Dispersion Relation For Longitudinal Waves Research Articles

Thermal effects on longitudinal waves instabilities in rotating beam-plasma interaction

The thermal effects of charged particles on longitudinal wave instabilities in the interaction of a relativistic rotating electron beam with a magnetized plasma are studied using the standard linearized Vlasov theory. The dispersion relations of longitudinal waves are obtained in two different frequency limits, i.e., when the wave frequency is either much lower or much higher than the electron gyro-frequency. Considering these dispersion relations and finding the growth rates of the instabilities, it is found that for the smaller wave frequency, the instability is not limited to a specific temperature range and the growth rate increases with the beam velocity, while for the larger wave frequency, not only the instability occurs in a specific temperature range, but also the growth rate decreases with the relativistic beam velocity contrary to nonrelativistic beam velocities.

Practical dispersion relations for strongly coupled plasma fluids

Very simple explicit analytical expressions are discussed, which are able to describe the dispersion relations of longitudinal waves in strongly coupled plasma systems such as one-component plasma and weakly screened Yukawa fluids with a very good accuracy. Applications to other systems with soft pairwise interactions are briefly discussed.

Longitudinal elastic wave control by pre-deforming semi-linear materials.

An incremental wave superimposed on a pre-deformed hyper-elastic material perceives an elastic media with the instantaneous modulus of the current material. This offers a new route with a broadband feature to control elastic waves by purposely creating finite deformation field. This study proves that the governing equation of a semi-linear material under a symmetric pre-deformation condition maintains the form invariance for longitudinal wave, so the longitudinal wave control can be made by transformation method without the constraint condition on principle stretches; however, this is not the case for shear waves. Therefore, pre-deforming a semi-linear material provides a potential method for treating longitudinal and shear waves differently. Examples with elastic wave control and band structure shift through pre-deforming a semi-linear material are provided to illustrate this finding. Finally, a one-dimensional spring lattice is proposed to mimic a semi-linear material, and the dispersion relation for longitudinal waves in a sandwich structure with such spring lattice is shown to be invariant during elongation, confirming the result found based on a homogeneous semi-linear material. These results may stimulate researches on designing new hyper-elastic microstructures as well as designing new devices based on pre-deformed hyper-elastic materials.

Unified Concept of Effective One Component Plasma for Hot Dense Plasmas.

Orbital-free molecular dynamics simulations are used to benchmark two popular models for hot dense plasmas: the one component plasma (OCP) and the Yukawa model. A unified concept emerges where an effective OCP (EOCP) is constructed from the short-range structure of the plasma. An unambiguous ionization and the screening length can be defined and used for a Yukawa system, which reproduces the long-range structure with finite compressibility. Similarly, the dispersion relation of longitudinal waves is consistent with the screened model at vanishing wave number but merges with the OCP at high wave number. Additionally, the EOCP reproduces the overall relaxation time scales of the correlation functions associated with ionic motion. In the hot dense regime, this unified concept of EOCP can be fruitfully applied to deduce properties such as the equation of state, ionic transport coefficients, and the ion feature in x-ray Thomson scattering experiments.

Dispersion relation of longitudinal waves in liquid He-4 in the framework of quantum macroscopic equations derived from Bohm’s potential

He-4 is known to become superfluid at very low temperatures. This effect is now generally accepted to be connected with BEC (Bose–Einstein Condensation). The dispersion relation of pressure waves in superfluid He-4 has been determined at 1.1 °K by Yarnell et al., and exhibits a non monotonic behavior–with a maximum and a minimum–usually explained in terms of excitations called rotons, introduced by Landau. In the present work an attempt is made to describe the phenomenon within the Bohmian interpretation of QM. To this end, the effects of the intermolecular potential, taken to be essentially of the Lennard-Jones type modified to account for molecule finiteness, are included as a Vlasov-type self-consistent field. A dispersion relation is found, that is in quite good agreement with Yarnell’s curve.

Acoustic Band Gap Extension in One-Dimensional Solid/Fluid Phononic Crystal Heterostructure

The propagation of elastic longitudinal waves in one-dimensional (1D) phononic crystals (PNCs) consisting of alternating solid and fluid media is comprehensively analyzed in theory. We demonstrate the acoustic band gap (ABG) structure determined by the dispersion relation for longitudinal waves at normal incidence. According to the band structure, we design a sub-PNC by setting a reasonable thickness ratio of fluid and solid media, and then form a phononic heterostructure by merging this PNC and other PNC designed in advance. We have shown that the wide band gap exists in such a phononic heterostructure for elastic longitudinal waves at normal incidence. For oblique incidence, the wide band gap shifts towards high frequency regions, meanwhile a low-frequency band gap is split.

Computation of generalized and exact dispersion relations for longitudinal plasma waves in nonextensive statistics and the effects of the nonextensivity on the oscillation modes and damps

We have derived generalized dispersion relations for longitudinal waves in collisionless thermal plasma using linear Vlasov-Poisson kinetic model and nonextensive distributions for electrons. The Maxwellian limit of the dispersion relations, where the q-nonextensive parameter tends to one, is calculated. The generalized dispersion relations are reduced to polynomials for some specific values of q. The well-known modes of oscillations such as the Langmuir and electron acoustic waves have been obtained by solving the dispersion relations. Some new modes of oscillation are also found. Finally, the dependence of the oscillation modes and damps on q is discussed.

Fluctuations in collisional plasma in the presence of an external electric field

The theory of large-scale fluctuations in a plasma is used to calculate the correlations functions of electron and ion density with regard to particle collisions described within the Bhatnagar-Gross-Krook (BGK) model and the presence of a constant external electric field. The changes of plasma particle distribution functions due to an external electric field and their influence on the plasma dielectric response are taken into account. The dispersion relations for longitudinal waves in such a plasma are studied in details. It is shown that external electric field can lead to the ion-acoustic wave instability and anomalous growth of the fluctuation level. Detailed numerical studies of the general relations for electron number density fluctuations are performed and the effect of external electric field on the fluctuation spectra is studied.

Validation of the non-local elastic shell model for studying longitudinal waves insingle-walled carbon nanotubes

This study deals with the dispersion of longitudinal waves in a single-walled armchaircarbon nanotube, focusing on the effect of the microstructure of the carbon nanotube onthe wave dispersion. To reveal the effect analytically, the carbon nanotube is modelled as anon-local elastic cylindrical shell in the study. The dynamic equation of the non-localelastic cylindrical shell is established and the corresponding dispersion relation oflongitudinal waves is derived. These analytic results are verified via molecular dynamicssimulations, based on the Terroff–Brenner potential, for the propagation of variouslongitudinal waves in three single-walled armchair carbon nanotubes. The moleculardynamics simulations indicate that the longitudinal wave dispersion predicted by the modelof the non-local elastic cylindrical shell shows a good agreement with that ofmolecular dynamics simulations in a wide frequency range up to the terahertz region.They also show that both the microstructure of the carbon nanotube and thecoupling between the longitudinal wave and the radial motion play an importantrole in the dispersion of longitudinal waves in a single-walled armchair carbonnanotube.

Wave equations, dispersion relations, and van Hove singularities for applications of doublet mechanics to ultrasound propagation in bio- and nanomaterials.

A fundamental mathematical framework for applications of Doublet Mechanics to ultrasound propagation in a discrete material is introduced. A multiscale wave equation, dispersion relation for longitudinal waves, and shear waves are derived. The van Hove singularities and corresponding highest frequency limits for the Mth-order wave equations of longitudinal and shear waves are determined for a widely used microbundle structure. Doublet Mechanics is applied to soft tissue and low-density polyethylene. The experimental dispersion data for soft tissue and low-density polyethylene are compared with results predicted by Doublet Mechanics and an attenuation model based on a Kramers-Kronig relation in classical continuum mechanics.

Thermal effects in the ultrarelativistic two-stream instability

The dispersion relation for longitudinal waves in a one-dimensional ultrarelativistic plasma is calculated. Analytical and numerical results for the growth rate and frequency of the two-stream instability are presented as a function of the energy spread in the denser stream when the dilute stream is cold. The case of energy spreads in both beams is investigated numerically: it is found that relatively small energy spreads in both streams can lead to suppression of the instability.

Extension of Doughterty's model Fokker–Planck equation for a plasma

A generalization of the Appleton–Hartree equation is made to include the effects of energy-dependent electron-neutron collisions, Coulomb encounters and spatial dispersion. The frequency of electromagnetic waves propagating in a magneto-plasma is sufficiently high that the ion motion may be neglected compared with the electron motion. The present analysis extends Dougherty (1963, 1964) to include the simultaneous effect on wave propagation of Coulomb forces, spatial dispersion and energy-dependent electron-neutral collisions, where one or more of these effects can have a significant influence on circularly polarized waves propagating at frequencies near electron cyclotron resonance. The electrical conductivity tensor is expressible in terms of appropriate velocity moments of the electron distribution function. The electron velocity distribution function is determined by expanding the inverse of the differential operator of the linearized kinetic equation in a small parameter ε2', where in one case ε2 is the ratio of Coulomb collision frequency to signal frequency, and in the second case ε2 is the ratio of electron-neutral collision frequency to signal frequency. For wave propagation along the magnetic field, the dispersion relations for right-hand and left-hand circularly polarized waves, and also the dispersion relation for longitudinal waves, are solved numerically, and graphs are presented to show the effects of collisionless damping, velocity dependent electron-neutral collisions and Coulomb collisions.

Convective and Absolute Instabilities in Semiconductors Exhibiting Negative Differential Mobility

It is shown that, in a semiconductor exhibiting negative differential mobility, the coupling between diffusion effects and the tendency for space-charge accumulation can lead to temporal growth rather than spatial amplification if the dielectric relaxation frequency exceeds a certain threshold value. This conclusion results from an analysis of the dispersion relation for longitudinal waves in the semiconductor. The criterion for temporal growth is a condition for absolute instability, as opposed to convective instability, which would indicate rather spatial amplification.

Dispersion relation for longitudinal waves in a relativistic plasmas

Imre, Buti and others studied the longitudinal waves from the general dispersion relation. Here we have derived the dispersion relation for the longitudinal waves in relativistic plasmas, a result which cannot easily be brought in a physically more convenient form.

Effect of radiation reaction on plasma waves

The integro-differential equation of motion for a point electron is used in the Vlasov equation to obtain a dispersion relation for longitudinal waves.

Wave Propagaton in a Partly Ionized Gas. II

Simple dispersion relations for longitudinal waves are derived for plasmas satisfying the hydrodynamic equations subject to the conditions: 1010 < Nn0 < 1016 cm−3; 109 < Ne0 < 1014 cm−3; Te ≫ Ti,n; Ti,n < 104 °K < Te. In addition, the exact solutions to the dispersion equation are obtained (for arbitrary number densities and temperatures) in the low-frequency limit, and the damping factor for the low-frequency acoustic mode is calculated.

Excitation of Longitudinal Waves in a Bounded Collisionless Plasma

The relativistic form of the Vlasov equation is used to derive an integral equation for an externally produced electric field oscillating with frequency ω in a plasma bounded by two specularly reflecting walls. A tentative solution is constructed from exact solutions of the half-space problem. It is shown that the exact equation is satisfied by the tentative solution up to terms which are exponentially small in the number of Debye lengths across the plasma. It happens that the bounded plasma acts like a medium with a dielectric constant κ=[1−ωp2/ω2(1−5KT/2mc2)], unless the driving frequency lies in the small band ωp2(1−5KT/2mc2)<ω2<ωp2(1+KT/2mc2). When the driving frequency satisfies the inequality, then the plasma also supports standing waves which satisfy the relativistic dispersion relation for longitudinal waves.