Water Wave Problem Research Articles

Water wave scattering by a rectangular trench

Assuming linear theory, the two-dimensional problem of water wave scattering by a rectangular submarine trench is reinvestigated here employing the multiterm Galerkin approximations involving ultraspherical Gegenbauer polynomials for solving the integral equations arising in the mathematical analysis. Because of the geometrical symmetry of the rectangular trench about the $$y$$ -axis, the problem is split into two separate problems involving symmetric and antisymmetric potential functions. Very accurate numerical estimates for the reflection and transmission coefficients for various values of different parameters are obtained, and these are seen to satisfy the energy identity. These coefficients are computed numerically and depicted graphically against the wave number in a number of figures. Some figures available in the literature drawn using different mathematical methods and laboratory experiments are also recovered following the present analysis, thereby confirming the correctness of the results presented here. It is also observed that the reflection and transmission coefficients depend significantly on the width of the trench.

Doppler effects of an oscillating line source in shear flow with a free surface

The linearised water-wave radiation problem for the oscillating 2D submerged source in an inviscid shear flow with a free surface is investigated analytically. There is a nonzero surface velocity. The depth is infinite and the vorticity is uniform. The amplitudes radiated from the source are calculated analytically. Due to Doppler effects, there may be up to four different emitted waves, and there is resonance with zero group velocity and infinite amplitude.

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.

Water wave scattering by an elastic plate floating in an ocean with a porous bed

The problem of water wave scattering by a thin horizontal elastic plate (semi-infinite as well as finite) floating on an ocean of uniform finite depth in which the ocean bed is composed of porous material of a specific type is analyzed. The method of eigenfunction expansion is used in the mathematical analysis and the quantities of physical interest, namely the reflection and transmission coefficients, are obtained. Numerical estimates for these coefficients are obtained for different values of the parameter describing the porosity of the ocean bed and for different edge conditions of the elastic plate. The edge conditions considered here involve (i) a free edge, (ii) a simply-supported edge and (iii) a built-in edge. From the numerical results it is observed that for free edge condition, the porosity of the ocean bed has little effect on the reflection and transmission coefficient for both the cases of semi-infinite and finite elastic plate. The energy identity related to reflection and transmission coefficients in a porous bed is derived and is used as a partial check on the correctness of the numerical results for the semi-infinite elastic plate.

A new nonlinear equation in the shallow water wave problem

In this paper, a new nonlinear equation describing shallow water waves with the topography of the bottom directly taken into account is derived. This equation is valid for weakly nonlinear, dispersive, and long wavelength limits. Some examples of soliton motion for various bottom shapes obtained in numerical simulations according to the derived equation are presented.

On the particle motion in geophysical deep water waves traveling over uniform currents

We describe a family of exact Gerstner-type solutions for the geophysical equatorial deep water wave problem in the f f -plane approximation. These Gerstner-type waves are two-dimensional and travel with constant speed over a uniform horizontal current. The particle paths in the presence and absence of the Coriolis force are also analyzed in dependence of the current strength.

Spectral gaps for the linear surface wave model in periodic channels

Summary We consider the linear water-wave problem in a periodic channel which consists of infinitely many identical containers connected with apertures of width � . Motivated by applications to surface wave propagation phenomena, we study the band-gap structure of the continuous spectrum. We show that for small apertures there exists a large number of gaps and also find asymptotic formulas for the position of the gaps as � → 0: the endpoints are determined within corrections of order � 3/2 . The width of the first bands is shown to be O(� ). Finally, we give a sufficient condition which guarantees that the spectral bands do not degenerate into eigenvalues of infinite multiplicity.

Spectrum of the linear water model for a two‐layer liquid with cuspidal geometries at the interface

We show that the linear water wave problem in a bounded liquid domain may have continuous spectrum, if the interface of a two‐layer liquid touches the basin walls at zero angle. The reason for this phenomenon is the appearance of cuspidal geometries of the liquid phases. We calculate the exact position of the continuous spectrum. We also discuss the physical background of wave propagation processes, which are enabled by the continuous spectrum. Our approach and methods include constructions of a parametrix for the problem operator and singular Weyl sequences.

On the Cauchy problem for gravity water waves

We are interested in the system of gravity water waves equations without surface tension. Our purpose is to study the optimal regularity thresholds for the initial conditions. In terms of Sobolev embeddings, the initial surfaces we consider turn out to be only of~$C^{3/2+\epsilon}$-class for some $\epsilon>0$ and consequently have unbounded curvature, while the initial velocities are only Lipschitz. We reduce the system using a paradifferential approach.

Steady Periodic Water Waves with Unbounded Vorticity: Equivalent Formulations and Existence Results

In this paper we consider the steady water wave problem for waves that possess a merely \(L_r\)-integrable vorticity, with \(r\in (1,\infty )\) being arbitrary. We first establish the equivalence of the three formulations – the velocity formulation, the stream function formulation, and the height function formulation – in the setting of strong solutions, regardless of the value of \(r\). Based upon this result and using a suitable notion of weak solution for the height function formulation, we then establish, by means of local bifurcation theory, the existence of small-amplitude capillary and capillary–gravity water waves with an \(L_r\)-integrable vorticity.

Capillary–Gravity Water Waves with Discontinuous Vorticity: Existence and Regularity Results

In this paper we construct periodic capillarity-gravity water waves with an arbitrary bounded vorticity distribution. This is achieved by reexpressing, in the height function formulation of the water wave problem, the boundary condition obtained from Bernoulli's principle as a nonlocal differential equation. This enables us to establish the existence of weak solutions of the problem by using elliptic estimates and bifurcation theory. Secondly, we investigate the a priori regularity of these weak solutions and prove that they are in fact strong solutions of the problem, describing waves with a real-analytic free surface. Moreover, assuming merely integrability of the vorticity function, we show that any weak solution corresponds to flows having real-analytic streamlines.

Homogenization in the problem of long water waves over a bottom site with fast oscillations

The system of equations of gravity surface waves is considered in the case where the basin’s bottom is given by a rapidly oscillating function against a background of slow variations of the bottom. Under the assumption that the lengths of the waves under study are greater than the characteristic length of the basin bottom’s oscillations but can be much less than the characteristic dimensions of the domain where these waves propagate, the adiabatic approximation is used to pass to a reduced homogenized equation of wave equation type or to the linearized Boussinesq equation with dispersion that is “anomalous” in the theory of surface waves (equations of wave equation type with added fourth derivatives). The rapidly varying solutions of the reduced equation can be found (and they were also found in the authors’ works) by asymptotic methods, for example, by the WKB method, and in the case of focal points, by the Maslov canonical operator and its generalizations.

The NLS Approximation Makes Wrong Predictions for the Water Wave Problem in Case of Small Surface Tension and Spatially Periodic Boundary Conditions

The nonlinear Schrodinger (NLS) equation describes small modulations in time and space of a spatially and temporally oscillating wave packet advancing in a laboratory frame. It has first been derived for the so called water wave problem in 1968 and the proof that it makes correct predictions has been recently the subject of intensive research. It is the purpose of this paper to show that in general this approximation is not valid without further assumptions. We construct a counter example showing that the NLS approximation makes wrong predictions for the water wave problem in case of small surface tension and spatially periodic boundary conditions.

On the Hamiltonian structure of the planar steady water-wave problem with vorticity

We consider the stream-function formulation of the hydrodynamic problem for steady rotational water waves both with and without surface tension. A natural Lagrangian formulation is presented from which (different) Hamiltonian formulations for the two cases are derived by duality theory in the spirit of the Legendre–Fenchel transform. The treatment is systematic and clarifies a recent ad hoc approach by Kozlov and Kuznetsov [7].

Existence of capillary-gravity water waves with piecewise constant vorticity

In this paper we construct small-amplitude periodic capillary-gravity water waves with a piecewise constant vorticity distribution. They describe water waves traveling on superposed linearly sheared currents that have different vorticities. This is achieved by associating to the height function formulation of the water wave problem a diffraction problem where we impose suitable transmission conditions on each line where the vorticity function has a jump. The solutions of the diffraction problem, found by using local bifurcation theory, are the desired solutions of the hydrodynamical problem.

A Spatial Dynamics Theory for Doubly Periodic Travelling Gravity-Capillary Surface Waves on Water of Infinite Depth

In this article we show how Kirchgassner’s spatial dynamics approach can be used to construct doubly periodic travelling gravity-capillary surface waves on water of infinite depth. The hydrodynamic problem is formulated as a reversible Hamiltonian system in which an arbitrary horizontal spatial direction is the time-like variable and the infinite-dimensional function space consists of wave profiles which are periodic (with fixed period) in a second, different horizontal direction. The imaginary part of the spectrum of the linearised Hamiltonian vector field consists of essential spectrum at the origin and a finite number of eigenvalues whose distribution is described geometrically. Periodic solutions to the spatial Hamiltonian system are detected using Iooss’s generalisation of the reversible Lyapunov centre theorem; these solutions correspond to doubly periodic solutions to the travelling water-wave problem. For a generic choice of the periodic domain there exist values of the physical parameters at which a doubly periodic wave with this domain exists.

Structural stability for the splash singularities of the water waves problem

In this paper we show a structural stability result for water waves. The main motivation for this result is that we aim to exhibit a water wave whose interface starts as a graph and ends in a splash. Numerical simulations lead to an approximate solution with the desired behaviour. The stability result will conclude that near the approximate solution to water waves there is an exact solution.

Small-amplitude equatorial water waves with constant vorticity: Dispersion relations and particle trajectories

We consider the two-dimensional equatorial water-wave problem with constant vorticity in the $f$-plane approximation. Within the framework of small-amplitude waves, we derive the dispersion relations and we find the analytic solutions of the nonlinear differential equation system describing the particle paths below such waves. We show that the solutions obtained are not closed curves. Some remarks on the stagnation points are also provided.

Oscillatory line source for water waves in shear flow

The linearized water-wave radiation problem for the oscillating 2D submerged source in an inviscid shear flow with a free surface is investigated analytically. The vorticity is uniform, with zero velocity at the free surface. Then there will be at most two emitted waves, and no Doppler effects. Exact far-field waves are derived, with radiation conditions applied at infinity. An upstream wave will always exist, whereas the downstream wave exists only when the angular frequency of oscillation exceeds the vorticity. The wave radiation problem is solved also for oscillating vortex and dipoles. The amplitudes and energy fluxes are calculated.

A Regularity Result for the Incompressible Euler Equation with a Free Interface

We address the local existence of solutions of the 2D and 3D water wave problems. For the space dimension three, we consider the irrotational datum u 0 and prove that the local in time existence holds for initial velocities belonging to H 2.5+? , where ?>0 is arbitrary. For the space dimension two, the data does not need to be irrotational. We prove the local in time existence when u 0 belongs to H 2+? and $\operatorname{curl}u_{0}$ to H 1.5+? , where ?>0 is arbitrary.