General Dispersion Equation Research Articles

Magneto-thermo-elastic waves in an infinite perfectly conducting elastic solid with energy dissipation

The generalized thermo-elasticity theory, i.e., Green and Naghdi (G-N) III theory, with energy dissipation (TEWED) is employed in the study of time-harmonic plane wave propagation in an unbounded, perfectly electrically conducting elastic medium subject to primary uniform magnetic field. A more general dispersion equation with complex coefficients is obtained for coupled magneto-thermo-elastic wave solved in complex domain by using the Leguerre’s method. It reveals that the coupled magneto-thermoelastic wave corresponds to modified dilatational and thermal wave propagation with finite speeds modified by finite thermal wave speeds, thermo-elastic coupling, thermal diffusivity, and the external magnetic field. Numerical results for a copper-like material are presented.

Global estimates of maximal operators generated by dispersive equations

Let $Tf(x, t) = e^{itφ(D)f}$ be the solution of a general dispersive equation with phase function φ and initial data $f$ in a Sobolev space. When the phase φ has a suitable growth condition and the initial data f has an angular regularity, we prove global and local L^p estimates for maximal operators generated by T. Here we do not assume the radial symmetry for the initial data. These results reveal some sufficient conditions on initial data for the boundedness of maximal operators in contrast to the negative results of [28]. We also prove a weighted L^2 maximal estimate, which is an extension of [19] to nonradial initial data.

An Updated Review of General Dispersion Relation for Conditionally and Unconditionally Stable FDTD Algorithms

We present a general numerical dispersion equation which is applicable to all known conditionally and unconditionally stable finite-difference time-domain (FDTD) algorithms on staggered rectangular grid, including Yee's FDTD, wavelet-based FDTD, extended curl FDTD, alternating direction implicit (ADI)-FDTD, Crank-Nicolson (CN)-FDTD, Crank-Nicolson split-step (CNSS)-FDTD, and their modifications of higher order spatial stencils. The real part of the complex eigenvalue of the total amplification matrix defines and distinguishes the dispersion relation for each individual scheme. Easy-to-check conditions are provided, under which the numerical dispersion of a particular time-domain scheme is governed by the proposed dispersion equation. These conditions are on the amplification matrix eigenvalues. The proposed dispersion equation includes each considered dispersion relation as a special case, and presents itself a general governing equation to estimate 3-D numerical dispersion of the aforementioned schemes in the frame of plane waves.

Correct initial boundary value problems for dispersive equations

This paper deals with correctness of initial boundary value problems for general dispersive equations of finite odd orders. For the Kawahara and KdV equations we prove existence, uniqueness and stability of strong global solutions in a bounded domain for different signs of a coefficient of the highest derivative as well as their asymptotics when the coefficient of the higher-order derivative in the Kawahara equation approaches zero.

Dispersion of multi-stream instability in quantum magnetized hot plasmas

Using the quantum hydrodynamic (QHD) equations with magnetic field on the Wigner–Maxwell system, the general dielectric tensor and dispersion equation for quantum plasmas were derived. Dispersion relations of one-, two-stream and beam-plasma instabilities in uniform quantum magnetized plasmas are investigated through the new dielectric tensor. The magnetic field which is parallel to the fluid velocity does not work on stream instabilities. The quantum and thermal effects have remarkable impact on two-stream instability. The critical wave number for beam-plasma instability with quantum effects correction is given too.

Investigation of Dispersion Characteristics in Coaxial Disk-Loaded Slow-Wave Structures With Both Symmetric and Asymmetric Modes

Coaxial disk-loaded waveguide has been widely used in magnetically insulated transmission line oscillators. Although the TM <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">00</sub> mode is its main mode, the asymmetric mode might be excited by a slight asymmetry of the structure, asymmetric emission of the beam, or both. Therefore, research on the asymmetric mode in coaxial disk-loaded waveguide is necessary. The general dispersion equation of the coaxial disk-loaded slow-wave structures is obtained using the field match theory for the first time. Numerical calculation is carried out based on the dispersion function. The theoretical calculated results agreed well with the simulated results of the MAFIA code and experiment.

Dispersion relation for surface-localized magnetic polaritons and magnetostatic waves in antiferromagnetic superlattice

In addition to the general dispersion equation for surface-localized magnetic polaritons and magnetostatic waves, which propagate in the system antiferromagnetic superlattice–antiferromagnet, we derive a simplified result for the long-wavelength limit λ ⪢ L ( L is the period of superlattice) when the superlattice is found to behave like an anisotropic bulk medium (effective-medium approach). The dispersion curves and frequency region of the existence of the surface magnetic polaritons and magnetostatic waves are presented numerically for several values of the external magnetic field and for different antiferromagnetic materials.

Wave properties of computational TLM models of metamaterials with negative refractive index

AbstractWe derive and compare the wave properties of transmission‐line‐based negative refractive index metamaterials and their transmission line matrix (TLM) models. The TLM networks are computational replica of reactively loaded periodic structures that support backward waves and constitute artificial media with negative refractive index. After a brief review of the TLM algorithms for three types of nodes, namely the 2D Shunt Node, the 3D Expanded (or Distributed) Node, and the 3D Symmetrical Condensed Node, we formulate and solve the general dispersion equations in 2D and 3D metamaterials based on these nodes. We compare the results with the dispersion properties of both idealized homogeneous metamaterials and of L‐C loaded periodic transmission line structures that have the same cell size as their computational TLM counterparts. Copyright © 2006 John Wiley &amp; Sons, Ltd.

An investigation of periodic waveguides with axial and azimuthal corrugations

The existing studies about the dispersion characteristics of the sinusoidal rippled wall cylindrical waveguides mostly center on one-dimensionally (1-D) corrugated waveguides. Sometimes two-dimensional (2-D) corrugated waveguides are needed to provide better selective and resonance properties not achievable with 1-D periodic structures. The authors of this paper improve the method of calculating the waveguide axially periodic by expanding the slow-wave structure's boundary function in Fourier series, so that it can be adopted in periodic waveguides with axial and azimuthal corrugations. In addition, the general dispersion equations for the investigation of 2-D periodic waveguide are presented in this paper. The advantage of the method lies in its universality, providing solution to the dispersion characteristics for most of the periodic waveguides with axial and azimuthal corrugations. By the method suggested here, the paper analyzes the dispersion characteristics and coupling impedance of a new rib-loaded disk-loaded waveguide and other periodic waveguides.

Thermoelastic wave propagation in a rotating elastic mediumwithout energy dissipation

A study is made of the propagation of time‐harmonic plane thermoelastic waves of assigned frequency in an infinite rotating medium using Green‐Naghdi model (1993) of linear thermoelasticity without energy dissipation. A more general dispersion equation is derived to examine the effect of rotation on the phase velocity of the modified coupled thermal dilatational shear waves. It is observed that in thermoelasticity theory of type II (Green‐Naghdi model), the modified coupled dilatational thermal waves propagate unattenuated in contrast to the classical thermoelasticity theory, where the thermoelastic waves undergo attenuation (Parkus, Chadwick, and Sneddon). The solutions of the more general dispersion equation are obtained for small thermoelastic coupling by perturbation technique. Cases of high and low frequencies are also analyzed. The rotation of the medium affects both quasielastic dilatational and shear wave speeds to the first order in ω for low frequency, while the quasithermal wave speed is affected by rotation up to the second power in ω. However, for large frequency, rotation influences both the quasidilatational and shear wave speeds to first order in ω and the quasithermal wave speed to the second order in 1/ω.

Magneto‐viscoelastic plane waves in rotating media in thegeneralized thermoelasticity II

A study is made of the propagation of time‐harmonic magneto‐thermoviscoelastic plane waves in a homogeneous electrically conducting viscoelastic medium of Kelvin‐Voigt type permeated by a primary uniform external magnetic field when the entire medium rotates with a uniform angular velocity. The generalized thermoelasticity theory of type II (Green and Naghdi model) is used to study the propagation of waves. A more general dispersion equation for coupled waves is derived to ascertain the effects of rotation, finite thermal wave speed of GN theory, viscoelastic parameters and the external magnetic field on the phase velocity, the attenuation coefficient, and the specific energy loss of the waves. Limiting cases for low and high frequencies are also studied. In absence of rotation, external magnetic field, and viscoelasticity, the general dispersion equation reduces to the dispersion equation for coupled thermal dilatational waves in generalized thermoelasticity II (GN model), not considered before. It reveals that the coupled thermal dilatational waves in generalized thermoelasticity II are unattenuated and nondispersive in contrast to the thermoelastic waves in classical coupled thermoelasticity (Chadwick (1960)) which suffer both attenuation and dispersion.

Low-frequency instability of circularly polarized microwave-pulsed-field-produced plasmas

The plasma produced by a high-power pulsed circularly polarized microwave field which is much weaker than atomic fields is investigated in the nonrelativistic limit. It is shown that the resulting electron distribution function is nonequilibrium and anisotropic. Furthermore, the general dispersion equation is derived and analyzed to study low-frequency ion oscillations. It will be shown that there is an instability in the aforementioned system only when the ion density exceeds a critical value.

On the new modes of planetary-scale electromagnetic waves in the ionosphere

Abstract. Using an analogy method the frequencies of new modes of the electromagnetic planetary-scale waves (with a wavelength of 103 km or more), having a weather forming nature, are found at different ionospheric altitudes. This method gives the possibility to determine spectra of ionospheric electromagnetic perturbations directly from the dynamic equations without solving the general dispersion equation. It is shown that the permanently acting factor-latitude variation of the geomagnetic field generates fast and slow weakly damping planetary electromagnetic waves in both the E- and F-layers of the ionosphere. The waves propagate eastward and westward along the parallels. The fast waves have phase velocities (1–5)km s–1 and frequencies (10–1–10–4), and the slow waves propagate with velocities of the local winds with frequencies (10–4–10–6)s–1 and are generated in the E-region of the ionosphere. Fast waves having phase velocities (10-1500)km s–1 and frequencies (1–10–3)s–1 are generated in the F-region of the ionosphere. The waves generate the geomagnetic pulsations of the order of one hundred nanoTesla by magnitude. The properties and parameters of the theoretically studied electromagnetic waves agree with those of large-scale ultra-low frequency perturbations observed experimentally in the ionosphere. Key words. Ionosphere (ionospheric disturbances; waves propagation; ionosphere atmosphere interactions)

Magnetoelastic plane waves in rotating media in thermoelasticity of type II (G‐N model)

A study is made of the propagation of time‐harmonic plane waves in an infinite, conducting, thermoelastic solid permeated by a uniform primary external magnetic field when the entire medium is rotating with a uniform angular velocity. The thermoelasticity theory of type II (G‐N model) (1993) is used to study the propagation of waves. A more general dispersion equation is derived to determine the effects of rotation, thermal parameters, characteristic of the medium, and the external magnetic field. If the primary magnetic field has a transverse component, it is observed that the longitudinal and transverse motions are linked together. For low frequency (χ ≪ 1, χ being the ratio of the wave frequency to some standard frequency ω∗), the rotation and the thermal field have no effect on the phase velocity to the first order of χ and then this corresponds to only one slow wave influenced by the electromagnetic field only. But to the second order of χ, the phase velocity, attenuation coefficient, and the specific energy loss are affected by rotation and depend on the thermal parameters cT, cT being the nondimensional thermal wave speed of G‐N theory, and the thermoelastic coupling εT, the electromagnetic parameters εH, and the transverse magnetic field RH. Also for large frequency, rotation and thermal field have no effect on the phase velocity, which is independent of primary magnetic field to the first order of (1/χ) (χ ≫ 1), and the specific energy loss is a constant, independent of any field parameter. However, to the second order of (1/χ), rotation does exert influence on both the phase velocity and the attenuation factor, and the specific energy loss is affected by rotation and depends on the thermal parameters cT and εT, electromagnetic parameter εH, and the transverse magnetic field RH, whereas the specific energy loss is independent of any field parameters to the first order of (1/χ).

Instability, Dynamics, and Morphology of Thin Slipping Films

Based on the linear stability and nonlinear simulations, we show that the surface instability, dynamics, and morphology of supported thin liquid films are profoundly altered by the presence of slippage on the substrate. A general dispersion equation for flow in slipping thin films is derived and simplified to identify three different regimes of slippage (weak, moderate, and strong) and obtain the length and time scales of instability in them. For illustration, the ubiquitous van der Waals interactions have been employed. Different regimes of slip-flow can be predicted based on a nondimensional parameter, xi, which is a function of slip length, film thickness, intermolecular potential, and interfacial tension. Two distinct transitions from weak to moderate slip and from moderate to strong slip occur at xiT1 approximately 0.01 and xiT2 approximately 500, respectively. More specifically, a decrease in film thickness causes transitions from weak to moderate to strong slip regime. Even a weak slippage causes faster breakup of a thin film, whereas slippage beyond a transition value (slip length, bT1) increases the length scale of instability and reduces the number density of holes compared to the nonslipping case. Strong slippage produces holes faster, and the holes are fewer in number and have less developed rims. The exponents for the length scale (lambdam infinity h0n; h0 is film thickness) and time scale of instability (tr infinity h0m) change nonmonotonically with slippage (for nonretarded van der Waals instability, n E (1.25, 2), m E (3, 6)). Retardation in van der Waals potential increases the exponents (n E (1.5, 2.5), m E (5, 8)). The initial stage of evolution of a slipping film, simulated based on nonlinear equations, follows the length scale and time scale of instability, close to the prediction of linear analysis. It is hoped that the present analysis will help in better interpretation of thin film experiments, in estimation of slippage, and in the determination of intermolecular forces from the length and time scales of the instability.

Relativistic kinetic theory of electromagnetic waves in equilibrium magnetized plasma. General dispersion equations

The relativistic kinetic theory of parallel propagating electromagnetic waves in a magnetized equilibrium plasma is presented. On the basis of relativistic Vlasov–Maxwell equations, a general explicit dispersion relation is derived by a correct analytical continuation for all complex frequencies of electromagnetic waves.PACS Nos.: 52.25.Dg, 52.25.Xz, 52.27.Ep, 52.27.Ny, 52.35.Hr, 52.35.Mw, 52.35.Py

General dispersion equation of exchange spin waves in an antiferromagnetic superlattice

AbstractA general dispersion equation for exchange spin waves propagating in a general direction in an antiferromagnetic superlattice is derived by the recurrence relations technique. The elementary unit cell of the superlattice under consideration consists of N different antiferromagnetic layers. The results are illustrated numerically for a particular choice of parameters. (© 2003 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Effects of coupling on plasmon modes and drift-induced instabilities in a pair of cylindrical nanotubes

We develop a theory of collective plasma excitations in a pair of parallel nonoverlapping cylindrical nanotubes. A general dispersion equation is obtained as a function of the angular momentum and linear momentum transfer. We use our results to calculate the phase velocity of the plasmon excitations when current flows parallel to the axis of the nanotube in either the same or neighboring nanotube as the plasma excitations. We demonstrate that the less energetic undamped plasmon excitations may have a phase velocity which is smaller than the Fermi velocity. The plasmons with the largest frequencies have phase velocities which decrease with increasing wave vector. We discuss the instabilities which may arise when the drift velocity of the electrons lies within a range which is determined by the phase velocity for the plasmon modes of the pair of cylindrical nanotubes.

``Cold'' dispersion relation of corrugated waveguide filled with plasma immersed in a finite magnetic field

A general dispersion equation of a partially filled plasma corrugated waveguide immersed in a finite magnetic field is presented. When the guiding magnet B0→∞ or 0, this equation can be reduced to the results obtained in previous works.

Effects of Ion Temperature and Inertia on Kinetic Alfvén Waves**The project supported by National Natural Science Foundation of China (Grant Nos. 10075076 and 49990452), Ministry of Science and Technology of China (Grant No. G20000784093), and Special Fund for Authors of National Excellent Ph. D. Dissertations of the Chinese Academy of Sciences

Kinetic Alfvén wave (KAW) has been an interesting topic for discussion extensively in the fields of laboratory, space, and astrophysical plasmas. A general dispersion equation is derived from the exact two-fluid model in this paper. Based on this dispersion equation, dispersion of Alfvén waves near the Alfvén frequency are discussed for arbitrary values of wavelength, where υA is the Alfvén velocity, and is the wavenumber parallel to the ambient magnetic field. For the short wavelength cases of , where and are the ion inertial length and gyrofrequency, respectively, our dispersion relations are appropriate for discussing effects of the ion temperature and inertia on KAWs. The present results show that both the ion temperature and inertia can affect considerably the behaviors of KAWs in propagation, resonance, and polarization. In particular, our results may be a great help to understanding some salient features of the low-frequency (in comparison with the ion gyrofrequency ) electromagnetic fluctuations frequently observed by the FREJA and FAST satellites in the auroral zone of the Earth's ionosphere and magnetosphere.