scholarly journals Different forms for nonlinear standing waves in deep water

1994 ◽  
Vol 272 ◽  
pp. 135-156 ◽  
Author(s):  
Peter J. Bryant ◽  
Michael Stiassnie
Keyword(s):  
Standing Wave ◽  
Deep Water ◽  
Nonlinear Boundary ◽  
Standing Waves ◽  
Linear Instability ◽  
Multiple Forms ◽  
Local Maxima ◽  
Progressive Waves ◽  
The Stability ◽  
Nonlinear Standing Waves

Multiple forms for standing waves in deep water periodic in both space and time are obtained analytically as solutions of Zakharov's equation and its modification, and investigated computationally as irrotational two-dimensional solutions of the full nonlinear boundary value problem. The different forms are based on weak nonlinear interactions between the fundamental harmonic and the resonating harmonics of 2, 3,…times the frequency and 4, 9,…respectively times the wavenumber. The new forms of standing waves have amplitudes with local maxima at the resonating harmonics, unlike the classical (Stokes) standing wave which is dominated by the fundamental harmonic. The stability of the new standing waves is investigated for small to moderate wave energies by numerical computation of their evolution, starting from the standing wave solution whose only initial disturbance is the numerical error. The instability of the Stokes standing wave to sideband disturbances is demonstrated first, by showing the evolution into cyclic recurrence that occurs when a set of nine equal Stokes standing waves is perturbed by a standing wave of a length equal to the total length of the nine waves. The cyclic recurrence is similar to that observed in the well-known linear instability and sideband modulation of Stokes progressive waves, and is also similar to that resulting from the evolution of the new standing waves in which the first and ninth harmonics are dominant. The new standing waves are only marginally unstable at small to moderate wave energies, with harmonics which remain near their initial amplitudes and phases for typically 100–1000 wave periods before evolving into slowly modulated oscillations or diverging.

On the observability of finite-depth short-crested water waves

1996 ◽  
Vol 322 ◽  
pp. 1-19 ◽  
Author(s):  
M. Ioualalen ◽  
A. J. Roberts ◽  
C. Kharif
Keyword(s):  
Standing Wave ◽  
Water Waves ◽  
Numerical Study ◽  
Standing Waves ◽  
Finite Depth ◽  
Two Dimensional ◽  
Wave Fields ◽  
Progressive Waves ◽  
Wave Limit ◽  
Narrow Bands

A numerical study of the superharmonic instabilities of short-crested waves on water of finite depth is performed in order to measure their time scales. It is shown that these superharmonic instabilities can be significant-unlike the deep-water case-in parts of the parameter regime. New resonances associated with the standing wave limit are studied closely. These instabilities ‘contaminate’ most of the parameter space, excluding that near two-dimensional progressive waves; however, they are significant only near the standing wave limit. The main result is that very narrow bands of both short-crested waves ‘close’ to two-dimensional standing waves, and of well developed short-crested waves, perturbed by superharmonic instabilities, are unstable for depths shallower than approximately a non-dimensional depth d= 1; the study is performed down to depth d= 0.5 beyond which the computations do not converge sufficiently. As a corollary, the present study predicts that these very narrow sub-domains of short-crested wave fields will not be observable, although most of the short-crested wave fields will be.

On the form of the highest progressive and standing waves in deep water

1973 ◽  
Vol 331 (1587) ◽  
pp. 445-456 ◽  
Keyword(s):  
Free Surface ◽  
Numerical Integration ◽  
Gravity Wave ◽  
Standing Wave ◽  
Deep Water ◽  
Standing Waves ◽  
Infinite Depth ◽  
Single Term

The form of a progressive gravity wave on deep water, which generally must be found by numerical integration (Michell 1893) is shown to be approximated with remarkable accuracy by a single term. Six consecutive waves are transformed conformally so as to surround the point corresponding to infinite depth. The free surface then corresponds closely to the boundary of a hexagon. In a similar way the profile of a standing wave is closely approximated to by transforming four consecutive waves conformally and taking the profile as the boundary of a square. The profile agrees closely with that calculated by Penney & Price (1952) and with the experiments of Taylor (1953).

scholarly journals CRITICAL SIZES OF STANDING NUCLEAR BURNING WAVES IN SYSTEMS WITH EXTERNAL CONTROL

2020 ◽  
pp. 16-22
Author(s):  
Yu.Ya. Leleko ◽  
V.V. Gann
Keyword(s):  
Mathematical Modeling ◽  
Standing Wave ◽  
Standing Waves ◽  
Spherical Shape ◽  
External Control ◽  
The Core ◽  
Mcnpx Code ◽  
Nuclear Burning ◽  
The Stability ◽  
Stability Limits

The theory of standing waves of nuclear combustion in reactors having a flat, cylindrical or spherical shape of the core is developed. A spherical standing wave occurs when nuclear burning propagates radially from the center of the sphere, and 238UO2 fuel moves to the center and is removed from the system. The stability limits of standing waves of nuclear burning are investigated. It has been established that for standing waves there are minimum (critical) sizes at which they exist. Mathematical modeling of standing waves using the MCNPX code was carried out and critical sizes of standing waves of various symmetries were determined.

Stability of standing waves for a class of quasilinear Schrödinger equations

2012 ◽  
Vol 23 (5) ◽  
pp. 611-633 ◽  
Author(s):  
JIANQING CHEN ◽  
YONGQING LI ◽  
ZHI-QIANG WANG
Keyword(s):  
Standing Wave ◽  
Schrodinger Equation ◽  
Standing Waves ◽  
Mathematical Physics ◽  
Schrödinger Equations ◽  
Quasilinear Schrödinger Equation ◽  
Stability And Instability ◽  
The Stability ◽  
Image Position ◽  
Physical Phenomena

This paper is concerned with the stability and instability of standing waves for the quasilinear Schrödinger equation of the form which has been derived in many models from mathematical physics. We find the exact threshold depending upon the interplay of quasilinear and nonlinear terms that separates stability and instability. More precisely, we prove that for α ∈ and odd p ∈ , when $1 < p < 4\alpha -1 +{4\over N}$, the standing wave is stable, and when $4\alpha -1 +{4\over N} \leq p < 2\alpha\cdot 2^\ast -1$ (where $2\alpha\cdot 2^\ast = \frac{4N\alpha}{N-2}$ for N ≥ 3 and 2 α ċ 2* = +∞ for N = 2), the standing wave is strongly unstable. Our results show that the quasilinear term 2 α(△|φ|2α)|φ|2α−2φ makes the standing waves more stable, which is consistent with the physical phenomena.

scholarly journals Long wavelength bifurcation of gravity waves on deep water

1980 ◽  
Vol 101 (3) ◽  
pp. 567-581 ◽  
Author(s):  
P. G. Saffman
Keyword(s):  
Gravity Waves ◽  
Deep Water ◽  
Finite Amplitude ◽  
Stokes Wave ◽  
Long Wavelength ◽  
Modulated Waves ◽  
Progressive Waves ◽  
The Stability ◽  
Permanent Form ◽  
Modulation Wavelength

Conditions are found for the appearance of non-uniform progressive waves of permanent form from a long-wave modulation of a finite-amplitude Stokes wave on deep water. The waveheight at which the modulated waves can occur is a very slowly decreasing function of the modulation wavelength for values up to 150 times the original wavelength. Some qualitative remarks are made about the problem of determining the stability of the new waves.

scholarly journals Almost limiting short-crested gravity waves in deep water

2010 ◽  
Vol 646 ◽  
pp. 481-503 ◽  
Author(s):  
MAKOTO OKAMURA
Keyword(s):  
Gravity Waves ◽  
Deep Water ◽  
Fundamental Mode ◽  
Incident Angle ◽  
Standing Waves ◽  
Resonant Mode ◽  
Two Dimensional ◽  
Progressive Waves ◽  
Harmonic Resonance

We investigate the properties of almost limiting short-crested gravity waves with harmonic resonance for various incident angles. When the incident angle is less than 47.5°, the enclosed crest angle in non-resonant limiting waves is 90°, which corresponds to that in standing waves. In contrast, when the incident angle exceeds 47.5°, the enclosed crest angle in non-resonant limiting waves is 120°, which corresponds to that in two-dimensional progressive waves. The enclosed crest angle is 90° in resonant limiting waves for all incident angles. The crest becomes flatter than the trough in resonant limiting waves if the fundamental mode has a different sign from its harmonic resonant mode. Bifurcation of short-crested waves is also investigated.

On the Variational Approach to the Stability of Standing Waves for the Nonlinear Schrödinger Equation

2004 ◽  
Vol 4 (4) ◽  
Author(s):  
H. Hajaiej ◽  
C.A. Stuart
Keyword(s):  
Periodic Function ◽  
Standing Wave ◽  
Variational Approach ◽  
Schrodinger Equation ◽  
Standing Waves ◽  
Orbital Stability ◽  
Focus Attention ◽  
Asymptotically Periodic ◽  
Linear Problems ◽  
The Stability

AbstractWe consider the orbital stability of standing waves of the nonlinear Schrdinger equationby the approach that was laid down by Cazenave and Lions in 1992. Our work covers several situations that do not seem to be included in previous treatments, namely,(i) g(x, s) − g(x, 0) → 0 as |x| → ∞ for all s ≥ 0. This includes linear problems.(ii) g(x, s) is a periodic function of x ∈ ℝ(iii) g(x, s) is asymptotically periodic in the sense that g(x, s) − gFurthermore, we focus attention on the form of the set that is shown to be stable and may be bigger than what is usually known as the orbit of the standing wave.

Interactions between steady and oscillatory convection in mushy layers

2010 ◽  
Vol 645 ◽  
pp. 411-434 ◽  
Author(s):  
PETER GUBA ◽  
M. GRAE WORSTER
Keyword(s):  
Standing Waves ◽  
Mushy Layer ◽  
Two Dimensional ◽  
Oscillatory Convection ◽  
Mushy Layers ◽  
Theoretical Predictions ◽  
Convection Patterns ◽  
The Stability ◽  
Nonlinear Solutions ◽  
Steady Convection

We study nonlinear, two-dimensional convection in a mushy layer during solidification of a binary mixture. We consider a particular limit in which the onset of oscillatory convection just precedes the onset of steady overturning convection, at a prescribed aspect ratio of convection patterns. This asymptotic limit allows us to determine nonlinear solutions analytically. The results provide a complete description of the stability of and transitions between steady and oscillatory convection as functions of the Rayleigh number and the compositional ratio. Of particular focus are the effects of the basic-state asymmetries and non-uniformity in the permeability of the mushy layer, which give rise to abrupt (hysteretic) transitions in the system. We find that the transition between travelling and standing waves, as well as that between standing waves and steady convection, can be hysteretic. The relevance of our theoretical predictions to recent experiments on directionally solidifying mushy layers is also discussed.

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