Two-dimensional Water Waves Research Articles

An alternative method to study wave scattering by semi-infinite inertial surfaces

A new method to solve the boundary value problem arising in the study of scattering of two-dimensional surface water waves by a discontinuity in the surface boundary conditions is presented in this paper. The discontinuity arises due to the floating of two semi-infinite inertial surfaces of different surface densities. Applying Green's second identity to the potential functions and appropriate Green's functions, this problem is reduced to solving two coupled Fredholm integral equations with regular kernels. The solutions to these integral equations are used to determine the reflection and the transmission coefficients. The results for the reflection coefficient are presented graphically and are compared to those obtained earlier using other research methods. It is observed from the graphs that the results computed from the present analysis match exactly with the previous results.

Steady Water Waves in the Presence of Wind

In this paper we develop an existence theory for small amplitude, steady, two-dimensional water waves in the presence of wind in the air above. The presence of the wind is modeled by a Kelvin--Helmholtz type discontinuity across the air-water interface, and a corresponding jump in the circulation of the fluids there. We consider both fluids to be inviscid, with the water region being irrotational and of finite depth. The air region is considered with constant vorticity in the case of infinite depth and with a general vorticity profile in the case of a finite, lidded atmosphere.

The second variation of the stream function energy of water waves with vorticity

We compute the second variation of the stream function energy of two-dimensional steady free surface gravity water waves with vorticity in the stream function formulation. We prove that for nonpositive vorticity the second variation of the stream function energy at extreme waves with Stokes corner asymptotics cannot be nonnegative in any small neighbourhood of a given isolated stagnation point. The particular form of our second variation suggests however the possibility that certain singularities in the case of nonzero vorticity might be constructible as minimizers of the stream function energy.

The Stokes conjecture for waves with vorticity

We study stagnation points of two-dimensional steady gravity free-surface water waves with vorticity.We obtain for example that, in the case where the free surface is an injective curve, the asymptotics at any stagnation point is given either by the “Stokes corner flow” where the free surface has a corner of 120°, or the free surface ends in a horizontal cusp, or the free surface is horizontally flat at the stagnation point. The cusp case is a new feature in the case with vorticity, and it is not possible in the absence of vorticity.In a second main result we exclude horizontally flat singularities in the case that the vorticity is 0 on the free surface. Here the vorticity may have infinitely many sign changes accumulating at the free surface, which makes this case particularly difficult and explains why it has been almost untouched by research so far.Our results are based on calculations in the original variables and do not rely on structural assumptions needed in previous results such as isolated singularities, symmetry and monotonicity.

Regularity for steady periodic capillary water waves with vorticity

In the following, we prove new regularity results for two-dimensional steady periodic capillary water waves with vorticity, in the absence of stagnation points. Firstly, we prove that if the vorticity function has a Hölder-continuous first derivative, then the free surface is a smooth curve and the streamlines beneath the surface will be real analytic. Furthermore, once we assume that the vorticity function is real analytic, it will follow that the wave surface profile is itself also analytic. A particular case of this result includes irrotational fluid flow where the vorticity is zero. The property of the streamlines being analytic allows us to gain physical insight into small-amplitude waves by justifying a power-series approach.

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.

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.

Reflection and transmission of water waves in a two‐layer fluid flowing through a channel with undulating bed

AbstractWithin the framework of two‐dimensional linear water wave theory, we consider the problem of normal surface wave propagation over small undulations in a channel flow consisting of a two‐layer fluid with the upper layer bounded by a fixed wall, considered as an approximation to the free surface, and the lower one by a bottom surface which has small undulations. The effects of surface tension at the surface of separation is neglected. Assuming irrotational motion, a perturbation analysis is employed to calculate the first order corrections to the velocity potentials in the two‐layer fluid by using Fourier transform appropriately and also to calculate the reflection and transmission coefficients in terms of integrals involving the shape function c (x) representing the bottom undulation. Two special forms of the shape function are considered for which explicit expressions for reflection and transmission coefficients are evaluated. For the specific case of a patch of sinusoidal ripples having the same wave number throughout, the reflection coefficient up to the first order is an oscillatory function in the quotient of twice the interface wave number and the ripple wave number. When this quotient is one, the theory predicts a resonant interaction between the bed and the interface, and the reflection coefficient becomes a multiple of the number of ripples. But the theory breaks down at resonance and it is applicable only to infinitesimal reflection when the reflection coefficient cannot assume large value, and away from resonance. Hence, the results demonstrated here is valid for up to the near resonant cases only. Again, when a patch of sinusoidal ripples having two different wave numbers is considered, the resonant interaction between the bed and the interface attains in the neighborhood of two (singular) points along x‐axis (when the ripple wave numbers of the bottom undulation become twice as large as interface wave number). The theoretical observations are presented in graphical form.

Steady Water Waves with Multiple Critical Layers

We construct small-amplitude periodic water waves with multiple critical layers. In addition to waves with arbitrarily many critical layers and a single crest in each period, multimodal waves with several crests and troughs in each period are found. The setting is that of steady two-dimensional finite-depth gravity water waves with vorticity.

A geometric approach to generalized Stokes conjectures

We consider Stokes' conjecture concerning the shape of the extremal two-dimensional water wave. By new geometric methods including a nonlinear frequency formula, we prove Stokes' conjecture in the original variables. Our results do not rely on structural assumptions needed in previous results such as isolated singularities, symmetry and monotonicity. Part of our results extends to the mathematical problem in higher dimensions.

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.

Generation of two-dimensional steep water waves on finite depth with and without wind

This paper reports on a series of numerical simulations designed to investigate the action of wind on steep waves and breaking waves generated through the mechanism of dispersive focusing on finite depth. The dynamics of the wave packet propagating without wind at the free surface are compared to the dynamics of the packet propagating in the presence of wind. Wind is introduced in the numerical wave tank by means of a pressure term, corresponding to the modified Jeffreys' sheltering mechanism. The wind blowing over a strongly modulated wave group due to the dispersive focusing of an initial long wave packet increases the duration and maximal amplitude of the steep wave event. These results are coherent with those obtained within the framework of deep water. However, steep wave events are less unstable to wind perturbation in shallow water than in deep water. Furthermore, a comparison between experimental and numerical wave breaking is presented in the absence of wind. The numerical simulations show that the wind speeds up the wave breaking and amplifies slightly the wave height. The wall pressure during the runup of the steep wave event on a vertical wall is also investigated and a comparison between experimental and numerical results is provided.

Scattering of internal waves in a two-layer fluid flowing through a channel with small undulations

Using two-dimensional linear water wave theory, we consider the problem of normal water wave (internal wave) propagation over small undulations in a channel flow consisting of a two-layer fluid in which the upper layer is bounded by a fixed wall, an approximation to the free surface, and the lower one is bounded by a bottom surface that has small undulations. The effects of surface tension at the surface of separation is neglected. Assuming irrotational motion, a perturbation analysis is employed to calculate the first-order corrections to the velocity potentials in the two-layer fluid by using Green’s integral theorem in a suitable manner and the reflection and transmission coefficients in terms of integrals involving the shape function c(x) representing the bottom undulation. Two special forms of the shape function are considered for which explicit expressions for reflection and transmission coefficients are evaluated. For the specific case of a patch of sinusoidal ripples having the same wave number throughout, the reflection coefficient up to the first order is an oscillatory function in the quotient of twice the interface wave number and the ripple wave number. When this quotient approaches one, the theory predicts a resonant interaction between the bed and the interface, and the reflection coefficient becomes a multiple of the number of ripples. High reflection of the incident wave energy occurs if this number is large. Again, when a patch of sinusoidal ripples having two different wave numbers for two consecutive stretches is considered, the interaction between the bed and the interface near resonance attains in the neighborhood of two (singular) points along the x-axis (when the ripple wave number of the bottom undulation become approximately twice as large as the interface wave number). The theoretical observations are presented in graphical form.

Two-dimensional steady edge waves. Part II: Solitary waves

We prove existence for two-dimensional solitary steady water waves propagating along the beach. The proof relies upon recent results for the periodic case, in combination with Sobolev estimates for edge wave solutions.

Two-dimensional steady edge waves. Part I: Periodic waves

We prove existence and uniqueness for two-dimensional steady water waves propagating along the beach. For small periodic shoreline data, global solutions vanishing in the seaward direction are found. In addition, we prove a priori properties of solutions, well-adapted to the physical background.

Stratified Steady Periodic Water Waves

This paper considers two-dimensional stratified water waves propagating under the force of gravity over an impermeable flat bed and with a free surface. We prove the existence of a global continuum of classical solutions that are periodic and traveling. These waves, moreover, can exhibit large density variation, speed, and amplitude.

Recent progress on particle trajectories in steady water waves

We survey recent results on particle trajectories within steady two-dimensional water waves. Particular emphasis is placed on the linear and exact mathematical theory of periodic and symmetric waves, and the effects of a (possibly rotational) background current. The different results vindicate and detail the classical Stokes drift, and also show the transition of orbits when waves propagate into running water. The classical approximation, depicting the trajectories as closed ellipses, is shown to be a mathematical rarity.

Explicit equations for two-dimensional water waves with constant vorticity

Governing equations for two-dimensional inviscid free-surface flows with constant vorticity over arbitrary nonuniform bottom profile are presented in exact and compact form using conformal variables. An efficient and very accurate numerical method for this problem is developed.

Scattering of surface waves over an uneven sea-bed

This work is concerned with the scattering of a train of progressive waves by a small undulation of the bottom of a laterally unbounded sea, using two-dimensional linear water wave theory. Assuming irrotational motion, a simplified perturbation analysis is employed to obtain the first-order corrections to the velocity potential by using the Green’s integral theorem in a suitable manner, and the reflection and transmission coefficients in terms of integrals involving the shape of the function c ( x ) representing the bottom undulation. Particular forms of the shape function are considered and the integrals for reflection and transmission coefficients are evaluated for these shape functions. For a varying sinusoidal sea bottom when Bragg resonance occurs, the reflection coefficient becomes a multiple of the number of ripples, and high reflection of the incident wave energy occurs if this number is large. We present the result in graphical form for one, three and five ripples in the patch. The results find absolute agreement with available results.

Symmetry of steady periodic gravity water waves with vorticity

We prove that steady periodic two-dimensional rotational gravity water waves with a monotone surface profile between troughs and crests have to be symmetric about the crest, irrespective of the vorticity distribution within the fluid.