Two-dimensional Water Wave Problem Research Articles

Nonlinear two-dimensional water waves with arbitrary vorticity

We consider the two-dimensional water-wave problem with a general non-zero vorticity field in a fluid volume with a flat bed and a free surface. The nonlinear equations of motion for the chosen surface and volume variables are expressed with the aid of the Dirichlet-Neumann operator and the Green function of the Laplace operator in the fluid domain. Moreover, we provide new explicit expressions for both objects. The field of a point vortex and its interaction with the free surface is studied as an example. In the small-amplitude long-wave Boussinesq and KdV regimes, we obtain appropriate systems of coupled equations for the dynamics of the point vortex and the time evolution of the free surface variables.

Validity of the Nonlinear Schrödinger Approximation for the Two-Dimensional Water Wave Problem With and Without Surface Tension in the Arc Length Formulation

We consider the two-dimensional water wave problem in an infinitely long canal of finite depth both with and without surface tension. In order to describe the evolution of the envelopes of small oscillating wave packet-like solutions to this problem the Nonlinear Schr\"odinger equation can be derived as a formal approximation equation. In recent years, the validity of this approximation has been proven by several authors for the case without surface tension. In this paper, we rigorously justify the Nonlinear Schr\"odinger approximation for the cases with and without surface tension by proving error estimates over a physically relevant timespan in the arc length formulation of the two-dimensional water wave problem. The error estimates are uniform with respect to the strength of the surface tension, as the height of the wave packet and the surface tension go to zero.

Two-dimensional wave radiation and diffraction problems in a flat or sloping seabed environment

Two-dimensional water wave problems are investigated in an environment with a flat or sloping rigid seabed adopting a continuous Rankine source method. All the fluid domain surfaces, that is the free, body and seabed surfaces, are discretized using continuous panels. These panels are positioned exactly on the fluid boundary surfaces and no desingularization technique is required. A new seabed source panel distribution method is developed to accommodate both symmetric and asymmetric seabed profiles. To validate the numerical model comparisons are made with published findings from other mathematical models and experimental data. The presence of a sloped seabed alters the symmetry of the fluid domain, causing wave reflection and shoaling, and therefore, significantly affects the hydrodynamic characteristics of water wave problems. The influence of these topographies on the responses in all three degrees of freedom (heave, sway and roll) of a rigid floating body are investigated and discussed accounting for wave radiation and diffraction problems.

The motion of a freely floating cylinder in the presence of a wall and the approximation of resonances

A linear theory, based on wide-spacing and high-frequency approximations, is developed to describe resonant behaviour in two-dimensional water-wave problems involving a freely floating half-immersed cylinder in the presence of a vertical rigid wall. The theory is not able to describe the lowest-frequency resonance, but otherwise yields explicit approximations for the locations of resonances in the complex plane and for their corresponding residues. Two problems are investigated in detail: the time-domain motion following a vertical displacement of the cylinder from equilibrium, and the time-harmonic motion of a cylinder that is excited by an incident plane wave.

Justification of the Nonlinear Schrödinger Equation for the Evolution of Gravity Driven 2D Surface Water Waves in a Canal of Finite Depth

In 1968 V.E. Zakharov derived the Nonlinear Schrodinger equation for the two-dimensional water wave problem in the absence of surface tension, that is, for the evolution of gravity driven surface water waves, in order to describe slow temporal and spatial modulations of a spatially and temporarily oscillating wave packet. In this paper we give a rigorous proof that the wave packets in the two-dimensional water wave problem in a canal of finite depth can be approximated over a physically relevant timespan by solutions of the Nonlinear Schrodinger equation.

An extended wide-spacing approximation for two-dimensional water-wave problems in infinite depth

This paper considers the wide-spacing approximation when applied to twodimensional linear problems involving the scattering and radiation of water waves by structures in a fluid of infinite depth. The approximation is set into the context of a formal expansion procedure and an extended approximation is obtained. The extended approximation significantly improves the accuracy of calculations for submerged structures when compared with the standard approximation procedure based on plane-wave interactions.

A note on the computation of a Green’s function for short wavelength water waves over a plane incline

A recently developed Green’s function for two-dimensional water wave problems on a plane beach was given in a form requiring (in part) direct numerical quadrature of a complex valued integral on the Bromwich line ℜs=12,ℑ(s)∈[−∞,∞]. These calculations would fail numerically for waves of sufficiently short wave-length.For special angles of inclination a more robust method is presented here which has no wavelength restrictions.

Solitary wave solutions of the high-order KdV models for bi-directional water waves

An approach, which allows us to construct specific closed-form solitary wave solutions for the KdV-like water-wave models obtained through the Boussinesq perturbation expansion for the two-dimensional water wave problem in the limit of long wavelength/small amplitude waves, is developed. The models are relevant to the case of the bi-directional waves with the amplitude of the left-moving wave of O( ϵ) ( ϵ is the amplitude parameter) as compared with that of the right-moving wave. We show that, in such a case, the Boussinesq system can be decomposed into a system of coupled equations for the right- and left-moving waves in which, to any order of the expansion, one of the equations is dependent only on the (main) right-wave elevation and takes the form of the high-order KdV equation with arbitrary coefficients whereas the second equation includes both elevations. Then the explicit solitary wave solutions constructed via our approach may be treated as the exact solutions of the infinite-order perturbed KdV equations for the right-moving wave with the properly specified high-order coefficients. Such solutions include, in a sense, contributions of all orders of the asymptotic expansion and therefore may be considered to a certain degree as modelling the solutions of the original water wave problem under proper initial conditions. Those solitary waves, although stemming from the KdV solitary waves, possess features found neither in the KdV solitons nor in the solutions of the first order perturbed KdV equations.

The effect of surface tension on trapped modes in water-wave problems

In this paper the effect of surface tension is considered on two two-dimensional water-wave problems involving pairs of immersed bodies. Both models, having fluid of infinite depth, support localized oscillations, or trapped modes, when capillary effects are excluded. The first pair of bodies is surface-piercing whereas the second pair is fully submerged. In the former case it is shown that the qualitative nature of the streamline shape is unaffected by the addition of surface tension in the free surface condition, no matter how large this parameter becomes. The main objective of this paper, however, is to study the submerged body problem. For this case it is found, by contrast, that there exists a critical value of the surface tension above which it is no longer possible to produce a completely submerged pair of bodies which support trapped modes. This critical value varies as a function of the separation of the two bodies. It can be inferred from this that surface tension does not always play a qualitatively irrelevant role in the linear water-wave problem.

Spatial Dynamics Methods for Solitary Gravity-Capillary Water Waves with an Arbitrary Distribution of Vorticity

This paper presents existence theories for several families of small-amplitude solitary-wave solutions to the classical two-dimensional water-wave problem in the presence of surface tension and with an arbitrary distribution of vorticity. Moreover, the established local bifurcation diagram for irrotational solitary waves is shown to remain qualitatively unchanged for any choice of vorticity distribution. The hydrodynamic problem is formulated as an infinite-dimensional Hamiltonian system in which the horizontal spatial direction is the timelike variable. A center-manifold reduction technique is employed to reduce the system to a locally equivalent Hamiltonian system with a finite number of degrees of freedom. Homoclinic solutions to the reduced system, which correspond to solitary water waves, are detected by a variety of dynamical systems methods.

Variational principles in water-wave problems (wave radiation conditions and their variational treatment)

To discuss water-wave problems in unlimited waters, it is important to know what type of wave radiation condition should be placed on a virtual surface corresponding to infinity. For this kind of problem, the Sommerfeld radiation condition is well known. In this article, the condition is extended to treat a case with an incident wave. Furthermore, a more general wave radiation condition is introduced from a different point of view. The above-mentioned wave radiation conditions are introduced into the variational principles of the Kelvin, Hellinnger–Reissner, and Dirichlet type. The Dirichlet-type variational principles are then used in numerical calculations for bending waves in a bar, and the effectiveness of the wave radiation conditions and the variational principles is shown. The numerical results for one-dimensional water-wave problems are then given. As expected, the region required for the numerical solution is reduced drastically compared with that required by the Sommerfeld-type formulation. Furthermore, the amplitude of the diverging wave is obtained in the process of reaching the variational solution. Finally, two-dimensional water-wave problems are briefly discussed.

The long-wave limit for the water wave problem I. The case of zero surface tension

The Korteweg-de Vries equation, Boussinesq equation, and many other equations can be formally derived as approximate equations for the two-dimensional water wave problem in the limit of long waves. Here we consider the classical problem concerning the validity of these equations for the water wave problem in an infinitely long canal without surface tension. We prove that the solutions of the water wave problem in the long-wave limit split up into two wave packets, one moving to the right and one to the left, where each of these wave packets evolves independently as a solution of a Korteweg-de Vries equation. Our result allows us to describe the nonlinear interaction of solitary waves. © 2000 John Wiley & Sons, Inc.

A finite element method for two-dimensional water-wave problems

In this paper, the authors treat the free-surface waves generated by a moving disturbance with a constant speed in water of finite and constant depth. Specifically, the case when the disturbance is moving with the critical speed is investigated. The water is assumed inviscid and its motion irrotational. The surface tension is neglected. It is well-known that the linear theory breaks down when a disturbance is moving with the critical speed. As a remedy to overcome the invalid linear theory, approximate non-linear theories have been applied with success in the past, i.e. Boussinesq and Korteweg de Vries equations, for example. In the present paper, the authors describe a finite element method applied to the non-linear water-wave problems in two dimensions. The present numerical method solves the exact non-linear formulation in the scope of potential theory without any additional assumptions on the magnitude of the disturbances. The present numerical results are compared with those obtained by other approximate non-linear theories. Also presented are the discussions on the validity of the existing approximate theories applied to two types of the disturbances, i.e. the bottom bump and the pressure patch on the free-surface at the critical speed. Copyright © 1999 John Wiley & Sons, Ltd.

A note on uniqueness in the linearized water-wave problem

The uniqueness theorem of Simon & Ursell (1984), concerning the linearized two-dimensional water-wave problem in a fluid of infinite depth, is extended in two directions. First, we consider a two-dimensional geometry involving two submerged symmetric bodies placed sufficiently far apart that they are not confined in the vertical right angle having its vertex on the free surface as the theorem of Simon & Ursell requires. A condition is obtained guaranteeing the uniqueness outside a finite number of bounded frequency intervals. Secondly, the method of Simon & Ursell is generalized to prove uniqueness in the axisymmetric problem for bodies violating John's condition provided the free surface is a connected plane region.

Scattering of water waves by a submerged disc using a hypersingular integral equation

The three-dimensional interaction between water waves and a submerged disc, in deep water, is considered. The problem is reduced to a hypersingular integral equation over the surface of the disc. The integral equation is solved numerically using an expansion-collocation method, generalizing a method used previously by Parsons and Martin for several two-dimensional water-wave problems. This method is shown to be very effective: it incorporates the known behaviour near the edge of the disc and it permits all hypersingular (finite-part) inegrals to be evaluated analytically. Numerical results are presented, with emphasis on the scattering properties of the submerged disc.

Uniqueness in linearized two-dimensional water-wave problems

A geometrical condition sufficient for uniqueness in two-dimensional time-periodic linear water-wave problems is derived, and some examples are given. The technique can be seen as a generalization of work by John (1950), who established uniqueness for surface-piercing bodies that have the property that vertical lines down from the free surface do not intersect the body.The paper is in two parts. Part 1 provides a review of current knowledge on the topic of uniqueness, and gives a simple form of John's proof. Part 2 describes the recent progress, in which the ideas of John's proof are extended so that uniqueness can be demonstrated for surface-piercing bodies that do not satisfy John's geometrical criterion. In fact the new technique proves uniqueness for a large class of problems involving floating bodies, submerged bodies and multiple-body systems. However, the present work still does not constitute a general proof of uniqueness for all configurations.

METHOD OF ANALYSES FOR TWO-DIMENSIONAL WATER WAVE PROBLEMS

One of the most powerful tools to analyze the boundary-value problems in water wave motion is the Green's function. However, to derive the Green's function which satisfies the imposed boundary conditions is sometimes difficult or impossible, especially in variable water depth. In this paper, a simple method of numerical analyses for two-dimensional boundary-value problems of small amplitude waves is proposed, and the wave transformation by fixed horizontal cylinders as an example of fixed boundaries, the wave transformation by and the motion of a cylinder floating on water surface as example of oscillating boundaries and the wave transformation by permeable seawall and breakwater as example of permeable boundaries are calculated and compared with experimental results.

<i>A priori</i> estimates for two-dimensional water waves with angled crests

We consider the two-dimensional water wave problem in the case where the free interface of the fluid meets a vertical wall at a possibly non-right angle; our problem also covers interfaces with angled crests. We assume that the fluid is inviscid, incompressible, and irrotational, with no surface tension and with air density zero. We construct a low-regularity energy and prove a closed energy estimate for this problem. Our work differs from earlier work in that, in our case, only a degenerate Taylor stability criterion holds, with − ∂P ∂n > 0, instead of the strong Taylor stability criterion − ∂P ∂n > c > 0.