Showing papers on "Dispersion relation published in 1973"

Decay of MHD waves by phase mixing

Abstract: It is well known, [1–6], that the linearized equations of motion of ideal MHD possess a continuous spectrum which leads to damping of propagating waves through phase mixing. We show how this arises by examining the dispersion relation for plasmas with non-uniform profiles and comparing the results with those of a sharp boundary model. In this paper the special case of the non-uniform sheet-pinch is examined in order to present the mathematical details as clearly as possible. It is shown that as a result of the non-uniformity the frequency of the waves is a complex quantity having a real and imaginary part. The corresponding eigenfunctions and their mathematical pathology are discussed.

Lattice Dynamics of Gold

Abstract: The complete phonon dispersion relations for gold in the high-symmetry directions have been measured at room temperature by the coherent inelastic scattering of neutrons. It is found that the forces in gold are not homologous with the other noble metals, the frequencies of gold lying appreciably higher than those "scaled" from copper and silver. An analysis of the data in terms of different force-constant models reveals that a general tensor force is required for the first-neighbor interaction, whereas for neighbors beyond the first either general tensor or axially symmetric forces give an excellent fit to the data. The axially symmetric model alone does not adequately describe the data even when forces extending to ninth-nearest neighbors are included in the fit. In addition, simple screened-pseudopoential models were fit to the data and these results also indicate the need for the first-neighbor interaction to be general. Frequency distribution functions and related thermodynamic quantities were calculated from the various force-constant models. The Debye temperature ${\ensuremath{\Theta}}_{C}$ versus temperature curves obtained show an anomaly at low temperatures consistent with the ${\ensuremath{\Theta}}_{C}(T)$ obtained from specific-heat measurements. The relation between this anomaly and the character of the dispersion curves is given.

Spatial instability of a jet

Abstract: The instability of a circular cylindrical jet of liquid in air is studied on the assumption that the wavenumber k of the disturbance is complex while its frequency σ is real. This implies that the disturbance grows with distance along the jet, but that it does not grow with time. The occurence of such disturbances is called spatial instability, in contrast to the temporal instability studied by Rayleigh and others, in which k is real and σ is complex. It is found that there are infinitely many unstable modes for the axially symmetric case and also for each of the asymmetric cases. In the case of high velocity jets, one of these modes for the symmetric case corresponds to the mode Rayleigh found. However, it is not the most rapidly growing mode. Both analytical and numerical solutions of the dispersion equation are given for k as a function of σ and of the dimensionless jet velocity.

Decay of MHD Waves by Phase Mixing 2. The Theta_Pinch in Cylindrical Geometry

Abstract: The dispersion relation for stable waves in ideal MHD plasmas with diffuse profiles has been calculated. The non-uniformity in the density and magnetic field profiles gives rise to a very strong damping. Phase mixing is the mechanism responsible for the damping and can be seen to occur due to the presence of the continuous spectrum of the MHD equations of motion. The special case of the linear θ-pinch is examined in detail. In particular, damping coefficients for torsional Alfvén waves are computed which agree very well with a rather wide range of experimental findings.

Elastic waves in free anisotropic plates

Abstract: The mathematical formalism for obtaining dispersion relations for acoustic waves in plates of arbitrary anisotropy is outlined, and dispersion curves for propagation in a (001)‐cut cubic plate are presented. These results are compared to the uncoupled SV and P modes which, in turn, are related to the slowness curves for bulk waves. This approach provides an explanation for the behavior of the computed dispersion curves, and it also provides a means of approximating plate wave dispersion curves from the behavior of the slowness curves. The relationship of plate waves to surface waves is also explored for directions in which pseudosurface waves are known to propagate.

Spectra of Strong Langmuir Turbulence

Abstract: Strong Langmuir turbulence is described in terms of a random set of blobs of selftrapped plasma waves. The interaction of these blobs leads to the generation of power spectra $〈{|{E}_{k}|}^{2}〉\ensuremath{\propto}{k}^{\ensuremath{-}2}$ that agree with the results of one-dimensional computer simulation.

Acoustic surface waves in piezoelectric materials with periodic metal strips on the surface

Abstract: Dispersion relations for acoustic surface waves in piezoelectrics with periodic metal strips applied to the surface are developed. Particular emphasis is given to the modification in effective piezoelectric coupling due to strip geometry, and the details of the dispersion relations around the stop bands. The general results are applied to discuss devices such as multistrip directional couplers and strip-coupled amplifiers.

Phase matching in second-harmonic generation using artificial periodic structures

Abstract: The use of artificial periodic structures, consisting of spatial modulations of the linear and nonlinear susceptibilities of a nonlinear optical medium, to achieve phase matching in second-harmonic generation is analyzed. Dispersion relations and approximate formulas for the second-harmonic fields generated under various conditions are obtained and used to evaluate the experimental situation.

Field Observations of Edge Waves

Abstract: EDGE waves are solutions for wave motion trapped against a shoaling beach. Their amplitude varies sinusoidally along the shore and diminishes rapidly seawards from the shoreline. Eckart1 has shown that there is a family of edge wave solutions of the shallow water equations. Each solution is characterized by a mode number which gives the number of zero crossings in the seaward decay of amplitude; each mode has a dispersion relation linking the longshore wavenumber with the edge wave frequency.

Dispersion and Cyclotron Damping of Pure Ion Bernstein Waves

Abstract: Ion Bernstein waves are excited by a long wire in the center of a potassium $Q$-machine plasma column. Because the wave vector is very nearly perpendicular to the magnetic field, we observe pure ion Bernstein waves, in contrast to neutralized ion Bernstein waves which were previously observed. The frequency bands for the cyclotron damping are so narrow that we can distinguish the isotopes of potassium.

Theory of metal‐insulator‐metal tunneling for a simple two‐band model

Abstract: The one‐band effective mass approximation for the dispersion relationship in the barrier of a metal‐insulator‐metal (MIM) tunnel junction fails if the barrier height is a considerable portion of the band gap of the insulator. This paper reports on results when the more realistic dispersion relation of Franz and Kane is used and displays the sensitivity of the tunnel effect to the band model. We illustrate some examples of the energy distribution of the current. Characteristic features of tunneling through trapezoidal‐like barriers will be discussed, including maxima in the logarithmic derivative curves, in the thermal I‐V characteristics, and the zero‐bias offset of the conductance minimum. Calculations are performed by integrating the tunnel equation numerically, and analytical approximations for this equation are also presented from which the effect of various parameters on the tunneling characteristic can be estimated.

General remarks concerning theories dealing with scattering and diffraction in random media

Abstract: The theories to be considered in this review deal mostly with the effect of random media on the propagation of electromagnetic or other waves traversing them. The purpose aimed at thus consists of finding connections between the statistical properties of these waves and the given statistical properties of the medium itself. The characteristics of this medium are to be fixed by one or more wave equations or by equivalent dispersion relations. The waves may be investigated as such, considering the wave function Ψ(r, t) of one or more associated scalar or vector components, or, more generally, the theory may refer to related quantities, e.g., the squared modulus |Ψ|2 which fixes an energy density or, as in quantum mechanics, a probability density.

Electric field and plasma density oscillations due to the high‐frequency Hall current two‐stream instability in the auroral E region

Abstract: Observations of oscillating electric fields and plasma density fluctuations have been made in the E region of the earth's auroral ionosphere during a major magnetospheric substorm. By means of double probe and barium cloud measurements of the ambient dc electric field and Langmuir probe measurements of the plasma density, the plasma was known to be unstable to the high-frequency Hall current two-stream instability. The altitude range, frequency dependence, polarization, and amplitude of the fluctuations were in excellent agreement with the theory for this instability. In addition, simultaneous measurements of the plasma density and electric field fluctuations allowed determination of the real part of the dispersion relation at low frequencies. The waves were found to propagate with no dispersion at a velocity of about 500 m/sec in a reference frame fixed to the earth, a result consistent with an acoustic wave explanation for the phenomenon and implying that nonlinear effects stabilize the wave at a speed less than the electron streaming velocity.

Excitation Spectrum of Magnetic Domain Walls

Abstract: The eigenvalue equation for the normal modes of excitation of a moving infinite planar magnetic domain wall in an infinite material having the most general second-rank tensor anisotropy is presented. The eigenvalues and eigenfunctions of both the spin waves in the presence of the moving wall and the excitation of the wall itself are given to first order in velocity. Cylindrical-domain resonance experiments are proposed to test for the existence of the excitation modes and their effect on domain propagation. The dispersion relation for the wall excitation modes is found to imply a new material requirement for high mobility.

New Method in the Theory of Surface Polaritons

Abstract: A new method for the treatment of surface polaritons is presented. It is based on a variant of the Ewald-Oseen extinction theorem, which holds for any point inside the medium. Surface polaritons correspond to those solutions of Maxwell equations which are subject to the Ewald-Oseen extinction theorem with no incident field. The method is quite general being applicable to any kind of medium, and it leads immediately to surface-polariton dispersion relations as well as to the structure of the fields associated with them. The power of the present method is demonstrated by applying it to (i) an isotropic nonmagnetic medium having different shapes; (ii) an anisotrpic nonmagnetic medium; (iii) and isotropic but magnetic medium; and (iv) an isotropic nonmagnetic, but spatially dispersive, medium. Surface-polariton response functions are also calculated using the extinction theorem in the above form. A specialized form of extinction theorem is used to discuss surface polaritons without retardation. Finally, a variation of the present approach is used to discuss Brewster modes.

Hydrodynamics of plasma in a strong high-frequency field

Abstract: The article reviews the theory of a number of phenomena occurring in a plasma situated in a strong high-frequency field. The analysis is based on a relatively simply hydrodynamic collisionless plasma model. Principal attention is paid to the stability of the plasma in the strong high-frequency field. A single dispersion equation is used to consider different modifications of parametric instabilities (decay instabilities, parametric resonance, transparent-plasma instability of the stimulated scattering type). Also discussed are a number of problems in the equilibrium and dynamics of a plasma in a strong high-frequency field, particularly nonlinear penetration and reflection of electromagnetic waves from a plasma, stationary nonlinear waves, and spreading and contraction of a plasma layer under the influence of the pressure of a high-frequency field.

Dispersion Curves for the Generalized Bernstein Modes

Abstract: Dispersion curves are plotted for the extraordinary branch of the electron- and ion-cyclotron harmonic waves propagating perpendicularly to the static magnetic field in a non-relativistic, hot Maxwellian plasma, without invoking the electrostatic approximation. It is found that, except in the vicinity of the cyclotron harmonics and the hybrid resonances, either the cold-plasma or the electrostatic approximation are accurate representations of the exact solution. The hybrid resonances of the cold-plasma model become monotonically shrinking regions of low group velocity as the temperature is increased, till all discernible evidence of these resonances disappears as the parameters corresponding to the thermonuclear plasmas are approached.

Lattice Dynamics of Calcite

Abstract: The phonon dispersion relations for the lowest-ten external branches, and for wave vectors along the optic axis, ahve been measured in calcite, CaC${\mathrm{O}}_{3}$, at room temperature, by means of inelastic-neutron-scattering techniques. The results are shown to be consistent with a simple shell model containing ten adjustable parameters.

Linear Contribution to Spatial Dispersion in the Spin-Wave Spectrum of Ferromagnets

Abstract: The implications of magnetic symmetry on the spin-wave spectrum of ferromagnets are examined. Contrary to the usual result, the spin-wave dispersion relation is found to contain a term linear in the wave vector for a definite set of magnetic symmetries. This linear dispersion is shown to be a consequence of antisymmetric exchange.

Line shapes for low frequency depolarized (VH) light scattering from fluids. Coupling of rotational and translational motion

Abstract: A generalized theory for depolarized (VH) light scattering from fluids of nonspherical rigid molecules at low frequencies is presented The dispersion equation for the part of the spectrum which can have fine structures (the autocorrelation function of αk x z) is reduced to a simple quadratic equation given the condition that the internal angular momentum of the molecules decays much faster than the anisotropic polarizability and the transverse linear momentum The solutions of the quadratic equation and the spectrum of the autocorrelation function of αk x z are similar to those of a linear damped harmonic oscillator Thus, the depolarized spectrum has different line shapes depending on the experimental situation (eg, temperature, wave vector, etc)

New dispersion relations and their application to partial-wave amplitudes

Abstract: After reviewing the commonly used dispersion relations, a systematic investigation of more generalized dispersion relations on parametrized curves in the Mandelstam plane fors-u crossing-symmetric amplitudes is made with the aim of obtaining dispersion relations which receive contributions from all three channels, however, in such a way that knowledge of the absorptive parts is only required in regions well inside the various Lehmann ellipses. In addition we require that the dispersion relations receive no contributions from kinematic singularities arising from the parametrization and that they allow partial-wave projections to be made in a relatively simple manner. It is found that dispersion relations on hyperbolic curves in the Mandelstam plane are a natural solution of the problem. The dispersion relations are written in a remarkably simple form similar to the usual fixed-t dispersion relation but with an additionalt-channel contribution. As an interesting application, we derive generalized partial-wave dispersion relations for elastic pion-nucleon scattering, where the left-hand cut contribution is explicitly given by convergent partial-wave series in the crossed channels.

Parametric excitation and suppression of drift waves by lower hybrid fields

Abstract: The effect of a large-amplitude, spatially uniform electric field at lower hybrid frequency, applied across the static magnetic field, on the dispersion relations for low-frequency drift waves ( omega << omega /sub ci/) and drift cyclotron waves ( omega approx equal omega /sub ci/) in a collisionless plasma is investigated. It is found that the lower hybrid field can parametrically excite or suppress the drift waves. Expressions are derived for the threshold electric field required for such effects. A particularly interesting result is the possibility of exciting some new macroscopic instabilities when wave-particle resonances are ignored. The results of this investigation are likely to be relevant to some of the recently proposed schemes using lower hybrid waves for heating plasma ions up to fusion temperatures. (auth)

Internal waves near the turning point

Abstract: A uniformly valid solution of internal wave equations is presented, using Langer's method, for an arbitrary density profile. The dispersion relation and mean square values of the velocity components are obtained. Particular emphasis is placed on the turning point problem and on estimating the errors involved in the approximations in a recent paper by Garrett and Munk (1972). Expressions are given for various spectral ratios of interest.

Ion streaming instabilities with application to collisionless shock wave structure

Abstract: The electromagnetic dispersion relation for two counterstreaming ion beams of arbitrary relative strength flowing parallel to a dc magnetic field is derived. The beams flow through a stationary electron background and the dispersion relation in the fluid approximation is unaffected by the electron thermal pressure. The dispersion relation is solved with a zero net current condition applied and the regions of instability in the k-U space (U is the relative velocity between the two ion beams) are presented. The parameters are then chosen to be applicable for parallel shocks. It was found that unstable waves with zero group velocity in the shock frame can exist near the leading edge of the shock for upstream Alfven Mach numbers greater than 5.5. It is suggested that this mechanism could generate sufficient turbulence within the shock layer to scatter the incoming ions and create the required dissipation for intermediate strength shocks.

Trapped ion instability in plasmas with magnetic shear

Abstract: The radial structure of the dissipative trapped ion mode is studied in order to determine the effect of shear and the possibility of finding ranges of tokamak operation which are stable to this mode even in the presence of a temperature gradient. It is found that owing to the presence of increasing numbers of resonant untrapped ions near the mode rational surfaces, where lq is an integer, the radial wavelengths are reduced and the modes are strongly Landau damped. These effects increase with increasing shear and with increasing values of l, the toroidal mode number. Numerical estimates of the growth rates and marginal stability conditions are obtained from eigenmode solutions of the radial differential equation and from algebraic dispersion relations which simulate the effects of the radial structure. It is concluded that if (r /q )d q / d r exceeds one, then the temperature threshold for instability is raised by approximately a factor of two from that of the no shear case. It is also proposed that even i...