical procedure for time domain nonlinear surface wave calculations

A numerical procedure based upon Boundary Integral Method for time domain wave-making problems is developed. The free-surface boundary condition is expressed in a Lagrangian notation to follow the free-surface's motion in time. The corresponding material derivative term is approximated by a finite difference expression, and the velocity terms are extrapolated in time for the completion of the formulations. Solutions with a linear free-surface condition and a non-linear free-surface condition are obtained at small time increment values. Modelling of surface wave problems are studied in two dimensions as well as in three dimensions, and comparisons are made to results based on linear theory as well as to published experimental results. Good agreements between the numerical solutions and measured values are found. Applications are on the simulations of surface waves in a towing tank, ship's bow wave calculation, and wave diffraction problem for a vertical, surface piercing circular cylinder. For the modelling of the three-dimensional wave diffraction problem, results at high wave amplitude are restricted because of the use of quadrilateral elements. The near cylinder region of the free-surface is not considered to be well represented because of the coarse element size. Wave forces calculated on the vertical cylinder are found to be affected by the modeled tank length. When the simulated wave length is comparable to the wave tank's length, numerical results are found to be less than the experimental measurements. However, when the wave length is much shorter than the wave tank's length, solutions are obtained with very good precision.

Transient development of surface waves in the presence of porous wave makers

The initial-value problem of surface waves generated by a harmonically oscillating porous wave maker immersed in an incompressible fluid of infinite horizontal extent but finite constant depth is prescribed. By the use of generalized functions and asymptotic analyses, it was possible to get both the steady-state solution and the transient solution without the need of imposing a radiation condition. Two limiting cases are also investigated: in the first the fluid is of infinite depth, while in the other the fluid is shallow.

Wave‐source expansions for water of infinite depth in the presence of surface tension

In this paper two expansions are obtained by contour integration methods for the velocity potential describing two-dimensional time-harmonic surface waves due to a free-surface wave source on water of infinite depth in the presence of surface tension. First the series expansion at r = 0 is found and then the asymptotic expansion as Kr®¥, where K is the wave number for progressive waves and r the radial distance from the source. The corresponding expansions for the more important submerged wave source in terms of the radial distance from the image source in the free surface may then easily be deduced. The latter are required in a number of surface wave problems, particularly those of a short-wave asymptotic nature, and are also relevant in obtaining expansions for finite constant depth.

INFINITE ELEMENTS FOR DYNAMIC PROBLEMS: PART 2

Survey of period infinite element developments The first infinite elements for periodic wave problems, as stated in Part 1, were developed by Bettess and Zienkiewicz, the earliest publication being in 1975. These applications were of ‘decay function’ type elements and were used in surface waves on water problems. This was soon followed by an application by Saini et al., to dam‐reservoir interaction, where the waves are pressure waves in the water in the reservoir. In this case both the solid displacements and the fluid pressures are complex valued. In 1980 to 1983 Medina and co‐workers and Chow and Smith successfully used quite different methods to develop infinite elements for elastic waves. Zienkiewicz et al. published the details of the first mapped wave infinite element formulation, which they went on to program, and to use to generate results for surface wave problems. In 1982 Aggarwal et al. used infinite elements in fluid‐structure interaction problems, in this case plates vibrating in an unbounded fluid. In 1983 Corzani used infinite elements for electric wave problems. This period also saw the first infinite element applications in acoustics, by Astley and Eversman, and their development of the ‘wave envelope’ concept. Kagawa applied periodic infinite wave elements to Helmholtz equation in electromagnetic applications. Pos used infinite elements to model wave diffraction by breakwaters and gave comparisons with laboratory photogrammetric measurements of waves. Good agreement was obtained. Huang also used infinite elements for surface wave diffraction problems. Davies and Rahman used infinite elements to model wave guide behaviour. Moriya developed a new type of infinite element for Helmholtz problem. In 1986 Yamabuchi et al. developed another infinite element for unbounded Helmholtz problems. Rajapalakse et al. produced an infinite element for elastodynamics, in which some of the integrations are carried out analytically, and which is said to model correctly both body and Rayleigh waves. Imai et al. gave further applications of infinite elements to wave diffraction, fluid‐structure interaction and wave force calculations for breakwaters, offshore platforms and a floating rectangular caisson. Pantic et al. used infinite elements in wave guide computations. In 1986 Cao et al. applied infinite elements to dynamic interaction of soil and pile. The infinite element is said to be ‘semi‐analytical’. Goransson and Davidsson used a mapped wave infinite element in some three dimensional acoustic problems, in 1987. They incorporated the infinite elements into the ASKA code. A novel application of wave infinite elements to photolithography simulation for semiconductor device fabrication was given by Matsuzawa et al. They obtained ‘reasonably good’ agreement with observed photoresist profiles. Häggblad and Nordgren used infinite elements in a dynamic analysis of non‐linear soil‐structure interaction, with plastic soil elements. In 1989 Lau and Ji published a new type of 3‐D infinite element for wave diffraction problems. They gave good results for problems of waves diffracted by a cylinder and various three dimensional structures.

Magnetic Electron Drift Vortex Modes in General Equilibria

AbstractThe equation and dispersion relation describing the magnetic electron drift vortex mode in a generalized equilibrium are presented. The results are useful for investigating related surface wave problems as well as for identifying the various forms of the magnetic electron drift vortex mode.

Explicit expressions of Barnett-Lothe tensors and their associated tensors for orthotropic materials

The three Barnett-Lothe tensorsS, H, L, appear very often in the solutions to two-dimensional anisotropic elasticity problems. So do their associated tensorsS^(θ),H^(θ),L^(θ)\hat S\left ( \theta \right ), \hat H\left ( \theta \right ), \hat L\left ( \theta \right )for line forces and dislocations andS(υ),H(υ),L(υ)S\left ( \upsilon \right ), H\left ( \upsilon \right ), L\left ( \upsilon \right )in the problem of surface waves. Explicit expressions of the components of these tensors are derived and presented for orthotropic materials in which the planes of material symmetry coincide with the coordinate planes. With minor modifications, the results forS, H, LandS^(θ),H^(θ),L^(θ)\hat S\left ( \theta \right ), \hat H\left ( \theta \right ), \hat L\left ( \theta \right )can be applied to orthotropic materials in which only thex3=0{x_3} = 0plane coincides with one of the planes of material symmetry.

Surface waves induced by uniform flow over a bulging bed

Abstract This study mainly relates the gravity‐surface‐waves generated when a uniform flow passes through a bulging bed. This is done firstly by treating the linear surface waves generated when a uniform flow passing through single wave bed, then, secondly, by making use of the fundamental solution obtained as mentioned above, and coordinating it with the Fourier integral and contour integral, so that the gravity surface wave generated by the more generalized and simple wave bed can be treated. With respect to the analytical solution, the major premise of this assumption is still limited to the uniform and irrotational flow, and at the same time neglecting the influence of surface tension, in order to obtain the analytical result of the flow field. It is important to propose possible methods to analyze the surface wave generated by any form of bulging bed, so that the problem of gravity surface waves generated by the bulging bed can be reasonably solved by the analytical method.

Asymptotic analysis of surface waves due to oscillatory wave maker

The initial value problem of surface waves generated by a harmonically oscillating vertical wave-maker immersed in an infinite incompressible fluid of finite constant depth is presented. The resulting motion is investigated using the generalized function method, and an asymptotic analysis for large times and distances is given for the free surface elevation.

A boundary‐type finite element model for water surface wave problems

AbstractA new combinative method of boundary‐type finite elements and boundary solutions is presented to study wave diffraction‐refraction and harbour oscillation problems. The numerical model is based on the mild‐slope equation. The key feature of this method is that the discretized matrix equation can be formulated only by the calculation of a line integral, since the interpolation equation which satisfies the governing equation in each element is used. The numerical solutions are compared with existing analytical, experimental, observed and other numerical results. The present method is shown to be an effective and accurate method for water surface wave problems.

A new approximate solution to the surface wave problem

Recent measurements of wave kinematics, showing that the horizontal velocity under the trough is in absolute value greater than the velocity under the crest, can not be explained by the Stokes theories up to and including the fifth order theory which are normally used by the offshore industry in the design process. This has led to a reconsideration of the surface wave problem. By applying a Green function technique a solution is obtained which reflects these recent experimental results on wave kinematics. This solution and the Stokes higher order solutions represent different approximate solutions to the surface wave problem. It turns out, however, that this new approximate solution gives a better fit to the measurements than do the approximate solutions obtained from the Stokes higher order theories. In the present paper deep water waves are discussed, but the method of solution can just as well be applied to the finite water depth case.

BOUNDARY TYPE FINITE ELEMENT METHOD FOR SURFACE WAVE PROBLEMS

A new finite element method for the analysis of surface wave problems is presented in this paper. The characteristic of this method is that the interpolation equation is selected to satisfy the Helmholtz equation in each element. This follows that the variational functional to be minimized can be formulated as in the form that all integrations are limited just on the boundary of the element. The numerical solutions obtained are compared with the analytical and observed results. From these comparative studies, it is concluded that the present method provides a useful and valuable tool for the analysis of surface wave problems.

The high-frequency radiation of water waves by oscillating bodies

The aim of this work is to extend the known asymptotic forms, in heave and sway, of the wave amplitude radiated by bodies at high frequency. Both two- and three-dimensional geometries will be considered, the prototype problems being the circular cylinder and sphere respectively, each with its centre in the mean free surface. The method is, in principle, applicable to other geometries by way of the solution of certain potential problems much simpler than the finite-frequency surface-wave problem.

Finite difference method for 2-D and 3-D nonlinear free surface wave problems

A finite difference method is developed for the numerical modelling of the 2-D and 3-D unsteady potential flow generated by transient disturbances on the free surface, on which the nonlinear boundary conditions are fully satisfied. The unknown function is computed with an iteration scheme processing in a transformed time-invariant space. After the velocity is calculated, the location of the free surface is renewed and so is the value of velocity on it. The boundary-value problem of the governing equation is then solved at the next time step. The present method incorporates the FFT. Consequently, a tri-diagonal equation system is obtained which could be readily solved. The feasibility of this method has been demonstrated by 2-D and 3-D examples corresponding to different initial disturbances.

Finite difference method of transinet nonlinear free surface wave problems

A finite difference method is developed for computing the two-dimensional transient potential flow generated by an impulse on the free surface. Both the dynamic and kinematic free surface conditions are considered in nonlinear version. the primary features of the present paper include the use of special coordinates transformations so that the geometry of the flow field is transformed into a time-invariant region, presents an iteration process, by which the velocity potential is computed as the solution of a Poisson equation, the application of fast Fourier transform (FFT) technique results in a tri-diagonal system of equations which can be readily solved by the Thomas algorithm, the computing time is significantly reduced. Thus an efficient technique for handling the transient potential problems is well justified. The feasibility of the present method has been verified by two examples including different initial disturbances respectively.

Irregular frequencies and iterative methods in the solution of steady surface-wave problems in hydrodynamics

Two linearized problems in free-surface hydrodynamics are discussed. The first concerns flow due to a submerged line vortex in a running stream, and the second investigates three-dimensional flow about a moving pressure distribution at the surface of the fluid. Closed-form solutions to both linearized problems are well known, and therefore are not of interest; however, it is shown that the solution of either problem by a boundary-integral technique utilizing “simple” (Rankine) sources as fundamental singular solutions leads to Fredholm integral equations of the second kind, for which the irregular frequencies do not occur discretely, but as a continuum. Consequently, Neumann-type iteration schemes for the solution of these equations necessarily diverge for any Froude number. Ramifications of this result in the attempted numerical solution of the corresponding non-linear problems are discussed, and the convergence difficulties encountered by Hess [1] are analyzed.

Seismic waves in a stratified half-space -- IV: P--SV wave decoupling and surface wave dispersion

Summary. In a region of smoothly varying seismic parameters, the seismic displacements can be transformed into portions depending principally on the P and SV parts of the wavefield. Although this representation has previously been used for body wave studies, it is also very suitable for understanding surface wave problems. With the aid of this transformation and the representation of Rayleigh wave dispersion in terms of reflection coefficients at the free-surface and from the stratification the behaviour of the Rayleigh wavefield with slowness can be related to the velocity structure. This allows the character of Rayleigh waves to be related to that of Love waves. There is very close correspondence when P-waves are evanescent throughout the stratification. In this case we can represent the Rayleigh wave behaviour in terms of a scalar eigenfunction which has simple, though approximate, orthogonality properties.

The Theory of Perturbed Surface Waves and the Long Wave Limit of Lattice Models

This paper discusses three related problems. (1) A semi-infinite elastic medium in which the unperturbed homogeneous bulk material does not extend right up to the surface either because of surface perturbations or else because of a thin overlayer of a different material. If the thickness of this surface region is much smaller than the wavelengths of interest, then the problem can be cast as a modification of the boundary conditions. (2) An analytical discussion of the long wave limit of surface perturbed lattice models and the connection with the above theory of surface perturbed surface waves. (3) A mathematical formalism here developed to solve the perturbed surface wave problem, given the perturbed boundary conditions.