Canonical equations for almost periodic, weakly nonlinear gravity waves

Wave Motion ◽  
1985 ◽  
Vol 7 (5) ◽  
pp. 473-485 ◽  
Author(s):  
A.C. Radder ◽  
M.W. Dingemans
Keyword(s):  
Gravity Waves ◽  
Almost Periodic ◽  
Weakly Nonlinear ◽  
Nonlinear Gravity

On steady three-dimensional deep water weakly nonlinear gravity waves

Wave Motion ◽  
1982 ◽  
Vol 4 (2) ◽  
pp. 113-125 ◽  
Author(s):  
Yan-Chow Ma
Keyword(s):  
Gravity Waves ◽  
Deep Water ◽  
Three Dimensional ◽  
Weakly Nonlinear ◽  
Nonlinear Gravity

scholarly journals Oscillatory spatially periodic weakly nonlinear gravity waves on deep water

1988 ◽  
Vol 191 (-1) ◽  
pp. 341 ◽  
Author(s):  
Juana A. Zufiria
Keyword(s):  
Gravity Waves ◽  
Deep Water ◽  
Weakly Nonlinear ◽  
Spatially Periodic ◽  
Nonlinear Gravity

A high-order spectral method for the study of nonlinear gravity waves

1987 ◽  
Vol 184 ◽  
pp. 267-288 ◽  
Author(s):  
Douglas G. Dommermuth ◽  
Dick K. P. Yue
Keyword(s):  
Gravity Waves ◽  
Mode Coupling ◽  
Arbitrary Order ◽  
Travelling Wave ◽  
Computational Effort ◽  
Zakharov Equation ◽  
Long Time ◽  
Nonlinear Free Surface ◽  
High Order Spectral Method ◽  
Nonlinear Gravity

We develop a robust numerical method for modelling nonlinear gravity waves which is based on the Zakharov equation/mode-coupling idea but is generalized to include interactions up to an arbitrary order M in wave steepness. A large number (N = O(1000)) of free wave modes are typically used whose amplitude evolutions are determined through a pseudospectral treatment of the nonlinear free-surface conditions. The computational effort is directly proportional to N and M, and the convergence with N and M is exponentially fast for waves up to approximately 80% of Stokes limiting steepness (ka ∼ 0.35). The efficiency and accuracy of the method is demonstrated by comparisons to fully nonlinear semi-Lagrangian computations (Vinje & Brevig 1981); calculations of long-time evolution of wavetrains using the modified (fourth-order) Zakharov equations (Stiassnie & Shemer 1987); and experimental measurements of a travelling wave packet (Su 1982). As a final example of the usefulness of the method, we consider the nonlinear interactions between two colliding wave envelopes of different carrier frequencies.

On resonant over-reflexion of internal gravity waves from a Helmholtz velocity profile

1979 ◽  
Vol 90 (1) ◽  
pp. 161-178 ◽  
Author(s):  
R. H. J. Grimshaw
Keyword(s):  
Velocity Profile ◽  
Gravity Waves ◽  
Second Harmonic ◽  
Phase Speed ◽  
Internal Gravity Waves ◽  
Finite Amplitude ◽  
Weakly Nonlinear ◽  
Velocity Discontinuity ◽  
Interface Displacement ◽  
Boussinesq Fluid

A Helmholtz velocity profile with velocity discontinuity 2U is embedded in an infinite continuously stratified Boussinesq fluid with constant Brunt—Väisälä frequency N. Linear theory shows that this system can support resonant over-reflexion, i.e. the existence of neutral modes consisting of outgoing internal gravity waves, whenever the horizontal wavenumber is less than N/2½U. This paper examines the weakly nonlinear theory of these modes. An equation governing the evolution of the amplitude of the interface displacement is derived. The time scale for this evolution is α−2, where α is a measure of the magnitude of the interface displacement, which is excited by an incident wave of magnitude O(α3). It is shown that the mode which is symmetrical with respect to the interface (and has a horizontal phase speed equal to the mean of the basic velocity discontinuity) remains neutral, with a finite amplitude wave on the interface. However, the other modes, which are not symmetrical with respect to the interface, become unstable owing to the self-interaction of the primary mode with its second harmonic. The interface displacement develops a singularity in a finite time.

scholarly journals The role of constant vorticity on weakly nonlinear surface gravity waves

Wave Motion ◽  
2020 ◽  
pp. 102702
Author(s):  
M.A. Manna ◽  
S. Noubissie ◽  
J. Touboul ◽  
B. Simon ◽  
R.A. Kraenkel
Keyword(s):  
Gravity Waves ◽  
Surface Gravity ◽  
Weakly Nonlinear ◽  
Surface Gravity Waves ◽  
Constant Vorticity ◽  
Nonlinear Surface

ASYMPTOTICAL ANALYSIS OF ONE DIMENSIONAL GAS DYNAMICS EQUATIONS

2001 ◽  
Vol 6 (1) ◽  
pp. 117-128 ◽  
Author(s):  
A. Krylovas ◽  
R. Čiegis
Keyword(s):  
Gas Dynamics ◽  
Numerical Experiments ◽  
Method Of Averaging ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
Resonance Case ◽  
Gas Dynamics Equations ◽  
Weakly Nonlinear ◽  
One Dimensional ◽  
Asymptotical Analysis

A method of averaging is developed for constructing a uniformly valid asymptotic solution for weakly nonlinear one dimensional gas dynamics systems. Using this method we give the averaged system, which disintegrates into independent equations for the non‐resonance systems. Conditions of the resonance for periodic and almost periodic solutions are presented. In the resonance case the averaged system is solved numerically. Some results of numerical experiments are given.

scholarly journals Steep capillary-gravity waves in oscillatory shear-driven flows

2009 ◽  
Vol 640 ◽  
pp. 131-150 ◽  
Author(s):  
SHREYAS V. JALIKOP ◽  
ANNE JUEL
Keyword(s):  
Surface Tension ◽  
Gravity Waves ◽  
Pitchfork Bifurcation ◽  
Nonlinear Regime ◽  
Radius Of Curvature ◽  
Square Root ◽  
Capillary Length ◽  
Weakly Nonlinear ◽  
Gravitational Forces ◽  
Steep Waves

We study steep capillary-gravity waves that form at the interface between two stably stratified layers of immiscible liquids in a horizontally oscillating vessel. The oscillatory nature of the external forcing prevents the waves from overturning, and thus enables the development of steep waves at large forcing. They arise through a supercritical pitchfork bifurcation, characterized by the square root dependence of the height of the wave on the excess vibrational Froude number (W, square root of the ratio of vibrational to gravitational forces). At a critical valueWc, a transition to a linear variation inWis observed. It is accompanied by sharp qualitative changes in the harmonic content of the wave shape, so that trochoidal waves characterize the weakly nonlinear regime, but ‘finger’-like waves form forW≥Wc. In this strongly nonlinear regime, the wavelength is a function of the product of amplitude and frequency of forcing, whereas forW<Wc, the wavelength exhibits an explicit dependence on the frequency of forcing that is due to the effect of viscosity. Most significantly, the radius of curvature of the wave crests decreases monotonically withWto reach the capillary length forW=Wc, i.e. the lengthscale for which surface tension forces balance gravitational forces. ForW<Wc, gravitational restoring forces dominate, but forW≥Wc, the wave development is increasingly defined by localized surface tension effects.

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