Steady water waves with vorticity: an analysis of the dispersion equation

2014 ◽  
Vol 751 ◽  
Author(s):  
V. Kozlov ◽  
N. Kuznetsov ◽  
E. Lokharu
Keyword(s):  
Dispersion Equation ◽  
Gravity Waves ◽  
Water Waves ◽  
Liouville Problem ◽  
Finite Depth ◽  
Stokes Waves ◽  
General Dispersion ◽  
Sturm Liouville ◽  
Necessary And Sufficient ◽  
Appl Math

AbstractTwo-dimensional steady gravity waves with vorticity are considered on water of finite depth. The dispersion equation is analysed for general vorticity distributions, but under assumptions valid only for unidirectional shear flows. It is shown that for these flows (i) the general dispersion equation is equivalent to the Sturm–Liouville problem considered by Constantin & Strauss (Commun. Pure Appl. Math., vol. 57, 2004, pp. 481–527; Arch. Rat. Mech. Anal., vol. 202, 2011, pp. 133–175), (ii) the condition guaranteeing bifurcation of Stokes waves with constant wavelength is fulfilled. Moreover, a necessary and sufficient condition that the Sturm–Liouville problem mentioned in (i) has an eigenvalue is obtained.

ON POSITIVE EIGENFUNCTIONS OF STURM-LIOUVILLE PROBLEM WITH NONLOCAL TWO‐POINT BOUNDARY CONDITION

2007 ◽  
Vol 12 (2) ◽  
pp. 215-226 ◽  
Author(s):  
Sigita Pečiulytė ◽  
Artūras Štikonas
Keyword(s):  
Boundary Condition ◽  
Nonlocal Boundary ◽  
Sufficient Conditions ◽  
Liouville Problem ◽  
Point Boundary ◽  
Existence Domain ◽  
Classical Boundary ◽  
Sturm Liouville ◽  
Necessary And Sufficient ◽  
Nonlocal Boundary Problem

Positive eigenvalues and corresponding eigenfunctions of the linear Sturm‐Liouville problem with one classical boundary condition and another nonlocal two‐point boundary condition are considered in this paper. Four cases of nonlocal two‐point boundary conditions are analysed. We get positive eigenfunctions existence domain for each case of these problems. This domain depends on the parameters of the nonlocal boundary problem and it gives necessary and sufficient conditions for existing positive eigenvalues with positive eigenfunctions.

scholarly journals Three-dimensional surface gravity waves of a broad bandwidth on deep water

2021 ◽  
Vol 926 ◽  
Author(s):  
Yan Li
Keyword(s):  
Deep Water ◽  
Water Waves ◽  
Order Approximation ◽  
Three Dimensional ◽  
Finite Depth ◽  
Second Order ◽  
Leading Order ◽  
Broad Bandwidth ◽  
Stokes Waves ◽  
Deep Water Waves

A new nonlinear Schrödinger equation (NLSE) is presented for ocean surface waves. Earlier derivations of NLSEs that describe the evolution of deep-water waves have been limited to a narrow bandwidth, for which the bound waves at second order in wave steepness are described in leading-order approximations. This work generalizes these earlier works to allow for deep-water waves of a broad bandwidth with large directional spreading. The new NLSE permits simple numerical implementations and can be extended in a straightforward manner in order to account for waves on water of finite depth. For the description of second-order waves, this paper proposes a semianalytical approach that can provide accurate and computationally efficient predictions. With a leading-order approximation to the new NLSE, the instability region and energy growth rate of Stokes waves are investigated. Compared with the exact results based on McLean (J. Fluid Mech., vol. 511, 1982, p. 135), predictions by the new NLSE show better agreement than by Trulsen et al. (Phys. Fluids, vol. 12, 2000, pp. 2432–2437). With numerical implementations of the new NLSE, the effects of wave directionality are investigated by examining the evolution of a directionally spread focused wave group. A downward shift of the spectral peak is observed, owing to the asymmetry in the change rate of energy in a more complex manner than that for uniform Stokes waves. Rapid oblique energy transfers near the group at linear focus are observed, likely arising from the instability of uniform Stokes waves appearing in a narrow spectrum subject to oblique sideband disturbances.

Steady water waves with vorticity: spatial Hamiltonian structure

2013 ◽  
Vol 733 ◽  
Author(s):  
Vladimir Kozlov ◽  
Nikolay Kuznetsov
Keyword(s):  
Dynamical Systems ◽  
Gravity Waves ◽  
Water Waves ◽  
Hamiltonian Structure ◽  
Finite Depth ◽  
Two Dimensional

AbstractSpatial dynamical systems are obtained for two-dimensional steady gravity waves with vorticity on water of finite depth. These systems have Hamiltonian structure and Hamiltonian is essentially the flow–force invariant.

An approximation for the highest gravity waves on water of finite depth

1998 ◽  
Vol 372 ◽  
pp. 45-70 ◽  
Author(s):  
E. A. KARABUT
Keyword(s):  
Gravity Waves ◽  
Water Waves ◽  
Finite Depth ◽  
Simple Calculation ◽  
Ideal Incompressible Fluid ◽  
Finite Amplitude ◽  
Steady Flows ◽  
Heavy Fluid ◽  
Flat Bottom ◽  
New Approach

Planar steady gravity waves of finite amplitude at the surface of an ideal incompressible fluid above a flat bottom are studied theoretically. A new approach to the construction of some steady flows of heavy fluid with a partially free surface is proposed. The hypothesis is suggested and justified that these flows are close to gravity waves. For the case of the highest waves a one-parameter family of exact solutions describing free boundary flows above a flat bottom and under two uneven symmetrically located caps is derived. This family of solutions gives an approximation to the highest water waves in moderate to shallow water depths, enabling relatively simple calculation of their properties.

scholarly journals Symmetry breaking in periodic and solitary gravity-capillary waves on water of finite depth

1987 ◽  
Vol 184 ◽  
pp. 183-206 ◽  
Author(s):  
Juan A. Zufiria
Keyword(s):  
Gravity Waves ◽  
Water Waves ◽  
Hamiltonian Formulation ◽  
Finite Depth ◽  
Capillary Waves ◽  
Bifurcation Structure ◽  
Low Gravity ◽  
Bifurcation Tree ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Model

A weakly nonlinear model is developed from the Hamiltonian formulation of water waves, to study the bifurcation structure of gravity-capillary waves on water of finite depth. It is found that, besides a very rich structure of symmetric solutions, non-symmetric Wilton's ripples exist. They appear via a spontaneous symmetrybreaking bifurcation from symmetric solutions. The bifurcation tree is similar to that for gravity waves. The solitary wave with surface tension is studied with the same model close to a critical depth. It is found that the solution is not unique, and that further non-symmetric solitary waves are possible. The bifurcation tree has the same structure as for the case of periodic waves. The possibility of checking these results in low-gravity experiments is postulated.

scholarly journals Non-existence of three-dimensional travelling water waves with constant non-zero vorticity

2014 ◽  
Vol 746 ◽  
Author(s):  
E. Wahlén
Keyword(s):  
Gravity Waves ◽  
Water Waves ◽  
Elliptic Equations ◽  
Three Dimensional ◽  
Unique Continuation ◽  
Finite Depth ◽  
Capillary Waves ◽  
Liouville Type ◽  
Liouville Type Results

AbstractWe prove that there are no three-dimensional bounded travelling gravity waves with constant non-zero vorticity on water of finite depth. The result also holds for gravity–capillary waves under a certain condition on the pressure at the surface, which is satisfied by sufficiently small waves. The proof relies on unique continuation arguments and Liouville-type results for elliptic equations.

scholarly journals Modified Babenko’s Equation For Periodic Gravity Waves On Water Of Finite Depth

2019 ◽  
Vol 72 (4) ◽  
pp. 415-428
Author(s):  
E Dinvay ◽  
N Kuznetsov
Keyword(s):  
Surface Tension ◽  
Gravity Waves ◽  
Water Waves ◽  
Surface Profile ◽  
Finite Depth ◽  
Parametric Representation ◽  
Two Dimensional ◽  
Free Surface Profile ◽  
The Mean ◽  
New Equation

Summary A new operator equation for periodic gravity waves on water of finite depth is derived and investigated; it is equivalent to Babenko’s equation considered in Kuznetsov and Dinvay (Water Waves, 1, 2019). Both operators in the proposed equation are nonlinear and depend on the parameter equal to the mean depth of water, whereas each solution defines a parametric representation for a symmetric free surface profile. The latter is a component of a solution of the two-dimensional, nonlinear problem describing steady waves propagating in the absence of surface tension. Bifurcation curves (including a branching one) are obtained numerically for solutions of the new equation; they are compared with known results.

Almost-highest gravity waves on water of finite depth

2002 ◽  
Vol 13 (1) ◽  
pp. 67-93 ◽  
Author(s):  
DMITRI V. MAKLAKOV
Keyword(s):  
Gravity Waves ◽  
Water Waves ◽  
Finite Depth ◽  
Maximum Slope ◽  
Free Surfaces ◽  
Asymptotic Formulae ◽  
Steep Waves ◽  
Second Period ◽  
Wave Properties ◽  
Limiting Value

The paper presents a method of computing periodic water waves based on solving an integral equation by means of discretization and automatically finding the mesh on which the functions to be found are approximated by the best way. The power of the method to describe ‘bad functions’ well makes it possible to reproduce all the main results of asymptotic theory for the almost-highest waves (Longuet-Higgins & Fox, 1977, 1978, 1996) by a direct numerical simulation. The method is able to compute two full periods of the oscillations of wave properties for all wave height-to-length ratios. The end of the second period corresponds to the wave steepness that achieves 99.99997% of the limiting value. So, the validity of the asymptotic formulae by Longuet-Higgins & Fox is proved for the steep waves of any finite depth. The refined value of the maximum slope of the free-surfaces is found to be 30.3787°.

scholarly journals Approximate solution for Euler equations of stratified water via numerical solution of coupled KdV system

2003 ◽  
Vol 2003 (63) ◽  
pp. 3979-3993 ◽  
Author(s):  
A. A. Halim ◽  
S. P. Kshevetskii ◽  
S. B. Leble
Keyword(s):  
Euler Equations ◽  
Water Waves ◽  
Waveguide Mode ◽  
Liouville Problem ◽  
Initial Boundary ◽  
Boussinesq Approximation ◽  
Initial Boundary Problem ◽  
Stream Functions ◽  
Sturm Liouville ◽  
Background State

We consider Euler equations with stratified background state that is valid for internal water waves. The solution of the initial-boundary problem for Boussinesq approximation in the waveguide mode is presented in terms of the stream function. The orthogonal eigenfunctions describe a vertical shape of the internal wave modes and satisfy a Sturm-Liouville problem. The horizontal profile is defined by a coupled KdV system which is numerically solved via a finite-difference scheme for which we prove the convergence and stability. Together with the solution of the Sturm-Liouville problem, the stream functions give the internal waves profile.

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