A note on the divergence effect and the Lagrangian-mean surface elevation in periodic water waves

1988 ◽  
Vol 189 ◽  
pp. 235-242 ◽  
Author(s):  
M. E. Mcintyre
Keyword(s):  
Free Surface ◽  
Velocity Field ◽  
Gravity Waves ◽  
Water Waves ◽  
Wave Amplitude ◽  
Exact Expression ◽  
Mean Velocity ◽  
Surface Gravity Waves ◽  
Generalized Lagrangian ◽  
Divergence Effect

Longuet-Higgins’ exact expression for the increase in the Lagrangian-mean elevation of the free surface due to the presence of periodic, irrotational surface gravity waves is rederived from generalized Lagrangian-mean theory. The raising of the Lagrangian-mean surface as wave amplitude builds up illustrates the non-zero divergence of the Lagrangian-mean velocity field in an incompressible fluid.

Surfing surface gravity waves

2017 ◽  
Vol 823 ◽  
pp. 316-328 ◽  
Author(s):  
Nick E. Pizzo
Keyword(s):  
Free Surface ◽  
Gravity Waves ◽  
Deep Water ◽  
Water Waves ◽  
Water Wave ◽  
Phase Speed ◽  
Surface Gravity ◽  
Simple Criterion ◽  
Surface Gravity Wave ◽  
Surface Gravity Waves

A simple criterion for water particles to surf an underlying surface gravity wave is presented. It is found that particles travelling near the phase speed of the wave, in a geometrically confined region on the forward face of the crest, increase in speed. The criterion is derived using the equation of John (Commun. Pure Appl. Maths, vol. 6, 1953, pp. 497–503) for the motion of a zero-stress free surface under the action of gravity. As an example, a breaking water wave is theoretically and numerically examined. Implications for upper-ocean processes, for both shallow- and deep-water waves, are discussed.

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’.

Surface Gravity Waves

2017 ◽  
Author(s):  
John A. Adam
Keyword(s):  
Gravity Waves ◽  
Water Waves ◽  
Water Surface ◽  
Wave Pattern ◽  
The Body ◽  
Surface Gravity ◽  
Shallow Water Waves ◽  
Ship Waves ◽  
Surface Gravity Waves ◽  
Basic Fluid

This chapter deals with the underlying mathematics of surface gravity waves, defined as gravity waves observed on an air–sea interface of the ocean. Surface gravity waves, or surface waves, differ from internal waves, gravity waves that occur within the body of the water (such as between parts of different densities). Examples of gravity waves are wind-generated waves on the water surface, as well tsunamis and ocean tides. Wind-generated gravity waves on the free surface of the Earth's seas, oceans, ponds, and lakes have a period of between 0.3 and 30 seconds. The chapter first describes the basic fluid equations before discussing the dispersion relations, with a particular focus on deep water waves, shallow water waves, and wavepackets. It also considers ship waves and how dispersion affects the wave pattern produced by a moving object, along with long and short waves.

scholarly journals Quantum Mechanical and Optical Analogies in Surface Gravity Water Waves

Fluids ◽  
2019 ◽  
Vol 4 (2) ◽  
pp. 96 ◽  
Author(s):  
Georgi Gary Rozenman ◽  
Shenhe Fu ◽  
Ady Arie ◽  
Lev Shemer
Keyword(s):  
Quantum Mechanics ◽  
Gravity Waves ◽  
Water Waves ◽  
Water Wave ◽  
Wave Packets ◽  
Theoretical Models ◽  
Surface Gravity ◽  
Finite Width ◽  
Self Healing ◽  
Surface Gravity Waves

We present the theoretical models and review the most recent results of a class of experiments in the field of surface gravity waves. These experiments serve as demonstration of an analogy to a broad variety of phenomena in optics and quantum mechanics. In particular, experiments involving Airy water-wave packets were carried out. The Airy wave packets have attracted tremendous attention in optics and quantum mechanics owing to their unique properties, spanning from an ability to propagate along parabolic trajectories without spreading, and to accumulating a phase that scales with the cubic power of time. Non-dispersive Cosine-Gauss wave packets and self-similar Hermite-Gauss wave packets, also well known in the field of optics and quantum mechanics, were recently studied using surface gravity waves as well. These wave packets demonstrated self-healing properties in water wave pulses as well, preserving their width despite being dispersive. Finally, this new approach also allows to observe diffractive focusing from a temporal slit with finite width.

Laboratory observations of mean flows under surface gravity waves

2007 ◽  
Vol 573 ◽  
pp. 131-147 ◽  
Author(s):  
S. G. MONISMITH ◽  
E. A. COWEN ◽  
H. M. NEPF ◽  
J. MAGNAUDET ◽  
L. THAIS
Keyword(s):  
Gravity Waves ◽  
Stokes Drift ◽  
Bottom Boundary Layer ◽  
Mean Velocity ◽  
Surface Gravity Waves ◽  
End Conditions ◽  
Different Types ◽  
Stokes Waves ◽  
Rotational Waves ◽  
Gerstner Waves

In this paper we present mean velocity distributions measured in several different wave flumes. The flows shown involve different types of mechanical wavemakers, channels of differing sizes, and two different end conditions. In all cases, when surface waves, nominally deep-water Stokes waves, are generated, counterflowing Eulerian flows appear that act to cancel locally, i.e. not in an integral sense, the mass transport associated with the Stokes drift. No existing theory of wave–current interactions explains this behaviour, although it is symptomatic of Gerstner waves, rotational waves that are exact solutions to the Euler equations. In shallow water (kH ≈ 1), this cancellation of the Stokes drift does not hold, suggesting that interactions between wave motions and the bottom boundary layer may also come into play.

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.

scholarly journals An exact theory of nonlinear waves on a Lagrangian-mean flow

1978 ◽  
Vol 89 (4) ◽  
pp. 609-646 ◽  
Author(s):  
D. G. Andrews ◽  
M. E. Mcintyre
Keyword(s):  
Water Waves ◽  
Finite Amplitude ◽  
Mean Flow ◽  
Radiation Stress ◽  
Initial State ◽  
Mean Velocity ◽  
Generalized Lagrangian ◽  
Flow Evolution ◽  
The Mean ◽  
The Waves

An exact and very general Lagrangian-mean description of the back effect of oscillatory disturbances upon the mean state is given. The basic formalism applies to any problem whose governing equations are given in the usual Eulerian form, and irrespective of whether spatial, temporal, ensemble, or ‘two-timing’ averages are appropriate. The generalized Lagrangian-mean velocity cannot be defined exactly as the ‘mean following a single fluid particle’, but in cases where spatial averages are taken can easily be visualized, for instance, as the motion of the centre of mass of a tube of fluid particles which lay along the direction of averaging in a hypothetical initial state of no disturbance.The equations for the Lagrangian-mean flow are more useful than their Eulerian-mean counterparts in significant respects, for instance in explicitly representing the effect upon mean-flow evolution of wave dissipation or forcing. Applications to irrotational acoustic or water waves, and to astrogeophysical problems of waves on axisymmetric mean flows are discussed. In the latter context the equations embody generalizations of the Eliassen-Palm and Charney-Drazin theorems showing the effects on the mean flow of departures from steady, conservative waves, for arbitrary, finite-amplitude disturbances to a stratified, rotating fluid, with allowance for self-gravitation as well as for an external gravitational field.The equations show generally how the pseudomomentum (or wave ‘momentum’) enters problems of mean-flow evolution. They also indicate the extent to which the net effect of the waves on the mean flow can be described by a ‘radiation stress’, and provide a general framework for explaining the asymmetry of radiation-stress tensors along the lines proposed by Jones (1973).

The Square Root Approximation to the Dispersion Relation of the Axially-symmetric Electron Wave on a Cylindrical Plasma

1997 ◽  
Vol 52 (10) ◽  
pp. 709-712
Author(s):  
V.M. Babović ◽  
B.A. Aničin ◽  
D. M. Davidović
Keyword(s):  
Dispersion Relation ◽  
Surface Waves ◽  
Gravity Waves ◽  
Water Surface ◽  
Bessel Functions ◽  
Exact Expression ◽  
Square Root ◽  
Axially Symmetric ◽  
Electron Wave ◽  
Surface Gravity Waves

Abstract This paper suggests the use of a simple square root approximation to the dispersion relation of axially-symmetric electron surface waves on cylindrical plasmas. The point is not merely to substitute the exact expression for the dispersion relation which involves a number of Bessel functions with a more tractable analytical approximant, but to cast the dispersion relation in a form useful in the comparison with other waves, such as water surface gravity waves and the associated tide-rip effect. The square root form of the dispersion relation is also of help in the analysis of surfactron plasmas, as it directly predicts a linear roll-off of electron density in the discharge.

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