Energy characteristics of spin waves with noncollinear group and phase velocities in an infinite ferrite medium

2016 ◽  
Author(s):  
Локк ◽  
Edvin Lokk
Keyword(s):  
Velocity Vector ◽  
Uniform Magnetic Field ◽  
Spin Waves ◽  
Angular Width ◽  
Energy Characteristics ◽  
Phase Velocities ◽  
Inflection Points ◽  
Isotropic Media ◽  
Group And Phase Velocities ◽  
Vector Orientation

Group velocity vector orientation ψ versus wave vector orientation  ´ is calculated for spin waves propagating in an infinite ferrite medium magnetized to saturation by uniform magnetic field H0. It is found that dependence ψ( ´) can have both extremum points and inflection points, in which derivative dψ/d is equal to zero. It means, that superdirected spin wave beams can arise in this medium. In the plane, containing vector H0, the angular width of such beams is not change during propagation and in the perpendicular plane the angular width of these beams, as in the isotropic media, is equal to λ/D.

Group and Phase Velocities of a Travelling Disturbance in the F Region of the Ionosphere

Nature ◽  
1961 ◽  
Vol 191 (4784) ◽  
pp. 157-157 ◽  
Author(s):  
L. H. HEISLER ◽  
J. D. WHITEHEAD
Keyword(s):  
Phase Velocities ◽  
F Region ◽  
Group And Phase Velocities

Nonlinear wave equation with intrinsic wave particle dualism

1976 ◽  
Vol 54 (13) ◽  
pp. 1383-1390 ◽  
Author(s):  
J. J. Klein
Keyword(s):  
Wave Equation ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Gordon Equation ◽  
Classical Mechanics ◽  
Carrier Wave ◽  
Sine Gordon Equation ◽  
Wave Speeds ◽  
Phase Velocities ◽  
Group And Phase Velocities

A nonlinear wave equation[Formula: see text]derived from the sine–Gordon equation is shown to possess a variety of solutions, the most interesting of which is a solution that describes a wave packet travelling with velocity uc modulating a carrier wave travelling with velocity uc. The envelop and carrier wave speeds agree precisely with the group and phase velocities found by de Broglie for matter waves. No spreading is exhibited by the soliton, so that it behaves exactly like a particle in classical mechanics. Moreover, the classically computed energy E of the disturbance turns out to be exactly equal to the frequency ω of the carrier wave, so that the Planck relation [Formula: see text] is automatically satisfied without postulating a particle–wave dualism.

scholarly journals Formation of electron holes in the long-time evolution of the bump-on-tail instability

2017 ◽  
Vol 35 (4) ◽  
pp. 706-721 ◽  
Author(s):  
Magdi Shoucri
Keyword(s):  
Distribution Function ◽  
Phase Space ◽  
Electron Density ◽  
Plasma Frequency ◽  
Unstable Wave ◽  
Trapped Particles ◽  
Phase Velocities ◽  
Inflection Points ◽  
Electron Holes ◽  
Fine Resolution

AbstractAn Eulerian Vlasov code is applied for the numerical solution of the one-dimensional Vlasov–Poisson system of equations for electrons, and with ions forming an immobile background. We study the non-linear evolution of the bump-on-tail instability in the case when the system length L is greater than the wavelength λ of the unstable mode, with a beam density of 10% of the total density, nb = 0.1. We follow the growth and the saturation of an initially unstable wave perturbation, and the formation of a traveling Bernstein–Greene–Kruskal (BGK) mode, which evolves out of the instability. This first stage is followed by sidebands growing from round-off errors which develop and disrupt the BGK equilibrium. In the excited spectrum, mode coupling is mediated by the oscillating resonant particles and results in the electric energy of the system flowing to the longest wavelengths (inverse cascade), and reaching in the asymptotic state a steady state with constant amplitude oscillation modulated by the persistent oscillation of the trapped particles. Coherent phase-space electron holes are formed, which are localized phase-space regions of reduced density on trapped electron orbits, where the electron density is lower than the surrounding plasma electron density. The distribution function evolves to a shape with stationary inflection points of zero slope, at the phase velocities of the excited waves. The longest wavelengths show oscillations at frequencies below the plasma frequency, with phase velocities higher than that of the injected beam, which can accelerate electrons to energies in excess of the initial beam energy. The present work makes a connection between the formation of electron holes, the existence of inflection points of zero slopes in the electron distribution function at the phase velocities of the dominant waves, and at frequencies below the plasma frequency. A fine resolution grid is used in the Eulerian Vlasov code in the phase space and time to allow an accurate calculation of the time history of the system and of the dynamic and oscillation of the trapped particles in the low-density regions of the phase space, and of those particles at the separatrix regions of the vortex structures which evolve periodically between trapping and untrapping states and which can only be accurately studied using a fine-resolution phase-space grid.

Effective and efficient approaches for calculating seismic ray velocity and attenuation in viscoelastic anisotropic media

Geophysics ◽  
2020 ◽  
Vol 86 (1) ◽  
pp. C19-C35
Author(s):  
Jianlu Wu ◽  
Bing Zhou ◽  
Xingwang Li ◽  
Youcef Bouzidi
Keyword(s):  
Velocity Vector ◽  
Hamiltonian Function ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Quality Factors ◽  
Approximate Methods ◽  
Slowness Vector ◽  
Homogeneous Complex ◽  
Isotropic Media ◽  
Complex Ray

In viscoelastic anisotropic media, the elastic moduli, slowness vector, phase, and ray velocity are all complex-valued quantities in the frequency domain. Solving the complex eikonal equation becomes computationally complex and time-consuming. We have developed two approximate methods to effectively calculate the ray velocity vector, attenuation, and quality factor in viscoelastic transversely isotropic media with a vertical symmetry axis (VTI) and in orthorhombic (ORT) anisotropy. The first method is based on the perturbation theory (PER) under the assumption of a homogeneous complex ray vector, which is obtained by applying the elastic background and viscoelastic perturbations to the real and imaginary components of the modulus tensor, respectively. The perturbations of the slowness vectors of the three wave modes (qP, qSV, and qSH) are determined through the vanishing Hamiltonian function. The second method is derived by applying a real slowness direction (RSD) to the inhomogeneous complex slowness vector and then approximately calculating the complex ray velocity vector with the condition of the homogeneous complex vector. The numerical results verify that the two approaches can produce accurate ray velocity vector, attenuation, and quality factors of the qP-wave in viscoelastic VTI and ORT media. The RSD method can yield high accuracies of ray velocity for the qSV- and qSH-wave in viscoelastic VTI models even at triplication of the qSV wavefronts, as well as qS1 and qS2 in a weak ORT medium ([Formula: see text] > 20), except for near the cusp of the qS1 wavefronts (errors approximately 6%) where the PER has more than 10% error.

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