Dispersion Equation For Waves Research Articles

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.

Surface waves with simple exponential transverse decay at a biaxial bicrystalline interface

The dispersion equation for surface waves--with simple transverse exponential decay at the interface of identical biaxial crystals with a relative twist about the axis normal to the interface and propagating along a bisector of the angle between the crystallographic configurations on either side of the interface--has several solutions of which only one is physical. The selected type of surface wave is possible only for a restricted range of the twist angle, which depends on the ratio of the maximum and the minimum of the principal refractive indexes and the angle between the optic ray axes.

Waves and diffusion on the sea surface

The frequency spectrum of surface elevations in the presence of wind waves is well known. On this basis, one can estimate the frequency spectrum of vertical velocities in sea-surface waves. Owing to liquid incompressibility, the spectrum of horizontal velocities should have the same frequency dependence. The use of the dispersion equation for waves on the surface of a heavy liquid allows one to obtain to the spatial spectrum of velocities. Therefore, one can estimate the spatial structure function of the velocity field. For short waves and large depths, the structure function increases as r1/2, where r is the distance between the points of observations. For long waves and shallow depths h, this increase is proportional to r. The coefficient of turbulent mixing K(r) of pollution spots of size r on the sea surface is now estimated as the product of the spot size and the rms difference of velocities. As a result, depending on r and h, the exponent in the rn dependence of K(r) may vary between 1.25 and 1.5. This outcome provides an explanation for a scatter in the values of the exponent n, a phenomenon that has been observed by many experimentalists.

Ab-initio thermal physics and Cr-isotopic fractionation of MgCr2O4

Ab-initio thermophysical properties and isotopic separative effects in MgCr2O4 were computed with the CRYSTAL03 code. All calculations were carried out within the LCAO approach (Linear Combination of Atomic Orbitals) by using the Hartree-Fock and the hybrid B3LYP density functional methods. Vibrational frequencies have been computed at the long-wavelength limit corresponding to the center of the Brillouin zone (k → 0). Since thermodynamic functions depend on the phonon dispersion relation, the ab-initio based vibrational analysis of the optical modes has been coupled with the semiempirical Kieffer model for the acoustic modes, which is based on a sine wave dispersion equation. All computed magnitudes (zero point energy, internal energy, Helmholtz free energy, optical vibrational modes, thermal corrections to internal energy, enthalpy and Gibbs free energy, thermal pressure, bulk modulus and its P and T derivatives, entropy, isochoric and isobaric heat capacity, and isotopic separative effects) are consistent with direct observations and/or reasonable estimates, within the accuracy of the method. The computed separative effects are similar to those observed in other Cr3+-bearing substances. The crystal field seems to have no influence on the separative effect for an isotopomeric couple with respect to an isolated gaseous couple with Cr3+ as central cation.

Dynamics of an interface with adsorption layer between two fluids

The dynamics of a flat interface with adsorption layer between two viscous fluids is studied on the basis of the linearized Navier-Stokes equations. The dispersion equation for interfacial waves involves surface tension and elasticity moduli of the interface, besides gravitational acceleration. The displacement of the interface due to a force density applied at the interface is characterized by a susceptibility tensor. Wave excitation due to a monochromatic plane wave source or a monochromatic line source located at some distance from the interface is discussed.

Screw vibrations localized on charged dislocations in semiconductor crystals

Hole-gas screw vibrations localized on charged dislocations are shown to exist in semiconductor crystals. The dispersion equations for localized waves are derived. The frequency ranges separating the frequencies of localized and volume waves are found.

Tailoring the near-field guiding properties of magnetic metamaterials with two resonant elements per unit cell

A theoretical and experimental study of magnetic metamaterials with unit cells containing two resonant elements is presented. The properties of these structures, consisting of split rings, are governed by strongly anisotropic magnetic coupling between individual elements. This coupling leads to propagation of slow magnetoinductive waves in the vicinity of the resonant frequency. The wavelength of magnetoinductive waves is much smaller than the free-space wavelength of the electromagnetic radiation. This opens up the possibility of manipulating the near field on a subwavelength scale. We develop a theoretical formulation for coupled chains of metamaterial elements allowing the tailoring of their guiding properties in the near field. In a comprehensive analysis modes of coupled waveguides supporting forward and/or backward waves are identified and the corresponding hybridization mechanisms for dispersion equations of magnetoinductive waves are determined. Analytical predictions are verified both experimentally and numerically on a variety of coupled waveguides. The approach can be employed for the design of near-field manipulating devices.

Love waves in double porosity media

Based on the control equations for double-porosity media extended from Biot's theory, the propagation of Love waves is discussed in this paper. Dispersion equations of Love waves in double porosity medium are obtained by utilizing the usual assumption of Love waves in elastic solid with the coupling mass coefficient ρ 23 and coupling permeability term k (12) being equal to zero. The dispersion and attenuation properties of Love waves in double porosity medium are analyzed. The approximate limit of the Love wave speed is also given. The effects of fracture porosity and fracture permeability on the behavior of the propagation of Love waves are investigated in detail.

Collinear interaction of light with space-charge waves in a semiconductor waveguide

The dispersion equation for space-charge waves (SCWs) that couples the wave vector of a traveling wave with frequency and the parameters of a semiconductor waveguide layer is obtained. Numerical analysis of the co-and contradirectional SCW-optical interaction for guided TE and TM modes in a waveguide structure based on n-GaAs is performed. It is demonstrated that the power exchange between the TE modes is more effective than the power exchange between the TM modes. The intervals of the waveguide layer thickness with a relatively high efficiency of the mode transformation are determined.

A criterion for the existence of spin waves in polarized gases in a wide temperature range

We derive a kinetic equation for a polarized paramagnetic gas that is exact in the Boltzmann (binary) approximation. The equation is written in a compact form and applies to both Fermi and Bose gases in a wide temperature range as long as the Boltzmann approximation remains applicable. The derived equation is used to analyze the conditions for the propagation of spin waves in polarized Fermi and Bose gases. We deduce a universal criterion for the propagation of weakly damped spin waves in a wide temperature range. The criterion is reduced to the condition that the real parts of the particle zero-angle scattering amplitudes (or T matrices) be much larger than their imaginary parts. We derive dispersion equations for spin waves at high and low gas temperatures and show that spin waves can propagate in both these limiting cases.

Dispersion equations for field-aligned cyclotron waves in axisymmetric magnetospheric plasmas

Abstract. In this paper, we derive the dispersion equations for field-aligned cyclotron waves in two-dimensional (2-D) magnetospheric plasmas with anisotropic temperature. Two magnetic field configurations are considered with dipole and circular magnetic field lines. The main contribution of the trapped particles to the transverse dielectric permittivity is estimated by solving the linearized Vlasov equation for their perturbed distribution functions, accounting for the cyclotron and bounce resonances, neglecting the drift effects, and assuming the weak connection of the left-hand and right-hand polarized waves. Both the bi-Maxwellian and bi-Lorentzian distribution functions are considered to model the ring current ions and electrons in the dipole magnetosphere. A numerical code has been developed to analyze the dispersion characteristics of electromagnetic ion-cyclotron waves in an electron-proton magnetospheric plasma with circular magnetic field lines, assuming that the steady-state distribution function of the energetic protons is bi-Maxwellian. As in the uniform magnetic field case, the growth rate of the proton-cyclotron instability (PCI) in the 2-D magnetospheric plasmas is defined by the contribution of the energetic ions/protons to the imaginary part of the transverse permittivity elements. We demonstrate that the PCI growth rate in the 2-D axisymmetric plasmasphere can be significantly smaller than that for the straight magnetic field case with the same macroscopic bulk parameters.

THEORETICAL AND EXPERIMENTAL STUDIES OF LONGITUDINAL ULTRASONIC GUIDED WAVE PROPAGATION CHARACTERISTICS IN WATER-FILLED PIPES

Longitudinal ultrasonic guide wave dispersion equation of inviscid liquid-filled pipe is analyzed theoretically. Propagation characteristics of longitudinal modes in water-filled steel pipe and water cylinder with rigid boundary are studied. Except a mode, other modes in wall-filled steel pipe are the coupling of the modes of steel pipe in vacuum and of water cylinder with rigid boundary. To the same mode in water-filled steel pipe, its propagation characteristics vary greatly with frequency range. Furthermore, longitudinal modes are excited and received in steel pipes filled with and without water experimentally. Results show that propagation characteristics, such as mode branching and group velocity decrease, occur in water-filled steel pipe. When frequency band of longitudinal mode in water-filled steel pipe is in the range of one unperturbed L(0, 2) mode branching of steel pipe in vacuum, the mode has the ability of long distance propagation, and the signal shape and amplitude are retained as it travels long distances for its low dispersion. As a result, the kind of longitudinal mode is suitable to defect detection of long distance water-filled pipe.

Propagation of Love Waves in an Inhomogeneous Fluid Saturated Porous Layered Half-Space with Properties Varying Exponentially

Based on Biot's theory for transversely isotropic fluid saturated porous media, the complex dispersion equation for Love waves in a transversely isotropic fluid-saturated porous layered half-space is derived with the consideration of the inhomogeneity of the layer. The equation is solved by an iterative method. Detailed numerical calculation is presented for an inhomogeneous fluid-saturated porous layer overlying a purely elastic half-space. The dispersion and attenuation of Love waves are discussed. In addition, the upper and lower bounds of Love wave speed are also explored.

An alternative dispersion equation for water waves over an inclined bed

A classical approach to extending the validity of Airy's dispersion relation for surface gravity waves by Friedrichs (1948) to gentle slopes (of special inclinations) is here re-examined with extended small-slope asymptotics using the full linear harmonic function theory combined with the method of steepest descent. A new dispersion relation emerges that appears to give significantly increased accuracy over sloping beds when tested on the plane beach problem with various forms of the mild-slope equation (MSE) and global error reductions of the order 50% are noted in some 'from deep to shore' computations. Unlike the classical formula, the new formula predicts a discontinuous wavenumber at a place where the bottom slope is discontinuous. Preliminary tests examining the reflection coefficient with the basic (early version) MSE over ramp-type profiles indicate that this is not a major problem and numerical results using wavenumber calculated by the new dispersion relation are qualitatively similar to those of the modified MSE (MMSE) developed in Chamberlain & Porter (1995). When the new formula is applied (with mass conservation) to the MMSE on the ramp, results are almost identical to those of a full linear model for inclines having a gradient up to 8:1. It is also shown that the dominant asymptotic analysis, responsible for the new formula, is valid for all slope angles a < n/2 and not just the special angles considered by Friedrichs.

How can one solve exactly some problems in plasma theory

A review of a technique for the application of a new Lambert function are outlined. The problems discussed include equilibrium electric charge of a solitary body in plasma, nonlinear ion-acoustic waves, the equilibrium configuration of a neutralized electric beam, the dynamics of shaping virtual cathodes in a medium with viscous friction, and the study of the dispersion equation for electron plasma waves in the framework of generalized Boltzmann kinetics.

Electron Cyclotron Absorption of Microwaves with Small Propagation Angle in Dense Plasmas

We derive an approximate dispersion equation for slow waves with frequency near electron cyclotron frequency, propagating quasi-longitudinally in plasmas. This equation is studied both numerically and within the framework of perturbation theory. The obtained cyclotron-absorption coefficients do not have singularities in the limiting case of strictly longitudinal propagation and ensure correct transition to well-known expressions for the case of large propagation angles.

Magnetohydrodynamic waves within the medium separated by the plane shock wave or rotational discontinuity

Abstract. Characteristics of small amplitude plane waves within the medium separated by the plane discontinuity into two half spaces are analysed. The approximation of the ideal one-fluid magnetohydrodynamics (MHD) is used. The discontinuities with the nonzero mass flux across them are mainly examined. These are fast or slow shock waves and rotational discontinuities. The dispersion equation for MHD waves within each of half space is obtained in the reference frame connected with the discontinuity surface. The solution of this equation permits one to determine the wave vectors versus the parameter cp, which is the phase velocity of surface discontinuity oscillations. This value of cp is common for all MHD waves and determined by an incident wave or by spontaneous oscillations of the discontinuity surface. The main purpose of the study is a detailed analysis of the dispersion equation solution. This analysis let us draw the following conclusions. (I) For a given cp, ahead or behind a discontinuity at most, one diverging wave can transform to a surface wave damping when moving away from the discontinuity. The surface wave can be a fast one or, in rare cases, a slow, magnetoacoustic one. The entropy and Alfvén waves always remain in a usual homogeneous mode. (II) For certain values of cp and parameters of the discontinuity behind the front of the fast shock wave, there can be four slow magnetoacoustic waves, satisfying the dispersion equation, and none of the fast magnetoacoustic waves. In this case, one of the four slow magnetoacoustic waves is incident on the fast shock wave from the side of a compressed medium. It is shown that its existence does not contradict the conditions of the evolutionarity of MHD shock waves. The four slow magnetoacoustic waves, satisfying the dispersion equation, can also exist from either side of a slow shock wave or rotational discontinuity. (III) The expressions determining the polarisation of the MHD waves are derived in the reference frame connected with the discontinuity surface. This form of presentation is much more convenient in investigating the interaction of small perturbations with MHD discontinuities. It is shown that the perturbations of the velocity and magnetic field associated with the surface magnetoacoustic wave have the elliptic polarisation. Usually the planes of polarisation for the perturbations of the velocity and magnetic field are not coincident with each other. Keywords. Space plasma physics (Discontinuities; Shock waves) – Interplanetary physics (Discontinuities; Interplanetary shocks) – Magnetospheric physics (Solar windmagnetosphere interactions)

Nonlinear inertial and kinetic Alfven waves

A two‐fluid (multifluid) model is used to study nonlinear kinetic (inertial) Alfven waves (solitons, oscillitons, periodic waves). The stationary wave dispersion equation yields the range of the wave speeds in which solitary waves are possible. Kinetic Alfven waves propagating almost perpendicular to the magnetic field exhibit a maximum of the phase speed (a positive dispersion followed by a negative one) which modifies the properties of solitons. At velocities greater than the maximum phase speed, soliton structure exists containing an oscillating core. Such structures, called oscillitons, also exist for other wave modes whose dispersion curves (ω − k) have an inflection point, e.g., the magnetoacoustic (whistler) mode, which gives rise to an extremum in the phase velocity. In a multifluid approach for a bi‐ion plasma, the main properties of oscillatory kinetic Alfven and magnetoacoustic waves are similar to those in a proton‐electron plasma. Assessment of kinetic effects (the Vlasov equations) shows that a maximum of the phase speed of kinetic Alfven wave remains but shifts to smaller k, and therefore the existence of oscilliton‐type solutions with embedded coherent oscillations at f ≤∼ 0.8 fp (fp is the proton gyrofrequency) is expected. In a multi‐ion Maxwellian plasma, a maximum moves further to smaller k (f ≤∼ 0.8fi, where fi is the heavy ion gyrofrequency) and even can cease when oxygen ions are injected to a proton‐electron plasma. The possibility of applying these results to observations carried out on the auroral field lines and near the magnetopause is discussed.

Acoustic Instability in Inhomogeneous Gas-Discharge Plasma

The possibility of amplification of acoustic waves in inhomogeneous atomic dc gas-discharge plasma (positive column of glow discharge) is investigated theoretically. This amplification is caused by the transfer of thermal energy from electrons to colder electron gas under conditions of elastic electron-atom collisions. It is demonstrated that the inclusion of nonuniform distribution of equilibrium temperature, neutral gas concentration, and electron concentration over the plasma column cross section, similar to the case of uniform distribution, cannot result in acoustic instability and, consequently, in the interpretation of the experimental results in sound amplification in weakly ionized plasma. The dispersion equation for sound waves in inhomogeneous glow-discharge plasma is derived and numerically analyzed in detail. Comparison of the theoretical and experimental results enables one to identify the reasons for their discrepancy and to actually reveal the basic principles essential for constructing a theory which will make it possible to quantitatively and adequately describe the experimental data on the amplification of sound in weakly ionized atomic plasma.

Dispersion of Surface Waves in a Periodically Laminated Piezoelectric Half-Space with Liquid Upper Layer

An approach is proposed to set up the dispersion equations for surface waves propagating through a periodically laminated piezoelectric medium, with the upper layer being a perfect compressible fluid. The approach is based on the formalism of Hamiltonian periodic systems. The dispersion equations derived are valid for an arbitrary law of variation in properties with periodicity coordinate. The influence of the liquid layer and inhomogeneity of the piezoelectric medium on the dispersion spectra of surface waves is studied