Gravity-capillary Water Research Articles

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.

Second-order Stokes wave solutions for gravity capillary water waves in three-layer dendity-stratified fluid

In this paper, gravity-capillary water waves in a three-layer stratified fluid are investigated by using a perturbation method, and the second-order asymptotic solutions of the velocity potentials and the second-order Stokes solutions of the associated elevations of the gravity-capillary water waves are presented based on the small amplitude wave theory. As expected, both the first-order and second-order solutions derived depend on not only the depth and density of the three-layer fluid but also the surface tension.

Dynamics of gravity–capillary solitary waves in deep water

AbstractThe dynamics of solitary gravity–capillary water waves propagating on the surface of a three-dimensional fluid domain is studied numerically. In order to accurately compute complex time-dependent solutions, we simplify the full potential flow problem by using surface variables and taking a particular cubic truncation possessing a Hamiltonian with desirable properties. This approximation agrees remarkably well with the full equations for the bifurcation curves, wave profiles and the dynamics of solitary waves for a two-dimensional fluid domain, and with higher-order truncations in three dimensions. Fully localized solitary waves are then computed in the three-dimensional problem and the stability and interaction of both line and localized solitary waves are investigated via numerical time integration of the equations. There are many solitary wave branches, indexed by their finite energy as their amplitude tends to zero. The dynamics of the solitary waves is complex, involving nonlinear focusing of wavepackets, quasi-elastic collisions, and the generation of propagating, spatially localized, time-periodic structures akin to breathers.

Local bifurcation and regularity for steady periodic capillary–gravity water waves with constant vorticity

We study periodic capillary–gravity waves at the free surface of water in a flow with constant vorticity over a flat bed. Using bifurcation theory the local existence of waves of small amplitude is proved even in the presence of stagnation points in the flow. We also derive the dispersion relation. Moreover, we prove a regularity result for the free surface.

A Free Boundary Approach to Two-Dimensional Steady Capillary Gravity Water Waves

We prove that suitable weak solutions of the two-dimensional steady capillary gravity water wave problem (even in the presence of vorticity) are smooth. On a technical level, solutions are closely related to critical points of the Mumford–Shah functional, so that our main task is to exclude cusps pointing into the water phase. We prove that cusps do not exist, and that the free surface is smooth.

On uniqueness in the problem of gravity–capillary water waves above submerged bodies

In this paper, we consider the two-dimensional linear problem of wave–body interaction with surface tension effects being taken into account. We suggest a criterion for unique solvability of the problem based on symmetrization of boundary integral equations. The criterion allows us to develop an algorithm for detecting non-uniqueness (finding trapped modes) for given geometries of bodies; examples of numerical computation of trapped modes are given. We also prove a uniqueness theorem that provides simple bounds for the possible non-uniqueness parameters.

Particle trajectories in linear periodic capillary and capillary–gravity water waves

Surface tension plays a significant role as a restoration force in the setting of small-amplitude waves, leading to pure capillary and gravity-capillary waves. We show that within the framework of linear theory, the particle paths in a periodic gravity-capillary or pure capillary wave propagating at the surface of water over a flat bed are not closed.

Do true elevation gravity–capillary solitary waves exist? A numerical investigation

This paper extends the numerical results of Hunter & Vanden-Broeck (1983) and Vanden-Broeck (1991) which were concerned with studies of solitary waves on the surface of fluids of finite depth under the action of gravity and surface tension. The aim of this paper is to answer the question of whether small-amplitude elevation solitary waves exist. Several analytical results have proved that bifurcating from Froude number F = 1, for Bond number τ between 0 and 1/3, there are families of ‘generalized’ solitary waves with periodic tails whose minimum amplitude is an exponentially small function of F−1. An open problem (which, for τ sufficiently close to 1/3, was recently proved by Sun 1999 to be false) is whether this amplitude can ever be zero, which would give a truly localized solitary wave.The problem is first addressed in terms of model equations taking the form of generalized fifth-order KdV equations, where it is demonstrated that if such a zero-tail-amplitude solution occurs, it does so along codimension-one lines in the parameter plane. Moreover, along solution paths of generalized solitary waves a topological distinction is found between cases where the tail does vanish and those where it does not. This motivates a new set of numerical results for the full problem, formulated using a boundary integral method, namely to probe the size of the tail amplitude as τ varies for fixed F > 1. The strong conclusion from the numerical results is that true solitary waves of elevation do not exist for the steady gravity–capillary water wave problem, at least for 9/50 < τ < 1=3. This finding confirms and explains previous asymptotic results by Yang & Akylas.

An existence theory for three-dimensional periodic travelling gravity–capillary water waves with bounded transverse profiles

This article presents a rigorous existence theory for three-dimensional gravity–capillary water waves which are uniformly translating and periodic in one horizontal spatial direction x and have a nontrivial transverse profile in the other z. The hydrodynamic equations are formulated as an infinite-dimensional Hamiltonian system in which z is the time-like variable, and a centre-manifold reduction technique is applied to demonstrate that the problem is locally equivalent to a finite-dimensional Hamiltonian system of ordinary differential equations. A family of straight lines C 1, C 2,… in an appropriate two-dimensional parameter space is identified at which the number of purely imaginary eigenvalues of the linear problem changes: at each point on one of these lines two real eigenvalues become purely imaginary by passing through zero. There are also codimension-two points: the line C k intersects each of the lines C k+1 , C k+2 ,… in precisely one point. General statements concerning the existence of waves which are periodic or quasiperiodic in z are made by applying standard tools in Hamiltonian-systems theory to the reduced equations. Moreover, a critical curve in parameter space is found at which a two-dimensional Stokes wave and a three-dimensional wave with a spatially localised and exponentially decaying transverse profile simultaneously bifurcate from the uniform flow. This curve is piecewise linear: it contains one line segment from each of C 1, C 2,… .

A spatial dynamics approach to three-dimensional gravity-capillary steady water waves

This paper contains a rigorous existence theory for three-dimensional steady gravity-capillary finite-depth water waves which are uniformly translating in one horizontal spatial direction x and periodic in the transverse direction z. Physically motivated arguments are used to find a formulation of the problem as an infinite-dimensional Hamiltonian system in which x is the time-like variable, and a centre-manifold reduction technique is applied to demonstrate that the problem is locally equivalent to a finite-dimensional Hamiltonian system. General statements concerning the existence of waves which are periodic or quasiperiodic in x (and periodic in z) are made by applying standard tools in Hamiltonian-systems theory to the reduced equations.A critical curve in Bond number–Froude number parameter space is identified which is associated with bifurcations of generalized solitary waves. These waves are three dimensional but decay to two-dimensional periodic waves (small-amplitude Stokes waves) far upstream and downstream. Their existence as solutions of the water-wave problem confirms previous predictions made on the basis of model equations.

Trapped modes supported by submerged obstacles

In this paper, trapped modes are shown to exist for a pair of bodies that are submerged in deep water. An inverse procedure is used whereby a flow that represents a local oscillation is constructed and closed streamlines that surround the singularities of the flow are interpreted as body boundaries. The velocity potential for the flow is constructed from a symmetric combination of horizontal and vertical dipoles. This combination of singular solutions of Laplace's equation satisfies the free-surface boundary condition, and the spacing and relative amplitudes of the singularities are related in such a way that there is no radiation of waves to infinity. The existence of closed streamlines that surround the singularities is demonstrated by investigating how the structure of the streamlines varies as the parameters of the flow change.

Traveling Two and Three Dimensional Capillary Gravity Water Waves

The main results of this paper are existence theorems for traveling gravity and capillary gravity water waves in two dimensions, and capillary gravity water waves in three dimensions, for any periodic fundamental domain. This is a problem in bifurcation theory, yielding curves in the two dimensional case and bifurcation surfaces in the three dimensional case. In order to address the presence of resonances, the proof is based on a variational formulation and a topological argument, which is related to the resonant Lyapunov center theorem.

A global investigation of solitary-wave solutions to a two-parameter model for water waves

The model equationformula herearises as the equation for solitary-wave solutions to a fifth-order long-wave equation for gravity–capillary water waves. Being Hamiltonian, reversible and depending upon two parameters, it shares the structure of the full steady water-wave problem. Moreover, all known analytical results for local bifurcations of solitary-wave solutions to the full water-wave problem have precise counterparts for the model equation.At the time of writing two major open problems for steady water waves are attracting particular attention. The first concerns the possible existence of solitary waves of elevation as local bifurcation phenomena in a particular parameter regime; the second, larger, issue is the determination of the global bifurcation picture for solitary waves. Given that the above equation is a good model for solitary waves of depression, it seems natural to study the above issues for this equation; they are comprehensively treated in this article.The equation is found to have branches of solitary waves of elevation bifurcating from the trivial solution in the appropriate parameter regime, one of which is described by an explicit solution. Numerical and analytical investigations reveal a rich global bifurcation picture including multi-modal solitary waves of elevation and depression together with interactions between the two types of wave. There are also new orbit-flip bifurcations and associated multi-crested solitary waves with non-oscillatory tails.

Experiments on ripple instabilities. Part 2 Selective amplification of resonant triads

Resonant three-wave interactions among capillary–gravity water waves are studied experimentally using a test wavetrain and smaller background waves (noise) generated mechanically in a channel. The spectrum of the background waves is varied from broad-banded to one with discrete components. When the noise spectrum is broad-banded, the test wavetrain amplifies all waves in its low-frequency band of allowable triads B[ell ], as anticipated from RIT (resonant interaction theory). When the noise spectrum has a discrete component in the high-frequency band of allowable triads Bh, the test wavetrain selectively amplifies a triad with two waves from B[ell ], contrary to expectations based on RIT. (Although, in accordance with RIT, no waves in Bh are amplified.) We conjecture that the mechanism for selective amplification comprises a sequence of exceedingly weak, higher-order interactions, normally neglected in RIT. This sequence allows the small amount of energy in a discrete spectral component to cascade to two waves in B[ell ], which then amplify, as anticipated from RIT, and dominate all other waves in B[ell ]. The conjectured sequence of nonlinear interactions is tested using both frequency and wave-vector data, which are obtained by in situ probes and by remote sensing of the water surface with a highspeed imaging system. Our predictions of selective amplification, as well as its absence, are consistent with all of the experiments presented herein and in Part 1. Selective amplification occurs for signal-to-noise (amplitude) ratios as large as 200, and its effects are measurable within ten wavelengths of the wavemaker. When selective amplification occurs, it has a profound impact on the long-time evolution of a ripple wavetrain.

The effect of capillarity and resonant interactions on the second-order Doppler spectrum of radar sea echo

Previous work on the second-order contributions of radar sea echo by Hasselmann and Barrick is generalized to include the effect of capillarity (i.e., surface tension) in the hydrodynamic part of the transfer coefficient, the lossy dielectric properties of the sea, and the angle of incidence and the polarization of the electromagnetic radiation. The present theoretical results apply to the backscattering of electromagnetic radiation from water waves that are gravity or gravity capillary waves. The introduction of surface tension in the analysis allows for second-order resonant interactions of the gravity capillary wave components. To include resonant interactions of gravity waves, the analysis must be extended to higher order. In general, resonant interactions produce higher-order contributions in the Doppler spectrum at the frequency of the first-order Bragg line that represent the nonstationary and non-Gaussian properties of the water surface. Furthermore, because of the different dispersion relation of gravity capillary water waves, the secondary lines of the second-order Doppler spectrum now are shifted toward the Bragg frequency and 2½ times the Bragg frequency. Numerical results are presented from our generalized theory for various radar and sea (water surface) parameters.