Showing papers on "Dispersion relation published in 1977"

Electromagnetic propagation in periodic stratified media. I. General theory

Abstract: The propagation of electromagnetic radiation in periodically stratified media is considered. Media of finite, semi-infinite, and infinite extent are treated. A diagonalization of the unit cell translation operator is used to obtain exact solutions for the Bloch waves, the dispersion relations, and the band structure of the medium. Some new phenomena with applications to integrated optics and laser technology are presented.

Effects of Finite Plasma Beta on the Lower-Hybrid-Drift Instability,

Abstract: The local dispersion relation for the lower-hybrid-drift (LHD) instability is derived and analyzed, taking into account the finite-beta effects associated with transverse electromagnetic perturbations as well as with resonant and nonresonant electron-orbit modifications due to magnetic-field gradients The influence of finite-beta effects on the LHD instability is calculated in a fully self-consistent manner for arbitrary values of electron-ion temperature ratio, local beta, cross-field ExB velocity/ion thermal speed ratio, and other plasma parameters Stability properties are investigated analytically for the case of cold electrons, and the local dispersion relation is solved numerically in the parameter regime of most interest for high-density plasma pinches The results show that for all parameter regimes studied, the net effect of finite plasma beta is to reduce the maximum growth rate of the LHD instability, although the details can vary, depending on the plasma parameters Except in the limit where the electron/ion temperature ratio tends to zero, it is found that there is a critical value of plasma beta above which the LHD instability is completely stabilized

Importance of physical dispersion in surface wave and free oscillation problems: Review

Abstract: Physical dispersion resulting from anelasticity is investigated from the point of view of linear viscoelastic models and causality relations. It is concluded that inasmuch as Q in the earth's mantle is nearly independent of frequency, at least in the seismic frequency band, a dispersion relation in the form of C(ω) = C(ω_r)[1 + (1/πQ_m) In (ω/ω_r)] must be used for correcting the effect of physical dispersion arising from anelasticity. (Here C(ω) is the phase velocity of either body waves, surface waves, or free oscillations, ω is the angular frequency, ωr is the reference angular frequency, and Q_m is the path average Q for body waves or Q of a surface wave or a mode of angular frequency ω; for surface waves and free oscillations, C(ω_r) should be understood as the phase velocity at ω computed by using the elastic moduli at ω = ω_r.) The values of Q outside the seismic frequency band affect mainly the absolute value of the phase velocity but do not affect significantly the relative dispersion within the seismic frequency band. Even if the microscopic mechanism of attenuation is nonlinear, this dispersion relation can be used if departure from elasticity is relatively small, so that the signal can be approximated by a superposition of propagating harmonic waves. Since surface wave and free oscillation Q is 100–500 for fundamental modes, a correction of 0.5–1.5% must be made for joint interpretation of body wave and surface wave data. This correction is nearly 1 order of magnitude larger than the uncertainties associated with these data and are therefore very significant. When this correction is made, the discrepancy between the observed surface wave phase velocities and free oscillation periods and those predicted by the Jeffreys or Gutenberg model becomes much smaller than has previously been considered.

Electromagnetic propagation in periodic stratified media. II. Birefringence, phase matching, and x-ray lasers *

Abstract: The theory of electromagnetic Bloch waves in periodic stratified media is applied to the problems of birefringence and group velocity in these media. The relevance of periodic media to phase matching in nonlinear mixing experiments and to laser action in the x-ray region is discussed.

Optical properties of rough surfaces: General theory and the small roughness limit

Abstract: The diffraction of electromagnetic waves from the rough surface of a material of finite permittivity is examined for the case where the wavelength of the incident radiation is comparable to the dimensions of the surface roughness. Two methods of calculation studied are the Rayleigh method and the extinction-theorem integral-equation method. The latter is shown to have a unique exact solution. This property is, in turn, used to show how the Rayleigh method can be modified to give convergent results. The extinction theorem is also used to reduce the Rayleigh equations to a simpler form. These reduced equations, which are extremely convenient to use in the case of small roughness, are applied in this case to find perturbative expressions for the reflected field and for the surface-plasmon dispersion relation.

Parametric Instabilities Due to Lower-Hybrid Radio Frequency Heating of Tokamak Plasmas

Abstract: The linear theory of parametric instabilities relevant to radiofrequency heating near the lower‐hybrid frequency of tokamak‐type plasmas is presented. The dispersion relations are analyzed numerically to all orders of the ion Larmor radius. The complete range of unstable spectra is obtained for 1<ω0/ωlh<5, for both deuterium and hydrogen plasmas. As the pump wave propagates from the edge of the plasma toward its interior, the transition from resonant decay to decay into quasi‐modes is demonstrated. The effects of inhomogeneities upon the threshold for parametric decay, such as density gradients, finite pump width, and magnetic shear, are obtained. The relevance of these results to recent tokamak experiments is discussed briefly.

Transition prediction and linear stability theory

Abstract: Linear stability theory is used in computing the amplitude ratio for other than two-dimensional instability waves The wave motion is obtained from the ray equations of kinematic wave theory, and the amplitude ratio by simply integrating the spatial amplification rate of the parallel flow theory along a ray Both the temporal and spatial theories are examined for two- and three-dimensional incompressible and two-dimensional compressible boundary layers The dispersion relation is most directly obtained with the temporal theory, but the magnitude and direction of the group velocity have to be computed to give the spatial amplification rate, and then only approximately The spatial theory gives the spatial amplification rate directly, but only after the direction of the group velocity is known Transition prediction methods, divided into amplitude-density and amplitude methods, are discussed

On the nonlinear theory for gravity waves on the ocean's surface. Part II : Interpretation and applications

Abstract: In a previous paper (Weber and Barrick, 1977), a generalization of Stokes’ perturbational technique permitted us to obtain solutions to higher orders for gravity-wave parameters for an arbitrary, two-dimensional periodic surface. In particular, the second-order wave-height correction and the third-order dispersion relation correction were derived there. In this paper, we interpret and apply those solutions in a variety of ways. First of all, we interpret the dispersion relation (and its higher order corrections) physically, as they relate to the phase velocity of individual ocean wave trains. Second, the validity of the two results derived previously is established by comparisons in the appropriate limiting cases with classical results available from the literature. It is shown how the solutions—derived for periodic surface profiles—can be generalized to include random wave fields whose average properties are to be specified. Then a number of examples of averaged higher order wave parameters, are...

Studies of backscattered sea return with a CW, dual-frequency, X-band radar

Abstract: A coherent, CW, dual-frequency, X -band radar was used to study microwave sea return from the Chesapeake Bay. It is shown that the product of the backscattered fields depends strongly on long surface wave properties. In particular, a sharp line is found in the product power spectrum whose frequency is that of the water wave whose wavelength is in resonance with the spatial period of the beat frequency between the two transmitted signals and whose wave vector is parallel to the horizontal line of sight. Thus, gravity wave dispersion relations can be obtained with the system. Furthermore, the degree of modulation of short waves by long ones is given by the intensity of the line. A broad background corresponding to the convolution of the single-frequency Doppler spectra is also seen in the product power spectrum. These results are shown to be interpretable by composite surface scattering theory.

Electromagnetic ion-cyclotron wave growth rates and their variation with velocity distribution function shape

Abstract: The sensitivity of electromagnetic ion-cyclotron wave growth rates to the details of the shape of proton velocity distribution functions is explored. For this purpose two different forms of bi-Lorentzian for the proton distribution functions were adopted. The growth rates for the two types of bi-Lorentzians and the biMaxwellians for the beam (hot) protons are compared. Although the growth rates for the three shapes depend on the velocity moments of the different velocity distributions in a similar way, their magnitudes were found to vary considerably.

Spin Waves in a Heusler Alloy Cu2MnAl

Abstract: The spin waves in a Heusler alloy Cu 2 MnAl were investigated at several temperatures by neutron inelastic scattering. The spin wave dispersion relations along three principal directions were systematically explored across the whole magnetic Brillouin zone at 4 K. The results were analysed using the Heisenberg Hamiltonian with long range magnetic interactions. At T =493 K ( T / T c =0.78), the renormalization of the spin wave energy at large momentum transfer regions was found to be very small. This result is interpreted in terms of the temperature dependence of the s - d interaction at large distances.

Mode conversion and tunneling in an inhomogeneous plasma

Abstract: A theory is presented describing mode conversion and tunneling in a warm, collisionless, toroidal plasma near the first ion‐cyclotron harmonic frequency when the waves are propagating perpendicular to the static magnetic field. The coupling region may be characterized by a fourth‐order differential equation with one turning point. Employing the method of Laplace integrals and the method of matched asymptotic expansions, transmission, reflection, and conversion coefficients are calculated assuming a fast wave is incident from either side of the coupling region. For a fast wave incident from the low magnetic field side, transmission, reflection, and conversion occur, while for a fast wave incident from the high magnetic field side, only transmission and conversion occur with no reflection, independent of the thickness of the tunneling layer, which implies complete conversion of the fast wave to the slow wave as the tunneling layer becomes thick. The problem of cavity resonance is discussed.

Dielectric theory of interacting excitonic resonances

Abstract: The dielectric function is developed for two interacting excitonic resonances with spatial dispersion. The longitudinal-transverse energy splitting and the reflectivity of these resonances depend very strongly on the energy separation between the resonances. It is shown that simple interpretations of reflection spectra of degenerate resonances split by external fields may yield incorrect oscillator strengths or polarizabilities and resonance frequency shifts. The energy versus wave-vector polariton dispersion relation is obtained and plotted for the materials CdS and ZnO as examples. Effective refractive indices are derived to be used in the generalized multilayer Fresnel formula for oblique incidence of light polarized parallel and perpendicular to the plane of incidence.

Lower hybrid parametric instabilities—Nonuniform pump waves and tokamak applications

Abstract: Electrostatic lower hybrid ’’pump’’ waves are often launched into tokamak plasmas by structures (e.g., waveguides) whose dimensions are considerably smaller than characteristic plasma sizes. Such waves propagate in well‐defined resonance cones and give rise to parametric instabilities driven by electron E×B velocities. The finite size of the resonance cone region determines the threshold for both convective quasi‐mode decay instabilities and absolute instabilities. The excitation of absolute instabilities depends on whether a traveling or standing wave pump model is used; traveling wave pumps require the daughter waves to have a definite frequency shift. Altogether, parametric instabilities driven by E×B velocities occur for threshold fields significantly below the threshold for filamentation instabilities driven by pondermotive forces. Applications to tokamak heating show that nonlinear effects set in when a certain power‐per‐wave‐launching port is exceeded. For sufficiently high powers, these instabilities will occur in the low‐density edge region of a tokamak. They are characterized by a daughter wave frequency 10% below the pump wave frequency, in agreement with experimental observations.

Unified Formalism of Lower Hybrid Parametric-Instabilities in Plasmas

Abstract: A unified theory of parametric instabilities in the lower hybrid frequency region is formulated using the drift kinetic equation for electrons and the Vlasov equation for unmagnetized ions. The three wave decay process into various decay channels (viz., two lower hybrid waves, lower hybrid and ion acoustic waves) are examined and compared for various pump frequencies and powers. Nonlinear scattering rates by electrons and ions (i.e., nonlinear Landau damping) are evaluated and compared with previous calculations. It is seen that the dominant channel of decay is through ion acoustic and lower hybrid waves when Te/2Ti≳4 and through lower hybrid waves and lower hybrid quasi‐modes when Te/2Ti<4.

Convective and absolute instability of baroclinic eddies

Abstract: The long time response to an arbitrary unstable disturbance in a two layer model on an f-plane is sought. It has been found that depending on the ratio of the shear to the average speed of the mean flow two types of baroclinic instabilities exist: convective and absolute. When the system supports convective instabilities the long time response to an initial pulse excitation decreases with time at a fixed point in space. When such a system is excited by a wave maker the steady state frequency of response of the system corresponds to a spatially amplifying wave oscillating with the frequency of the wave maker. If the dispersion relation yields a saddle point of the frequency in the wave number complex plane with positive imaginary part of the frequency the system supports absolute instabilities. The response of the system at any point in space excited by an arbitrary signal grows exponentially with time at a rate determined by the properties of the system at the saddle point. This response is diffe...

Study of drift-wave turbulence by microwave scattering in a toroidal plasma

Abstract: A study of drift wave turbulence by microwave scattering technique was carried out in a toroidal plasma confinement device, the FM-1 spherator. The principal results are; (a) Observation of the linear dispersion relation of drift waves in the high magnetic shear condition. The linear dispersion relation is followed up to weak turbulence state (..gamma.. approximately less than ..omega..). (b) An experimental demonstration of the stabilization of the drift waves with an increase of the shear strength. In particular, the ion mass dependence of the observed marginal stability criterion for drift waves (..omega.. approximately ..omega../sub */ was in reasonably good agreement with theoretical expectations. Further, very low frequency (..omega.. much less than ..omega../sub */) fluctuations were enhanced when the shear strength was lowered. (c) A detailed study of the behavior of drift waves in a weak shear condition where a state of isotropic turbulence was observed. The turbulent density fluctuations showed a strong dependence on frequency, but no dependence on wave number (somewhat in sharp contrast to theoretical expectations). The possible dependence of the anomalous loss process on the observed drift wave turbulence is also discussed.

Variational method for electromagnetic waves in a magneto-plasma

Abstract: 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.

Subband Structures of N-Channel Inversion Layers on III–V Compounds –A Possibility of the Gate Controlled Gunn Effect–

Abstract: Multi-subband structures of n-channel inversion layers on the surface of p-type III–V compounds are calculated by a variational method. Nonparabolicity is taken into account in the bulk dispersion relation of the \(\varGamma\)-valley. When the surface electron density is low, electrons occupy only the subbands in the \(\varGamma\)-valley, while when it exceeds certain critical value, most electrons occupy the ground subband in the second minimum valleys. From this behavior, the working condition of the Gunn effect in the system can be changed by applying the gate voltage in the MOS structure. The critical surface electron density depends on the charge density in the depletion layer and the surface orientation, but in case of GaAs, it is about 7×10 12 cm -2 for typical operating conditions.

Classical electrodynamics of non-specular dielectric surfaces

Abstract: The effect of spatial dispersion on the electrodynamics of surfaces is studied for a phenomenological model which describes the surface scattering of the elementary excitations in terms of a parameter p, the specular surface being the case p = 1. Surface impedance, reflectivity and surface mode dispersion relation are obtained for arbitrary p without resorting to additional boundary conditions. It is thus shown that the problem is sufficiently defined if the model, including the effect of the surface, is specified. Various forms of additional boundary conditions used in the literature are discussed and characterized as models corresponding to particular values of p. The different roles of the longitudinal and transverse dielectric constants are discussed in physical terms.

Deformational instability of a plane interface with transfer of matter. Part 1.—Non-oscillatory critical states with a linear concentration profile

Abstract: General laws of conservation of mass and momentum are formulated for a moving and arbitrarily deformed interface in local equilibrium with the adjacent, immiscible bulk liquids in which a third component is distributed. The equations are linearised for the case of small deformations from a plane interface.The stability of a plane interface with a perpendicular, linear concentration gradient with respect to deformation is investigated by means of linear, hydrodynamic stability theory. A general dispersion relation between the complex time constant of perturbation ω and the wavenumber k is obtained. An explicit solution for the exchange of stability boundary is found. For spontaneous, interfacial deformation to occur, the diffusion has to be directed from the liquid with the smallest value for the diffusion coefficient of the third component to the liquid with the greatest. The direction of the gravitational field has no importance in the linear theory. All wavenumbers below a certain critical number kcr will be unstable for a fixed difference of slopes of the concentration profiles. On the other hand, when k is fixed, the difference of slopes has to exceed an instability threshold which increases with the bulk and surface viscosities and the diffusion coefficients, and decreases with increasing value of the coupling coefficient between the surface pressure and the surface concentration of the third component.

Two-Dimensional Crystallization of the Electrons in MOS Structures Induced by Strong Magnetic Field

Abstract: The phonon dispersion relation of the Wigner lattice in MOS inversion layers under strong magnetic field is investigated by the effective Hamiltonian for the center coordinates of the cyclotron motion. The transverse phonon velocity is calculated in the self-consistent harmonic approximation as a function of the temperature for various carrier concentrations and for various values of the impurity potential strength. The phase diagram is obtained by plotting the critical concentration for the vanishing of the real solution of the phonon velocity. The calculated melting concentration is compared with the observed immobile carrier density in the Si MOS.

Eighth velocity of light

Abstract: In dispersive media the phase velocity, group velocity, energy velocity, signal velocity, relativistic velocity constant, and ratio‐of‐units velocity are usually not useful concepts for wave packets. The centrovelocity has been suggested as a measure which overcomes many of the objections to the first six. We describe yet another, based on the cross‐correlation of the original and received wave packets, which is shown to be useful in time‐of‐flight measurements in weakly and strongly dispersive media; absorption and amplification are readily accommodated. Applications for specific examples of wave packets in magnetoplasmas are presented.

Interfacial electrohydrodynamic instability in normal electric field

Abstract: Surface‐coupled electrohydrodynamic instability at the interface between miscible fluid components with identical permittivity, mass density, and viscosity but disparate conductivities, as stressed by an equilibrium normal electric field, is described in the instantaneous relaxation limit Long wave, inertia‐dominated and short wave, viscous‐dominated regimes of the dispersion relation are identified Experimental studies, using freon 113 as a base liquid, compare observed interfacial displacements as functions of time with growth rates predicted by the model Correlation varies from almost perfect to experimental growth only 01 as fast as predicted by theory, with all observed motions in the inertia‐dominated regime Instability dynamics at the scale of interfacial conductivity distribution are documented by allowing diffusion before electric field application

Dispersion relations for complex reflectivities

Abstract: A simplified derivation of the dispersion relations connecting the phase and amplitude of the normal optical reflectivity of a vacuum-matter interface is presented. The non-Kramers-Kronig form of the expression for reflectivity in terms of phase is shown to be a consequence of the fact that amplitudes are determined only up to a multiplicative constant by the phase. The phase, however, is uniquely determined by the amplitude.

Effects of Metal on the Dispersion Relation of Magnetostatic Surface Waves

Abstract: The dispersion relation of magnetostatic surface waves in a ferromagnetic slab placed between two metal plates is analyzed. The dispersion characteristics are affected considerably by the normalized spacings A and B between the sample and the metals (Fig. 1). When A+B is finite, the frequency at which the wave number ζ approaches zero becomes {(ΩH+1)[ΩH+1/(A+B+1)]}1/2, where ΩH=Hi/4πMs, Hi is the internal magnetic field and 4πMs is the saturation magnetization. This frequency varies with A+B from [ΩH(ΩH+1)]1/2 to ΩH+1. The existence of this frequency has been verified by experiment for A=B.

Measurement of ultrasonic dispersion by phase comparison of continuous harmonic waves

Abstract: The method of phase comparison of continuous waves is applied to determine the dispersion relation, phase, and group velocities as a function of frequency of dispersive materials. A combination of the variable frequency method and the variable path‐length method is found necessary to eliminate any uncertainty in the dispersion relation determination. Experiments are performed on specimens of various thickness. A constraint equation can be derived since the dispersion relation is unique and independent of the specimen thickness. This equation provides a procedure for determining the absolute number of wavelengths in the specimen. Measurements in unidirectional, fiber‐reinforced boron–epoxy specimens show good agreement with results reported previously.