scholarly journals On essential and continuous spectra of the linearized water-wave problem in a finite pond

2010 ◽  
Vol 106 (1) ◽  
pp. 141 ◽  
Author(s):  
Sergey A. Nazarov ◽  
Jari Taskinen
Keyword(s):  
Boundary Condition ◽  
Spectral Problem ◽  
Essential Spectrum ◽  
Water Waves ◽  
Water Wave ◽  
Water Surface ◽  
Wave Problem ◽  
Water Wave Problem ◽  
Linearized Theory ◽  
Cuspidal Edge

We show that the spectrum of the Laplace equation with the Steklov spectral boundary condition, in the connection of the linearized theory of water-waves, can have a nontrivial essential component even in case of a bounded basin with a horizontal water surface. The appearance of the essential spectrum is caused by the boundary irregularities of the type of a rotational cusp or a cuspidal edge. In a previous paper the authors have proven a similar result for the Steklov spectral problem in a bounded domain with a sharp peak.

scholarly journals An alternative approach to study irrotational periodic gravity water waves

2021 ◽  
Vol 72 (4) ◽  
Author(s):  
Biswajit Basu ◽  
Calin I. Martin
Keyword(s):  
Free Surface ◽  
Water Waves ◽  
Water Wave ◽  
Wave Problem ◽  
Water Wave Problem ◽  
Stagnation Points ◽  
Alternative Approach ◽  
New Formulation ◽  
Bounded Below ◽  
Surface Dynamic

AbstractWe are concerned here with an analysis of the nonlinear irrotational gravity water wave problem with a free surface over a water flow bounded below by a flat bed. We employ a new formulation involving an expression (called flow force) which contains pressure terms, thus having the potential to handle intricate surface dynamic boundary conditions. The proposed formulation neither requires the graph assumption of the free surface nor does require the absence of stagnation points. By way of this alternative approach we prove the existence of a local curve of solutions to the water wave problem with fixed flow force and more relaxed assumptions.

Transmission of water waves through small apertures

1971 ◽  
Vol 49 (1) ◽  
pp. 65-74 ◽  
Author(s):  
E. O. Tuck
Keyword(s):  
Transmission Coefficient ◽  
Asymptotic Expansions ◽  
Water Waves ◽  
Water Wave ◽  
Approximate Formula ◽  
Matched Asymptotic Expansions ◽  
Wave Problem ◽  
Water Wave Problem ◽  
Method Of Solution

A method of solution is proposed for flows through small apertures in otherwise impermeable barriers. This method, which is an application of the method of matched asymptotic expansions, is used to solve a specific water-wave problem, yielding an approximate formula for the transmission coefficient.

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

1997 ◽  
Vol 342 ◽  
pp. 199-229 ◽  
Author(s):  
ALAN R. CHAMPNEYS ◽  
MARK D. GROVES
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Water Waves ◽  
Water Wave ◽  
Model Equation ◽  
Global Bifurcation ◽  
Solitary Wave Solutions ◽  
Wave Problem ◽  
Water Wave Problem ◽  
Wave Solutions

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.

scholarly journals Transverse instability of solitary-wave states of the water-wave problem

2001 ◽  
Vol 439 ◽  
pp. 255-278 ◽  
Author(s):  
T. J. BRIDGES
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Water Waves ◽  
Water Wave ◽  
Explicit Calculation ◽  
Geometric Condition ◽  
Wave Problem ◽  
Water Wave Problem ◽  
Transverse Instability ◽  
Stability And Instability

Transverse stability and instability of solitary waves correspond to a class of perturbations that are travelling in a direction transverse to the direction of the basic solitary wave. In this paper we consider the problem of transverse instability of solitary waves for the water-wave problem, from both the model equation point of view and the full water-wave equations. A new universal geometric condition for transverse instability forms the backbone of the analysis. The theory is first illustrated by application to model PDEs for water waves such as the KP equation, and then it is applied to the full water-wave problem. This is the first theory proposed for transverse instability of solitary waves of the full water-wave problem. The theory suggests the introduction of a new functional for water waves, whose importance is suggested by the mathematical structure. Without explicit calculation, the theory is used to argue that the basic class of solitary waves of the water-wave problem, which bifurcate at Froude number unity, are likely to be stable to transverse perturbations, even at large amplitude.

Singularities in water waves and the Rayleigh–Taylor problem

2010 ◽  
Vol 651 ◽  
pp. 211-239 ◽  
Author(s):  
M. A. FONTELOS ◽  
F. DE LA HOZ
Keyword(s):  
Numerical Simulation ◽  
Water Waves ◽  
Water Wave ◽  
Asymptotic Methods ◽  
Logarithmic Spiral ◽  
Capillary Waves ◽  
Wave Problem ◽  
Water Wave Problem ◽  
Taylor Problem ◽  
Regularizing Effects

We describe, by means of asymptotic methods and direct numerical simulation, the structure of singularities developing at the interface between two perfect, inviscid and irrotational fluids of different densities ρ1 and ρ2 and under the action of gravity. When the lighter fluid is on top of the heavier fluid, one encounters the water-wave problem for fluids of different densities. In the limit when the density of the lighter fluid is zero, one encounters the classical water-wave problem. Analogously, when the heavier fluid is on top of the lighter fluid, one encounters the Rayleigh–Taylor problem for fluids of different densities, with this being the case when one of the densities is zero for the classical Rayleigh–Taylor problem. We will show that both water-wave and Rayleigh–Taylor problems develop singularities of the Moore-type (singularities in the curvature) when both fluid densities are non-zero. For the classical water-wave problem, we propose and provide evidence of the development of a singularity in the form of a logarithmic spiral, and for the classical Rayleigh–Taylor problem no singularities were found. The regularizing effects of surface tension are also discussed, and estimates of the size and wavelength of the capillary waves, bubbles or blobs that are produced are provided.

scholarly journals Existence and amplitude bounds for irrotational water waves in finite depth

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170094 ◽  
Author(s):  
Florian Kogelbauer
Keyword(s):  
Water Waves ◽  
Water Wave ◽  
Wave Speed ◽  
Finite Depth ◽  
Relative Mass ◽  
Wave Problem ◽  
Nonlinear Water Waves ◽  
Water Wave Problem ◽  
Pseudo Differential Equation ◽  
Nonlinear Solutions

We prove the existence of solutions to the irrotational water-wave problem in finite depth and derive an explicit upper bound on the amplitude of the nonlinear solutions in terms of the wavenumber, the total hydraulic head, the wave speed and the relative mass flux. Our approach relies upon a reformulation of the water-wave problem as a one-dimensional pseudo-differential equation and the Newton–Kantorovich iteration for Banach spaces. This article is part of the theme issue ‘Nonlinear water waves’.

scholarly journals Regularity of rotational travelling water waves

2012 ◽  
Vol 370 (1964) ◽  
pp. 1602-1615 ◽  
Author(s):  
Joachim Escher
Keyword(s):  
Water Waves ◽  
Water Wave ◽  
Travelling Wave ◽  
Capillary Waves ◽  
Wave Profile ◽  
Wave Problem ◽  
Water Wave Problem ◽  
Common Assumption ◽  
Infinite Depth ◽  
Stagnation Points

Several recent results on the regularity of streamlines beneath a rotational travelling wave, along with the wave profile itself, will be discussed. The survey includes the classical water wave problem in both finite and infinite depth, capillary waves and solitary waves as well. A common assumption in all models to be discussed is the absence of stagnation points.

scholarly journals Approximate Method for Studying the Waves Propagating along the Interface between Air-Water

2011 ◽  
Vol 2011 ◽  
pp. 1-21 ◽  
Author(s):  
M. M. Khader ◽  
R. F. Al-Bar
Keyword(s):  
Water Waves ◽  
Water Wave ◽  
Decomposition Method ◽  
Fluid Layer ◽  
Wave Packets ◽  
Nonlinear Problems ◽  
Wave Problem ◽  
Water Wave Problem ◽  
Kdv Equations ◽  
Laplace Decomposition Method

This paper is devoted to consider the approximate solutions of the nonlinear water wave problem for a fluid layer of finite depth in the presence of gravity. The method of multiple-scale expansion is employed to obtain the Korteweg-de Vries (KdV) equations for solitons, which describes the behavior of the system for free surface between air and water in a nonlinear approach. The solutions of the water wave problem split up into two wave packets, one moving to the right and one to the left, where each of these wave packets evolves independently as the solutions of KdV equations. The solution of KdV equations is obtained analytically by using a reliable modification of Laplace decomposition method (LDM), namely, the modified Laplace decomposition method (MLDM) is presented. This procedure is a powerful tool for solving large amount of nonlinear problems. The proposed method provides the solution as a series which may converge to the exact solution of the problem. Also, the convergence analysis of the proposed method is given. Finally, we observe that the elevation of the water waves is in form of traveling solitary waves. The horizontal and vertical of the velocity components have nonlinear characters.

Korteweg–de Vries type equations for waves propagating along the interface between air–water

2008 ◽  
Vol 86 (12) ◽  
pp. 1427-1435 ◽  
Author(s):  
A M Abourabia ◽  
M A Mahmoud ◽  
G M Khedr
Keyword(s):  
Water Waves ◽  
Water Wave ◽  
Fluid Layer ◽  
Wave Packets ◽  
Multiple Scale ◽  
Finite Depth ◽  
Wave Problem ◽  
Water Wave Problem ◽  
Kdv Equations ◽  
De Vries

We present solutions of the water wave problem for a fluid layer of finite depth in the presence of gravity and surface tension. The method of multiple scale expansion is employed to obtain the Korteweg–de Vries (KdV) equations for solitons, which describes the behavior of the system for the free surface between air and water in a nonlinear approach. The solutions of the water wave problem split up into two wave packets, one moving to the right and one to the left, where each of these wave packets evolves independently as the solutions of the KdV equations. The solutions of the KdV equations are obtained analytically by using the tanh-function method. The dispersion relations of the model KdV equations are studied. Finally, we observe that the elevation of the water waves are in the form of traveling solitary waves. The horizontal and vertical velocities, and the phase diagrams of the velocity components have a nonlinear characters.PACS No.: 47.11.St

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