Showing papers on "Group velocity published in 1997"
TL;DR: In this paper, a new technique for single-station phase velocity measurements is developed and applied to a large number of globally recorded Rayleigh and Love waves in the period range 35-150 s. The method is based on phase-matched filter theory and iteratively suppresses the effect of interfering overtones by minimizing residual dispersion.
Abstract: A new technique for making single-station phase velocity measurements is developed and applied to a large number of globally recorded Rayleigh and Love waves in the period range 35–150 s. The method is based on phase-matched filter theory and iteratively suppresses the effect of interfering overtones by minimizing residual dispersion. The model surface wave signal is described by its amplitude and apparent phase velocity, both of which are parameterized in terms of smooth B-spline functions of frequency. A misfit function is constructed which represents the difference between the model and observed waveforms, and the optimal spline coefficients are estimated in an iterative misfit minimization algorithm. In order to eliminate cycle skips in the measurements of phase at short periods, the waveforms are first matched at long periods, and the frequency range is gradually extended to include higher frequencies. The application of the algorithm to records from the Global Seismographic Network, using earthquakes in the Harvard centroid-moment tensor catalog, results in the determination of more than 50,000 high-quality dispersion curves. The observed variations in measured dispersion for pairwise similar paths are used to estimate realistic uncertainties in the data. Phase delays at discrete periods are inverted for global maps of variations in phase velocity expanded in spherical harmonics up to degree 40. A realistic resolution test indicates that structures are well recovered up to at least degree 20. The new phase velocity maps explain 70–96% of the observed variance in phase residuals, reflecting the high internal consistency of the dispersion measurements.
394 citations
Abstract: The nonlinear optics of plasmas at relativistic intensities are analyzed using only the physically intuitive processes of longitudinal bunching of laser energy, transverse focusing of laser energy, and photon acceleration, together with the assumption of conservation of photons, i.e., the classical action. All that is required are the well-known formula for the phase and group velocity of light in plasma, and the effects of the ponderomotive force on the dielectric function. This formalism is useful when the dielectric function of the plasma is almost constant in the frame of the light wave. This is the case for Raman forward scattering (RFS), envelope self-modulation (SM), relativistic self-focusing (SF), and relativistic self-phase modulation (SPM). In the past, the growth rates for RFS and SPM have been derived in terms of wave-wave interactions. Here we rederive all of the aforementioned processes in terms of longitudinal bunching, transverse focusing, and photon acceleration. As a result, the physical mechanisms behind each are made clear and the relationship between RFS and envelope SM is made explicitly clear. This allows a single differential equation to be obtained which couples RFS and SM, so that the relative importance between each process can now be predicted for given experimental conditions.
160 citations
TL;DR: In this paper, high-resolution, fully threedimensional, unsteady simulations of the interaction of a tornado vortex with the surface were performed in an attempt to answer questions about the character of turbulent transport in this unique flow.
Abstract: High-resolution, fully three-dimensional, unsteady simulations of the interaction of a tornado vortex with the surface were performed in an attempt to answer questions about the character of turbulent transport in this unique flow. The authors demonstrate that sufficient resolution was achieved for the particular physical conditions of their example that the time-averaged velocity and pressure distributions showed little sensitivity in the region of maximum velocities to either finer resolution or modified subgrid turbulent model. The time-averaged velocity distributions show the maximum velocity values occurring within 50 m of the surface. The instantaneous velocity distributions show the turbulence dominated by a relatively small number of strong secondary vortices spiralling around the main vortex with the maximum instantaneous velocities typically one-third larger than the maximum time-averaged velocity. These eddies are centered a little inside of the cone of maximum mean swirl velocity and spiral around the mean vortex at velocities less than the average maximum velocity. Statistical analysis of the velocity fluctuations induced by the secondary vortices shows that the turbulent transport of angular momentum is predominantly inward at low levels, allowing the inner recirculating flow to acquire values of angular momentum of up to 30% of that provided by the inflow boundary conditions, thus enhancing the surface intensification of the velocities.
157 citations
TL;DR: In this paper, a traveling-wave photodetector (TWPD) is modeled by a terminated section of transmission line with a position-dependent photocurrent source propagating on it at the optical group velocity.
Abstract: Photodetector efficiency decreases as bandwidth increases, Bandwidth-efficiency limitations of traveling-wave photodetectors (TWPDs) are substantially greater than those of lumped-element photodetectors because the velocity-mismatch bandwidth limitation is independent of device length. TWPDs can be long for high efficiency without significantly compromising bandwidth. The TWPD is modeled by a terminated section of transmission line with a position-dependent photocurrent source propagating on it at the optical group velocity. A wave model for the transmission line confirms the accuracy of an equivalent-circuit model for electrical wave propagation. The velocity-mismatch impulse and frequency response are determined by absorption coefficient and wave velocities rather than junction capacitance and load resistance. The velocity-mismatch bandwidth limitations can be written in a simple form which elucidates the factors affecting device response, A discretized periodic TWPD is described by the same equations as the fully distributed version. This more complicated device offers additional degrees of freedom in design and potentially improved performance.
129 citations
Journal Article•
TL;DR: In this paper, the steady-state displacements of an Euler-Bernoulli beam on an elastic half-space due to a uniformly moving constant load are determined using the concept of the equivalent stiffness of the half space.
Abstract: The steady-state displacements of an Euler-Bernoulli beam on an elastic half-space due to a uniformly moving constant load are determined using the concept of the equivalent stiffness of the half-space. The displacements are calculated for four different cases of beam and half-space parameters. The displacements in each case are derived for five relevant load velocities. The lowest is small with respect to the shear (S)-wave velocity in the half-space, the subsequent velocities are near the lowest critical velocity, between the critical velocity and the Rayleigh (R)-wave velocity, between the R-wave anal S-wave velocity and larger than S-wave velocity. The beam displacement under the load is also determined for each case for all load velocities. Near the critical velocity the effect of an external viscous damping along the beam on the displacement under the load is studied.
111 citations
TL;DR: In this paper, a high order accurate finite difference method for direct numerical simulation of coherent structure in the mixing layers is presented, and the reason for oscillation production in numerical solutions is analyzed.
98 citations
TL;DR: In this paper, the authors used Doppler lidar observations of trapped lee waves to measure the rate at which their horizontal wavelengths change, including a detailed analysis of two events in which such changes are clearly documented.
Abstract: Although considerable understanding of mountain waves has been gained with the aid of the steady-state assumption, it is clear that mountain waves evolve over time. Group velocity arguments indicate that this evolution can occur in less than 1 h. This study uses observations of trapped lee waves to measure the rate at which their horizontal wavelengths change, including a detailed analysis of two events in which such changes are clearly documented. In one case, Doppler lidar observations show a steady increase in horizontal wavelength of 6% h−1 over 4 h and clearly illustrates the relationship between the wave clouds and wave motions. In a second case, visible satellite imagery reveals an increase in wavelength of 14% h−1, which is related to temporal changes in vertical air motions measured by wind profilers within the wave field. Hourly vertical profiles of wind and virtual temperature measured by radio acoustic sounding systems (RASS) and wind profilers reveal important changes in the wave env...
94 citations
TL;DR: In this paper, an analytical expression for the velocity distributions of molecules ejected in laser ablation is proposed based on the results of molecular dynamics simulations, which provides a consistent description of the axial and radial velocity distributions.
Abstract: Based on the results of molecular dynamics simulations, we propose an analytical expression for the velocity distributions of molecules ejected in laser ablation. The Maxwell-Boltzmann distribution on a stream velocity, commonly used to describe the measured velocity distributions, is modified to account for a range of stream velocities in the ejected plume. The proposed distribution function provides a consistent description of the axial and radial velocity distributions. The function has two parameters that are independent of the desorption angle and have clear physical meaning, namely, the temperature of the plume and the maximum stream velocity or velocity of the plume propagation.
92 citations
TL;DR: In this article, the maximum radiation power was achieved under group synchronism conditions, when the electron bunch translational velocity coincides with the group velocity of the wave propagating in the waveguide.
Abstract: Intense microwave pulses (several hundreds of kilowatts) of ultrashort duration (less than 0.5 ns) were obtained from an ensemble of electrons rotating in a uniform magnetic field. The comparison with theoretical simulations proves that this emission can be interpreted as cyclotron superradiance. The maximum radiation power was achieved under group synchronism conditions, when the electron bunch translational velocity coincides with the group velocity of the wave propagating in the waveguide.
92 citations
TL;DR: In this paper, it was shown that asymmetric solitary waves of the form suggested by the nonlinear Schrodinger (NLS) theory in fact are not possible and that an infinity of symmetric and asymmetric wave solution families comprising two or more NLS solitary wavepackets bifurcate at finite values of the amplitude parameter.
Abstract: Symme tric gravity–capillary solitary waves with decaying oscillatory tails are known to bifurcate from infinitesimal periodic waves at the minimum value of the phase speed where the group velocity is equal to the phase speed. In the small-amplitude limit, these solitary waves may be interpreted as envelope solitons with stationary crests and are described by the nonlinear Schrodinger (NLS) equation to leading order. In line with this interpretation, it would appear that one may also co nstruct asymmetric solitary waves by shifting the carrier oscillations relative to the envelope of a symmetric solitary wave. This possibility is examined here on the basis of the fifth-order Korteweg–de Vries (KdV) equation, a model for g ravity–capillary waves on water of finite depth when the Bond number is close to 1/3. Using techniques of exponential asymptotics beyond all orders of the NLS theory, it is shown that asymmetric solitary waves of the form suggested by the NLS theory in fact are not possible. On the other hand, an infinity of symmetric and asymmetric solitary-wave solution families comprising two or more NLS solitary wavepackets bifurcate at finite values of the amplitude parameter. The asymptotic results are consistent with numerical solutions of the fifth-order KdV equation. Moreover, the asymptotic theory suggests that such multi-packet gravity–capillary solitary waves also exist in the full water-wave problem near the minimum of t he phase speed.
85 citations
TL;DR: The performance of velocity estimates from two different coherent Doppler lidars in the weak signal regime is compared with the predictions of computer simulations for pulse accumulation from 1 to 100 pulses.
Abstract: In the weak signal regime coherent Doppler lidar velocity estimates are characterized by a localized distribution around the true mean velocity and a uniform distribution of random outliers over the velocity search space. The performance of velocity estimators is defined by the standard deviation of the good estimates around the true mean velocity and the fraction of random outliers. The quality of velocity estimates is improved with pulse accumulation. The performance of velocity estimates from two different coherent Doppler lidars in the weak signal regime is compared with the predictions of computer simulations for pulse accumulation from 1 to 100 pulses.
TL;DR: In this article, it was shown that the wave group velocity along the shock normal can be approximately equal to the shock velocity: such waves are not rapidly convected away from the shock, and can thus grow to a high level.
Abstract: The standard model of cosmic ray electron acceleration requires electrons to be pre-accelerated to mildly relativistic energies. It has been suggested that energy transfer from waves, generated by gyrotropic ions reflected from quasi-perpendicular shocks, could provide the necessary pre-acceleration. The distribution of shock-reflected ions at any upstream point is more likely to consist of two beams rather than a gyrotropic ring. Wave excitation in the presence of both types of ion distribution is studied. It is shown that gyrotropic or beam ions, reflected from shocks associated with supernova remnants, can excite waves capable of accelerating electrons to beyond the required injection energies. The wave group velocity along the shock normal can be approximately equal to the shock velocity: such waves are not rapidly convected away from the shock, and can thus grow to a high level. Moreover, waves satisfying this condition which also have phase velocities parallel to the magnetic field ranging from the electron thermal speed to relativistic speeds are excited in high Mach number shocks with a low ratio of electron plasma frequency to cyclotron frequency. Bulk electrons can then be accelerated to the required energies within the region in which shock-reflected ions are present. This is consistent with a suggestion, based on a comparison between Wolf-Rayet stars and radio supernovae, that there exists a threshold perpendicular shock speed (between 1 and 3 per cent of the speed of light) above which the efficiency of electron injection increases by several orders of magnitude.
TL;DR: In this paper, it was shown that resonant interactions cannot occur unless the inclination of the slope to the horizontal, α, is less than the velocity vector of the wave group velocity vector, β, and interactions generate a nonzero Eulerian upslope flow, which is exactly balanced by a Stokes drift, so leading to zero Lagrangian up slope flow.
Abstract: Incident internal waves and those reflected from a uniform slope interact at second order. These interactions are considered for incident waves traveling obliquely to the slope in a uniformly stratified rotating fluid. It is found that (i) resonant interactions cannot occur unless the inclination of the slope to the horizontal, α, is less than the inclination of the wave group velocity vector to the horizontal, β; (ii) interactions generate a nonzero Eulerian upslope flow, which is exactly balanced by a Stokes drift, so leading to zero Lagrangian upslope flow; (iii) a Stokes drift parallel to the isobaths is generated by the reflecting waves, provided the incident waves are not in a plane normal to the slope. This drift is in addition to the possible Eulerian flows along the slope, which are in geostrophic balance with a correspondingly perturbed density field and are zero in a nonrotating flow. The order of magnitude of the drift is typically 0.01 m s−1. These secondary flows do not depend on di...
TL;DR: In this article, the authors used Rayleigh wave tomographic inversion to obtain group velocity maps of East Asia (40° E −160° E and 20° N −70° N).
Abstract: —Rayleigh waves are used in a tomographic inversion to obtain group velocity maps of East Asia (40° E–160° E and 20° N–70° N). The period range studied is 30 to 70 seconds. Seismograms used for this study were recorded at CDSN stations, at a temporary broadband seismic array in Tibet, at several SRO stations, and Kirnos-equipped stations established in Asia by the former Soviet Union, in Siberia, in the Sakhalin and in Mongolia. Altogether more than 1200 paths were available in the tomographic inversion. The study area includes the Angara craton, the geologically ancient core of Asia, and the subsequently accreted units, the Altaids (a Paleozoic collision complex), the Sino-Korean platform (a chain of Archaen terranes separated by belts of active structures), the south China platform (a collage of Precambrian, Paleozoic and Mesozoic metamorphic and igneous terranes), as well as the Tibetan plateau (an active tectonic feature created in late Cenozoic through collision of the Indian subcontinent and the Asian continent). Many of these main units are recognizable in the tomographic images as distinctive units; Tibet appears as a prominent low velocity (about −15% from the average) structure, with western and central Tibet often appearing as the areas with the lowest velocities, the Central Asian fold-belt, and the Angara craton are consistently high group velocity areas. Some lesser tectonic features are also recognizable. For example, Lake Baikal is seen as a high velocity feature at periods greater than 40 seconds. However, the high group velocity feature does not stop near the southern end of Lake Baikal; it extends south-southwestward across Mongolia. The North China Plain, a part of the platform where extensional tectonics dominate, is an area of high velocities as a result of relatively thin crust. The south China block, the least tectonically active region of China, is generally an area of high velocity. For periods longer than 40 seconds, a NNE trending high group velocity gradient clearly exists in eastern China; the velocities are noticeably higher in the east. From the group velocity maps, average dispersion curves at twelve locations were determined and inverted to obtain velocity structures. Main results of group velocity inversion include: (1) a Tibetan crust of around 60 km thick, with low crustal and upper mantle shear velocities, at 3.3 km/s and 4.2 km/s, respectively; (2) with the Moho constrained at 40–43 km, the Angara craton and the Central Asian foldbelt have a VS in excess of 4.6 km/s; (3) relatively low shear velocities are obtained for tectonically active areas. In many parts of the study area, where Precambrian basement is exposed, the process in the crust and upper mantle due to recent tectonic activities have modified the crust and upper mantle velocity structures under the Precambrian terranes, they are no longer underlain by high velocity crust and mantle.
TL;DR: In this article, the dynamical evolution of an ultrashort pulse, whose initial temporal envelope is infinitely smooth with compact support, is considered as it propagates through a temporally dispersive, attenuative medium characterized by two resonance lines.
Abstract: The dynamical evolution of an ultrashort pulse, whose initial temporal envelope is infinitely smooth with compact support, is considered as it propagates through a temporally dispersive, attenuative medium characterized by two resonance lines. As the propagation distance increases, the accuracy of the popular group velocity description decreases monotonically, whereas the accuracy of the mathematically well-defined asymptotic description and its derivative energy velocity description increase.
TL;DR: In this article, the stability and accuracy of the advection phase calculation of the Cubic Interpolated Propagation scheme, which solves the universal hyperbolic equation, were examined using Fourier analysis.
TL;DR: In this paper, the authors used a theoretical model that accounts for the renormalization of scattering by the coupling between neighboring scatterers, quantitatively predicting the scattering delay that causes the strong frequency dependence of these velocities seen in their experiments.
Abstract: Measurements of the diffusive transport of multiply scattered ultrasonic waves show that the energy velocity is very similar in magnitude and frequency dependence to the group velocity Our data are accurately described using a theoretical model that accounts for the renormalization of scattering by the coupling between neighboring scatterers, quantitatively predicting the scattering delay that causes the strong frequency dependence of these velocities seen in our experiments This gives a unified physical picture of the velocities of energy transport by both diffusive and ballistic waves [S0031-9007(97)04300-7]
01 Jan 1997
TL;DR: In this article, it was shown that the wave group velocity along the shock normal can be approximately equal to the shock velocity: such waves are not rapidly convected away from the shock, and can thus grow to a high level.
Abstract: The standard model of cosmic ray electron acceleration requires electrons to be pre-accelerated to mildly relativistic energies. It has been suggested that energy transfer from waves, generated by gyrotropic ions reflected from quasi-perpendicular shocks, could provide the necessary pre-acceleration. The distribution of shock-reflected ions at any upstream point is more likely to consist of two beams rather than a gyrotropic ring. Wave excitation in the presence of both types of ion distribution is studied. It is shown that gyrotropic or beam ions, reflected from shocks associated with supernova remnants, can excite waves capable of accelerating electrons to beyond the required injection energies. The wave group velocity along the shock normal can be approximately equal to the shock velocity: such waves are not rapidly convected away from the shock, and can thus grow to a high level. Moreover, waves satisfying this condition which also have phase velocities parallel to the magnetic field ranging from the electron thermal speed to relativistic speeds are excited in high Mach number shocks with a low ratio of electron plasma frequency to cyclotron frequency. Bulk electrons can then be accelerated to the required energies within the region in which shock-reflected ions are present. This is consistent with a suggestion, based on a comparison between Wolf-Rayet stars and radio supernovae, that there exists a threshold perpendicular shock speed (between 1 and 3 per cent of the speed of light) above which the efficiency of electron injection increases by several orders of magnitude.
TL;DR: In this article, the impact of mean flow variability on finite-amplitude trapped mountain lee waves is investigated by conducting two-dimensional mountain wave simulations for a set of idealized, time-dependent background flows.
Abstract: The impact of mean-flow variability on finite-amplitude trapped mountain lee waves is investigated by conducting two-dimensional mountain wave simulations for a set of idealized, time-dependent background flows. The lee-wave patterns generated by these time-dependent flows depend on two factors: 1) the degree to which the transition in the background flow changes the amplitude of the stationary trapped lee wave and 2) the difference between the group velocities of the trapped waves generated before and after the transition. When the transition in the background flow significantly reduces the amplitude of the stationary lee wave, the lee-wave pattern generated prior to the transition gradually drifts downstream away from the mountain or back over the mountain, depending on the sign of this wave packet’s group velocity after the transition. When the transition in the background flow changes the resonant wavelength while leaving the lee-wave amplitude relatively unchanged, the lee-wave train develop...
TL;DR: In this paper, the critical velocities of a harmonically varying point load moving uniformly along an elastic layer are determined as a function of the load frequency and the critical depths of the layer were determined as function of load velocity.
Abstract: The critical (resonance) velocities of a harmonically varying point load moving uniformly along an elastic layer are determined as a function of the load frequency. It is shown that resonance occurs when the velocity of the load is equal to the group velocity of the waves generated by the load. The critical depths of the layer are determined as function of the load velocity in the case the load frequency is proportional to the load velocity. This is of importance for high-speed trains where the loading frequency of the train wheel excitations is mainly determined by the ratio between the train velocity and the distance between the sleepers (ties). It is shown that the critical depths are decreasing with increasing train velocity. It is concluded that the higher the train velocity, the more important are the properties of the ballast and the border between the ballast and the substrate.
TL;DR: In this article, the spatial pattern of kinetic Alfven waves excited by a current-modulating disk whose dimension a, transverse to the confining magnetic field, is comparable to the ion sound gyroradius cs/Ωi, where cs is the sound speed and Ωi the ion cyclotron frequency.
Abstract: This analytical study illustrates the spatial pattern of kinetic Alfven waves excited by a current-modulating disk whose dimension a, transverse to the confining magnetic field, is comparable to the ion sound gyroradius cs/Ωi, where cs is the sound speed and Ωi the ion cyclotron frequency. The radial structure of the wave azimuthal magnetic field is found to consist of four regions: a Bessel function behavior for r
TL;DR: In this paper, a wave approach is adopted to determine the relationship between energy density and energy flow, which leads to a differential equation similar to the heat conduction equation in steady state conditions.
TL;DR: In this article, the energy of translating tensioned beams and fluid transporting pipes under fixed, simply supported and free boundary conditions are analyzed in a generalized manner. But the energy transfer between the translating continua and the boundary supports is not considered.
TL;DR: In this article, the group velocity of frustrated transverse magnetic modes in an optical waveguide including a metal is shown to be negative when the power flow inside the metal is larger than the power flows inside the dielectric medium, while the phase velocity remains positive.
TL;DR: In this article, the authors investigated diffusion and coagulation in a random, isotropic flow in the presence of hydrodynamic interactions, interparticle forces, and Brownian diffusion.
Abstract: Diffusion and coagulation are investigated in a random, isotropic flow in the presence of hydrodynamic interactions, interparticle forces, and Brownian diffusion. Different strain and rotation rate time scales characterize the velocity field and the particles are assumed small compared with the characteristic length of the flow, so that the velocity field is linear in the vicinity of the particles. The pair probability equation for the relative motion of two particles is written in terms of a diffusion tensor and a drift velocity. This technique is valid in the limit of small strain, i.e., when the product of the characteristic velocity gradient and time scale of the fluctuating velocity gradient is small. A consequence of the drift velocity is that, at steady state in a noncoagulating system, the pair probability distribution is nonuniform when hydrodynamic interactions are included, and there is a higher probability of particle pairs at close proximity. The pair probability conservation equation is used...
TL;DR: Comparison between the theoretical simulation and experimental data demonstrates that the competition between third- and second-order nonlinear phenomena strongly modifies the structure of the pulses generated.
Abstract: Temporal and spectral characteristics of pulses resulting from second-harmonic generation of 120-fs amplified Ti:sapphire laser pulses up to 0.1 mJ at a wavelength of 815 nm in type I KDP crystal were experimentally and theoretically analyzed. Widely different behaviors were observed, according to the sign of the phase mismatch. Comparison between the theoretical simulation and experimental data demonstrates that the competition between third- and second-order nonlinear phenomena strongly modifies the structure of the pulses generated.
TL;DR: In this paper, a new definition of the energy transport velocity for waves in dispersive media is proposed, which can never exceed the vacuum speed of light, not even in the case of superluminal group velocities.
TL;DR: In this article, the authors investigated the transport of classical waves in strongly scattering media using ultrasonic techniques, allowing them to measure both the ballistic and scattered components of the wave field, and they found that the ballistic propagation is dramatically slowed down by scattering resonances, although the group velocity remains well-defined.
Abstract: The transport of classical waves in strongly scattering media is investigated using ultrasonic techniques, allowing us to measure both the ballistic and scattered components of the wave field. We find that the ballistic propagation is dramatically slowed down by scattering resonances, although the group velocity remains well-defined. The propagation of the scattered waves is also strongly affected by resonant scattering, and is shown to be well described by using the diffusion approximation. A model based on the generalized coherent potential approximation gives a quantitative explanation of the experimental data.
TL;DR: In this paper, an asymmetric dual-core fiber, with opposite signs of the dispersion in its two cores, was shown to support bright gap solitons, which is a new nonlinear optical medium, the fabrication of which is technologically feasible.
Abstract: We demonstrate that an asymmetric dual-core fiber, with opposite signs of the dispersion in its two cores, supports bright gap solitons. Such a fiber is a new nonlinear optical medium, the fabrication of which is technologically feasible. There are three nontrivial features of these new gap solitons: (i) Their exponential tails may oscillate as they decay; (ii) the energy in the normal-core component may be equal to or larger than the energy in the anomalous-core one, despite the obvious fact that the normal-dispersion core, in isolation, cannot support any bright solitons; and (iii) a part of the finite gap available for soliton solutions remains empty, having no solitons inside it. In addition, we show that such fibers can also support weakly delocalized solitons, which sit on a small-amplitude, nonvanishing, continuous-wave background. We also investigate the case in which the dispersion in both cores is anomalous but in which the cores are asymmetric and have different magnitudes of dispersion. In this case the general picture of the soliton solutions, including their bifurcations and stability, is qualitatively the same as when one has equal dispersions in the two cores, but with an asymmetry produced by a phase-velocity mismatch. These studies have been carried out by a combination of an analysis of the dispersion relation for the linearized system, the variational approximation for the full nonlinear one, and direct numerical methods.
TL;DR: In this paper, a three-phase (gas-liquid-solid) fluidized bed (0.152 m ID × 2.5 m high) is considered, and the effects of gas velocity, liquid velocity and particle size on particle dispersion have been determined.