scholarly journals No steady water waves of small amplitude are supported by a shear flow with a still free surface

2013 ◽  
Vol 717 ◽  
pp. 523-534 ◽  
Author(s):  
Vladimir Kozlov ◽  
Nikolay Kuznetsov
Keyword(s):  
Free Surface ◽  
Shear Flow ◽  
Gravity Waves ◽  
Water Waves ◽  
Small Amplitude ◽  
Finite Depth ◽  
Coordinate Frame ◽  
Two Dimensional ◽  
Horizontal Shear ◽  
Positive Coefficients

AbstractThe two-dimensional free-boundary problem describing steady gravity waves with vorticity on water of finite depth is considered. It is proved that no small-amplitude waves are supported by a horizontal shear flow whose free surface is still, that is, it is stagnant in a coordinate frame such that the flow is time-independent in it. The class of vorticity distributions for which such flows exist includes all positive constant distributions, as well as linear and quadratic ones with arbitrary positive coefficients.

scholarly journals New exact relations for steady irrotational two-dimensional gravity and capillary surface waves

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170220 ◽  
Author(s):  
Didier Clamond
Keyword(s):  
Free Surface ◽  
Gravity Waves ◽  
Water Waves ◽  
Two Dimensional ◽  
Physical Plane ◽  
Theme Issue ◽  
Nonlinear Water Waves ◽  
Dimensional Gravity ◽  
Irrotational Motion ◽  
Exact Relations

Steady two-dimensional surface capillary–gravity waves in irrotational motion are considered on constant depth. By exploiting the holomorphic properties in the physical plane and introducing some transformations of the boundary conditions at the free surface, new exact relations and equations for the free surface only are derived. In particular, a physical plane counterpart of the Babenko equation is obtained. This article is part of the theme issue ‘Nonlinear water waves’.

Note on the scattering of water waves by a vertical barrier

1966 ◽  
Vol 62 (3) ◽  
pp. 507-509 ◽  
Author(s):  
W. E. Williams
Keyword(s):  
Free Surface ◽  
Water Waves ◽  
Small Amplitude ◽  
Two Dimensional ◽  
Vertical Barrier ◽  
Alternative Approach

Introduction. In this note an alternative approach is presented to the problem of the scattering of small amplitude two-dimensional water waves by a fixed barrier, one edge of the barrier lying in the free surface of the water. This problem was first solved by Ursell ((1)) and generalizations of the problem have been considered by John ((2)) and Lewin ((3)).

Stationary gravity waves on non-uniform free streams: jet-like streams

1975 ◽  
Vol 77 (2) ◽  
pp. 415-438 ◽  
Author(s):  
D. H. Peregrine ◽  
Ronald Smith
Keyword(s):  
Free Surface ◽  
Gravity Waves ◽  
Small Amplitude ◽  
Wave Number ◽  
Explicit Solutions ◽  
Stationary Waves ◽  
Two Dimensional ◽  
Large Wave ◽  
Surface Gravity Waves ◽  
Large Wave Number

AbstractThe basic state considered in this paper is a parallel flow of a jet-like character with the centre of the jet being at or near a free surface which is horizontal. Stationary surface gravity waves may exist on such a flow, and a number of examples are looked at for small amplitude waves. Explicit solutions are given for ‘top-hat’ profile jets and for two-dimensional flows. Asymptotic solutions are developed for stationary waves of large wave-number.

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.

scholarly journals Small amplitude capillary-gravity waves in a channel of finite depth

1989 ◽  
Vol 31 (2) ◽  
pp. 142-160 ◽  
Author(s):  
M. C. W. Jones
Keyword(s):  
Integral Equation ◽  
Free Surface ◽  
Gravity Waves ◽  
Small Amplitude ◽  
Unsolved Problem ◽  
Equations Of Motion ◽  
Finite Depth ◽  
Wave Problem ◽  
Infinite Depth ◽  
Existence And Multiplicity

Introductory Remarks. Recently a number of studies (Chen & Saffman [2], Jones & Toland [7,11], Hogan [5]) have been made of periodic capillary-gravity waves which form the free surface of an ideal fluid contained in a channel of infinite depth. However, little work appears to have been done on the corresponding problem when the depth is finite. The most significant contributions appear to be those of Reeder & Shinbrot [9], Barakat & Houston [1] and Nayfeh [8] all of whom confined themselves to Wilton ripples (see §1.3). Yet there are sound reasons why such a study should be made. For quite apart from the unsolved problem regarding the type of capillary-gravity waves which may occur at finite depths, the consideration of the finite depth problem may be regarded as a first step in the study of solitary capillary-gravity waves. In this paper, a new integral equation for the infinite depth problem, due to J. F. Toland and the author, is adapted to be of use in tackling the finite depth problem. Using this we obtain results for the exact equations of motion which answer rigorously the questions of existence and multiplicity of small amplitude solutions of the periodic capillary-gravity wave problem of finite depth.

scholarly journals On bounds and non-existence in the problem of steady waves with vorticity

2015 ◽  
Vol 765 ◽  
Author(s):  
V. Kozlov ◽  
N. Kuznetsov ◽  
E. Lokharu
Keyword(s):  
Free Surface ◽  
Gravity Waves ◽  
Surface Profile ◽  
Finite Depth ◽  
Two Dimensional ◽  
Vorticity Distribution ◽  
Unidirectional Flow ◽  
Wave Solutions ◽  
Free Surface Profile ◽  
Parallel Shear Flow

AbstractFor the problem describing steady gravity waves with vorticity on a two-dimensional unidirectional flow of finite depth the following results are obtained. (i) Bounds are found for the free-surface profile and for Bernoulli’s constant. (ii) If only one parallel shear flow exists for a given value of Bernoulli’s constant, then there are no wave solutions provided the vorticity distribution is subject to a certain condition.

scholarly journals The Froude number for solitary water waves with vorticity

2015 ◽  
Vol 768 ◽  
pp. 91-112 ◽  
Author(s):  
Miles H. Wheeler
Keyword(s):  
Free Surface ◽  
Shear Flow ◽  
Froude Number ◽  
Simple Proof ◽  
Upper Bound ◽  
Water Waves ◽  
Wave Speed ◽  
Critical Value ◽  
Arbitrary Distribution ◽  
Two Dimensional

We consider two-dimensional solitary water waves on a shear flow with an arbitrary distribution of vorticity. Assuming that the horizontal velocity in the fluid never exceeds the wave speed and that the free surface lies everywhere above its asymptotic level, we give a very simple proof that a suitably defined Froude number $F$ must be strictly greater than the critical value $F=1$. We also prove a related upper bound on $F$, and hence on the amplitude, under more restrictive assumptions on the vorticity.

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.

scholarly journals On periodic geophysical water flows with discontinuous vorticity in the equatorial f -plane approximation

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170096 ◽  
Author(s):  
Calin Iulian Martin
Keyword(s):  
Free Surface ◽  
Water Waves ◽  
Small Amplitude ◽  
Wave Speed ◽  
Laminar Flows ◽  
Local Bifurcation ◽  
Two Dimensional ◽  
Theme Issue ◽  
Nonlinear Water Waves ◽  
Water Flows

We are concerned here with geophysical water waves arising as the free surface of water flows governed by the f -plane approximation. Allowing for an arbitrary bounded discontinuous vorticity, we prove the existence of steady periodic two-dimensional waves of small amplitude. We illustrate the local bifurcation result by means of an analysis of the dispersion relation for a two-layered fluid consisting of a layer of constant non-zero vorticity γ 1 adjacent to the surface situated above another layer of constant non-zero vorticity γ 2 ≠ γ 1 adjacent to the bed. For certain vorticities γ 1 , γ 2 , we also provide estimates for the wave speed c in terms of the speed at the surface of the bifurcation inducing laminar flows. This article is part of the theme issue ‘Nonlinear water waves’.

A note on the uniqueness of small-amplitude water waves travelling in a region of varying depth

1979 ◽  
Vol 86 (3) ◽  
pp. 511-519 ◽  
Author(s):  
G. F. Fitz-Gerald ◽  
R. H. J. Grimshaw
Keyword(s):  
Linear Theory ◽  
Gravity Waves ◽  
Water Waves ◽  
Small Amplitude ◽  
Velocity Potential ◽  
Surface Gravity ◽  
Two Dimensional ◽  
Physical Interest ◽  
Surface Gravity Waves ◽  
Convexity Conditions

The two-dimensional, irrotational, linear theory used in the investigation of the propagation of monochromatic surface gravity waves in a region of varying depth is considered. Uniqueness of the velocity potential is established for bottom profiles satisfying certain convexity conditions. These include the majority of profiles of physical interest.

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