scholarly journals Nonlinear waves on the surface of a fluid covered by an elastic sheet

2013 ◽  
Vol 733 ◽  
pp. 394-413 ◽  
Author(s):  
Luc Deike ◽  
Jean-Claude Bacri ◽  
Eric Falcon
Keyword(s):  
Dispersion Relation ◽  
Wave Field ◽  
Nonlinear Waves ◽  
Random Noise ◽  
Finite Size ◽  
Wave Turbulence ◽  
Transverse Motion ◽  
Elastic Sheet ◽  
Theoretical Hypothesis ◽  
Noise Forcing

AbstractWe experimentally study linear and nonlinear waves on the surface of a fluid covered by an elastic sheet where both tension and flexural waves occur. An optical method is used to obtain the full space–time wave field, and the dispersion relation of the waves. When the forcing is increased, a significant nonlinear shift of the dispersion relation is observed. We show that this shift is due to an additional tension of the sheet induced by the transverse motion of a fundamental mode of the sheet. When the system is subjected to a random-noise forcing at large scales, a regime of hydroelastic wave turbulence is observed with a power-law spectrum of the scale, in disagreement with the wave turbulence prediction. We show that the separation between relevant time scales is well satisfied at each scale of the turbulent cascade as expected theoretically. The wave field anisotropy, and finite size effects are also quantified and are not at the origin of the discrepancy. Finally, the dissipation is found to occur at all scales of the cascade, contrary to the theoretical hypothesis, and could thus explain this disagreement.

Nonlinear Effects in a Viscous Wave Field Excited by a Finite-Size Plate

2019 ◽  
Vol 64 (10) ◽  
pp. 1418-1423
Author(s):  
V. A. Pavlov ◽  
A. S. Pavlovskii ◽  
N. G. Semenova
Keyword(s):  
Wave Field ◽  
Finite Size ◽  
Nonlinear Effects

Stability of strong electromagnetic waves in overdense plasmas against long-wavelength perturbations

1985 ◽  
Vol 33 (2) ◽  
pp. 285-301 ◽  
Author(s):  
F. J. Romeiras ◽  
G. Rowlands
Keyword(s):  
Dispersion Relation ◽  
Nonlinear Waves ◽  
Electromagnetic Waves ◽  
Modulation Depth ◽  
Longitudinal Direction ◽  
Transverse Component ◽  
Ion Density ◽  
Long Wavelength ◽  
The Stability ◽  
Two Parameters

We consider the stability against long-wavelength small parallel perturbations of a class of exact standing wave solutions of the equations that describe an unmagnetized relativistic overdense cold electron plasma. The main feature of these nonlinear waves is a circularly polarized transverse component of the electric field periodically modulated in the longitudinal direction. Using an analytical method developed by Rowlands we obtain a dispersion relation valid for long-wavelength perturbations. This dispersion relation is a biquadratic equation in the phase velocity of the perturbations whose coefficients are very complicated functions of the two parameters used to define the nonlinear waves: the normalized ion density and a quantity related to the modulation depth. This dispersion relation is discussed for the whole range of the two parameters revealing, in particular, the existence of a region in parameter space where the nonlinear waves are stable.

The statistical mechanics of vortex—acoustic ion wave turbulence

1980 ◽  
Vol 23 (3) ◽  
pp. 545-566
Author(s):  
M. J. Giles
Keyword(s):  
Statistical Mechanics ◽  
Wave Field ◽  
Sound Field ◽  
Vortex State ◽  
Energy Functional ◽  
Wave Turbulence ◽  
Indirect Interaction ◽  
Canonical Distribution ◽  
Generating Functional ◽  
The Mean

The equilibrium statistical mechanics of electrostatic ion wave turbulence is studied within the framework of a continuum ion flow with adiabatic electrons. Attention is drawn to the fact that the wave field consists in general of two components, namely ion-acoustic and ion vortex modes. It is shown that the latter can significantly affect the equilibria on accoant of their ability both to emit and to scatter ion sound. Exact equilibria for the vortex—acoustic wave field are given in terms of a canonical distribution and the correlation functions are expressed in terms of a generating functional. A nonlinear transformation of the wave field, which removes the vortex-acoustic interaction energy to lowest order in the strength of the coupling, while preserving the phase space volume element, is then introduced. This enables the Feynman-Hibbs variational principle to be used to obtain an approximate generating functional based on a ‘trial’ energy functional, which is a quadratic in the new variables. Detailed calculations are carried out for the case in which the dominant coupling is an indirect interaction of the vortex modes mediated by the sound field. An equation for the spectrum of the vortex modes is obtained for this case, which is shown to possess a simple exact solution. This solution shows that the spectrum of fluctuations changes considerably as the total energy increases. At low levels of excitation the solution reduces to equipartition of energy. At higher levels the indirect interaction becomes significant and the formation of a condensed vortex state is possible when the mean energy exceeds the electron thermal energy. It is suggested that condensed vortex states could occur in the plasma sheet of the earth's magnetosphere and it is shown that the predicted ratio of the mean ion energy to the mean electron energy is consistent with the trend of observed values.

scholarly journals A class of reduced-order models in the theory of waves and stability

2016 ◽  
Vol 472 (2186) ◽  
pp. 20150703 ◽  
Author(s):  
C. J. Chapman ◽  
S. V. Sorokin
Keyword(s):  
Dispersion Relation ◽  
Asymptotic Analysis ◽  
Wave Field ◽  
Elastic Waves ◽  
Physical Space ◽  
Psi Function ◽  
Logarithmic Correction ◽  
Reduced Order Models ◽  
Partial Fraction ◽  
Reduced Order

This paper presents a class of approximations to a type of wave field for which the dispersion relation is transcendental. The approximations have two defining characteristics: (i) they give the field shape exactly when the frequency and wavenumber lie on a grid of points in the (frequency, wavenumber) plane and (ii) the approximate dispersion relations are polynomials that pass exactly through points on this grid. Thus, the method is interpolatory in nature, but the interpolation takes place in (frequency, wavenumber) space, rather than in physical space. Full details are presented for a non-trivial example, that of antisymmetric elastic waves in a layer. The method is related to partial fraction expansions and barycentric representations of functions. An asymptotic analysis is presented, involving Stirling's approximation to the psi function, and a logarithmic correction to the polynomial dispersion relation.

The Interaction of Nonlinear Waves With the Submerged Caissons of a Gravity Based Structure

2008 ◽  
Author(s):  
Marios Christou ◽  
Jannicke S. Roos ◽  
Chris Swan ◽  
Ove T. Gudmestad
Keyword(s):  
Wave Field ◽  
Incident Wave ◽  
Nonlinear Waves ◽  
Wave Structure ◽  
Boundary Integral ◽  
Numerical Predictions ◽  
Water Region ◽  
Geometric Discontinuities ◽  
The Waves ◽  
Wave Structure Interaction

This paper concerns the numerical description of nonlinear waves propagating over the storage caissons of a gravity based structure. This process produces a steepening of the incident wave-field, which occurs when the waves propagate into the shallower water region above the storage caissons, resulting in the focussing of wave energy. A fully nonlinear Multiple-flux Boundary Element Method (MF-BEM) is applied to simulate this effect. The MF-BEM differs from traditional boundary integral approaches in two important respects: first, a multiple-flux approach is employed to overcome the problem of geometric discontinuities; and, second, no filtering, smoothing, re-gridding or redistribution of the nodes is performed at any stage during the simulations. These two aspects are believed to play an important role in accurately predicting the steepening of the incident wave-field. The numerical predictions are compared to new laboratory observations that examine the extent of this wave-structure interaction and, particularly, the steepening of the incident wave-field.

THE ANISOTROPIC CHARACTERS IN TWO-DIMENSIONAL LATTICE

2013 ◽  
Vol 27 (07) ◽  
pp. 1361010
Author(s):  
YANG YANG ◽  
MAI-MAI LIN ◽  
WEN-SHAN DUAN
Keyword(s):  
Dispersion Relation ◽  
Nonlinear Waves ◽  
Particle Displacement ◽  
Transverse Waves ◽  
Compressional Waves ◽  
Cubic Lattice ◽  
Simple Cubic ◽  
Simple Cubic Lattice ◽  
Special Cases ◽  
Linear And Nonlinear

The anisotropic characters of simple cubic lattice are investigated in this paper. Both the linear and nonlinear wave propagating in this lattice have been studied. The dispersion relation has been studied numerically. It is shown that the dispersion relation strongly depends on the directions of wave propagation. Generally, the direction of waves has the inclination angle α with respect to particle displacement. There are compressional waves α = 0 or transverse waves α = π/2 only for some special cases. The nonlinear waves in this lattice have also been studied. The anisotropic characters of this lattice for the nonlinear waves have also been shown. The compressional and transverse nonlinear solitons have also been studied. The characters of both solitons, such as amplitude and width, have been investigated.

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