Dispersion Relation Equation Research Articles

A novel error compensation method with the dispersion relation equation for the CIP method

We propose a novel compensation method based on a dispersion relation equation for the constrained interpolation profile (CIP) method. In our method, two additional operations are performed after the CIP calculation. One is to calculate the numerical attenuation constant $\widetilde\alpha$ and the numerical phase constant $\widetilde\beta$ from the dispersion relation equation using Newton-Raphson method. The other is product and sum operations to compensate for amplitude and phase from the calculated $\widetilde\alpha$ and the $\widetilde\beta$. As described in this paper, both the amplitude and the phase error have been compensated using our method.

Gyrotropic Effect on Surface Polariton Propagation

Surface polariton propagation between a gyrotropic medium with arbitrary magnetization direction and an isotropic medium is investigated. Assuming that both the permittivity and the permeability of the gyrotropic medium are anisotropic, the dispersion relation equation for the surface polaritons is derived, which is applied to specific configurations of the propagation direction of the surface polaritons, the magnetization and the interface. When the gyrotropic effect is sufficiently weak, which means that the permittivity and the permeability of the gyrotropic medium are nearly isotropic, the influence of the magnetization direction on the propagation characteristics of the surface polaritons, such as the emergence of nonreciprocity, is investigated in detail.

Nonresonant kinetic instabilities of a relativistic plasma in a uniform magnetic field: Longitudinal and transverse mode coupling effects

The stability properties of relativistic plasmas embedded in a uniform magnetic field are investigated for longitudinal and transverse modes and with coupling effects between these modes. The direction of wave propagation in the plasma is not necessarily either parallel or transverse to the ambient magnetic field. The basic dispersion relation equations are given for arbitrary propagation directions. Detailed examination is focused on perpendicular wave propagation in this paper. The concept of neutral points in wave number space, introduced by Harris [Phys. Rev. Lett. 2, 34 (1959)], is generalized to allow for the inclusion of ion effects and the effects of fluctuating magnetic fields. Starting from the relativistic conductivity tensor, an expansion procedure for low wave frequencies is used to determine the stability properties in the neighborhood of neutral points and in the frequency regime below the ion cyclotron frequency. The bulk plasma properties determine stability or instability but the mode structure is derivable only from a particle kinetic picture, as with Weibel [Phys. Rev. Lett. 2, 83 (1959)] instabilities, and not from a magnetohydrodynamic description. For monoenergetic plasma distribution functions of electrons and ions, as well as for electrons and positrons, numerical examples are given to illustrate the neutral points and the unstable wave number regimes.

Accurate hybrid mode analysis of hollow optical fibers

The dispersion relation equation of hybrid modes in hollow optical fiber is derived in the closed form by modeling the fiber in a three-layered structure and by rigorous hybrid mode analysis. The effective refractive indices of hybrid modes as a function of wavelength are accurately calculated. Modal characteristics and transverse field distributions of hollow optical fiber are precisely analyzed by use of the derived hybrid mode dispersion relation equation and the compact 2-D finite difference time domain method. In addition, the effects of the fiber dimension on propagating properties are presented. The number of hybrid modes that can exist in hollow optical fiber is calculated as a function of fiber dimension. We believe that this rigorous analysis is very accurate and provides precise computing methods to design and characterize hollow optical fiber based optical devices.

The Helium Valley: Comparison of Impulsive Solar Flare Ion Abundances and Gyroresonant Acceleration with Oblique Turbulence in a Hot Multi‐Ion Plasma

We present a detailed interpretation of the heavy ion abundance enhancements observed in impulsive —are energetic particles, in terms of the conditions for gyroresonant acceleration by moderately oblique waves in a hot solar coronal plasma. On the basis of a realistic coronal plasma containing its complete set of minor ions, we analyze —rst all parallel wave modes in terms of their dispersion relation, damping timescale, and condition for gyro- resonant acceleration of thermal ions, as a function of temperature. We identify the ii Helium Valley,ˇˇ the region in the frequency-wavenumber plane of strong wave damping by thermal 4He'2 ions, as crucial for explaining the observed abundances: any ions with charge-to-mass ratio in the neighborhood of 0.5 cannot be accelerated preferentially, relative to 4He'2. Then solving the dispersion relation equation for oblique waves in a hot e-p-He plasma, we discuss this general class of waves in terms of polarization and damping timescale. For waves propagating at moderate angles to the magnetic —eld our calculations indicate that the —rst harmonic n 1 ( hB / 90i), gyroresonance is dominant, and that the corresponding He valley narrows down for increasing angle h. Using this analysis, we calculate the limits of the He valley and investigate the preferential gyroresonant acceleration of heavy ions by moderately oblique waves in a solar coronal plasma. Only for ( hB / 90i) nearly perpendicular waves ( hB 90i), are higher order resonances important and regions of wave damping by interaction with thermal particles vanishingly narrow in frequency. We estimate the fraction of ions of each element outside the He valley as a function of temperature and compare the resulting enhanced abundances with the observed enhancements, for the case of a spec- trum of nonquasi-perpendicular waves, as produced by a cascading of the general turbulence. The results allow us to specify the range of possible temperatures for the source plasma of the accelerated particles to between D2.4 and D4.5 ) 106 K, i.e., comparable to active region (AR), but not to —aring gas, temperatures. This points to an acceleration of the ions taking place, either in the AR gas surround- ing the —are itself or within the —aring loop but before it became heated. Constraints are set on the typical time *t over which the ions are accelerated preferentially. We —nd times between D5 ) 10~4 and D3 ) 10~2 s (for our nominal plasma with density and —eld of

The Benjamin-Feir instability of a deep-water Stokes wavepacket in the presence of a non-uniform medium

The influence of a non-uniform medium on the Benjamin-Feir instability of weakly nonlinear deep-water waves has been investigated, and an approach via a suitable nonlinear Schrodinger equation was adopted. For the derivation of the relevant cubic Schrodinger equation. the approach of Yuen & Lake (1975) was followed and an applicable dispersion relation and energy equation was derived by the averaged Lagrangian technique. With the assumption that the lengthscale of current variation is much greater than the lengthscale of the wavepacket, a cubic Schrodinger equation with slowly varying coefficients is obtained. Three different examples of non-uniform media are treated: (i) waves on a current with variation along the stream; (ii) waves on a shear current; and (iii) long deep-water gravity waves interacting with shorter waves.

Electromagnetic-wave propagation in a bounded magnetoplasma

Dispersion diagrams (ωvs. k) are obtained by numerical solution of the dispersion relation and determinantal equation for electromagnetic waves propagation along the lines of force in a bounded, cold, uniform magnetoplasma for frequencies in the vicinity of the electron cyclotron frequency. The results for slow waves (ω/k<c) are compared with those given by the electrostatic theory. By using a combined numerical and analytical analysis the electrostatic theory is obtained from the electromagnetic one. A solution of the electromagnetic dispersion relation with azimuthal-mode indexm<0 is found to correspond to the right circularly polarized mode of the infinite magnetoplasma. It is shown that there is no counterpart to this mode in the azimuthally symmetricm=0 solution.

Coupled gravitational and electromagnetic waves in arbitrary space-times with magnetized plasmas and dust, I

We derive constraint-free, coupled wave equations for the propagation of coupled electromagnetic and gravitational waves traveling through a time-dependent inhomogeneous medium. The medium consists of an arbitrary gravitational field, dust, a cold two-fluid plasma, and an arbitrary magnetic field. In this first of two papers we apply a two-timing ansatz to the constraint-free system of wave equations. In the second paper, dispersion relation and transport equations are found by means of a WKB analysis.

Extensional vibrations of rectangular crystal plates

A system of one-dimensional equations governing the extensional (contour) modes of vibrations in rectangular crystal plates is derived from the Cauchy–Voigt two-dimensional equations of extensional motion of anisotropic, elastic plates. The system of equations is further divided into two groups of equations governing, separately, the symmetric (contour extensional) and antisymmetric (contour shear) modes of vibrations, by setting the elastic stiffness coefficients γ15 = γ35 = 0 in order to neglect the coupling effect between the two groups. Closed form solutions of contour extensional vibrations are obtained for rectangular crystal plates satisfying the traction-free conditions at four edges. Dispersion relation and frequency equation are obtained in explicit form for the first three contour extensional modes. Resonance frequencies, for various contour extensional modes, are computed as a function of the length-to-width ratio (a/c) of plates. Predicted results are compared with the detailed measurements for rectangular GT cuts with close agreement. Two-dimensional vibration patterns have also been calculated at resonances for different values of a/c ratio.

Variational method for electromagnetic waves in a magneto-plasma

A variational method is presented to derive the electromagnetic response for waves of a model plasma with arbitrary β in a magnetic field whose curvature simulated by a gravitational field. The variational method is particularly well suited for deriving the electromagnetic local dispersion relation and differential equations of long-wavelength modes. In particular, the result is applied to finding the critical β needed for stabilization of low-frequency drift waves with density and temperature gradients.

Electrostatic Oscillations and Feedback Control of a Lossfree Plasma

The general features of the electrostatic oscillations in a lossless plasma are discussed from the dispersion relation and energy equation. It is shown that the unstable electrostatic waves in lossless plasma have always zero energy density which is realized as a sum of a negative kinetic energy density and a positive electrostatic energy density. The feedback stabilization for reactive instabilities is reconsidered by a simple model, and a cold electron plasma--beam system is studied as a simple example. (auth)

Stability of a Relativistic Magneto-Gasdynamic Half-Jet

The stability of a relativistic half-jet of infinitely conducting, inviscid, compressible fluids with imbedded magnetic fields is considered. The general dispersion relation and eigenvalue equation are derived from the theory of special relativity. Numerical results show that the magnetic fields have a stabilizing effect and the effect is comparatively greater than that of an increase of the speed of sound. Moreover, disturbances in the vicinity of neutral oscillations correspond to supersonic, outgoing waves.