General Dispersion Equation Research Articles

Steady water waves with vorticity: an analysis of the dispersion equation

AbstractTwo-dimensional steady gravity waves with vorticity are considered on water of finite depth. The dispersion equation is analysed for general vorticity distributions, but under assumptions valid only for unidirectional shear flows. It is shown that for these flows (i) the general dispersion equation is equivalent to the Sturm–Liouville problem considered by Constantin & Strauss (Commun. Pure Appl. Math., vol. 57, 2004, pp. 481–527; Arch. Rat. Mech. Anal., vol. 202, 2011, pp. 133–175), (ii) the condition guaranteeing bifurcation of Stokes waves with constant wavelength is fulfilled. Moreover, a necessary and sufficient condition that the Sturm–Liouville problem mentioned in (i) has an eigenvalue is obtained.

Derivative free method for computing modes of multilayer planar waveguide

A technique to solve generalized dispersion equation of multilayer planar waveguide has been demonstrated to obtain all the expected guided modes. The solution is based on the derivative free method for computing the zeros of an analytical function in complex plane. The derivative free method extracts the roots which are very close to actual zeros of the function. Roots are further refined using the robust iteration method to achieve the desired accuracy. Application of the proposed method has been verified by solving the modes of a variety of structures including lossless structure, leaky structure, quantum well waveguide, active waveguide, ARROW waveguide and metal clad waveguide. The method is efficient and computes all modes of planar waveguide with high accuracy.

Love Wave in an Isotropic Half-Space with a Graded Layer

This work addresses the dispersion of Love wave in an isotropic homogeneous elastic half-space covered with a functionally graded layer. First, the general dispersion equations are given. Then, the approximation analytical solutions of displacement, stress and the general dispersion relations of Love wave in both media are derived by the WKBJ approximation method. The solutions are checked against numerical calculations taking an example of functionally graded layer with exponentially varying shear modulus and density along the thickness direction. The dispersion curves obtained show that a cut-off frequency arises in the lowest order vibration model.

OBLIQUE SURFACE JOSEPHSON PLASMA WAVES IN LAYERED SUPERCONDUCTORS

In this paper the theoretical analysis of the oblique surface Josephson plasma waves propagating along a dielectric‐layered superconductor interface is first represented. The case when the superconductor layers are perpendicular to the interface is considered. The oblique surface Josephson plasma waves are the surface electromagnetic waves propagating in the interface plane at an arbitrary angle to the superconductor layers. The electromagnetic field of the waves is the superposition of ordinary and extraordinary surface electromagnetic waves. The dispersion equation of the oblique surface Josephson plasma waves has been derived. This equation is the most general dispersion equation of the surface Josephson plasma waves in the case under study. The problem on the excitation of oblique surface Josephson plasma waves by means of the attenuated total reflection method is solved. It has been established that the process of the excitation of oblique surface Josephson plasma waves is accompanied by the onset of an additional reflected wave in the prism with the polarization, being orthogonal to the polarization of the incident wave. It has been found that the reflection coefficient of the exciting wave in the prism can vanish for some optimal parameters of the structure.

Smoothing properties of evolution equations via canonical transforms and comparison principle

This paper describes a new approach to global smoothing problems for dispersive and non-dispersive evolution equations based on the global canonical transforms and the underlying global microlocal analysis. For this purpose, the Egorov-type theorem is established with canonical transformations in the form of a class of Fourier integral operators, and their weighted L-2-boundedness properties are derived. This allows us to globally reduce general dispersive equations to normal forms in one or two dimensions. Then, a new comparison principle for evolution equations is introduced. In particular, it allows us to relate different smoothing estimates by comparing certain expressions involving their symbols. As a result, it is shown that the majority of smoothing estimates for different equations are equivalent to each other. Moreover, new estimates as well as several refinements of known results are obtained. The proofs are considerably simplified. A comprehensive analysis is presented for smoothing estimates for dispersive equations. Applications are given to the detailed description of smoothing properties of the Schrodinger, relativistic Schrodinger, wave, Klein-Gordon and other equations.

Non-Homogeneous Boundary Value Problems for Linear Dispersive Equations

While the non-homogeneous boundary value problem for elliptic, hyperbolic and parabolic equations is relatively well understood, there are still few results for general dispersive equations. We define here a convenient class of equations comprising the Schrödinger equation, the Airy equation and linear “Boussinesq type” systems, which is in some sense a generalization of strictly hyperbolic equations, and for which we define a generalized Kreiss–Lopatinskiĭ condition. From the construction of generalized Kreiss symmetrizers (adapted from hyperbolic theory) we deduce a priori estimates and well posedness for the pure boundary value problems (BVP) on a half-space associated to this class of equations. The initial boundary value problem (IBVP) is investigated too for the special case of the Schrödinger equation, and possible generalizations of the proof for other problems are indicated.

Kinetic Alfvén wave instability driven by a field-aligned current in high-βplasmas

Including the ion-gyroradius effect, a general low-frequency kinetic dispersion equation is presented, which simultaneously takes account of a field-aligned current and temperature anisotropy in plasmas. Based on this dispersion equation, kinetic Alfvén wave (KAW) instability driven by the field-aligned current, which is carried by the field-aligned drift of electrons relative to ions at a drift velocity V(D), is investigated in a high-β plasma, where β is the kinetic-to-magnetic pressure ratio in the plasma. The numerical results show that the KAW instability driven by the field-aligned current has a nonzero growth rate in the parallel wave-number range 0<k(z)<k(z)(u), where the growth rate reaches a maximum at k(z)∼k(z)(u)/2. On the other hand, in the perpendicular wave-number range 0<k(⊥)<k(⊥)(u) the growth rate monotonously decreases with the wave number k(⊥) from a maximum at k(⊥)=0 to zero at k(⊥)=k(⊥)(u). In particular, the upper boundaries of the growing ranges in the wave-number space, k(z)(u) and k(⊥)(u), as well as the growth rate increase with V(D), imply that there are wider growing ranges and larger growth rate for larger drift velocities. Including the temperature anisotropy of plasma particles, the result shows that the instability conditions for these two driven mechanisms are both modified considerably, in which the growing parametric ranges of KAWs widen. The results have potential importance in understanding the physics of space and astrophysical plasma active phenomena since the field-aligned current, also called the Birkeland current, is one of the most active factors in plasma phenomena.

New longitudinal waves in electron-positron-ion quantum plasmas

A general quantum dispersion equation for electron-positron(hole)-ion quantum plasmas is derived and studied for some interesting cases. In an electron-positron degenerate Fermi gas, with or without the Madelung term, a new type of zero sound waves are found. Whereas in an electron-hole plasmas a new longitudinal quantum waves are revealed, which have no analogies in quantum electron-ion plasmas. The excitation of these quantum waves by a low-density monoenergetic straight electron beam is examined. Furthermore, the KdV equation for novel quantum waves is derived and the contribution of the Madelung term in the formation of the KdV solitons is discussed.

Magnetic Surface Polariton in a Planar Biaxial Metamaterial with Dual Negative Magnetic Permeabilities

The magnetic surface polaritons (MSPs) mode rarely exists in natural materials, mainly due to their limited magnetic response. In order to understand the relationship between the MSPs mode and the limited magnetic response, we theoretically derive a general dispersion equation of the MSPs mode for two kinds of biaxial anisotropic media. The dispersion equation suggests the requirements of μx 1, and momentum conservation need to be satisfied, which guides us to design a planar biaxial magnetic metamaterials (PBMM) with two orthogonal negative permeabilities (i.e., μx < 0 and μy < 0). In addition, the results of the retrieval magnetic permeability and electric permittivity indeed point out that our PBMM satisfies the aforementioned requirements. Meanwhile, we investigate the mechanism of the magnetic field by a consideration of an induced current loop, and subsequently we demonstrate the artificial MSPs mode under transverse electric excitation by numerical simulation. After a numerical fit of simulation results, we find that the field of the MSPs mode is in good agreement with the analytical calculation of Maxwell’s equations. Moreover, we also simulate three unsatisfied aforementioned requirements as control conditions to verify the general dispersion equation of the MSPs mode.

Dispersion characteristics of transverse surface waves in piezoelectric coupled solid media with hard metal interlayer

The propagation of transverse surface waves in a piezoelectric layer/metal substrate system with one or multiple hard metal interlayer(s) is investigated analytically. The general dispersion equations for the existence of the waves are obtained in a simple mathematic form for class 6 mm piezoelectric materials. The presence of a hard metal interlayer can not only get rid of the undesired mode appearing in the case without an interlayer but shorten the existence range of the phase velocity within which a nonleaky but dispersive mode exists. The effects of the hard interlayer on the phase velocity can be used to manipulate the behavior of the waves and has implications in acoustic wave devices.

Landau damping of longitudinal oscillation in ultra-relativistic plasmas with nonextensive distribution

The generalized dispersion equation for longitudinal oscillation in an unmagnetized, collisionless, isotropic and relativistic plasma is derived in the context of nonextensive q-distribution. An analytical expression for the Landau damping is obtained in an ultra-relativistic regime, which is related to q-parameter. In the limit q → 1, the result based on the relativistic Maxwellian distribution is recovered. It is shown that the interactions between the wave and particles are stronger and the waves are more strongly damped for lower values of q-parameter. The results are explained by the increased number of superthermal particles or low velocity particles contained in the plasma with the nonextensive distribution.

Thermo elastic waves with thermal relaxation in isotropic micropolar plate

In the present investigation, we have discussed about the features of waves in different modes of wave propagation in an infinitely long thermoelastic, isotropic micropolar plate, when the generalized theory of Lord–Shulman (L–S) is considered. A more general dispersion equation is obtained. The different analytic expressions in symmetric and anti-symmetric vibration for short as well as long waves are obtained in different regions of phase velocities. It is found that results agree with that of the existing results predicted by Sharma and Eringen in the context of various theories of classical as well as micropolar thermoelasticity.

Dispersion relation of transverse oscillation in relativistic plasmas with non-extensive distribution

AbstractThe generalized dispersion equation for superluminal transverse oscillation in an unmagnetized, collisionless, isotropic and relativistic plasma with non-extensiveq-distribution is derived. The analytical dispersion relation is obtained in an ultra-relativistic regime, which is related toq-parameter and temperature. In the limitq→ 1, the result based on the relativistic Maxwellian distribution is recovered. Using the numerical method, we obtain the full dispersion curve that cannot be given by an analytic method. It is shown that the numerical solution is in good agreement with the analytical result in the long-wavelength and short-wavelength region for ultra-relativistic plasmas.

Dispersion relation of longitudinal oscillation in relativistic plasmas with nonextensive distribution

The generalized dispersion equation for longitudinal oscillation in an unmagnetized, collisionless, isotropic and relativistic plasma is derived in the context of nonextensive q-distribution. The analytical expression for the dispersion relation is obtained in an ultra-relativistic regime, which is related to the q-parameter and temperature. In the limit q→1, the result based on the relativistic Maxwellian distribution is recovered. Using the numerical simulation method, we obtain the full dispersion curve which cannot be given by an analytic method. It is shown that the numerical solution is in good agreement with the analytical result in the long-wavelength, short-wavelength and near light wave region for ultra-relativistic plasmas.

Effect of relaxation times and rotation on the propagation of plane waves in generalized magneto- thermo-viscoelasticity

The problem of wave propagation in the generalized dynamical theory of thermo-elasticity proposed by Green and Lindsay is applied to study the propagation of harmonically time dependent thermo-visco-elastic plane waves of assigned frequency in an infinite visco-elastic solid in a magnetic field. A more general dispersion equation is deduced to determine the effects of rotation, viscoelasticity and relaxation time on the phase velocity of the coupled waves. The perturbation technique has been employed to obtain the phase velocity and attenuation coefficient for small thermo-elastic couplings. Taking an appropriate material, the numerical values of the phase velocity of the waves are computed and the results are shown graphically to illustrate the problem. The results are compared with those obtained earlier. Key words: Plane waves, magneto-visco-elastic medium, phase velocity, attenuation coefficient perturbation technique.

Electromagnetic oscillations of the Earth's upper atmosphere (review)

Abstract. A complete theory of low-frequency MHD oscillations of the Earth's weakly ionized ionosphere is formulated. Peculiarities of excitation and propagation of electromagnetic acoustic-gravity, MHD and planetary waves are considered in the Earth's ionosphere. The general dispersion equation is derived for the magneto-acoustic, magneto-gravity and electromagnetic planetary waves in the ionospheric E- and F-regions. The action of the geomagnetic field on the propagation of acoustic-gravity waves is elucidated. The nature of the existence of the comparatively new large-scale electromagnetic planetary branches is emphasized.

Theoretical analysis on Love waves in a layered structure with a piezoelectric substrate and multiple elastic layers

A method is developed to analyze the existence and behavior of piezoelectric Love waves in a multilayered structure consisting of a piezoelectric substrate and multiple elastic layers which are isotropic, nonpiezoelectric materials. The acoustic waves and electric fields in the substrate and the layers are investigated. A general dispersion equation is derived to describe the existence of Love surface waves with respect to phase velocity as a function of normalized layer thickness. An iteration formula for XN is introduced to describe the mechanical action between the layers and the substrate at the interface. Another formula for ε¯LN, the equivalent permittivity of the wave-guide layers, is produced to describe the electric fields in the layers. The dispersion equation including a mass loading on the surface of the top layer is deduced, and a formula for calculating the mass sensitivity of the phase velocity is presented. We also find the dispersion equation with an electric shorted interface and introduce a formula for calculating the electromechanical coupling coefficient K2. Numerical results illustrate the phase velocity, the mass sensitivity of the phase velocity and the electromechanical coupling coefficient as functions of the normalized layer thickness for the Love waves in a layered structure with a polymethylmethacrylate (PMMA) layer and a sputtered SiO2 layer on a 90° rotated ST-cut quartz (ST-quartz) substrate.

Magneto-Acoustic Waves in Compressible Magnetically Twisted Flux Tubes

The oscillatory modes of a magnetically twisted compressible flux tube embedded in a compressible magnetic environment are investigated in cylindrical geometry. Solutions to the governing equations to linear wave perturbations are derived in terms of Whittaker’s functions. A general dispersion equation is obtained in terms of Kummer’s functions for the approximation of weak and uniform internal twist, which is a good initial working model for flux tubes in solar applications. The sausage, kink and fluting modes are examined by means of the derived exact dispersion equation. The solutions of this general dispersion equation are found numerically under plasma conditions representative of the solar photosphere and corona. Solutions for the phase speed of the allowed eigenmodes are obtained for a range of wavenumbers and varying magnetic twist. Our results generalise previous classical and widely applied studies of MHD waves and oscillations in magnetic loops without a magnetic twist. Potential applications to solar magneto-seismology are discussed.

Properties of Love waves in layered piezoelectric structures

Love waves propagating in a layered structure with an elastic layer deposited on a piezoelectric substrate are analytically investigated. We present a general dispersion equation that describes the properties of Love waves in the structure. A detailed discussion regarding the dispersion equation is presented, and the parameters for Love-mode sensors are also introduced. The properties of Love waves are illustrated by means of sample results for a layered structure with an SiO 2 layer sputtered on an ST-cut 90° X-propagating quartz substrate. Interestingly, we found that a threshold-normalized layer thickness existed for the fundamental Love mode in such a structure.

Lp–Lqestimates for dispersive equations and related applications

This paper is mainly concerned with the L p – L q estimates of solutions for a class of general dispersive equations under the condition ( H b ) with an index b ∈ ( 0 , 1 ] . To the end, the new pointwise decay estimates of oscillatory integrals related to the fundamental solutions are proved. If b = 1 , then the pointwise estimates are particularly consistent with the sharp results of the nondegenerate cases. Moreover, as an application of the L p – L q estimates, we also show that higher-order differential operator i P ( D ) + V ( x , D ) generates a fractionally integrated group on some L p ( R n ) , from which certain L p -estimates for the solutions of generalized Schrödinger equations are obtained.