The wave properties of matter and the zeropoint radiation field

The origin of the wave properties of matter is discussed from the point of view of stochastic electrodynamics. A nonrelativistic model of a charged particle with an effective structure embedded in the random zeropoint radiation field reveals that the field induces a high-frequency vibration on the particle; internal consistency of the theory fixes the frequency of this jittering at mc²/. The particle is therefore assumed to interact intensely with stationary zeropoint waves of this frequency as seen from its proper frame of reference; such waves, identified here as de Broglie's phase waves, give rise to a modulated wave in the laboratory frame, with de Broglie's wavelength and phase velocity equal to the particle velocity. The time-independent equation that describes this modulated wave is shown to be the stationary Schrödinger equation (or the Klein-Gordon equation in the relativistic version). In a heuristic analysis appled to simple periodic cases, the quantization rules are recovered from the assumption that for a particle in a stationary state there must correspond a stationary modulation. Along an independent and complementary line of reasoning, an equation for the probability amplitude in configuration space for a particle under a general potential V(x) is constructed, and it is shown that under conditions derived from stochastic electrodynamics it reduces to Schrödinger's equation. This equation reflects therefore the dual nature of the quantum particles, by describing simultaneously the corresponding modulated waveand the ensemble of particles.     

Confinement limit of a Dirac particle in two and three dimensions

Consider a particle that is in a stationary state described by the Dirac equation with a finite-range potential. In two and three dimensions the particle can be confined to an arbitrarily small spatial region. This is in contrast to the one-dimensional case in which the confinement region cannot be much narrower than the Compton wavelength.     

Frequency shifts of the sound field interference pattern in shallow water because of second-mode soliton-like internal waves

Numerical simulation is carried out to analyze the effect of an internal soliton of the second gravity mode on low-frequency sound propagation in an oceanic shelf region. The simulation is performed using the data of a full-scale experiment performed on the shelf of the South China Sea near Dongsha atoll, where the aforementioned solitons had been detected by stationary vertical thermistor arrays. The calculations take into account the effect of horizontal refraction of sound waves. It is assumed that a stationary acoustic track is oriented across the predominant propagation direction of internal waves. The results of simulation show that, when the soliton crosses the stationary track, some of the sound field modes are focused, whereas other modes are defocused. It is demonstrated that the soliton parameters can be adequately determined from the frequency shifts of the sound field interference pattern. However, such an estimate of the soliton parameters is only possible for a limited length of the stationary track for which the effect of horizontal refraction is sufficiently weak.     

Oblique waves lift the flapping flag

The flapping of the flag is a classical model problem for the understanding of fluid-structure interaction: How does the flat state lose stability? Why do the nonlinear effects induce hysteretic behavior? We show in this Letter that, in contrast with the commonly studied model, the full three-dimensional flag with gravity has no stationary state whose stability can be formally studied: The waves are oblique and must immediately be of large amplitude. The remarkable structure of these waves results from the interplay of weight, geometry, and aerodynamic forces. This pattern is a key element in the force balance which allows the flag to hold and fly in the wind: Large amplitude oblique waves are responsible for lift.     

Stationary Striations due to Interaction of Two Ionization Waves in Xenon Glow Discharge

Experimental observations on stationary striations in the positive column of xenon discharge are reported. Stationary striations are observed when two ionization waves exist simultaneously in the positive column at low pressure and high current region. These stationary striations are caused by nonlinear interference of two backward ionization waves of which frequencies are either equal or are in the ratio 1:2. The spatial intervals for the striated pattern are equal to the reciprocal of the difference between the wave-numbers of two ionization waves.     

Knudsen Gas in a Finite Random Tube: Transport Diffusion and First Passage Properties

We consider transport diffusion in a stochastic billiard in a random tube which is elongated in the direction of the first coordinate (the tube axis). Inside the random tube, which is stationary and ergodic, non-interacting particles move straight with constant speed. Upon hitting the tube walls, they are reflected randomly, according to the cosine law: the density of the outgoing direction is proportional to the cosine of the angle between this direction and the normal vector. Steady state transport is studied by introducing an open tube segment as follows: We cut out a large finite segment of the tube with segment boundaries perpendicular to the tube axis. Particles which leave this piece through the segment boundaries disappear from the system. Through stationary injection of particles at one boundary of the segment a steady state with non-vanishing stationary particle current is maintained. We prove (i) that in the thermodynamic limit of an infinite open piece the coarse-grained density profile inside the segment is linear, and (ii) that the transport diffusion coefficient obtained from the ratio of stationary current and effective boundary density gradient equals the diffusion coefficient of a tagged particle in an infinite tube. Thus we prove Fick’s law and equality of transport diffusion and self-diffusion coefficients for quite generic rough (random) tubes. We also study some properties of the crossing time and compute the Milne extrapolation length in dependence on the shape of the random tube.     

On the pressure of gravitational waves

A system of coupled point masses under the influence of gravitational waves is considered. By means of the geodesic deviation equation as the equation of motion it is shown, taking into account the second order small terms, that there exist forces which cause the acceleration of the system in the longitudinal direction. The longitudinal force is due to the fact that simultaneously with energy momentum is also absorbed from waves. It is proved directly on the basis of the equations of motion of the point masses that the energy and momentum absorbed by the test system obey the special relativistic relationship of a zero rest mass particle. The case when the Weber oscillator moves at a relativistic speed with respect to the source of gravitational waves is also examined. In this case, the absorption of energy and momentum by the Weber oscillator is much larger or smaller compared to the stationary situation.     

Sum-Frequency Generation from a Thin Spherical Layer: II. Analysis of Solution

We have analyzed the influence of the angle between the wave vectors of two incident waves (the opening angle) and of the type of the anisotropy of a nonlinear layer on the shape of the directivity pattern of a sum-frequency harmonic generated by the two plane elliptically polarized electromagnetic waves from a thin spherical optically nonlinear layer deposited on the surface of a dielectric spherical particle placed in a dielectric. Our analysis has shown that, if the radius of the thin spherical nonlinear layer is small, the shape of the directivity pattern will change significantly with increasing opening angle only for some types of anisotropy: the main lobes shift toward the direction that is opposite to the direction of the sum of the wave vectors of the incident waves. For three types of anisotropy, the directivity patterns have similar shapes. We have also found that, for one of the types of the anisotropy, the shape of the directivity pattern remains unchanged upon a change in the opening angle. The mathematical properties of functions describing the spatial distribution of the generated harmonic have been determined. In particular, it has been found that, upon incidence of linearly polarized waves on a thin nonlinear spherical layer that possesses either solely chiral or solely nonchiral nonlinear properties, linearly polarized radiation of the sum-frequency harmonic is generated.     

Axisymmetry Breaking to Travelling Waves in the Cylinder with Partially Heated Sidewall

The transition from an axisymmetric stationary flow to three-dimensional time-dependent flows is carefully studied in a vertical cylinder partially heated from the side, with the aspect ratio A = 2 and Prandtl number Pτ=0.021. The flow develops from the steady toroidal pattern beyond the first instability threshold, breaks the axisymmetric state at a Rayleigh number near 2000, and transits to standing or travelling azimuthal waves. A new result is observed that a slightly unstable flow pattern of standing waves exists and will transit to stable travelling waves after a long time evolution. The onset of oscillations is associated with a supercritical Hopf bifurcation in a system with O（2） symmetry.     

Solitary waves in a nonlinear oppositely directed coupler

The propagation of pulses in the system of two tunnel-coupled optical waveguides from optically nonlinear materials one of which has a negative refractive index, while the other one, positive, is investigated theoretically. The propagation of nonlinear waves in this structure is studied based on the model of coupled modes. For linear waves, this pair of coupled waveguides behaves as a mirror resulting in the change of direction of the energy flow upon penetration of radiation from one waveguide to the other. The solutions to the system of nonlinear equations describing the stationary propagation of the solitary wave, the gap soliton, in a particular direction are found. This soliton is formed by the coupled pair of wave packets each localized in the corresponding waveguide.     

Oscillatory patterns in a rotating aqueous suspension

Suspensions of granular material in glycerin-water mixtures agitated in horizontally aligned rotating tubes show a whole variety of patterns. The stationary pattern of a homogeneous distribution and a chain of rings have been investigated before. Here we report on two types of oscillatory states in the same system. For a certain range of the rotation frequency and sufficiently high viscosity traveling waves propagate with constant velocity back and forth along the tube in an almost homogeneous distribution of sedimenting particles. The transition from a stationary to the traveling-wave state is found to be an imperfect supercritical bifurcation. The dependence of the wave length and speed on the tube's rotation frequency and the dynamic viscosity of the fluid are determined. Experiments with low viscosities show no traveling waves but low-frequency oscillations, when the previously known chain of rings undergoes a secondary instability.     

Flow waves of hierarchical pattern formation induced by chemical waves: The birth, growth and death of hydrodynamic structures

We introduce a short review of chemically driven convection together with a series of our experiments on hydrodynamic instabilities induced by chemical waves excited in the batch reactor of a Belousov-Zhabotinsky reaction. Several unresolved phenomena are picked out and possible mechanisms are discussed extensively. Interesting features of these phenomena can be summarized as being caused by the ‘global and dynamic hydrodynamic pattern induced by chemical waves’. These chemically induced global pattern of hydrodynamic phenomena may not be simply explained by the reaction-diffusion-convection model based on Marangoni instability (surface tension-driven convection), which produces only a localized structure of the convection pattern. Observed flow waves show global and dynamic patterns of convection that generate a functional structure associated with hierarchical patterns appearing in the reaction-diffusion-convection system. In particular, we clarify the existence of a continuous stream of hydrodynamic flow with growing amplitude and its rotating direction. We find that the flow does not stabilize to a motionless state until the system has self-collapsed. This new picture of the flow waves requires a revision of the reaction-diffusion-convection model. The established flow structure can be regarded as a mixing and/or transport process to supply the substrate from the peripheral region to the centre of the chemical waves to sustain the reaction. This characteristic may be a function of the hierarchical structure. A new mechanism for the viscous-elastic feature of the gas-liquid interface is discussed in order to understand these curious phenomena of interest.     

Long-time behavior of sand ripples induced by water shear flow

The formation of sand ripples under water shear flow in a narrow annular channel and the approach of the ripple pattern towards a steady state were studied experimentally. Four results are obtained: i) The mean amplitude, the average drift velocity and the mean sediment transport rate of the evolving bed shape are strongly related. A quantitative characterization of this relation is given. ii) The ripple pattern reaches a stationary state with a finite ripple amplitude and wavelength. The time needed to reach the state depends on the shear stress and may be several days. iii) The onset of ripple formation is determined by the bed shear stress, but it seems neither to depend on the grain diameter nor on the depth of the water layer. iv) The ripple amplitude, drift velocity and sediment transport in this stationary state depend on the grain size. This dependency is neither captured by the particle Reynolds number nor by the Shields parameter: an empirical scaling law is presented instead.     

Pattern formation in reaction-diffusion system in crossed electric and magnetic fields

We consider a reaction-diffusion system in crossed electric and magnetic fields lying on the reaction plane. It is shown that a charge separation along the direction normal to the reaction plane resulting in a diffusional flux may cause a differential flow induced chemical instability and stationary pattern formation on a homogeneous steady state. This pattern is generically different from a Turing pattern modified by the crossed fields. The special role of magnetic field is emphasized. Our theoretical analysis is corroborated by numerical simulation on a reaction-diffusion system in three dimensions.     

Excitable optical waves in semiconductor microcavities

We demonstrate experimentally and theoretically the existence of excitable optical waves in semiconductor microcavities. Although similar to those observed in biological and chemical systems, these excitable optical waves are self-confined. This is due to a new dynamical scenario, where a stationary Turning pattern controls the propagation of waves in an excitable medium, thus bringing together the two paradigms of dynamical behavior (waves and patterns) in active media.     

One-Dimensional Steady Transport by Molecular Dynamics Simulation：Non-Boltzmann Position Distribution and Non-Arrhenius Dynamical Behavior

A non-equilibrium steady state can be characterized by a nonzero but stationary flux driven by a static external force. Under a weak external force, the drift velocity is difficult to detect because the drift motion is feeble and submerged in the intense thermal diffusion. In this article, we employ an accurate method in molecular dynamics simulation to determine the drift velocity of a particle driven by a weak external force in a one-dimensional periodic potential. With the calculated drift velocity, we found that the mobility and diffusion of the particle obey the Einstein relation, whereas their temperature dependences deviate from the Arrhenius law. A microscopic hopping mechanism was proposed to explain the non-Arrhenius behavior. Moreover, the position distribution of the particle in the potential well was found to deviate from the Boltzmann equation in a non-equilibrium steady state. The non-Boltzmann behavior may be attributed to the thermostat which introduces an effective "viscous" drag opposite to the drift direction of the particle.     

Particle paths in Stokes’ edge wave

We treat the particle motion in Stokes’ linear edge wave along a uniformly sloping beach. By a rotation of the coordinate frame, we show that there is no particle motion in the direction orthogonal to the sloping beach, and conclude that particles have a longshore drift in the direction of wave propagation which decreases with depth and distance from the shoreline. We discuss the application of this rotated coordinate frame to higher mode (Ursell) and weakly nonlinear (Whitham) edge waves, and show that the weakly nonlinear case is identical to that for two-dimensional deep-water Stokes waves.     

Initial condition effects in the evolution of a nonlinear dimer

The initial state analysis of the evolution of a nonlinear degenerate dimer shows that, in addition to the self-trapping transition, a new transition occurs while the particle is in the trapped region. This transition can be understood in part in terms of the behavior of a linear nondegenerate dimer, and is intimately related to the stationary states of the nonlinear dimer.     

Source bearing estimation in the Arctic Ocean using ice-mounted geophones

This paper investigates the use of geophones mounted on the surface of Arctic sea ice for estimating the bearing to acoustic sources in the water column. The approach is based on measuring ice seismic waves for which the direction of particle motion is oriented radially outward from the source. However, the analysis is complicated by the fact that sea ice supports several types of seismic waves, producing complex particle motion that includes significant nonradial components. To suppress seismic waves with transverse particle motion, seismic polarization filters are applied in conjunction with a straightforward rotational analysis (computation of particle-motion power as a function of angle). The polarization filters require three-dimensional (3D) measurements of particle motion, and apply theoretical phase relationships between vertical and horizontal components for the various waves types. In addition, the 180 degrees ambiguity inherent in the rotational analysis can be resolved with 3D measurements by considering particle motion in the vertical-radial plane. Arctic field trials were carried out involving two components. First, a hammer source was used to selectively excite the various ice seismic waves to investigate their propagation properties and relative importance in bearing estimation. Second, impulsive acoustic sources were deployed in the water column at a variety of bearings and ranges from 200-1000 m. For frequencies up to 250 Hz, source bearings are typically estimated to within an average absolute error of approximately 100.     

Laser saturation grating phenomena

Spatial hole burning of population differences by standing waves is considered in terms of Bragg-like gratings. This philosophy is used to explain increased saturation in two-mirror, single-mode laser operation with stationary active systems, modified mode coupling in the corresponding two-mode operation, an intensity dip that might occur in flowed standing wave lasers, bistable unidirectional ring laser operation, and increased saturation in distributed feedback lasers. The analysis is developed for both stationary media and for those moving with respect to the standing wave. The latter treatment is used to interpret washout of grating contributions in Doppler broadened media, and to determine level decay constants in stationary media. This last application constitutes a stationary system analog to the Doppler medium's Lamb dip spectroscopy and can be called saturation grating spectroscopy. Knowledge of the decay constants is particularly important in laser studies involving coherent mode couplings such as due to saturation grating scattering and population pulsations. It is further shown that the same equations result for probe and saturating waves propagating in the same direction. One then obtains a signal absorption with a heterodyne advantage. Inasmuch as diffusion affects the two methods quite differently, this phenomenon should be easily examined. Work supported in part by a U.S. Senior Scientist Award (administered by the Alexander von Humboldt Stiftung) and in part by the Space and Missile Systems Organization, Los Angeles, California.