Instability Of Standing Waves Research Articles

Standing wave instability in large area capacitive discharges operated within or near the gamma mode

Large-area capacitive discharges used for plasma deposition operate in a regime where both electromagnetic and secondary electron emission effects are important. The standing wave shortened wavelength in the presence of plasma depends on the sheath size, and in the γ mode, the secondary electron multiplication controls the sheath physics. Near the α-to-γ transition, and within the γ mode, the sheath width typically varies inversely with the discharge voltage, and large center-to-edge voltage (standing wave) ratios may exist. This can give rise to a standing wave instability, in which the central voltage of the discharge grows uncontrollably, for a given voltage excitation at the discharge edge. Using a simple model, we determine the discharge equilibrium properties, the linearized stability condition, and the nonlinear time evolution. For sufficiently large areas, we show that a discharge equilibrium no longer exists above a critical edge voltage at marginal stability.

Pattern selection for thermocapillary flow in rectangular containers in microgravity

The performance of Phase Change Material (PCM) devices in microgravity can be significantly improved by thermocapillary convection. However, the melting process in this case is affected by a series of instabilities and mode transitions due to the evolving size and shape of the liquid domain. We perform a numerical investigation of pattern selection for thermocapillary flow in ideal rectangular containers of liquid n-octadecane in microgravity and show how this can be applied to explain the more complex dynamics of melting PCMs. In particular, the locations of travelling and standing wave instabilities predicted in this way show very good agreement with numerical simulations of PCM melting.

Negative energy standing wave instability in the presence of flow

We study standing waves on the surface of a tangential discontinuity in an incompressible plasma. The plasma is moving with constant velocity at one side of the discontinuity, while it is at rest at the other side. The moving plasma is ideal and the plasma at rest is viscous. We only consider the long wavelength limit where the viscous Reynolds number is large. A standing wave is a superposition of a forward and a backward wave. When the flow speed is between the critical speed and the Kelvin–Helmholtz threshold the backward wave is a negative energy wave, while the forward wave is always a positive energy wave. We show that viscosity causes the standing wave to grow. Its increment is equal to the difference between the negative energy wave increment and the positive energy wave decrement.

Strong instability of standing waves for the Schrödinger-Poisson-Slater equation

本文研究Schrodinger-Poisson-Slater (SPS) 方程 i∂ ψ /∂ t +△ ψ -χ(| x | -1 *| ψ | 2 ) ψ + | ψ | p-1 ψ = 0, χ > 0, x ∈ R 3 , 当 p > 7/3 时, 运用变分法证明SPS 方程的基态驻波强不稳定, 并进一步用基态驻波解刻画SPS 方程 的解在有限时间爆破的门槛初始条件; 在 p = 7/3 时, 建立了基态驻波解的 L 2 范数的下界.

Instability of nonlinear standing waves in front of a vertical wall

The standing waves formed in front of a vertical breakwater in Gdansk North Port Harbour are examined. To simplify the wave-structure interaction problem, a laboratory experiment and mathematical model were designed. Standing waves in a limited space were generated under resonance conditions between a vertical wall and wave generator. Such slowly growing standing waves eventually become unstable and, consequently, create an impact impulse on the vertical wall. The dynamics of the vertical wall and hydrodynamics of the standing wave were measured and compared with a numerical model derived by variational calculus. The phenomenon of standing wave instability observed in nature was reproduced by laboratory experimentation and numerical simulation. The presented mechanism of breaking standing waves is more complicated in reality due to wave randomness and the multidirectional wave field.

Standing wave instabilities, breather formation and thermalization in a Hamiltonian anharmonic lattice

Modulational instability of travelling plane waves is often considered as the first step in the formation of intrinsically localized modes (discrete breathers) in anharmonic lattices. Here, we consider an alternative mechanism for breather formation, originating in oscillatory instabilities of spatially periodic or quasiperiodic nonlinear standing waves (SWs). These SWs are constructed for Klein-Gordon or Discrete Nonlinear Schrodinger lattices as exact time periodic and time reversible multibreather solutions from the limit of uncoupled oscillators, and merge into harmonic SWs in the small-amplitude limit. Approaching the linear limit, all SWs with nontrivial wave vectors (0 < Q < π) become unstable through oscillatory instabilities, persisting for arbitrarily small amplitudes in infinite lattices. The dynamics resulting from these instabilities is found to be qualitatively different for wave vectors smaller than or larger than π/2, respectively. In one regime persisting breathers are found, while in the other regime the system thermalizes.

Properties of the nucleon-nucleon interaction leading to a standing wave instability in symmetric nuclear matter

We examine a recently proposed nucleon-nucleon interaction, claimed by its authors both to be realistic and to lead to a standing-wave instability in symmetric nuclear matter. Contrary to these claims, we find that this interaction leads to a serious overbinding of ${}^{4}\mathrm{He},$ ${}^{16}\mathrm{O},$ and ${}^{40}\mathrm{Ca}$ nuclei when the Hartree-Fock method is properly applied. The resulting nuclear densities contradict the experimental data and all realistic Hartree-Fock results.

An analytical study of the Kelvin-Helmholtz instabilities of compressible, magnetized tangential velocity discontinuities with generalized polytrope laws

The linear Kelvin-Helmholtz instability of tangential velocity discontinuities in high velocity magnetized plasmas with isotropic or anisotropic pressure is investigated. A new analytical technique applied to the magnetohydrodynamic equations with generalized polytrope laws (for the pressure parallel and perpendicular to the magnetic field) yields the complete structure of the unstable, standing waves in the (inverse plasma beta, Mach number) plane for modes at arbitrary angles to the flow and the magnetic field. The stable regions in the (inverse plasma beta, propagation angle) plane are mapped out via a level curve analysis, thus clarifying the stabilizing effects of both the magnetic field and the compressibility. For polytrope indices corresponding to the double adiabatic and magnetohydrodynamic equations, the results reduce to those obtained earlier using these models. Detailed numerical results are presented for other cases not considered earlier, including the cases of isothermal and mixed waves. Also, for modes propagating along or opposite to the magnetic field direction and at general angles to the flow, a criterion is derived for the absence of standing wave instability—in the isotropic MHD case, this condition corresponds to (plasma beta) ≤ 1 \le 1 .

Standing wave instabilities in a chain of nonlinear coupled oscillators

We consider existence and stability properties of nonlinear spatially periodic or quasiperiodic standing waves (SWs) in one-dimensional lattices of coupled anharmonic oscillators. Specifically, we consider Klein–Gordon (KG) chains with either soft (e.g., Morse) or hard (e.g., quartic) on-site potentials, as well as discrete nonlinear Schrödinger (DNLS) chains approximating the small-amplitude dynamics of KG chains with weak inter-site coupling. The SWs are constructed as exact time-periodic multibreather solutions from the anticontinuous limit of uncoupled oscillators. In the validity regime of the DNLS approximation these solutions can be continued into the linear phonon band, where they merge into standard harmonic SWs. For SWs with incommensurate wave vectors, this continuation is associated with an inverse transition by breaking of analyticity. When the DNLS approximation is not valid, the continuation may be interrupted by bifurcations associated with resonances with higher harmonics of the SW. Concerning the stability, we identify one class of SWs which are always linearly stable close to the anticontinuous limit. However, approaching the linear limit all SWs with non-trivial wave vectors become unstable through oscillatory instabilities, persisting for arbitrarily small amplitudes in infinite lattices. Investigating the dynamics resulting from these instabilities, we find two qualitatively different regimes for wave vectors smaller than or larger than π/2, respectively. In one regime persisting breathers are found, while in the other regime the system rapidly thermalizes.

INFLUENCE OF GEOMETRICAL ASPECT RATIO ON THE OSCILLATORY MARANGONI CONVECTION IN LIQUID BRIDGESUM

Oscillatory Marangoni convection in silicone oil liquid bridges is investigated by three-dimensional, time-dependent numerical solutions of the model equations and by micro-scale experimentation. The field equations are numerically solved with three-dimensional control volume methods in a staggered cylindrical non-uniform grid. Two experimental configurations are utilized: (1) the two disks sustaining the bridges are made of copper, their temperatures are controlled with Peltier elements, the flow field in a vertical section is visualized by tracers illuminated by a laser light cut in the meridian plane and four fine temperature sensors are inserted axially from the hot disk into the liquid bridge; (2) the lower disk is made of copper and the upper one is made of transparent glass, heated by an electrical resistance, for visual measurements in a cross section orthogonal to the liquid bridge axis. The surface temperature distribution is measured by an infrared thermocamera. It is shown that the flow field organization, depending on the critical wave number, is related to the geometrical aspect ratio of the liquid bridge and that smaller is the aspect ratio, larger is the critical wave number and more complex the flow-field structure. For each aspect ratio considered, the flow field exhibits a first transition from the axi-symmetric steady to a standing wave instability model and then a second transition from the standing wave to the travelling wave regime. The influence of buoyancy effects on the oscillatory Marangoni flow organization is investigated by heating the liquid bridges either from above or from below.

Transient development of instabilities in bounded shear flow of granular materials

A study has been conducted on the analysis and fate of instabilities arising from rapid shear flow of a granular material. Specifically, this study is made through a detailed analysis of continuum rheological models, to examine whether the initiation and growth of the instabilities predicted by this analysis are consistent with the linear stability theory. It is found that rapidly sheared particles tend to form alternating bands or clusters of high and low solid concentrations. Two types of standing-wave instabilities, namely layering and stationary modes, are investigated. The fate of these density waves is tracked through transient integration of the macroscopic balance equations, and examined by a fast Fourier transform (FFT) analysis. In such a formulation, a few characteristic modes with different wave numbers have been identified. The temporal evolution of the intensity of fluctuating energy is investigated by analyzing the most dominant mode in the system. It is found that the structure and dynamics of the resulting waves can be quantitatively correlated with the power spectrum density of concentration fluctuations. The development of the layering mode is governed primarily by linear instability. On the other hand, clusters of particles of the stationary mode may stretch and interact with neighboring clusters. This new finding indicates that the transient development of a two-dimensional stationary disturbance may ultimately lead to a one-dimensional layered structure. The stationary and layering perturbations eventually evolve to a similar structure and the stationary perturbation leads the system to the layered state more rapidly than the layering mode.

An experimental study of tire standing waves

Small-scale test rigs for experimental investigations of tire standing wave instability are described. The rigs are used in experiments in which tire rotational speed is gradually increased past the critical speed at which standing waves appear around the circumference of the tire. Detailed results from one of the rigs are presented. Images grabbed from live video are processed to extract the spatial structure of the standing waves. Frequency and time domain vibrational data and temperature histories are also collected and presented. The spatial frequencies, strains, and logarithmic decrement for the standing wave are computed for four different tire pressures, and the results are shown to be in qualitative agreement with numerical results of previous researchers.

Regular and irregular regimes of binary fluid convection excited by parametric resonance

A set of two asymptotically correct amplitude equations for envelopes of two counterpropagating waves in a horizontal layer of a binary fluid, excited by parametric resonance, is derived and studied. The only stable stationary monochromatic solution possible is standing wave (SW). It can lose its stability in several ways. Sideband instability of SW, similar to that studied by Eckhaus, is possible here and results in either the onset of SW with different wavelength or an irregular regime. The transitions caused by sideband instability are studied numerically. A qualitative explanation of the irregular behavior is proposed.

Erratum: Cluster formation, standing waves, and stripe patterns in oscillatory active media with local and global coupling

In recent years, the effect of global coupling on spatiotemporal pattern formation in oscillatory media has attracted considerable interest. For the complex Ginzburg-Landau equation, modified by a global coupling term, we derive a criterion for cluster formation and discuss standing wave solutions with an intrinsic wavelength in the Benjamin-Feir stable parameter range. We argue that clustering expresses the dominance of global coupling. In two-dimensional media the interplay between the standing wave instability and anisotropic diffusion may generate stripe patterns coupled to a spatially uniform oscillating mode. Then, by direct numerical simulation, we show that the globally coupled reconstruction model—proposed by Krischer, Eiswirth, and Ertl for CO oxidation on Pt(110) [Surf. Sci. 900, 251 (1991)]—exhibits clustering and predicts the presence of standing waves with an intrinsic wavelength.

Cluster formation, standing waves, and stripe patterns in oscillatory active media with local and global coupling.

In recent years, the effect of global coupling on spatiotemporal pattern formation in oscillatory media has attracted considerable interest. For the complex Ginzburg-Landau equation, modified by a global coupling term, we derive a criterion for cluster formation and discuss standing wave solutions with an intrinsic wavelength in the Benjamin-Feir stable parameter range. We argue that clustering expresses the dominance of global coupling. In two-dimensional media the interplay between the standing wave instability and anisotropic diffusion may generate stripe patterns coupled to a spatially uniform oscillating mode. Then, by direct numerical simulation, we show that the globally coupled reconstruction model---proposed by Krischer, Eiswirth, and Ertl for CO oxidation on Pt(110) [Surf. Sci. 900, 251 (1991)]---exhibits clustering and predicts the presence of standing waves with an intrinsic wavelength.

On the instability of the flow in an oscillating tank of fluid

The instability of a viscous fluid inside a rectangular tank oscillating about an axis parallel to the largest face of the tank is investigated in the linear regime. The flow is shown to be unstable to both longitudinal roll and standing wave instabilities. The particular cases of low and high oscillation frequencies are discussed in detail. The relationship between the roll instability and convective or centrifugal instabilities in unsteady boundary layers is discussed. The eigenvalue problems associated with the roll and standing wave instabilities are solved using Floquet theory and a combination of numerical and asymptotic methods. The results obtained are compared to the recent experimental investigation of Bolton &amp; Maurer (1994) which indeed provided the stimulus for the present investigation.

An analytical study of the Kelvin–Helmholtz instabilities of compressible, magnetized, anisotropic, and isotropic tangential velocity discontinuities

An analytical study is made of the linear stability of tangential velocity discontinuities in anisotropic and isotropic compressible, magnetized plasmas. A new analytical technique leads to entirely new results for both cases, and recovers previously obtained numerical results for the isotropic case. For both the anisotropic and isotropic vortex sheets, the structure of unstable, standing wave modes is mapped out in the (inverse plasma beta, Mach number) plane for modes at arbitrary angles to the flow and the magnetic field. Solutions are obtained for unstable standing and traveling modes of the anisotropic vortex sheet propagating parallel or antiparallel to the magnetic field, and at small angles to these directions. Numerically computed unstable standing modes obtained previously for the isotropic case are recovered analytically and extended to more general configurations. Also, it is shown that modes propagating along or opposite to the magnetic field direction, and at a general angle to the flow, do not exhibit standing wave instability for (plasma beta)≤1, for the isotropic vortex sheet.

The stability of finite length circular cross-section pipes conveying inviscid fluid

This paper is concerned with the linear stability of the system composed of a finite, thin elastic tube, replacing a section of an infinite rigid pipe, which conveys inviscid compressible fluid in uniform subsonic motion, by finding asymptotic forms for the fluid pressures. The analysis is based on linearized, potential flow theory, the Flugge-Kempner shell equation and Galerkin's method. Results are presented for the case of a simply-supported connection between the flexible and the rigid tube. For tubes of large length to radius ratio, the analysis demonstrates the connection between the finite length standing wave instability and the travelling wave instability of infinite tubes. The relationship between the “beam mode” and higher mode instabilities is made clear and discussed in some detail. The wide range of applicability of the asymptotic results is demonstrated by comparison with previously published numerical work. The small length to radius forms are also given and higher order terms are obtained in both limits.