Showing papers on "Dispersion relation published in 1991"

A powerful local shear instability in weakly magnetized disks. I - Linear analysis. II - Nonlinear evolution

Abstract: A broad class of astronomical accretion disks is presently shown to be dynamically unstable to axisymmetric disturbances in the presence of a weak magnetic field, an insight with consequently broad applicability to gaseous, differentially-rotating systems. In the first part of this work, a linear analysis is presented of the instability, which is local and extremely powerful; the maximum growth rate, which is of the order of the angular rotation velocity, is independent of the strength of the magnetic field. Fluid motions associated with the instability directly generate both poloidal and toroidal field components. In the second part of this investigation, the scaling relation between the instability's wavenumber and the Alfven velocity is demonstrated, and the independence of the maximum growth rate from magnetic field strength is confirmed.

Plasma ionization by helicon waves

Abstract: The dispersion relation for helicon waves in a uniform, bounded plasma is derived with both collisional and Landau damping. It is shown that the latter can explain the very high absorption efficiency of helicon waves in plasma sources and can lead to plasma generators with a controlled primary electron energy. The wave pattern and other features of helicon waves are pointed out.

Analysis of the numerical error caused by the stair-stepped approximation of a conducting boundary in FDTD simulations of electromagnetic phenomena

Abstract: A rigorous analysis of the numerical error associated with the use of stair-stepped (saw-toothed) approximation of a conducting boundary for finite-difference time-domain (FDTD) simulations is presented. First, a dispersion analysis in two dimensions is performed to obtain the numerical reflection coefficient for a plane wave scattered by a perfectly conducting wall, tilted with respect to the axes of the finite-difference grid, under both transverse electric and transverse magnetic polarizations. The characteristic equation for surface waves that can be supported by such saw-tooth conducting surfaces is derived. This equation leads to expressions that show the dependence of the propagation constant along the boundary and the attenuation constant perpendicular to it on cell size and wavelength. Numerical simulations that demonstrate the effects predicted by the dispersion analysis are presented. >

Electromagnetic Processes in Dispersive Media

Abstract: Part I. Electromagnetic Fields in Vacuo: 1. Electromagnetic fields 2. Cartesian tensors 3. The stress tensor and multipole moments 4. Fourier transforms 5. Greens functions Part II. The Electromagnetic Responses of Media: 6. The response of a medium 7. General properties of response tensors 8. Analytic properties of response functions 9. Response tensors for some idealized media 10. Response tensors for plasmas Part III. Wave Properties: 11. Dispersion relations and polarization vectors 12. Waves in dielectrics and anisotropic crystals 13. Waves in plasmas 14. The polarization of transverse waves 15. Energetics and damping of waves Part IV. Theory of Emission Processes: 16. The emission formula 17. Emission by multipoles 18. The Larmor formula 19. Alternative treatment of emission processes Part V. Specific Emission Processes: 20. Cerenkov emission 21. Bremsstrahlung 22. Formal theory of gyromagnetic emission 23. Gyroemission by mildly relativistic electrons 24. Synchrotron emission 25. Scattering of waves by particles 26. Nonlinear emission processes Bibliographic notes Glossary of symbols Index.

Diskoseismology: Probing Accretion Disks. I. Trapped Adiabatic Oscillations

Abstract: The normal modes of acoustic oscillations within thin accretion disks which are terminated by an innermost stable orbit around a slowly rotating black hole or weakly magnetized compact neutron star are analyzed. The dominant relativistic effects which allow modes to be trapped within the inner region of the disk are approximated via a modified Newtonian potential. A general formalism is developed for investigating the adiabatic oscillations of arbitrary unperturbed disk models. The generic behavior is explored by way of an expansion of the Lagrangian displacement about the plane of symmetry and by assuming separable solutions with the same radial wavelength for the horizontal and vertical perturbations. The lowest eigenfrequencies and eigenfunctions of a particular set of radial and quadrupole modes which have minimum motion normal for the plane are obtained. These modes correspond to the standard dispersion relation of disk theory.

Landau damping in space plasmas

Abstract: The Landau damping of electrostatic Langmuir waves and ion-acoustic waves in a hot, isotropic, nonmagnetized, generalized Lorentzian plasma is analyzed using the modified plasma dispersion function. Numerical solutions for the real and imaginary parts of the wave frequency omega sub 0 - (i)(gamma) have been obtained as a function of the normalized wave number (k)(lambda sub D), where lambda sub D is the electron Debye length. For both particle distributions the electrostatic modes are found to be strongly damped at short wavelengths. At long wavelengths, this damping becomes less severe, but the attenuation of Langmuir waves is much stronger for a generalized Lorentzian plasma than for a Maxwellian plasma. It is concluded that Landau damping of ion-acoustic waves is only slightly affected by the presence of a high energy tail, but is strongly dependent on the ion temperature.

Bloch Wave Analysis of Dispersion and Pulse Propagation in Pure Distributed Feedback Structures

Abstract: The excitation and behaviour of monochromatic and pulsed optical Bloch waves in pure distributed feed-back structures are discussed and analysed. The Bloch wave approach, based on a detailed knowledge of the natural optical modes of the periodic structure, is complementary to the more commonly used coupled-wave approach. The inter-relationship between dispersion, field micro-structure and group velocity is discussed, and the effects of group-velocity and higher-order dispersion on pulse propagation treated. Questions about the usefulness of DFB structures for dispersion correction and soliton formation are addressed.

Temperature dependence of refractive index of SiO2 glass

Abstract: The refractive index of silica glass has been measured in the UV and visible regions from −165.4 to 83.3°C. The wavelength dispersion at each temperature was fitted by the three-term Sellmeier equation. Analysis of the Sellmeier equation shows that the temperature dependence of the refractive index originates from the temperature shift of the observed shift of the exciton peak at around 10.4 eV.

Hydrodynamic surface modes on concentrated polymer solutions and gels

Abstract: We present a coupled two‐fluid model for hydrodynamic surface modes on concentrated polymer solutions and polymer gels. This model is used to investigate such surface modes in the limit of strong coupling. The surface mode dispersion relation ω(k) and the structure factor S(k,ω) of thermally induced surface fluctuations are derived in this limit for materials with a general constitutive relation η(ω). We argue for the existence of several types of surface modes, including Rayleigh elastic waves, capillary waves, and overdamped modes resulting from viscous dissipation in the solvent as well as from polymer diffusion in the case of polymer solutions. The properties of surface modes are discussed separately for the cases of polymer gels and concentrated polymer solutions. Detailed predictions for the form of the normal modes ω(k), and the surface mode structure factor S(k,ω) are given and their dependence on relevant material properties are discussed in each case. Possible experimental scenarios are anticipa...

Theory of electrostatic fluid modes in a cold spheroidal non-neutral plasma.

Abstract: The normal modes of a magnetized spheroidally shaped pure ion plasma have recently been measured. Here the theory of these modes is presented. Although one might expect that a numerical solution is required (because the plasma dielectric is anisotropic and the plasma is inhomogeneous), the problem is actually separable in an unusual coordinate system. The result is a simple electrostatic fluid dispersion relation for modes in a cloud of any spheroidal shape.

Kramers-Kronig relations in optical data inversion.

Abstract: Some remarks are made on the use of Kramers-Kronig relations in optical data inversion. It is shown that symmetry relations imposed on the optical constant should be taken into account when modeling the tails of the absorption and extinction curves.

Origin of vector parasites in numerical Maxwell solutions

Abstract: Dispersion relations are derived for conventional finite-element (FE) and finite-difference (FD) approximations for four versions of the Maxwell equations in the plane: the double-curl equation; the vector Helmholtz equation; the penalty equation; and the primitive, coupled Maxwell curl equations. Comparison with their analytic counterparts reveals the presence and origin of vector parasites. For the double-curl and penalty methods, the dispersion relations are double-valued, admitting an extra, spurious dispersion surface of real-valued wavenumbers. As a result, low wavenumbers support well-resolved and poorly resolved vector parasites. The Helmholtz schemes have monotonic, single-valued dispersion relations for divergence-free physical modes. Specification of divergence-free boundary conditions is sufficient to guarantee the absence of parasites. The primitive schemes have single-valued but folded (nonmonotonic) dispersion relations, supporting poorly resolved vector parasites at low wavenumbers. Use of a staggered finite-difference grid eliminates these parasites and results in a dispersion relation identical to that for the Helmholtz scheme. In cases where vector parasites arise, the same essential weakness in the discretized form of either the first or cross-derivative is responsible. >

Parametric instabilities of circularly polarized large-amplitude dispersive Alfvén waves: excitation of obliquely-propagating daughter and side-band waves

Abstract: We investigate the parametric instabilities of a large-amplitude circularly polarized dispersive parallel-propagating Alfven wave. Our treatment is more general than that of previous derivations based on the two-fluid equations in that we allow for propagation of the unstable daughter and side-band waves at arbitrary angles to the background (DC) magnetic field. We present the characteristics of the decay and modulational instabilities as functions of propagation angle. We find, in addition to the well-known decay and modulational instabilities, that at oblique and perpendicular propagation there is another parametric instability, namely the filamentation instability, which is characterized by a broad band-width in wavenumber and which satisfies the condition Re (ω) ≪ γ. A second parametric process at oblique and perpendicular angles of propagation, which has not been reported before is also investigated, namely the parametric magneto-acoustic instability. This instability is distinct from the filamentation instability in that it is characterized by density perturbations with large real frequencies that satisfy the condition Re (ω) ≫ γ. Unlike the filamentation instability, the magneto-acoustic instability extends over a broad angular range, but has a very narrow band-width in wavenumber. We report the dispersive characteristics of the filamentation and magneto-acoustic instabilities as functions of plasma β dispersion η and pump amplitude η for arbitrary propagation angles.

Resonant acceleration of alpha particles by ion cyclotron waves in the solar wind

Abstract: Preferential acceleration of alpha particles interacting with left-hand polarized ion cyclotron waves is studied. It is shown that a small positive drift velocity between alpha particles and protons can lead to alpha particle velocities well in excess of the proton bulk velocity. During the acceleration process, which is assumed to take place at heliocentric distances less than 0.3 AU, the alpha particle drift velocity should exceed the proton bulk velocity, and then the gap which exists around the alpha particle gyrofrequency should disappear. It is also shown that for proton thermal anisotropies of the order of those observed in fast solar wind, the waves either grow or are not damped excessively, so that the waves can exist and might thus lead to the observed differential speeds. However, the way in which the alpha particles exceed the proton velocity remains unexplained.

Accuracy and Resolution of Surface Wave Inversion

Abstract: Shear wave velocity profiles of soils and pavements may be evaluated nondestructively using surface wave tests such as the Spectral Analysis of Surface Waves (SASW) method. In these tests, dispersion data are measured in situ and inverted using least squares techniques to obtain the shear wave velocity profile. This paper examines the influence of the number of dispersion points, the maximum wavelength, and the distribution of dispersion data with wavelength on the accuracy and resolution of the shear wave velocity profile. The results indicated that the best overall accuracy and resolution is obtained when the dispersion data is evenly distributed between the minimum and maximum wavelengths and the maximum wavelength is one to two times the maximum desired depth of the shear wave velocity profile. The number of points did not appear to significantly influence the inverted profile as long as the number of points remains greater than the number of layers in the inversion profile.

Dispersion relations and sum rules in nonlinear optics.

Abstract: We prove that dispersion relations similar to the Kramers-Kronig equations of linear optics can be obtained for the nonlinear-response function to all orders in the electric field. When energy dissipation is involved the dispersion relations obtained here concern the case in which nonlinearity on a probe beam is produced by external radiaton beams of any given frequency. Using the superconvergence theorem, we find a set of nonlinear sum rules. Some of them imply that the already known sum rules of linear optics---in particular, the Thomas-Reiche-Kuhn and the Alatarelli-Dexter-Nussenzweig-Smith sum rule---are true to all orders because all the nonlinear contributions vanish. Others do not have a linear counterpart and are specific to nonlinear optics. Implications of these results and possibilities of anomalous emission effects are dicussed.

Cloud formation by combined instabilities in galactic gas layers : evidence for a Q threshold in the fragmentation of shearing wavelets

Abstract: The growth of shearing wavelets in thick galactic gas disks is studied, including the magnetic Rayleigh-Taylor instability perpendicular to the plane, various degrees of thermal instability, and the gravitational instability. Growth rates are calculated numerically for a wide range of parameter values, giving an effective dispersion relation and mass distribution function, and an approximate dispersion relation is derived analytically for the epoch of peak growth. An extensive coverage of parameter space illustrates the relative insensitivity of the gaseous shear instability to the axisymmetric stability parameter Q. The fragmentation of shearing wavelets by self-gravitational collapse parallel to the wave crest is also considered.

Helicon waves and efficient plasma production

Abstract: Helicon waves generated by radio‐frequency (rf) waves are experimentally demonstrated to have the characteristics of Landau damping, as predicted theoretically, and fully ionized plasmas are realized by this efficient coupling of rf powers to plasmas. Excited waves are identified as a helicon wave by measuring wavelengths in the plasma along the magnetic field and comparing with the dispersion relation. Good agreement is found between experimental and theoretical results.

Analysis of reflectance data using the Kramers-Kronig Relations

Abstract: Kramers-Kronig Relations (KKR) are a well-known tool to interpret reflectance spectra by reconstructing the reflected wave's phase from its modulus with the help of a dispersion relation. However, a unscrupulous application is only possible in the case of semiinfinite media at perpendicular incidence.

Control of the radial electric field and of turbulent fluctuations in a tandem mirror plasma

Abstract: The characteristics of low frequency waves in the presence of × rotation of a tandem mirror plasma are investigated using the Fraunhofer diffraction method. The observed dispersion relations are in good agreement with those of drift waves including a Doppler shift due to the × rotation velocity. The effect of the radial electricfield on the drift waves is studied quantitatively by applying a bias voltage to the end plates of the tandem mirror. The fluctuation level is observed to depend on the radial electric field Er. The fluctuation has a maximum value when Er 0 and decreases with increasing Er, regardless of its sign. The radial confinement time estimated from the particle balance equation decreases as the fluctuation level increases. The dependence of the fluctuation level agrees with that evaluated from the quasi-linear theory of drift wave turbulence.

Dispersion relation for bianisotropic materials and its symmetry properties

Abstract: The dispersion relation for an arbitrary general bianisotropic medium is derived in Cartesian coordinates, in a form well suited to imposing the boundary conditions when dealing with layered media with planar and parallel interfaces. Special cases of practical interest are also considered. Eleven fundamental coefficient families are identified by considering in detail all the symmetries present in the dispersion relation. An ad hoc expression of the determinant of the sum of two 3*3 matrices permits the use of a simple procedure to obtain the coefficients of the dispersion equation. The discussed symmetry properties have general validity, and this technique to evaluate the coefficients may be useful in other fields of application where dispersion relations are of importance. >

傾斜機能材料平板におけるラム波伝播とその衝撃応答 : 第一報,解析手法

Abstract: The numerical methods which have been proposed by the present authors for Lamb waves in an anisotropic laminated plate and its transient responses are expanded for the wave propagation analysis of functionally gradient material (FGM) plates. The material properties of the plate change gradually in the thickness direction, and are anisotropic in the plane of the plate. The plate is divided into N plate elements. In the element, we assume that the material properties change linearly in the thickness direction, and that the displacement field is expressed by second-order polynomials. The principle of virtual work is used to develop approximate dynamic equilibrium equations. The dispersion relation and the mode shape of the Lamb waves are obtained by using the free boundary conditions. The energy velocities of the Lamb waves are formulated with the aid of the Rayleigh quotient. The method of Fourier transforms in conjunction with the modal analysis is used to determine the response of displacements. The formulation of the theory is described in this paper.

Surface polaritons in cylindrical optical fibers

Abstract: The properties of the polariton modes associated with a resonance in the dielectric function of an optical fiber are derived. Particular attention is given to the surface polaritons, whose energy densities and power flows are concentrated close to the interface of the fiber core and its cladding. The results are illustrated with a range of dispersion curves for the two most important cases, when the fiber dielectric function has the reststrahl and plasma forms. General expressions are given and illustrated graphically for the power flow and the power density. It is shown numerically that the energy velocity, or group velocity, can have opposite sign to the phase velocity of the surface polariton.

Parametric instabilities of circularly polarized large-amplitude dispersive Alfvén waves: excitation of parallel-propagating electromagnetic daughter waves

Abstract: We investigate the parametric decay and modulational instabilities of a large-amplitude circularly polarized dispersive Alfven wave. Our treatment is more general than that of previous derivations based on the two-fluid equations in that we allow for propagation of the unstable daughter waves at arbitrary angles to the background magnetic field, although our main concern in this paper is the exploration of new aspects of propagation parallel to the DC magnetic field. In addition to the well-known coupling of pump waves to electrostatic daughter waves, we find a new parametric channel where the pump wave couples directly to electromagnetic daughter waves. Excitation of the electromagnetic instability occurs only for modulation (k/k0 ≤ 1) and not for decay (k/k0 0) and right-hand (K < 0) polarization. For large k/k0, the electromagnetic channel dominates, while at lower values the electrostatic channel has a larger growth rate for modest values of dispersion, pump-wave amplitude and plasma β. Unlike the electrostatic modulational instability, the growth rate of the electromagnetic instability increases monotonically with increasing pump wave amplitude. This analysis confirms that, for decay, the dominant process is coupling to electrostatic daughter waves, at least for parallel propagation. For modulation, the coupling to electromagnetic daughter waves usually dominates, suggesting that the parametric modulational instability is really an electromagnetic phenomenon.

TE waves at an interface between linear gyromagnetic and nonlinear dielectric media

Abstract: The dispersion relations for low-frequency TE waves guided by the interface between semi-infinite non-linear and linear frequency-dependent gyromagnetic media have been investigated theoretically and numerically. It is shown that TE waves can propagate, with or without linear limits. The power dependence of the non-reciprocal behaviour of these waves is quite dramatic and, in principle, can lead to some interesting experimental applications. The variation of the wave index with the power flow, and the consequent interface nonlinearity, are found to be very frequency dependent.

Nonlinear wave scattering and electron beam relaxation

Abstract: The role played by nonlinear scattering during the relaxation of a warm electron beam is investigated through a numerical code based on kinetic equations The code encompasses the quasi-linear wave-electron interaction and wave-wave scattering off ion clouds Ions with velocities 2 nu sub i (nu sub i being the ion thermal velocity) are found to be the most efficient for scattering the Langmuir waves off their polarization clouds The transfer rate of the spectrum out of resonance with the beam is larger by a factor 3 compared to usual estimates The changes produced in the dispersion relation by the presence of the beam electrons dramatically alter the characteristics of the secondary spectrum In a late phase the classic condensate K of about 0 is depleted, with the formation of a new condensate in resonance with the flat-topped beam distribution, which follows from the fact that the mere presence of the beam electrons creates a minimum in the frequency-wave-number relation For strong and slow beams, the predictions of the code are found to be in excellent agreement with the results of the particle simulation if a dispersion relation that includes the beam is used

On nonlinear periodic drift waves

Abstract: Nonlinear periodic drift waves are investigated on the basis of a simple perturbation scheme for both the amplitude and inverse frequency. The coefficients for the generation of the forced harmonics are derived, a nonlinear dispersion relation is suggested and a criterion for the onset of the modulational instability is obtained. The results are compared with those obtained with the help of a standard KBM treatment. Moreover, cnoqidal drift waves are suggested and compared to an experimental observation.