Water Wave Problem Research Articles

Non-existence of cusps for a free-boundary problem for water waves

In Varvaruca and Weiss (2011), Varvaruca and Weiss eliminate the existence of cusps for a free-boundary problem for two-dimensional water waves under assumptions that hold for solutions for which \{u>0\} is a “strip-like” domain in the sense of Varvaruca (2008). In this paper, it is proven that cusps do not exist in the natural setting for these free-boundary problems. In particular, non-strip-like domains are also allowed. This qualitative result follows from quantitative results which, roughly speaking, give lower bounds on the “slope” at which the free boundary approaches a stagnation point. This builds upon recent work on non-existence of cusps in McCurdy and Naples (2022) for local minimizers.

Efficient numerical approximations for a nonconservative nonlinear Schrödinger equation appearing in wind‐forced ocean waves

AbstractWe consider a nonconservative nonlinear Schrödinger equation (NCNLS) with time‐dependent coefficients, inspired by a water waves problem. This problem does not have mass or energy conservation, but instead mass and energy change in time under explicit balance laws. In this paper, we extend to the particular NCNLS two numerical schemes which are known to conserve energy and mass in the discrete level for the cubic nonlinear Schrödinger equation. Both schemes are second‐order accurate in time, and we prove that their extensions satisfy discrete versions of the mass and energy balance laws for the NCNLS. The first scheme is a relaxation scheme that is linearly implicit. The other scheme is a modified Delfour–Fortin–Payre scheme, and it is fully implicit. Numerical results show that both schemes capture robustly the correct values of mass and energy, even in strongly nonconservative problems. We finally compare the two numerical schemes and discuss their performance.

Higher‐order integrable models for oceanic internal wave–current interactions

AbstractIn this paper, we derive a higher‐order Korteweg–de Vries (HKdV) equation as a model to describe the unidirectional propagation of waves on an internal interface separating two fluid layers of varying densities. Our model incorporates underlying currents by permitting a sheared current in both fluid layers, and also accommodates the effect of the Earth's rotation by including Coriolis forces (restricted to the Equatorial region). The resulting governing equations describing the water wave problem in two fluid layers under a “flat‐surface” assumption are expressed in a general form as a system of two coupled equations through Dirichlet–Neumann (DN) operators. The DN operators also facilitate a convenient Hamiltonian formulation of the problem. We then derive the HKdV equation from this Hamiltonian formulation, in the long‐wave, and small‐amplitude, asymptotic regimes. Finally, it is demonstrated that there is an explicit transformation connecting the HKdV we derive with the following integrable equations of a similar type: KdV5, Kaup–Kuperschmidt equation, Sawada–Kotera equation, and Camassa–Holm and Degasperis–Procesi equations.

General Solutions to the Navier-Stokes Equations for Incompressible Flow

<i>Waves are mainly generated by wind via the transfer of wind energy to the water through friction. When the wind subsides, the waves transition into swells and eventually dissipate. Friction plays a crucial role in the generation and dissipation of waves. Numerous wave theories have been developed based on the assumption of inviscid flow, but these theories are inadequate in explaining the transformation of waves into swells. This study addressed these limitations by analytically deriving general solutions to the Navier–Stokes equations. By expressing the velocity field as the product of a solution to the Helmholtz equation and a time-dependent univariate function, the Navier–Stokes equations are decomposed into an ordinary differential equation and the Euler equations, which are solved using tensor calculus. This paper provides solutions for viscous flow with shear currents when applied to the water wave problem. These solutions were validated through their application to the vorticity equation. The decay modulus of water waves was compared with experimental data, showing a significant degree of concordance. In contrast to other wave theories, this study clarified the process through which waves evolve into swells.</i>

On Small-Amplitude Asymmetric Water Waves

AbstractWe generalize the method used by Mæhlen and Seth (Asymmetric traveling wave solutions of the capillary–gravity Whitham equation, 2023, arxiv:2312.15343) used to prove the existence of small-amplitude asymmetric solutions to the capillary–gravity Whitham equation, so that it can be applied directly to a class of similar equations. The purpose is to prove or disprove the existence of asymmetric waves for the water wave problem or other model equations for water waves. Our main result in this paper is a theorem that gives both necessary and sufficient conditions for the existence of small-amplitude periodic asymmetric solutions for this class of equations. The result is then applied to an infinite depth capillary–gravity Whitham equation and an infinite depth capillary–gravity Babenko equation to show a nonexistence result of small-amplitude waves for these equations. This example also highlights the similarities between these equations suggesting the potential existence of small-amplitude asymmetric waves for the finite depth capillary–gravity Babenko equation.

On the stability of fully nonlinear hydraulic-fall solutions to the forced water wave problem

Two-dimensional free-surface flow over localised topography is examined, with the emphasis on the stability of hydraulic-fall solutions. A Gaussian topography profile is assumed with a positive or negative amplitude modelling a bump or a dip, respectively. Steady hydraulic-fall solutions to the full incompressible, irrotational Euler equations are computed, and their linear and nonlinear stability is analysed by computing eigenspectra of the pertinent linearised operator and by solving an initial value problem. The computations are carried out numerically using a specially developed computational framework based on the finite-element method. The Hamiltonian structure of the problem is demonstrated, and stability is determined by computing eigenspectra of the pertinent linearised operator. It is found that a hydraulic-fall flow over a bump is spectrally stable. The corresponding flow over a dip is found to be linearly unstable. In the latter case, time-dependent simulations show that ultimately, the flow settles into a time-periodic motion that corresponds to an invariant solution in an appropriately defined phase space. Physically, the solution consists of a localised large-amplitude wave that pulsates above the dip while simultaneously emitting nonlinear cnoidal waves in the upstream direction and multi-harmonic linear waves in the downstream direction.

Optimal control approach for moving bottom detection in one‐dimensional shallow waters by surface measurements

We consider the Boussinesq‐Peregrine (BP) system as described by Lannes [Lannes, D. (2013). The water waves problem: mathematical analysis and asymptotics (Vol. 188). American Mathematical Soc.], within the shallow water regime, and study the inverse problem of determining the time and space variations of the channel bottom profile, from measurements of the wave profile and its velocity on the free surface. A well‐posedness result within a Sobolev framework for (BP), considering a time dependent bottom, is presented. Then, the inverse problem is reformulated as a nonlinear PDE‐constrained optimization one. An existence result of the minimum, under constraints on the admissible set of bottoms, is presented. Moreover, an implementation of the gradient descent approach, via the adjoint method, is considered. For solving numerically both, the forward (BP) and its adjoint system, we derive a universal and low‐dissipation scheme, which contains non‐conservative products. The scheme is based on the FORCE‐ method proposed in [Toro, E. F., Saggiorato, B., Tokareva, S., and Hidalgo, A. (2020). Low‐dissipation centred schemes for hyperbolic equations in conservative and non‐conservative form. Journal of Computational Physics, 416, 109545]. Finally, we implement this methodology to recover three different bottom profiles; a smooth bottom, a discontinuous one, and a continuous profile with a large gradient. We compare with two classical discretizations for (BP) and the adjoint system. These results corroborate the effectiveness of the proposed methodology to recover bottom profiles.

Response of an Oscillating Water Column spanning a circle sector and embedded in a circular platform

This paper develops a semi-analytical solution of a water waves problem concerning the interaction between linear waves and an Oscillating Water Column (OWC) embedded into a circular platform. Unlike similar studies available in the literature, the proposed investigation concerns a realistic situation involving the use of an OWC having limited angular width. Such a configuration is likely to be employed in the arrangement of platforms in which the use of a large OWC is neither efficient nor feasible.The proposed solution is developed in the framework of the linear potential flow theory. The solution relies on the use of a domain decomposition approach involving eigen-function expansions of velocity potentials with unknown coefficients, which are calculated via a matching technique. The solution is utilized for conducting relevant parameter studies concerning the effects of the main geometrical parameters on the overall system performance and on the system hydrodynamic parameters.

Scattering of water waves by multiple rows of vertical thin barriers

The reflection and transmission of waves by a periodic array of thin fixed identical vertical barriers extending uniformly through the depth of the fluid is considered. The water wave problem, which also has analogues in acoustics and electromagnetics, involves scattering by a large finite number of equally spaced rows, each consisting of an array of barriers with a linear periodic arrangement extending to infinity. The main purpose of the work is to compare the results of two approximate methods of solution based on homogenisation techniques with exact results allowing us to determine when the results can be expected to be in good agreement. When no approximation is made, a method of solution (Discrete Model) describing the propagation of incident waves through the exact barrier arrangement is derived using Floquet–Bloch eigensolutions for the corresponding infinite periodic array. It is shown how the solution can be reduced to a simple pair of integral equations which results only from matching the region containing barrier arrays to the two exterior domains. Numerical solutions are, nevertheless, relatively complicated and two approximate models are developed which considerably simplify the numerical effort required. In one approximation (Continuum Model I) it is assumed only that the spacing between two adjacent rows of barriers is small and is complementary to the commonly-used wide-spacing approximation. The complication of the geometry is removed by replacing the governing equations and boundary conditions with an effective continuous medium and a simplified pair of integral equations now govern the solutions. In a second approximation (Continuum Model II), a low-frequency homogenisation approach based on a long-wave assumption is invoked and matching now leads to closed-form expressions for the reflection and transmission coefficients. Results concentrate on the comparison between different methods which allows us to assess the validity of each continuum model. This understanding will be important in developing approximate models that describe the operation of wave energy farms consisting of large arrays of oscillating flaps designed to absorb energy.

Global Bifurcation and Highest Waves on Water of Finite Depth

We consider the two-dimensional problem for steady water waves with vorticity on water of finite depth. While neglecting the effects of surface tension we construct connected families of large amplitude periodic waves approaching a limiting wave, which is either a solitary wave, the highest solitary wave, the highest Stokes wave or a Stokes wave with a breaking profile. In particular, when the vorticity is nonnegative we prove the existence of highest Stokes waves with an included angle of 120^circ . In contrast to previous studies, we fix the Bernoulli constant and consider the wavelength as a bifurcation parameter, which guarantees that the limiting wave has a finite depth. In fact, this is the first rigorous proof of the existence of extreme Stokes waves with vorticity on water of finite depth. Aside from the existence of highest waves, we provide a new result about the regularity of Stokes waves of arbitrary amplitude (including extreme waves). Furthermore, we prove several new facts about steady waves, such as a lower bound for the wavelength of Stokes waves, while also eliminating a possibility of the wave breaking for waves with non-negative vorticity.

Curved channels with constant cross sections may support trapped surface waves

Curved channels with constant cross sections are constructed which support a trapped surface wave. Since corresponding eigenvalues are embedded in the continuous spectrum of the water wave problem and therefore possess the natural instability, the construction procedure requires “fine-tuning” of several parameters in the (small) curvature of the channel as well as geometrical restrictions on the cross section. In particular, the mirror symmetry of the cross section with respect to the vertical axis disrupts the procedure, but examples of suitable non-symmetric cross sections are provided.

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.

Transverse Dynamics of Two-Dimensional Traveling Periodic Gravity–Capillary Water Waves

We study the transverse dynamics of two-dimensional traveling periodic waves for the gravity–capillary water-wave problem. The governing equations are the Euler equations for the irrotational flow of an inviscid fluid layer with free surface under the forces of gravity and surface tension. We focus on two open sets of dimensionless parameters (alpha ,beta ), where alpha and beta are the inverse square of the Froude number and the Weber number, respectively. For each arbitrary but fixed pair (alpha ,beta ) in one of these sets, two-dimensional traveling periodic waves bifurcate from the trivial constant flow. In one open set we find a one-parameter family of periodic waves, whereas in the other open set we find two geometrically distinct one-parameter families of periodic waves. Starting from a transverse spatial dynamics formulation of the governing equations, we investigate the transverse linear instability of these periodic waves and the induced dimension-breaking bifurcation. The two results share a common analysis of the purely imaginary spectrum of the linearization at a periodic wave. We apply a simple general criterion for the transverse linear instability problem and a Lyapunov center theorem for the dimension-breaking bifurcation. For parameters (alpha ,beta ) in the open set where there is only one family of periodic waves, we prove that these waves are linearly transversely unstable. For the other open set, we show that the waves with larger wavenumber are transversely linearly unstable. We also identify an open subset of parameters for which both families of periodic waves are transversely linearly unstable. For each of these transversely linearly unstable periodic waves, a dimension-breaking bifurcation occurs in which three-dimensional doubly periodic waves bifurcate from the two-dimensional periodic wave.

Evolution of tsunami-like isolated waves shoaling over a plane slope

Numerical experiments are conducted to simulate Gaussian-shaped waves shoaling over a uniformly sloping bed. A higher-order pseudo-spectral method is implemented to solve the fully nonlinear Euler water-wave problem for a variable depth. Unlike a solitary wave, the amplitude and wave breadth of a Gaussian waveform can be assigned independently. Therefore, it is convenient to use a Gaussian as a generic tsunami waveform. Although the rate of shoaling is consistent with Green's law only for small-amplitude waves propagating over intermediate sloping beds, the shoaling processes, in general, are complex and the behaviors cannot be accurately predicted by a single law. Several factors are involved in the complexity. When the slope is sufficiently mild, a Gaussian wave transforms into a series of solitons, hence leading to the adiabatic shoaling process. In some instances, the initial rate of shoaling even surpasses the adiabatic rate. This occurs when a soliton emerges from the initial conditions with an amplitude greater than the original Gaussian wave amplitude. In general, there is a trend of a faster shoaling rate when the nonlinearity effect of the Gaussian is greater than that of the corresponding solitary wave, whereas the rate of shoaling for Gaussian waves is slower when the frequency dispersion is greater than that of the solitary wave. The model is further applied to simulate the 2011 Heisei tsunami event, and the shoaling behavior is analyzed by comparing the numerical results with the recorded data in the field.

Liouville-type results for the time-dependent three-dimensional (inviscid and viscous) water wave problem with an interface

Under consideration here are time-dependent three-dimensional stratified water flows of finite depth with a free surface and an interface (playing the role of an internal wave and separating two layers of constant and different densities). The main outcome of the paper is that, under the assumption that the vorticity vectors in the two layers are constant, non-vanishing and parallel to each other, we prove that bounded solutions to the three-dimensional equations are essentially two-dimensional. More precisely, the free surface, the interface, the pressure and the velocity field present no variations in the direction orthogonal to the direction of motion. In addition, one of the horizontal components of the velocity field is constant throughout the flow and the vorticity also points in the direction orthogonal to the wave propagation direction. The rigid lid case is analyzed from the perspective of relaxed assumptions concerning the vorticity vectors. We also consider the viscous water wave problem case with normal and tangential stress conditions at the free surface and find that under similar conditions on the vorticity vectors, the conclusion about the two-dimensionality remains in place. Along the way, we also provide an explicit solution for the time-dependent three-dimensional inviscid water wave problem with an interface.

Symmetry of Periodic Traveling Waves for Nonlocal Dispersive Equations

Of concern is the a priori symmetry of traveling wave solutions of a general class of nonlocal dispersive equations , where is a Fourier multiplier operator with symbol . Our analysis includes both homogeneous and inhomogeneous symbols. We characterize a large class of symbols guaranteeing that periodic traveling wave solutions are symmetric under a mild assumption on the wave profile. In contrast with the classically imposed setting in the water wave problem which assumes traveling waves to have a unique crest and trough per period or a monotone structure near troughs, we formulate a reflection criterion which does not presuppose a monotone structure on the wave profile. Thereby, the reflection criterion enables us to treat a priori solutions with multiple crests of different size per period. Moreover, our result applies not only to smooth traveling wave solutions, but also to those with singular crests around which some cancellation structure appears, including in particular waves with peaks or cusps. The proof relies on a so-called touching lemma, which is related to a strong maximum principle for elliptic operators, and a weak form of the celebrated method of moving planes.

Water wave models using conformal coordinates

A review of nonlinear water wave problems in complex domains is presented. These include non-smooth highly variable bottom topographies and boundaries as for example forked channel regions. The physical and mathematical relevance of each case is discussed. The Schwarz–Christoffel mapping is the common modeling tool for these different problems. The mapping produces conformal coordinates which allow the problems’ dimension to be reduced and the weakly nonlinear partial differential system of equations to be simplified. Usually conformal mapping is adopted while preserving the Laplace operator. The main focus in the present review is to highlight some water wave problems where the underlying differential operators are not invariant under the conformal change of coordinates. The Schwarz–Christoffel mapping is used to produce a boundary fitted, curvilinear conformal coordinate system. The models reported include solitary waves on branching channels, and its reduction to nonlinear waves on graphs, as well as three dimensional surface-waves propagating over highly variable ridges. Of particular interest is nonlinear water waves on graphs, which has important applications and a scarce mathematical literature. Conformal mapping plays a crucial role regarding a systematic incorporation of forking angles into the equations, as not done before, and deducing a generalized compatibility condition at the graph nodes.

Wave Radiation from Bottom Vibrations in Open-Channel Flow with Uniform Vorticity

The linearized water-wave radiation problem for a 2D oscillating source in an inviscid shear flow with a free surface is investigated analytically. The singular source is located at the bottom, with constant fluid depth. The velocity of the basic flow varies linearly with depth, assuming uniform vorticity. The far-field surface waves radiated out from the 2D bottom source are calculated, based on Euler’s equation of motion with the application of radiation conditions. The present analysis extends the work by Tyvand and Sveen to include a nonzero surface flow. Doppler effects will arise in a system at rest with the undisturbed free surface. Resonance with zero group velocity will occur at a critical Froude number for the surface flow. In the presence of a basic surface flow, there are in total four radiated waves when the surface flow is subcritical. With supercritical flow, there are only two emitted waves, with downstream propagation. The critical Froude number depends on the surface velocity, the shear rate and the oscillation frequency of the bottom source.

Global bifurcation structure and some properties of steady periodic water waves with vorticity

This paper studies the classical water wave problem with vorticity described by the Euler equations with a free surface under the influence of gravity over a flat bottom. Based on fundamental work [5], we first obtain two continuous bifurcation curves which meet the laminar flow only one time by using the modified analytic bifurcation theorem. They are symmetric waves whose profiles are monotone between each crest and trough. Moreover, we show a connection between the concavity and convexity of wave profile and the monotonicity of the vertical velocity component v along the free surface. As an important application, we make up the missing major aspect on the behavior of v as mentioned in [4, Sect 4.4] for small amplitude waves. In addition, for favorable vorticity, we prove that the vertical displacement of water waves decreases with depth.

Hamiltonian Variational Formulation of Three-Dimensional, Rotational Free-Surface Flows, with a Moving Seabed, in the Eulerian Description

Hamiltonian variational principles have provided, since the 1960s, the means of developing very successful wave theories for nonlinear free-surface flows, under the assumption of irrotationality. This success, in conjunction with the recognition that almost all flows in the sea are not irrotational, raises the question of extending Hamilton’s principle to rotational free-surface flows. The Euler equations governing the bulk fluid motion have been derived by means of Hamilton’s principle since the late 1950s. Nevertheless, a complete variational formulation of the rotational water-wave problem, including the derivation of the free-surface boundary conditions, seems to be lacking until now. The purpose of the present work is to construct such a missing variational formulation. The appropriate functional is the usual Hamilton’s action, constrained by the conservation of mass and the conservation of fluid parcels’ identity. The differential equations governing the bulk fluid motion are derived as usually, applying standard methods of the calculus of variations. However, the standard methodology does not provide enough structure to obtain the free-surface boundary conditions. To overcome this difficulty, differential-variational forms of the aforementioned constraints are introduced and applied to the boundary variations of the Eulerian fields. Under this transformation, both kinematic and dynamic free-surface conditions are naturally derived, ensuring the Hamiltonian variational formulation of the complete problem. An interesting feature, appearing in the present variational derivation, is a dual possibility concerning the tangential velocity on the boundary; it may be either the same as in irrotational flow (no condition) or zero, corresponding to the small-viscosity limit. The deeper meaning and the significance of these findings seem to deserve further analysis.