Capillary gravity waves on the free surface of an inviscid fluid of infinite depth. Existence of solitary waves

1996 ◽  
Vol 136 (1) ◽  
pp. 1-19 ◽  
Author(s):  
Gérard Iooss ◽  
Pius Kirrmann
Keyword(s):  
Free Surface ◽  
Solitary Waves ◽  
Gravity Waves ◽  
Inviscid Fluid ◽  
Infinite Depth

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.

Surface instability of a dielectric fluid in an electric field

1968 ◽  
Vol 64 (4) ◽  
pp. 1203-1207 ◽  
Author(s):  
D. H. Michael
Keyword(s):  
Free Surface ◽  
Dispersion Relation ◽  
Gravity Waves ◽  
Normal Mode Analysis ◽  
Horizontal Layer ◽  
Mode Analysis ◽  
Dielectric Fluid ◽  
Inviscid Fluid ◽  
Static Displacement ◽  
The Stability

This paper is a sequel to a recent paper (1) in which the author discussed gravity waves on a horizontal layer of conducting fluid with a normal electrostatic field at the free surface. In this work results are given for waves in an incompressible dielectric fluid, in a similar configuration. Treating the dielectric as an inviscid fluid the stability of the system is first described in terms of the changes in potential energy in a small static displacement. The result so obtained is then confirmed by a normal mode analysis in which a dispersion relation is obtained for the inviscid model. The paper gives finally a discussion of the results for a viscous dielectric fluid, the main point of which is that, as in (1), in the transition from stable to unstable disturbances viscosity plays no part, and that the stability characteristics are the same as those for an inviscid dielectric fluid.

Unsteady flow over a submerged source with low Froude number

2014 ◽  
Vol 25 (5) ◽  
pp. 655-680 ◽  
Author(s):  
CHRISTOPHER J. LUSTRI ◽  
S. JONATHAN CHAPMAN
Keyword(s):  
Free Surface ◽  
Unsteady Flow ◽  
Froude Number ◽  
Gravity Waves ◽  
Surface Gravity ◽  
Infinite Depth ◽  
Surface Gravity Waves ◽  
Flow Past ◽  
Low Froude Number ◽  
Exponential Asymptotics

In the low-Froude number limit, free-surface gravity waves caused by flow past a submerged obstacle have amplitude that is exponentially small. Consequently, these cannot be represented using an asymptotic series expansion. Previous studies have considered linearized steady flow past a submerged source in infinite-depth fluids, in which exponential asymptotics were used to determine the behaviour of downstream longitudinal and transverse free-surface gravity waves. Here, unsteady flow past a submerged source in an infinite-depth fluid is investigated, with the free surface taken to be initially waveless. The source is taken to be weak, and the flow is linearized about the undisturbed solution. Exponential asymptotics are applied to determine the wave behaviour on the free surface in terms of the two-dimensional plan-view, in order to show how the free surface waves evolve over time and eventually tend to the steady solution.

Capillary–gravity waves produced by a wavemaker

1991 ◽  
Vol 224 ◽  
pp. 217-226 ◽  
Author(s):  
L. M. Hocking ◽  
D. Mahdmina
Keyword(s):  
Surface Tension ◽  
Contact Angle ◽  
Free Surface ◽  
Surface Waves ◽  
Gravity Waves ◽  
Contact Line ◽  
Dynamic Properties ◽  
Transient Motion ◽  
Infinite Depth ◽  
The Waves

Surface waves in a channel can be produced by the horizontal motion of a plane wavemaker at one end of the channel. The amplitude and the frequency of the waves depend on both surface tension and gravity, as well as on the condition imposed at the contact line between the free surface and the wavemaker. Some of the previous work on the generation of capillary–gravity waves has been based on the unjustified assumption that the slope of the free surface at the contact line can be prescribed. A more acceptable condition is one that relates the slope to the motion of the contact line relative to the wavemaker; in this way the dynamic properties of the contact angle can be incorporated. The waves generated by a plane wavemaker in fluid of infinite depth and in fluid of a depth equal to that of the wavemaker are determined. An important reason for including surface tension is that in its absence the transient motion initiated by an impulsive start is singular; when surface tension is included this singularity is removed.

scholarly journals A note on solitary waves with variable surface tension in water of infinite depth

2006 ◽  
Vol 48 (2) ◽  
pp. 225-235 ◽  
Author(s):  
E. Özuğurlu ◽  
J.-M. Vanden-Broeck
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Solitary Waves ◽  
Surface Boundary ◽  
Two Dimensional ◽  
Infinite Depth ◽  
Free Surface Boundary Condition ◽  
Variable Surface ◽  
Dimensional Gravity ◽  
Surface Boundary Condition

AbstractTwo-dimensional gravity-capillary solitary waves propagating at the surface of a fluid of infinite depth are considered. The effects of gravity and of variable surface tension are included in the free-surface boundary condition. The numerical results extend the constant surface tension results of Vanden-Broeck and Dias to situations where the surface tension varies along the free surface.

scholarly journals Transient development of capillary-gravity waves in a running stream

1973 ◽  
Vol 9 (3) ◽  
pp. 417-432 ◽  
Author(s):  
Kalyan Kumar Bagchi ◽  
Lokenath Debnath
Keyword(s):  
Free Surface ◽  
Steady State ◽  
Gravity Waves ◽  
Critical Speed ◽  
Shallow Depth ◽  
Stream Velocity ◽  
Inviscid Fluid ◽  
Generalized Fourier Transform ◽  
Free Surface Elevation ◽  
Initial Value

An initial value investigation is made of the propagation of capillary-gravity waves generated by an oscillating pressure distribution acting at the free surface of a running stream of finite, infinite, and shallow depth. The solution for the free surface elevation is obtained explicitly by using the generalized Fourier transform and its asymptotic expansion. It is found that the solution consists of both the steady state and the transient components. The latter decays asymptotically as t → ∞ and the ultimate steady state is attained. It is shown that the steady state consists of two or four progressive capillary-gravity waves travelling both upstream and downstream according as the basic stream velocity is less or greater than the critical speed. Special attention is given to the existence of the critical values associated with the running stream of finite, infinite, and shallow depth. A comparison is made between the unsteady wave motions in an inviscid fluid with or without surface tension.

Capillary–gravity waves of solitary type on deep water

1989 ◽  
Vol 200 ◽  
pp. 451-470 ◽  
Author(s):  
Michael S. Longuet-Higgins
Keyword(s):  
Free Surface ◽  
Lower Bound ◽  
Solitary Waves ◽  
Gravity Waves ◽  
Deep Water ◽  
Phase Speed ◽  
Numerical Calculations ◽  
Maximum Angle ◽  
Monotonically Decreasing Function

On physical grounds it was recently suggested that limiting capillary–gravity waves of solitary type may exist on the surface of deep water (Longuet-Higgins 1988). This paper describes accurate numerical calculations which support the conjecture. The limiting wave has a phase speed c = 0.9267 (gτ)¼. It is one of a family of solitary waves having speeds c [les ] 1.30 (gτ)¼. The maximum angle of inclination αmax of the free surface is a monotonically decreasing function of the speed c. Physical arguments suggest that αmax has a positive lower bound.

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’.

A note on the unsteady motion under gravity of a corner point on a free surface: a generalization of Stokes' theory

2008 ◽  
Vol 465 (2101) ◽  
pp. 165-173 ◽  
Author(s):  
D.J Needham ◽  
J Billingham
Keyword(s):  
Free Surface ◽  
Corner Point ◽  
Constant Pressure ◽  
Unsteady Motion ◽  
Constant Speed ◽  
Inviscid Fluid ◽  
Irrotational Motion

In this paper, we develop a theory based on local asymptotic coordinate expansions for the unsteady propagation of a corner point on the constant-pressure free surface bounding an incompressible inviscid fluid in irrotational motion under the action of gravity. This generalizes the result of Stokes and Michell relating to the horizontal propagation of a corner at constant speed.

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