Expressions For Phase Velocity Research Articles

Ion acoustic cnoidal waves and associated nonlinear ion flux in a warm ion plasma

Using periodic boundary conditions in the reductive perturbation method for ion acoustic cnoidal waves in a collisionless, unmagnetized plasma consisting of warm ions and hot isothermal electrons, evolution equations for the first order and second order potential are derived. Determining the nonsecular solution of these coupled equations, expressions for wave phase velocity and averaged nonlinear ion flux are obtained. Variations of wave phase velocity and averaged nonlinear ion flux are examined for different values of wave frequency and ion to electron temperature ratio (Ti∕Te). It is found that wave phase velocity increases with an increase in wave amplitude for a given value of frequency and temperature ratio. But for a given wave amplitude, the wave phase velocity increases as the temperature ratio increases or wave frequency decreases. The averaged nonlinear ion flux associated with the propagation of large amplitude cnoidal waves in plasma is found to be negative, which corresponds to back flow of ions. The magnitude of the ion flux increases with increase in wave amplitude as temperature ratio or wave frequency increases.

Wave Equation Depth Migration with Symmetric Non‐Stationary Phase Shift Method in Axi‐Symmetrical Anisotropic Media

AbstractA 3‐D dispersion equation for the transversely isotropic media with a tilted symmetry axis is derived from the qP wave phase velocity expression. The anisotropic phase shift operator can be obtained with the vertical wave number, which is analytically solved from the dispersion equation. Then the symmetric non‐stationary phase shift wave equation depth migration method in isotropic media is extended to anisotropic media. The macro velocity model used for describing anisotropic media is consistent with the current velocity estimation method and can simultaneously accommodate the isotropy, weak anisotropy and strong anisotropy. The symmetric nonstationary phase shift extrapolation operator can accommodate transverse variation of velocity as well as the anisotropic parameters. Although a 2‐D VTI example is present in the paper, the proposed algorithm can be used in 3‐D TI media as well.

Weak elastic anisotropy by perturbation theory

We demonstrate the advantages of adopting a wave-vector-based coordinate system (WCS) for the application of perturbation theory to derive and display approximate expressions for qP- and qS-wave polarization vectors, phase velocities, and group velocities in general weakly anisotropic media. The advantages stem from two important properties of the Christoffel equation when expressed in the WCS: (1) Each element of the Christoffel matrix is identical to a specific stiffness component in the WCS, and (2) the Christoffel matrix of an isotropic medium is diagonal in the WCS. Using these properties, one can easily identify the small components of the Christoffel matrix in the WCS for a weakly anisotropic medium. Approximate solutions to the Christoffel equation are then obtained by straightforward algebraic manipulations, which make our perturbation theory solution considerably simpler than previously published methods. We compare and contrast our solutions with those discussed by other workers. Numerical comparisons between the exact, first-order, and zero-order qS-wave polarization vectors illustrate the accuracy of our approximate formulas. The form of the WCS phase-velocity expressions facilitates the derivation of closed-form, first-order expressions for qP- and qS-wave group-velocity vectors, providing explicit formulas for the direction of propagation of seismic energy in general weakly anisotropic media. Numerical evaluation of our group-velocity expressions demonstrates their accuracy. We discuss problems with the approximate qS-wave group velocities and polarizations in neighboring directions of singularities. Standard methods are used to transform our solutions from the WCS to the acquisition coordinates, as illustrated by application to orthorhombic symmetry.

A study of ion-acoustic solitons and double layers in a multispecies collisionless weakly relativistic plasma

The effect on the propagation of ion-acoustic solitons and double layers has been studied in collisionless weakly relativistic plasma consisting of two-electron temperature with isothermal electrons and finite ion temperature. The Korteweg de-Vries (KdV) equation is derived for ion-acoustic solitons propagating in a collisionless plasma. This equation is solved in a stationary frame to obtain the expression for soliton phase velocity, soliton width and peak soliton amplitude. It is observed that these quantities are significantly influenced by the relativistic effect, ion temperature, low-temperature electron density and ratio of cold to hot electron temperatures. Many features expected from hot ion theory and two species electron plasmas automatically emerge. The analysis is further extended to higher order nonlinearity and modified Korteweg de-Vries (mKdV) equation is derived. Even though compressive and rarefactive ion-acoustic solitons are obtained, only rarefactive ion-acoustic double layers are obtained in the present investigation.

Practical VTI approximations: a systematic anatomy

Transverse isotropy (TI) with a vertical symmetry axis (VTI) often provides an appropriate earth model for prestack imaging of steep-dip reflection seismic data. Exact P-wave and SV-wave phase velocities in VTI media are described by complicated equations requiring four independent parameters. Estimating appropriate multiparameter earth models can be difficult and time-consuming, so it is often useful to replace the exact VTI equations with simpler approximations requiring fewer parameters. The accuracy limits of different previously published VTI approximations are not always clear, nor is it always obvious how these different approximations relate to each other. Here I present a systematic framework for deriving a variety of useful VTI approximations. I develop first a sequence of well-defined approximations to the exact P-wave and SV-wave phase velocities. In doing so, I show how the useful but physically questionable heuristic of setting shear velocities identically to zero can be replaced with a more precise and quantifiable approximation. The key here to deriving accurate approximations is to replace the stiffness a13 with an appropriate factorization in terms of velocity parameters. Two different specific parameter choices lead to the P-wave approximations of Alkhalifah (Geophysics 63 (1998) 623) and Schoenberg and de Hoop (Geophysics 65 (2000) 919), but there are actually an infinite number of reasonable parametrizations possible. Further approximations then lead to a variety of other useful phase velocity expressions, including those of Thomsen (Geophysics 51 (1986) 1954), Dellinger et al. (Journal of Seismic Exploration 2 (1993) 23), Harlan (Stanford Exploration Project Report 89 (1995) 145), and Stopin (Stopin, A., 2001. Comparison of v(θ) equations in TI medium. 9th International Workshop on Seismic Anisotropy). Each P-wave phase velocity approximation derived this way can be paired naturally with a corresponding SV-wave approximation. Each P-wave or SV-wave phase velocity approximation can then be converted into an equivalent dispersion relation in terms of horizontal and vertical slownesses. A simple heuristic substitution also allows each phase velocity approximation to be converted into an explicit group velocity approximation. From these, in turn, travel time or moveout approximations can also be derived. The group velocity and travel time approximations derived this way include ones previously used by Byun et al. (Geophysics 54 (1989) 1564), Dellinger et al. (Journal of Seismic Exploration 2 (1993) 23), Tsvankin and Thomsen (Geophysics 59 (1994) 1290), Harlan (89 (1995) 145), and Zhang and Uren (Zhang, F. and Uren, N., 2001. Approximate explicit ray velocity functions and travel times for P-waves in TI media. 71st Annual International Meeting, Society of Exploration Geophysicists, Expanded Abstracts, 106–109).

Flux distributions in stable criticality waves

An analysis is presented of so-called self-stabilizing criticality waves, as can be ignited in originally subcritical systems under the condition that the burn up dependent properties of the system satisfy certain requirements, as was shown in a previous publication for an analytically solvable model. For the solution of the analytically solvable model (as far as the asymptotic waves are concerned) a new approach was adopted, based on taking the flux-fluence space as a starting point. In this way the expressions for ignition condition, wave form, amplitude (i.e. reactor power) and phase velocity are obtained in a more systematic way than before. The analysis is extended to skew criticality waves by adopting cubic flux-fluence trajectories. From the parameters of these trajectories the parameters of the associated burn up functions and the wave properties are inferred. For a particular case the results are corroborated by a computer simulation.

Reflection of quasi-P and quasi-SV waves at the free and rigid boundaries of a fibre-reinforced medium

The propagation of plane waves in fibre-reinforced media is discussed. The expressions of phase velocities of quasi-P (qP) and quasi-SV (qSV) waves propagating in plane symmetry are obtained in terms of propagation vectors. We have established a relation from which the displacement vector can be obtained in terms of the propagation vector. Expressions for the reflection coefficients of qP and qSV waves are obtained. Numerical results of reflection coefficients are obtained and presented graphically. The partition of energy between qP and qSV waves reflected on free and rigid boundaries due to incident qP and qSV waves are also obtained and presented graphically.

The Waves on the Vortex Ring in HeII

Using the equations of vortex dynamics the waves on a vortex ring are studied both analytically and by computer simulations. At small amplitudes the results coincide. The dispersion relations and the expressions of phase and group velocities are determined. The free and forced oscillations are considered. At large amplitudes the shape of a vortex is strongly distorted and the ring motion changes its direction. The second approximation calculations and computer simulations agree in order of magnitude.

Influence of vibration on granular flowability and its mechanism of aided flow

Regarding flowing granular media as weak transverse isotropic media, the phase velocity expressions of wave P, wave SH and wave SV were deduced, the propagation characteristics of waves in flowing granular media were analyzed. The experiments show that vibration has great influence on granular fluidity. The wavefront of wave P is elliptic or closely elliptic, the wavefront of wave SH is elliptic, and the wavefront of wave SV is not elliptic. Wave propagation in the granular flowing field attenuates layer after layer. The theory and experiment both substantiate that the density difference is the key factor which leads to the attenuation of vibrating energy. In terms of characteristics of wave propagation one can deduce that vibrating waves have less influence on flowability of granules when the amplitude and frequency are small. However, when the amplitude and frequency increase gradually, the eccentricity of ellipsoid, the viscosity resistance and inner friction among granules, and shear intensity of granules decrease, and the loosening coefficient of granules increases, which shows the granules have better flowability.

Explicit Anisotropic P-wave Ray Velocity Functions

Sedimentary rocks encountered in seismic exploration frequently exhibit directional seismic wave velocity dependence. These anisotropic rocks often can be regarded as transversely isotropic (TI) media in which the ray velocity surface of SH-waves is elliptical, while those of P- and SV-waves are non-elliptical. Transversely isotropic phase velocity expressions exist for P-, SH- and SV-waves. Unfortunately, until now no explicit exact analytical ray velocity functions exist for P- or SV-waves propagating in TI media. In exploration seismology, explicit ray velocity functions are needed for efficient migration, velocity analysis and other seismic processing procedures.Approximate explicit analytical P-wave ray velocity functions in transversely isotropic media have now been developed. These ray velocity functions include a new dimensionless constant, A, which depends on elastic parameters C11, C33, C44, and C13 or Thomsen's parameters ε, δ, α0 and (β0. The accuracy of the functions has been tested for all the materials published by Thomsen (1986). From these ray velocity functions, the travel time between any two points in homogeneous transversely isotropic media may be calculated. Reflection travel times from a single horizontal reflector overlain by transversely isotropic media with vertical symmetrical axis (VTI) are also calculated and compared with those obtained by Tsvankin (1996). The results show that the proposed explicit ray velocity functions have high accuracy.

Linearized approximations for phase velocities of elastic waves in weaklyanisotropic media

This paper focuses on the development of linearized formulae for the phasevelocities of quasi-longitudinal (QL) and quasi-transverse (QT) waves in weaklyanisotropic media. The formulae for quasi-transverse waves derived fromfirst-order perturbation theory possess the property of rotational invariance in aplane perpendicular to the wave normal. Based on this property, a planarcoordinate system, describing the possible polarization of the two degeneratetransverse waves, can be conveniently selected. The relationships between thefirst-order formulae and other related existing approximate expressions areclarified. For a symmetry plane of orthorhombic media, we extended Gassmann'sformula for the QL-wave phase velocity in transversely isotropic media to QLand QT waves. Three forms of phase velocity expressions are discussed. Fornon-symmetry planes of weakly orthorhombic media, we propose a unified formfor the QL-wave phase velocity and in a very simple way derive the linearizedThomsen- and Gassmannn-like formulae. Furthermore we suggest QT-wave phasevelocity expressions which are linear functions of the elastic constantsassuming that wave propagation has a slight deviation from a symmetry plane oforthorhombic media. Finally, numerical results in a symmetry plane show thatthe set of first-order approximate formulae performs best for weaklyanisotropic materials.

Characteristic Velocities for Higher-Order Stokes Waves in Deep Water

The accuracy of asymptotic expressions for phase, wave energy, and wave action velocities for steady, plane gravity waves progressing in deep water and with no ambient currents is investigated. The calculations are to a large extent based on Fenton's fifth-order Stokes wave theory. Using the dispersion relation correct to sixth order for arbitrary water depth, the phase velocity is calculated to fourth order in wave steepness for infinitely deep water; this is in accordance with a number of previously found results. Potential and kinetic energy is calculated, which leads to the energy density correct to sixth order. Comparison with previous findings showed one discrepancy, which can however be explained. Also, the energy flux is found correct to sixth order, leading to a fourth-order expression for the wave energy velocity. From sixth-order action and action flux results, the wave action velocity is finally calculated correct to fourth order. Agreement with more accurate numerical results is excellent for all three velocities except near the wave of maximum steepness. Practical asymptotic formulas are introduced, enabling one to calculate the characteristic velocities explicitly from given wave height and period. Two new exact relations between integral properties for deep water with no current are presented.

A theory for vortex rossby‐waves and its application to spiral bands and intensity changes in hurricanes

AbstractIn this paper we examine further the physics of vortex axisymmetrization, with the goal of elucidating the dynamics of outward‐propagating spiral bands in hurricanes. the basic shysics is illustrated most simply for stable vorticity monopoles on an f‐plane. Unlike the dynamics of sheared disturbances in rectilinear shear flow, axisymmetrizing disturbances on a vortex are accompanied by outward‐propagating vortex Rossby‐waves whose restoring mechanism is associated with the radial gradient of storm vorticity. Expressions for both phase and group velocities are derived and verified; they confirm earlier speculations on the existence of vortex Rossbywaves in hurricanes. Effects of radially propagating vortex Rossby‐waves on the mean vortex are also analysed. In conjunction with sustained injection of vorticity near the radius of maximum winds, these results reveal a new mechanism of vortex intensification. the basic theory is then applied to a hurricane‐like vortex in a shallow‐water asymmetric‐balance model. the wave mechanics developed here shows promise in elucidating basic mechanisms of hurricane evolution and structure changes, such as the formation of secondary eye‐walls. Radar observations possessing adequate temporal resolution are consistent with the predictions of this work, though more refined observations are needed to quantify further the impact of mesoscale banded disturbances on the evolution of the hurricane vortex.

A MODEL FOR ESTIMATION OF ELECTROMECHANICAL CHARACTERISTICS OF PIEZOELECTRIC POLYMER COMPOSITE DISCS

The constitutive equations of piezoelectric polymer composite materials used for their electromechanical characterisation are complicated and their analytical solution is difficult. Earlier models assumed either stress or strain components to be constant. In this paper we present a model for electromechanical characterisation of piezoceramic polymer composites. The assumptions are (i) the samples have complete stress free boundary conditions at the polymer piezoelectric interfaces, (ii) the materials are lossless, (iii) the coordinate axes are pure mode propagation directions, and (iv) the electric potential and the displacements are assumed to be independent degrees of freedom. Based on these assumptions, expressions for impedance, phase velocities, and electromechanical coupling constant are developed for piezoelectric polymer composites of cylindrical geometry.

A model for estimation of electromechanical characteristics of piezoelectric polymer composites

Abstract The constitutive equations, of piezoelectric polymer composite materials used for their electromechanical characterisation are complicated and their analytical solution is difficult. Earlier models assumed either stress or strain components to be constant. In this paper we present a model for electromechanical characterisation of piezoceramic polymer composites. The assumptions are (i) the samples have complete stress free boundary conditions at the polymer piezoelectric interfaces, (ii) the materials are lossless, (iii) the co-ordinate axes are pure mode propagation directions and (iv) the electric potential and the displacements are assumed to be independent degrees of freedom. Based on these assumptions, expressions for impedance, phase velocities and electromechanical coupling constant are developed for piezoelectric polymer composites.

Group and phase velocities in the TLM-symmetrical-condensed node mesh

This paper presents the analytical expressions of group and phase velocities as a function of frequency in the TLM-symmetrical-condensed node mesh for the three fundamental directions. The direction /spl lsqb/1, 1, 1/spl rsqb/ happens to be more dispersive than the directions /spl lsqb/1, 1, 0/spl rsqb/ and /spl lsqb/1, 0, 0/spl rsqb/, giving the TLM-symmetrical-condensed formulation dispersion characteristics similar to those of the FD-TD algorithm with a Courant number of 1//spl radic/3.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Propagation coefficient in a superconducting asymmetrical parallel-plate transmission line with a buffer layer

A sinusoidal wave solution is found for a superconducting asymmetrical parallel-plate transmission line with a buffer layer, by using Maxwell’s and London’s equations, the two-fluid model, and the classical skin effect for the normal component of the current. Expressions for phase velocity and attenuation coefficient of the wave are derived. Values for the attenuation coefficient are computed for various combinations of material properties, and losses in the buffer layer are discussed.

Excitation of electrostatic mode waves by hot electrons and ions in the plasmasphere

The analytical solutions of the general dispersion equation for ion acoustic and electron acoustic waves in two component cold plasma penetrated by a hot electron and a hot ion component represented by appropriate velocity distribution functions in the presence of collisions have been obtained. The expression for phase velocity, critical drift velocity for onset of instability and damping/growth rate have been derived. Using the various plasma and field parameters corresponding to L = 4 in the plasmasphere, we have computed various results using analytical expressions and we find that our results are in good agreement with the numerical solution of the general dispersion equation. Various features of electrostatic waves excited by streaming energetic electrons from equatorial region of the plasmasphere downwards, and by streaming ions from the polar ionosphere upwards into the plasmasphere has been studied. The diagnostic uses of these waves in the investigations of plasmaspheric region has been discussed.

Analysis of the Dielectric Constant of a Two-Temperature Plasma

The dielectric constant of a homogeneous, fully ionized two-temperature plasma is investigated to determine: (1) the conditions under which ion waves will be found; (2) accurate values for the phase velocity and damping rate of the ion waves; (3) the validity of approximate expressions for the ion wave phase velocity and damping; (4) the behavior of the dynamic shielding factor when ion waves are possible.

Negative ion effects on whistler mode propagation

Expressions for group and phase velocity are derived for left- and right-hand circularly polarized waves propagating along the static magnetic field of a magnetoplasma composed of electrons, positive ions, and negative ions. It is shown that a small percentage of negative ions can have pronounced effects on the propagation characteristics of the right-hand wave near the negative ion gyrofrequency. In particular, the wave becomes evanescent in a frequency band just above the gyrofrequency, exhibits a ‘nose effect’ below the gyrofrequency, and above the stop band the wave's group velocity is strongly influenced by the negative ions. Some experimental observations consistent with negative ion effects are discussed.