General Dispersion Equation Research Articles

On EMEC Instability in Bi‐Kappa Distributed Space Plasmas Under the Influence of Parallel DC Electric Field

ABSTRACTEMEC waves driven by kinetic anisotropies and suprathermal particles are expected to play an important role at dissipative scales in the solar wind and planetary magnetospheres. Moreover, the correlation of EMEC waves with electric field magnitude and direction can be a tool to probe the inner magnetosphere. Thus, in this manuscript, we present the study of EMEC waves driven by temperature anisotropy in bi‐Kappa distributed space plasmas in the presence of a parallel DC electric field. The dispersion equation bearing the influence of electric field and suprathermal particles followed by the analytical expressions for wave frequency and wave growth rate is derived. The general dispersion equation is solved numerically to unveil the effect of suprathermal particles, temperature anisotropy, and the magnitude of the electric field on growth rate as well as the domain of unstable wave numbers, and the obtained numerical outcomes are compared with those of the bi‐Maxwellian model. Moreover, the maximum growth rates for EMEC waves have also been obtained.

ICE modes with high wave numbers

This work deals with high-k modes of the ion cyclotron emission (ICE)—the modes with frequencies close to ion cyclotron harmonics and wave numbers well exceeding those of fast magnetoacoustic modes (FMM). These modes exist due to finite Larmor radius of the ions. They are responsible for the ICE recently observed in the DIII-D tokamak [N. A. Crocker et al., Nucl. Fusion 62, 026023 (2022)]. The work is aimed at knowing the radial structure of high-k eigenmodes. Its analysis is relied on a comprehensive study carried out by employing a general dispersion equation for traveling waves, which allowed to group modes and determine conditions of their existence. Differential equations for various eigenmodes (standing waves) were obtained and analyzed. In particular, electrostatic modes and electromagnetic modes with various ratios of the longitudinal phase velocity to the electron thermal velocity were considered. A numerical code solving these equations was developed. It is found that high-k modes with sufficiently large poloidal mode numbers are located at the plasma periphery, and their amplitudes in the plasma core are very small. It is concluded that high-k modes are not an exceptional case, although presumably FMMs can explain most ICE experiments. They exist in plasmas with various magnitudes of β, representing different wave branches associated with finite Larmor radius of the ions. The mentioned experiment on DIII-D is discussed.

Anisotropy and dispersion of elastic waves in periodically multilayered anisotropic media

The propagation characteristics of elastic waves in general periodically multilayered media formed by alternate arrangement of two or more arbitrarily-anisotropic materials are comprehensively studied. Combining with the state-space formalism for obtaining the wave solutions to the constituent layers, the Floquet-Bloch theorem for providing the periodic relations between neighboring unit cells, the method of reverberation-ray matrix (MRRM) is employed to derive the general dispersion equations of the periodically multilayered anisotropic media. By solving the dispersion equations in cases of all possible sorts of wavenumber components, the complete anisotropic and dispersion characteristics of elastic waves can be provided. By numerical examples, the slowness and the phase velocity curves are provided to summarize the innovative anisotropic characteristics of elastic waves in three coordinate planes. The effects of the anisotropic parameters on the slowness/phase velocity curves have also been discussed. Moreover, all kinds of dispersion curves including the frequency-related and phase velocity-related curves along the directions perpendicular, parallel, and oblique to the layering are provided to reflect the general dispersion characteristics of elastic waves from various viewpoints. It is found that the innovative anisotropy characteristics of elastic waves in the periodically multilayered anisotropic media include that the slowness curves have periodicity in the thickness direction and the least positive period decreases with the increasing of frequency. As the frequency becomes big enough, the slowness/phase velocity curves show interactions between the modes within the overlapping neighboring periods. The common dispersion characteristics of elastic waves along various directions include the symmetry of wavenumber, the asymptotic of the wavelength and the phase velocity curves to the frequency lines corresponding to zero wavenumber, as well as the cutoff property of the phase velocities of the lowest three basis modes. The waves in oblique directions are special in that the mode transition and repulsion due to the zone folding effect of periodicity along thickness as well as the cutoff property of higher-than-third-order overtones along layering are simultaneously in effective.

On the decay property of the cubic fourth-order Schrödinger equation

In this short paper, we prove that the solution of the cubic fourth-order Schrödinger equation (4NLS) on R d \mathbb {R}^d ( 5 ≤ d ≤ 8 5 \leq d \leq 8 ) enjoys the same decay property as its linear solution does. This result is proved via a bootstrap argument based on the corresponding global result by Pausader [J. Funct. Anal. 256 (2009), pp. 2473–2517]. This result can be extended to more general dispersive equations (including some more 4NLS models) with scattering asymptotics.

Propagation of surface plasmons in a quantum collisional semiconductor plasma

In this work, the propagation of surface plasmons in a quantum collisional semiconductor plasma is theoretically investigated using the quantum hydrodynamic model. The general dispersion equation of surface plasmons is deduced and solved in the presence of collisional effects and quantum tunneling of electrons and holes. It is shown that collisions play a significant role in the decay or growth of surface plasmons. Moreover, it is found that the quantum tunneling of electrons and holes increases the phase velocity, the group velocity, and the growth rate of instability. It is also indicated that the growth rate of instability is decreased with the increase of the ratio of hole density to electron density and the ratio of electron effective mass to hole effective mass.

Kelvin–Helmholtz stability of rotating magnetoplasma with electron inertia

Abstract In this theoretical exploration, the stabilizing or destabilizing impacts of the rotation, electron inertia, and electrical resistivity on the Kelvin–Helmholtz stability in two-superimposed incompressible magnetized plasma fluids incorporating finite ion Larmor radius (FLR) correction and suspended dust particulates are studied. The linearized perturbation equations for the Kelvin–Helmholtz instability problems are determined based on the magnetohydrodynamic (MHD) model. The general dispersion equation is derived by using appropriate boundary conditions. By the numerical estimation, the finite ion Larmor radius does not have any significant role in the Kelvin–Helmholtz instability of the magnetoplasma medium. The graphical estimates reveal the destabilization impact of the resistivity and electron inertia on the Kelvin–Helmholtz hydrodynamic plasma fluid system. In this paper, graphical representations have also analyzed the effect of rotation on the Kelvin–Helmholtz stability growth rate with the variation of electron inertia and resistivity. This current analysis provides pertinent information about the significant involvement of this considered system in space and astrophysical structures.

Magnetoinductive waves in attenuating media

The capability of magnetic induction to transmit signals in attenuating environments has recently gained significant research interest. The wave aspect—magnetoinductive (MI) waves—has been proposed for numerous applications in RF-challenging environments, such as underground/underwater wireless networks, body area networks, and in-vivo medical diagnosis and treatment applications, to name but a few, where conventional electromagnetic waves have a number of limitations, most notably losses. To date, the effects of eddy currents inside the dissipative medium have not been characterised analytically. Here we propose a comprehensive circuit model of coupled resonators in a homogeneous dissipative medium, that takes into account all the electromagnetic effects of eddy currents, and, thereby, derive a general dispersion equation for the MI waves. We also report laboratory experiments to confirm our findings. Our work will serve as a fundamental model for design and analysis of every system employing MI waves or more generally, magnetically-coupled circuits in attenuating media.

Parametric resonance of acoustic and optical phonons in GaAs/GaAsAl quantum well in the presence of laser field

In this paper, we analytically investigated the possibility of parametric resonance of acoustic and optical phonons. We obtained a general dispersion equation for parametric amplification and transformation of phonons. The dispersions of the resonant acoustic phonon modes and the threshold amplitude of the field for acoustic phonon parametric amplification are obtained. The parametric amplification for acoustic phonons in a GaAs/GaAsAl quantum well can occur under the condition that the amplitude of the external electromagnetic field is higher than the threshold amplitude. Full Text: PDF ReferencesB. A. Glavin, V. A. Kochelap, T. L. Linnik, P. Walker, A. J. Kent and M. Henini, "Monochromatic terahertz acoustic phonon emission from piezoelectric superlattices", Journal of Physics: Conference Series, Vol 92, (2007). CrossRef O. A. C. Nunes, "Piezoelectric surface acoustical phonon amplification in graphene on a GaAs substrate", Journal of Applied Physics 115, 233715 (2014). CrossRef Yu. E. Lozovik, S. P. Merkulova, I. V. Ovchinnikov, "Sasers: resonant transitions in narrow-gap semiconductors and in exciton system in coupled quantum wells", Phys. Lett. A 282, 407-414, (2001). CrossRef R.P. Beardsley, A.V. Akimov, M. Henini and A.J. Kent, "Coherent Terahertz Sound Amplification and Spectral Line Narrowing in a Stark Ladder Superlattice", PRL 104, 085501, (2010). CrossRef Pascal Ruello, Vitalyi E. Gusev, "Physical mechanisms of coherent acoustic phonons generation by ultrafast laser action", Ultrasonics 56, 21-35, (2015). CrossRef L. Esaki, in Proc. 17th Int. Conf. Phys. Semiconductors, San Francisco, CA, Aug, J.D. Chadi and W.A. Harrison, Eds, Berlin: Springer- Verlag, 473, (1984). DirectLink W. Xu, F. M. Peeters and J. T. Devreese, "Electrophonon resonances in a quasi-two-dimensional electron system", Phys. Rev. B 48, 1562 (1993). CrossRef Tran Cong Phong, Nguyen Quang Bau, "Parametric resonance of acoustic and optical phonons in a quantum well", Journal of the Korean Physical Society, Vol. 42, No. 5, 647-651, (2003). DirectLink

Mathematical modelling of Love waves propagation in viscoelastic waveguide loaded with complex fluids

Love waves propagation in a viscoelastic waveguide loaded on its surface with viscoelastic fluids of finite thickness is investigated in this paper. The Maxwell and Kelvin-Voigt constitutive equations are employed in order to describe the fluid viscoelasticity. By solving the equations of motion in the different media (viscoelastic fluid, viscoelastic waveguide and elastic substrate) and imposing the suitable boundary conditions, an accurate and simple generalized complex dispersion equation is established for Love waves. Subsequently, a comparison is made with the published complex dispersion equations in the literature in some particular cases, and a very good agreement is showed. A detailed study was conducted by varying key parameters such as operating frequency, waveguide thickness and fluid thickness. The waveguide surface was subjected to various glycerol concentrations, with a wide range of dynamic viscosity, representing both Newtonian and viscoelastic behaviors. Theoretical analysis shows that to reasonably predict the characteristics responses of Love waves, the Maxwell fluid is more appropriate for low glycerol concentration and, the Kelvin-Voigt fluid is more suitable for high glycerol concentration and at high frequency. Results also evaluated the influence of layer thickness on the dispersion curves. The obtained results can be very useful in the design and optimization of Love wave fluid sensors.

МГД-волны в столкновительной плазме солнечной короны и земной ионосферы

We have studied MHD waves (Alfvén and fast compressional modes) in a homogeneous collisional three-component low-β plasma. The three-component plasma consists of electrons, ions, and neutrals with arbitrary ratio between collision frequencies and wave time scales. We have derived a general dispersion equation and relationships for phase velocity and collisional damping rates for MHD modes for various limiting cases: from weak collisions to a strong collisional coupling, and for longitudinal and oblique propagation. In a weak collision limit, the MHD eigen-modes are reduced to ordinary low-damping Alfvén and fast magnetosonic waves. For a partially ionized plasma with a strong collisional coupling of neutrals and ions, velocities of magnetosonic and Alfvén waves are substantially reduced, as compared to the Alfvén velocity in the ideal MHD theory. At a very low frequency, when neutrals and ions are strongly coupled, a possibility arises of weakly damping MHD modes, called “decelerated” MHD modes. These modes can be observed in the solar corona/chromosphere and in the F layer of the terrestrial ionosphere.

MHD WAVES IN THE COLLISIONAL PLASMA OF THE SOLAR CORONA AND TERRESTRIAL IONOSPHERE

We have studied MHD waves (Alfvén and fast compressional modes) in a homogeneous collisional three-component low-β plasma. The three-component plasma consists of electrons, ions, and neutrals with arbitrary ratio between collision frequencies and wave time scales. We have derived a general dispersion equation and relationships for phase velocity and collisional damping rates for MHD modes for various limiting cases: from weak collisions to a strong collisional coupling, and for longitudinal and oblique propagation. In a weak collision limit, the MHD eigen-modes are reduced to ordinary low-damping Alfvén and fast magnetosonic waves. For a partially ionized plasma with a strong collisional coupling of neutrals and ions, velocities of magnetosonic and Alfvén waves are substantially reduced, as compared to the Alfvén velocity in the ideal MHD theory. At a very low frequency, when neutrals and ions are strongly coupled, a possibility arises of weakly damping MHD modes, called “decelerated” MHD modes. These modes can be observed in the solar corona/chromosphere and in the F layer of the terrestrial ionosphere.

Auxiliary Top‐Illumination in the Kretschmann–Raether Configuration: A Theoretical Study on the Surface Plasmon Resonance Response

A new variation of surface plasmon resonance (SPR) sensors design with Kretschmann–Raether (KR)‐type configuration is presented. Using a rotational incident angle scanning platform, the conductor nanolayer is normally irradiated with an auxiliary unpolarized light beam from the dielectric side. Both illumination sources should have a coherence length much larger than their source/Au layer optical path. Beginning with a Lorentzian‐type oscillator model, a general dispersion equation for the metal nanolayer is developed that incorporates interband transitions and polychromaticity of the top‐illumination beam. SPR curves are then predicted based on the transfer‐matrix method. Simulation results show that the top‐illumination frequency and its relative amplitude with respect to the bottom laser beam may impart significant changes to the SPR curve. It is shown that the implementation of short‐wave ultraviolet light (180 nm) may considerably improve the performance of SPR sensors in terms of the figure‐of‐merit. Larger wavelengths generally lead to performance deterioration. Based on the transverse magnetic solutions of the wave equation, a theoretical interpretation of the results is presented that confirms the strong effect of top illumination on the magnitude of the decay lengths in the conductor/dielectric and the propagation constant.

On time-splitting methods for nonlinear Schrödinger equation with highly oscillatory potential

In this work, we consider the numerical solution of the nonlinear Schrödinger equation with a highly oscillatory potential (NLSE-OP). The NLSE-OP is a model problem which frequently occurs in recent studies of some multiscale dynamical systems, where the potential introduces wide temporal oscillations to the solution and causes numerical difficulties. We aim to analyze rigorously the error bounds of the splitting schemes for solving the NLSE-OP to a fixed time. Our theoretical results show that the Lie–Trotter splitting scheme is uniformly and optimally accurate at the first order provided that the oscillatory potential is integrated exactly, while the Strang splitting scheme is not. Our results apply to general dispersive or wave equations with an oscillatory potential. The error estimates are confirmed by numerical results.

Exponential decay and symmetry of solitary waves to Degasperis-Procesi equation

We improve the decay argument by Bona and Li (1997) [5] for solitary waves of general dispersive equations and illustrate it in the proof for the exponential decay of solitary waves to steady Degasperis-Procesi equation in the nonlocal formulation. In addition, we give a method which confirms the symmetry of solitary waves, including those of the maximum height. Finally, we discover how the symmetric structure is connected to the steady structure of solutions to the Degasperis-Procesi equation, and give a more intuitive proof for symmetric solutions to be traveling waves. The improved argument and new method above can be used for the decay rate of solitary waves to many other dispersive equations and will give new perspectives on symmetric solutions for general evolution equations.

Circumferential defect detection using ultrasonic guided waves

Purpose Quantitatively detecting surface defects in a circular annulus with high levels of accuracy and efficiency has been paid more attention by researchers. The purpose of this study is to investigate the theoretical dispersion equations for circumferential guided waves and then develop an efficient technique for accurate reconstruction of defects in pipes. Design/methodology/approach The methodology applied to determine defects in pipelines includes four steps. First, the theoretical work is carried out by developing the appropriate dispersion equations for circumferential guided waves in a pipe. In this phase, formulations of strain-displacement relations are derived in a general equidistant surface coordinate. Following that, a semi-analytical finite element method (SAFEM) is applied to solve the dispersion equations. Then, the scattered fields in a circular annulus are calculated using the developed hybrid finite element method and simulation results are in accord with the law of conservation of energy. Finally, the quantitative detection of Fourier transform (QDFT) approach is further enhanced to efficiently reconstruct the defects in the circular annuli, which have been widely used for engineering applications. Findings Results obtained from four numerical examples of flaw detection problems demonstrate the correctness of the developed QDFT approach in terms of accuracy and efficiency. Reconstruction of circumferential surface defects using the extended QDFT method can be performed without involving the analytical formulations. Therefore, the streamlined process of inspecting surface defects is well established and this leads to the reduced time in practical engineering tests. Originality/value In this paper, the general dispersion equations for circumferential ultrasonic guided waves have been derived using an equidistant surface coordinate and solved by the SAFEM technique to discover the relationship between wavenumber of a wave and its frequency. To reconstruct defects with high levels of accuracy and efficiency, the QDFT approach has been further enhanced to inspect defects in the annular structure.

Parametric resonance of acoustic and optical phonons in a doped semiconductor superlattice in the presence of a laser field

In this paper, we analytically investigated the possibility of parametric resonance of acoustic and optical phonons. We obtained a general dispersion equation for parametric amplification and transformation of phonons. The dispersions of the resonant acoustic phonon modes and the threshold amplitude of the field for acoustic phonon parametric amplification are obtained. The parametric amplification for acoustic phonons in a doped semiconductor superlattice can occur under the condition that the amplitude of the external electromagnetic field is higher than the threshold amplitude.

Effect of Interface Damage on Band Structures in a Periodic Multilayer Plate

The interface/interphase is modeled by distributed springs, and the degree of interface damage is characterized by the flexibility coefficient of the springs. Considering different degrees of damage at different interfaces, a general dispersion equation by the transfer matrix method was obtained. For the purpose of application, the transmission coefficient of finite-period phononic crystals is also derived. Additionally, the band structures and the transmission coefficients are calculated for different degrees of damage to confirm the interface effect on band gaps. A numerical simulation of the transmission coefficient is performed using the commercial FEM software Comsol Multiphysics. The numerical results obtained agree well with the analytical solution. Finally, based on a monotonic decreasing relationship between the degree of interface damage and the central frequency of band gaps and a monotonic relationship between the degree of interface damage and the amplitude of transmission coefficients, a new method how to evaluate the degree of interface damage theoretically is proposed.

On wave propagation in a rotating random micropolar generalized thermoelastic medium

This article examines the problem of wave propagation in a rotating random micropolar generalized thermoelastic medium. The entire frame of reference is assumed to be rotating with a uniform angular velocity where is a unit vector representing the direction of the axis of rotation. The smooth perturbation technique amenable to stochastic differential equations has been employed. The generalized dispersion equation has been derived and analyzed. The effects due to random variations of micropolar elastic and generalized thermal parameters have been observed. The effect of rotation on wave propagation is discussed. Change of phase speed occurs on account of random fluctuations of inhomogeneities of the medium. Attenuation coefficients for both random and nonrandom media have been computed both theoretically and numerically. Some illustrative graphs have been drawn to demonstrate the effects of rotation and generalized thermoelastic parameters on wave propagation in the micropolar medium.

Hydrodynamic Dispersion Coefficient of Urea in Silty Loam Soil

The modern theme of agriculture is not only to increase production but also to minimize undesirable environmental effects. Leaching of surface-applied fertilizer is the major source of groundwater pollution. Nitrogenous fertilizers are the most popular among the Indian farmers, which on leaching reach the groundwater in different forms (NH4-N, NO3-N, etc). NO3-N leaches faster than other types, remains in-reactive in groundwater, moves with the velocity of groundwater and contaminates it. Contamination arises when NO3-N accumulates in groundwater and consumed in high amount by humans and animals, may result in adverse health effects. For the study of contaminant transport phenomenon in porous medium, a general convection dispersion equation is used, in which dispersion coefficient is one of the primary parameters necessary to be determined for a particular soil. Keeping it in view a study was conducted to assess different available techniques to determine the dispersion coefficient with the help of soil columns having silty loam soil as soil medium. The value of the dispersion coefficient obtained for silty loam soil, by this method was equal to 0.00576 m2.

Surface wave in a Maxwell liquid-saturated poroelastic layer

An analytical approach of the propagation and attenuation of Love waves in a viscoelastic liquid-saturated poroelastic layer has been considered in this paper. The equations of motion have been formulated separately for different media under suitable boundary conditions at the interface of viscoelastic liquid, poroelastic layer and elastic substrate. Following Biot’s theory of poroelasticity, a new accurate and simple generalized dispersion equation has been established to design Love wave liquid sensors. The effect of liquid shear viscosity on the Love waves velocity has been studied. The influence of thickness and porosity of the waveguide layer has also been shown on the Love waves velocity and attenuation. The various investigations results can serve as benchmark solutions in design of liquid sensors and nondestructive testing.