Dispersion Of Plane Waves Research Articles

Fractional electrical impurity

We examine the localized mode and the transmission of plane waves across a capacitive impurity of strength Δ, in a 1D bi-inductive electrical transmission line where the usual discrete Laplacian is replaced by a fractional one characterized by a fractional exponent s. In the absence of the impurity, the plane wave dispersion is computed in closed form in terms of hypergeometric functions. It is observed that the bandwidth decreases steadily, as s decreases towards zero, reaching a minimum width at s = 0. The localized mode energy and spatial profiles are computed in closed form vía lattice Green functions. The profiles show a remnant of the staggered-unstaggered symmetry that is common in non-fractional chains. The width of the localized mode decreases with decreasing s, becoming completely localized at the impurity site at s = 0. The transmission coefficient of plane waves across the impurity is qualitatively similar to its non-fractional counterpart (s = 1), except at low s values ( s≪1 ). For a fixed exponent s, the transmission decreases with increasing Δ.

Magnon condensation in a dense nitrogen-vacancy spin ensemble

The feasibility of creating a Bose-Einstein condensate of magnons using a dense ensemble of nitrogen-vacancy spin defects in diamond is investigated. Through assessing a density-dependent spin-exchange interaction strength and the magnetic phase-transition temperature (${T}_{c}$) using the Sherrington-Kirkpatrick model, the minimum temperature-dependent concentration for magnetic self-ordering is estimated. For a randomly dispersed spin ensemble, the calculated average exchange constant exceeds the average dipole interaction strengths for concentrations approximately greater than 70 ppm, while ${T}_{c}$ is estimated to exceed 10 mK beyond 90 ppm, reaching 300 K at a concentration of approximately 450 ppm. On this basis, the existence of dipole-exchange spin waves and their plane-wave dispersion is postulated and estimated using a semiclassical magnetostatic description. This is discussed along with a ${T}_{c}$-based estimate of the four-magnon scattering rate, which indicates magnons and their condensation may be detectable in thin films for concentrations greater than 90 ppm.

Higher-Order Dispersion, Stability, and Waveform Invariance in Nonlinear Monoatomic and Diatomic Systems

Recent studies have presented first-order multiple time scale approaches for exploring amplitude-dependent plane-wave dispersion in weakly nonlinear chains and lattices characterized by cubic stiffness. These analyses have yet to assess solution stability, which requires an analysis incorporating damping. Furthermore, due to their first-order dependence, they make an implicit assumption that the cubic stiffness influences dispersion shifts to a greater degree than the quadratic stiffness, and they thus ignore quadratic shifts. This paper addresses these limitations by carrying-out higher-order, multiple scales perturbation analyses of linearly damped nonlinear monoatomic and diatomic chains. The study derives higher-order dispersion corrections informed by both quadratic and cubic stiffness and quantifies plane wave stability using evolution equations resulting from the multiple scales analysis and numerical experiments. Additionally, by reconstructing plane waves using both homogeneous and particular solutions at multiple orders, the study introduces a new interpretation of multiple scales results in which predicted waveforms are seen to exist over all space and time, constituting an invariant, multiharmonic wave of infinite extent analogous to cnoidal waves in continuous systems. Using example chains characterized by dimensionless parameters, numerical studies confirm that the spectral content of the predicted waveforms exhibits less growth/decay over time as higher-order approximations are used in defining the simulations' initial conditions. Thus, the study results suggest that the higher-order multiple scales perturbation analysis captures long-term, nonlocalized invariant plane waves, which have the potential for propagating coherent information over long distances.

Penetration and screening of perpendicularly launched electromagnetic waves through bounded supercritical plasma confined in multicusp magnetic field

The question of electromagnetic wave penetration and screening by a bounded supercritical (ωp>ω with ωp and ω being the electron-plasma and wave frequencies, respectively) plasma confined in a minimum B multicusp field, for waves launched in the k⊥Bo mode, is addressed through experiments and numerical simulations. The scale length of radial plasma nonuniformity (|ne/(∂ne/∂r)|) and magnetostatic field (Bo) inhomogeneity (|Bo/(∂Bo/∂r)|) are much smaller than the free space (λo) and guided wavelengths (λg). Contrary to predictions of plane wave dispersion theory and the Clemow–Mullaly–Allis (CMA) diagram, for a bounded plasma a finite propagation occurs through the central plasma regions where αp2=ωp2/ω2≥1 and βc2=ωce2/ω2⪡1(∼10−4), with ωce being the electron cyclotron frequency. Wave screening, as predicted by the plane wave model, does not remain valid due to phase mixing and superposition of reflected waves from the conducting boundary, leading to the formation of electromagnetic standing wave modes. The waves are found to satisfy a modified upper hybrid resonance (UHR) relation in the minimum B field and are damped at the local electron cyclotron resonance (ECR) location.

Counterpropagating nondiffracting beams through reflection gratings

We investigate the counterpropagation of paraxial non-diffracting optical beams through a medium hosting a bulk reflection grating in the quasi-Bragg matching condition. The impact of the relative magnitude of the Bragg detuning parameter and the grating depth on the plane wave dispersion relation allows us to identify three distinct regimes where counterpropagation and interaction of nondiffracting beams show qualitatively different features, encompassing longitudinally invariant, periodic or exponential intensity profiles. In one of the identified regimes the dispersion relation is not monotonic and the consequent "longitudinal degeneracy" allows the investigation of new class of nondiffracting beams characterized by a double spectral ring profile.

Sensitivity of structural models of composite material to structural length scales

It is shown that there are models in the micromechanics of composite materials (so-called structural models of the second order) that are sensitive to structural change from micro-to nanoscale, whereas the model of effective constants (structural model of the first order) in the mechanics of composites does not have such a property. The structural model of elastic mixture is used as an example to reveal a distinction between the descriptions of the dispersion of plane waves propagating in fibrous micro-and nanocomposites of similar structure

Finite element dispersion of cylindrical and spherical acoustic waves

A reference solution for cylindrical waves is obtained by WKB approximation. Finite element representations of both cylindrical and spherical waves satisfy the resolution-dependent plane wave dispersion relation, and approximate the desired decay rates independent of mesh resolution.

Non-linear dispersion of plane wave in granular media

A high-gradient elastic stress-strain model that accounts for the effects of gradient of strain and gradient of rigid body rotation is presented. The present model is distinct from its micromechanics approach; the constitutive coefficients are expressed as explicit functions of inter-particle stiffness and particle size. The stress-strain model is then applied to analyze the propagation of plane waves in granular media. The present stress-strain model leads to a phenomenon of non-linear dispersion of waves. The dispersion relations derived from the present high-gradient model for granular media are compared with those from crystal lattice models and non-local models. Links among the present model, crystal lattice model, and non-local model are discussed. According to the derived dispersion relations, the granular material is a natural filter, in which short-length waves and high-frequency waves cannot transmit through. The effects of inter-particle stiffness and particle size on the admissible range of wave lengths and frequencies are also discussed.

Duality transformations, Green dyadics and plane-wave solutions for a class of bianisotropic media

Duality transformations are defined as linear transformations between pairs of vector fields and sources. A class of bianisotropic media is defined so that a pair of duality transformations can be found which leave the media invariant (self dual). It is seen that for such media the permittivity, permeability and nonreciprocity parameters must be multiples of the same dyadic while the chirality parameter may be an arbitrary dyadic. Any source and field in such a medium can be decomposed in a pair of self-dual components which see the bianisotropic medium as two equivalent anisotropic media of affine-isotropic type. The Green dyadics in these two media can be solved through affine transformations and the Green dyadics of the original bianisotropic medium can be written in explicit analytic form. The plane-wave dispersion equation can be solved through a similar self-dual decomposition of fields.

Errata - Reflection function of a plane sound wave in a cylindrical tube

As dissipation and dispersion of plane waves in a cylindrical tube are proportional to the square root of frequency, the form of the impulse response can be obtained analytically. We analyse the reflection function at the input of the tube, i.e. the inverse Fourier transform of the reflection coefficient, especially the important distortions due to a numerical treatment and the property of stability through convolution. Using this property, the reflection function is determined for a sequence of tubes as well as for a connection of three tubes. An extension of the formulas to the case of rigid-walled closed cylindrical tube is given in Appendix A Etude de la fonction de reflexion a l'entree d'un tuyau et des importantes distorsions dues au traitement numerique ainsi que de la propriete de stabilite par convolution. Fonction de reflexion pour plusieurs tubes en serie et pour un branchement de 3 tubes

Reflection function of a plane sound wave in a cylindrical tube

As dissipation and dispersion of plane waves in a cylindrical tube are proportional to the square root of frequency, the form of the impulse response can be obtained analytically. We analyse the reflection function at the input of the tube, i.e. the inverse Fourier transform of the reflection coefficient, especially the important distortions due to a numerical treatment and the property of stability through convolution. Using this property, the reflection function is determined for a sequence of tubes as well as for a connection of three tubes. An extension of the formulas to the case of rigid-walled closed cylindrical tube is given in Appendix A Etude de la fonction de reflexion a l'entree d'un tuyau et des importantes distorsions dues au traitement numerique ainsi que de la propriete de stabilite par convolution. Fonction de reflexion pour plusieurs tubes en serie et pour un branchement de 3 tubes

Spiral waves in the Belousov-Zhabotinskii reaction

The beautiful spiral waves of oxidation in the Belousov-Zhabotinskii reaction are the source of many interesting and important questions about periods structures in excitable media. It has long been known that these spirals are similar to involutes of circles, at least some distance from the center, but until now, no way has been known to determine the correct wavelength and frequency. In this paper, we show that the parameters of a spiral wave can be viwed s eigenvalues of a problem with unique solution. The critical ingredients of the theory are the effects of curvature on the propagation of wavefronts in two-dimensional media, and the dispersion of plane waves Our analytical results are shown to be in good agreement with experimental data for the Belousov-Zhabotinskii reagent.

Control of elastic plane wave dispersion in two-dimensional finite element meshes

The control of discretization dispersion and spurious oscillations in finite element plane wave propagation problems is examined. Using a lumped mass spatial discretization and an explicit temporal integration, two-dimensional plane strain and plane stress meshes are considered. The control of the discretization errors in uniform meshes is very effective using a time step exactly equal to the stability limit. The magnitudes of spurious wave reflections in non-uniform meshes depend largely on the ratio of element lengths. Actual stress wave transmissions and reflections in composite bars are accurately predicted.

Plasma fluid equations in the high collision-frequency limit

The nature of the fluid approximation for plasmas in the high collision-frequency limit is examined in an effort to understand some of the limits of validity of the commonly used ideal magnetohydrodynamic equations. The plane wave dispersion relation for a homogeneous medium is evaluated. The dependence of the damping on the various dissipative terms is examined for each type of wave predicted by the dispersion relation. The effects of these terms are then compared, using the standard values for transport quantities with collision frequency larger than gyro-frequency. When the total dissipation is considered as a function of collision frequency, it is found that there is a minimum in the region where the collision and gyro-frequencies are about equal. This is the régime for which ideal magnetohydrodynamics is a good approximation. As collision frequency is increased, dissipation reaches a maximum when the wave is in resonance with the resistive skin depth, and then decreases into a régime where ordinary sound and light waves propagate. Lower collision frequencies, where thermal conduction and equilibration are important, are considered.

Linear theory of nonlocal elasticity and dispersion of plane waves

The equations of the nonlocal elasticity given in [1]and[2] are linearized. The dispersion relations are obtained for one dimensional plane waves. The nonlocal material moduli are determined to fit exactly the acoustical branch of elastic waves within one Brillouin zone in periodic one dimensional lattices.

Pulse Propagation in a Dispersive Medium with Moving Density Profile

Space-time ray methods are developed and employed to obtain asymptotic solutions for pulse propagation in media with slow spatial and temporal inhomogeneities. Emphasis is placed on cold isotropic plasmas whose plasma frequency ωp has a space-time dependence of the form (z−Vt), thereby simulating ordered profile motion. Asymptotic results are constructed directly by the ray method, are related to those in a spatially inhomogeneous but temporally invariant plasma by the Lorentz transformation, and are also compared with an exact closed form solution for the special case of exponential variation of ωp(z − Vt). The presentation stresses physical attributes of the propagation process through extensive use of space-time ray diagrams and plane wave dispersion surfaces.

Theory of mixtures for micromorphic materials—I: Balance laws

A micromorphic theory is developed for polycrystalline mixtures, granular composites, and fluid suspensions. By taking account of local motions independent of the average motion, the theory aims at extending the classical theory to include physical phenomena in which the wave length of the imposed motion is comparable to the average dimension of the intrinsic discontinuities in the material, e.g. the average grain size or inter-grain distance. For each constituent and for the mixture as a whole, balance laws are constructed for mass, microinertia moment, momentum, moment of momentum, energy, and the production of entropy. These balance equations are then reduced to the micropolar case in which the local motions are restricted to local rotations only. In Part II (forthcoming) we present a constitutive theory for micromorphic and micropolar mixtures and discuss the dispersion of plane waves.

Asymptotic theory of pulse compression in dispersive media

This paper deals with the long distance propagation of FM pulses in dispersive media. The treatment is based on an asymptotic theory which utilizes space-time rays and plane wave dispersion surfaces as well as integral representations. The discussion includes the propagation of prescribed FM signals in homogeneous media as well as synthesis of signals to achieve maximum pulse compression. Special attention is given to the caustics and foci exhibited by the space-time ray system when pulse compression takes place, and to the description of the field in the corresponding space-time transition regions where simple asymptotic constructs are inadequate. While the theory is presented for media with relatively arbitrary spatial and dispersive characteristics, detailed calculations are carried out for a homogeneous isotropic cold plasma for which results can be obtained in explicit form.

The coherent wave in a random uniaxial anisotropic medium

The coherent (mean) wave in an unbounded medium, characterized by random fluctuations superimposed on a deterministic background, is studied. Both the random fluctuations and the deterministic background are assumed to be uniaxial anisotropic. The procedure for obtaining the mean field is presented and generally follows the usual development of Dyson's equation (the diagrammatical representation of the integrodifferential equation for the mean field) except for modifications needed to treat the anisotropy. The use of these modifications allows the Feynman diagrammatical representation of Dyson's equation to remain unaltered except for the mass operator containing additional terms. An example is used to illustrate the procedure required to obtain an effective dielectric constant and the plane-wave dispersion relationship.

Rays, Dispersion Surfaces and Their Uses for Radiation and Diffraction Problems

In certain asymptotic parameter regimes, propagation, radiation and diffraction processes in a linear field may be described locally in terms of plane waves and schematized in terms of rays. Properties of space-time rays descriptive of time-dependent fields in lossless dispersive, isotropic or anisotropic media, and corresponding features of time-harmonic rays, are reviewed, with emphasis on the relation between rays and the plane wave dispersion surfaces characteristic of the medium. Graphical procedures are given for ray tracing in the presence of weak, continuous inhomogeneities in the medium properties; abrupt spatial or temporal medium changes ; and discontinuities exciting certain types of diffracted rays. These aspects are then related to integral representations for the field and to its asymptotic behavior characterized by saddle points and singularities in the integrands. Attention is given to transition regions wherein simple asymptotic methods fail and the ray description must be modified. The ...