Parametric Analysis of Complex Dispersion Curves for Flexural Lamb Waves in Layered Plates in the Low-Frequency Range

A combined asymptotical and iteration method is used to study dispersion curves for the case of dynamic bending of isotropically layered plates. Based on the explicit limit formulation of dispersion equation, asymptotics of roots are derived in closed form for large values of root moduli. The influence of elastic and geometric parameters of layers are analyzed. The existence of critical values of geometric parameters that correspond to change of the type of asymptotics is demonstrated. The errors of asymptotics are estimated, and an iterative method is proposed for calculating the exact values of roots in statics. A low-frequency long-wave asymptotics of complex dispersion curves is derived; its accuracy is the higher the lower the frequency and the greater the number of the curve are. It is also proved that each complex curve has a long flat segment, the length of which increases simultaneously with the number of curve. The dispersion curves themselves are also calculated by another specific iterative procedure. The fundamental bending mode is analyzed together with its purely imaginary sister. The existence of the additional purely imaginary curve at low frequency is proved. Examples of calculating the static roots and the dispersion curves for subcritical and supercritical values of geometrical parameters are presented, and the efficiency of the algorithm is estimated.     

On the Generalized Ornstein–Ulenbeck Process

The model generalized Ornstein–Ulenbeck process is developed on the basis of the anomalous-diffusion nature of random walk. Asymptotics of the basic statistical characteristics of this process are found. It is shown that the asymptotics are governed by the classical or anomalous diffusion laws if the central limit theorem is fulfilled or violated, respectively.     

Dispersion of doppleron-phonon modes in strong coupling regime

The dispersion equation for doppleron-phonon modes was constructed and solved analytically in the strong coupling regime. The Fermi surface model proposed previously for calculating the doppleron spectrum in an indium crystal was used. It was shown that in the vicinity of doppleron-phonon resonance, the dispersion curves of coupled modes form a gap qualitatively different from the one observed under helicon-phonon resonance: there is a frequency interval forbidden for existence of waves of definite circular polarization depending upon direction of the external DC magnetic field. The physical reason for it is interaction of the waves which have oppositely directed group velocities.     

On the Charge Form Factor and Eigenvalue of Fermion Energy in a Three-Dimensional Electrodynamics

A static field and self-energy of a particle are considered for a particle charge distributed throughout a 2 + 1-measurement space. The potential of the static field for r has the same asymptotics as for the delta form factor, provided an account is taken of the contribution from vacuum polarization; at the origin of coordinates, the above potential is regular. The proposed form factor allows a relation for the particle charge distribution to be derived in a closed form. The diagonal tension-tensor components of the particle-generated field are found to vanish and the particle field mass calculated using the classical method appears to be finite in the case where the proposed form factor is used. This mass coincides with that obtained through quantum calculations by the order of magnitude.     

Formation of vortex light fields of specified intensity for laser micromanipulation

method for calculating phase diffractive elements is proposed to form light fields with intensity described by flat curves. This method is based on the optics of spiral light beams and Gerchberg-Saxton iterative algorithm. The experimental data on the formation of fields and their use in manipulation with latex microparticles are reported.     

Resonance effects of the axisymmetrical mode propagation in composite cylindrical solids with nematic coatings

The dispersion curves for harmonic waves in composite cylindrical elastic solids with a coating, made of non traditional soft material (nematic elastomer) are investigated. The obtained earlier low frequency effective model of nematics is used with taking into account the anisotropy, viscosity and internal rotation of long molecules of material. The impedance matrices, dispersion equations and asymptotics of speeds and wavenumbers in the long-wave approximation are derived for the further numerical analysis. The obtained quasi resonance effects of the first and of the second kinds are discussed for waves of different polarization.     

Analysis of the dispersion spectrum of fluid-loaded anisotropic plates: leaky-wave branches

The supersonic complex-velocity versus real-frequency dispersion spectrum of the leaky waves in fluid-loaded anisotropic plates is discussed. Utilizing the sextic plate formalism provides approximate solutions for leaky-wave velocity in a form that reveals their basic features, such as the unique correspondence of the signs of its imaginary part and of the free-plate group velocity, the relation between the leakage and the rate of frequency dispersion, and the principal trends at low, high and near-cutoff frequencies in arbitrary anisotropic plates. A particular thrust of the study is the derivation of closed-form asymptotics for the fundamental leaky-wave velocity branch(es) at low frequency and for the continuum of leaky-wave branches near the fluid-coupled and fluid-uncoupled thickness resonances. Conditions for the asymptotics accuracy are analysed, and a comparison between an analytical approximation and exact numerical curves is presented for various cases.     

Temperature and pressure dependence of some thermal characteristics of magnesium

The theoretical values of the Debye-Waller factor, specific heat and the phonon dispersion relations as functions of temperature for magnesium are derived from a simple model of lattice dynamics proposed by Czachor for hexagonal crystals.. The pressure dependence of the dispersion curves is also derived. The theoretical results have been found to be in very good agreement with experimental data available.     

Comparison between the dispersion curves calculated in complex frequency and the minima of the reflection coefficients for an embedded layer

Analytical solutions of Lamb functions for symmetric and antisymmetric elastodynamic modes propagating within a solid layer embedded in an infinite medium are presented. Alternative theoretical analyses of such modes are performed, first in terms of the usual approach of harmonic heterogeneous plane waves (real frequency and complex slowness) and then in terms of transient homogeneous plane waves (complex frequency and real slowness). An example structure of a 0.1-mm-thick "alpha case" (an oxygen-rich phase of titanium that is relatively stiff) plate embedded in titanium is used for the study. A large difference between the usual dispersion curves calculated in real frequency and complex slowness and those calculated in complex frequency and real slowness is shown. Thus the choice between a spatial and a temporal parameter to describe the imaginary part of the guided waves is shown to be significant. The minima and the zeros of the longitudinal and shear plane-wave reflection coefficients are calculated and are compared with the dispersion curves. It is found that they do not match with the dispersion curves for complex slowness, but they do agree quite well with the dispersion curves for complex frequency. This implies that the complex frequency approach is better suited for the comparison of the modal properties with near-field reflection measurements.     

Wakefield radiation generated by an electron bunch in a rectangular dielectric waveguide

Vavilov-Cerenkov radiation generated by a relativistic electron bunch in a rectangular waveguide with a transverse-inhomogeneous dielectric filling is analyzed. A method is proposed for constructing an orthogonal basis of the transverse operator, which can be subsequently used for determining the wakefield of the relativistic bunch moving parallel to the waveguide axis. The dispersion equation for the structure is derived and the expressions for the wake field produced by such a bunch are obtained. The formalism described here forms the basis for calculating parameters of the accelerating structure for generator bunches of the Argonne Wakefield Accelerator (AWA) at the Argonne National Laboratory and the FACET complex of the SLAC accelerator. It is shown that using such structures, accelerating field gradients higher than 150 MV/m can be generated at frequencies 20?C35 GHz and exceeding 1 GeV/m in the frequency range ??1 THz.     

An alternative method for plotting dispersion curves

Solving the frequency equation and plotting the dispersion curves in problems of wave propagation in cylinders and plates, particularly when the material is anisotropic, are complicated tasks. The traditional numerical methods are usually based on determination of the zeros of the frequency equation by using an iterative find-root algorithm. In this paper, an alternative method is proposed which extracts the solution of the frequency equation in the form of dispersion curves from the three-dimensional illustration of the frequency equation. For this purpose, a three-dimensional representation of the real roots of the frequency equation is first plotted. The dispersion curves, which are the numerical solutions of the frequency equation, are then obtained by a suitable cut in the velocity-frequency plane. The advantages of this method include simplicity, high speed, low possibility of numerical error, and presentation of the results in a graphical form that promotes ease of interpretation. This method is not directly applicable to problems which incorporate high damping or leaky waves. However, if the damping is not very high, it could be a good estimate of the true dispersion curves.     

Propagation of SH-wave in a corrugated viscous sandy layer sandwiched between two elastic half-spaces

The present paper attempts to investigate the effect of sandiness, corrugated boundary surfaces, heterogeneity, and gravity on phase velocity and attenuation of SH-wave propagating in a viscous sandy layer with corrugated upper and lower boundary surfaces sandwiched between an upper heterogeneous elastic half-space and lower viscoelastic half-space under gravity. Heterogeneity associated with the upper half-space is due to exponentially varying density which is a function of depth; but the rigidity is constant. The closed form of dispersion relation is established and found to be in complex form. Real part and imaginary part of the dispersion relation correspond to dispersion curve and attenuation curve, respectively. One of the salient points of present study is the use of DEBYE asymptotic expansion to establish that the obtained dispersion relation is in well-agreement with the classical Love wave equation in isotropic case. The effect of presence and absence of corrugated boundary surfaces, measured by initial flatness parameter, on dispersion and attenuation curves have been meticulously examined. Moreover, the substantial effect of heterogeneity, sandiness, internal friction, and Biot’s gravity parameter on phase velocity and attenuation coefficient of SH-wave has been studied and demonstrated by means of graphical illustration and numerical computations.     

一种计算半导体力常数的简单途径——对GaAs(111)-2×2表面声子研究的应用

本文利用成键轨道近似简化电子能带结构能量的计算,导出了任意杂化下半导体力常数的解析表达式。通过对一些材料声子色散曲线的计算和实验结果的比较,表明该方法是简单有效的。作者将其应用于GaAs(111)-2×2表面声子的研究,得到了该表面的强局域声子模的色散曲线。 关键词：     

Spectral anomalies of waveguide electromagnetic modes in layered structures

A theoretical study of electromagnetic guided eigenwaves in two-and three-layered plates with metallized surfaces is accomplished. The appropriate dispersion equations are explicitly analyzed using the discretization first introduced by Mindlin in the theory of Lamb acoustic waves. It is shown that the dispersion branches of independent eigenmode families cross each other in the nodes of a grid formed by two infinite series of bond lines. The latter represent the dispersion curves for homogeneous plates with permittivities ɛ₂ or ɛ₂ coinciding with those for the layers of the waveguide. It is proved that, beyond nodes of the grid, the dispersion curves do not intersect bond lines, which thus provides definite “corridors” for these curves. The dispersion lines have a wavy (“zigzag”) form in the grid zone and remain smooth beyond the grid. The crossing branches have coinciding cutoff frequencies. In dimensionless coordinates “slowness (S) vs. frequency (f),” the branches S _l (f) have two asymptotic levels: S = √ɛ₁ and S = √ɛ₂. At the lower level, the spectrum forms a steplike terracing pattern with a progressive closing to the asymptote of a succession of dispersion curves, which change each other at this level with further going up to the next asymptote. An extension to anisotropic waveguides with layers made of uniaxial crystals is considered. The text was submitted by the authors in English.     

Dispersion properties of helical waves in radially inhomogeneous elastic media

In this paper, a method describing dispersion curve calculation for waves propagating in radially layered, inhomogeneous isotropic elastic waveguides is developed. Particular emphasis is placed on the helical waves with noninteger azimuthal wavenumbers, which can be potentially applied in such fields as nondestructive evaluation, acoustic tomography, etc., stipulating their practical importance. To solve the problem under consideration, the matrix Riccati equation is formulated for an impedance matrix. The use of the latter yields a simple form of the dispersion equation. Numerical computation of dispersion curves can encounter difficulties, which are due to potential singularities of the impedance matrix and the necessity to separate roots of the dispersion equation. These difficulties are overcome by employing the Cayley transform and invoking the parametric continuation method. The method developed by the authors is demonstrated by calculating dispersion diagrams in support of helical waves for several models of practical interest. Such computations for an inhomogeneous layer and its approximation by a set of homogeneous layers using a transfer matrix and Riccati equation methods revealed higher computational accuracy of the latter. Dispersion curves calculated for layers with different types of inhomogeneity demonstrated significant discrepancies at low frequencies.     

Guided modes in slab waveguides with a left handed material cover or substrate

We study guided modes propagating along a dielectric slab waveguide with a left handed material (LHM) cover or substrate. The dispersion relation is derived by using normalized waveguide parameters. An analytical method is then proposed to calculate the universal dispersion curves. Different from a slab waveguide with a LHM core, we find that guidance properties are strongly dependent on dielectric permittivity ε and magnetic permeability μ of the substrate and cover layers. For oscillating guided modes, fundamental zero order mode is not always absence, sometimes it exists in a restricted range of normalized propagation constant. First order mode behaves as other higher order modes and exists up to infinite high frequency. Higher order modes have no double degeneracy in the case of LHM cover layer. For surface guided modes, the existence and the type of the mode solutions with respect to different parameters are classified systematically and discussed in detail. Unlike a slab waveguide with a LHM core where the dispersion curve of TE₁ surface mode continues with that of oscillating TE₁ mode, the dispersion curve of TE₁ surface mode continues with that of oscillating TE₀ mode. It seems that the two different kinds of modes compensate each other to form one whole mode. Both TE and TM guided modes are discussed.     

Modified Form of Wigner Functions for Non-Hamiltonian Systems

Quantization of non-Hamiltonian systems （such as damped systems） often gives rise to complex spectra and corresponding resonant states, therefore a standard form calculating Wigner functions cannot lead to static quasiprobability distribution functions. We show that a modified form of the Wigner functions satisfies a ＊-genvalue equation and can be derived from deformation quantization for such systems.     

Asymptotics of dispersion curves in time-dependent problems of free viscous–inviscid interaction at transonic speeds

New high-frequency asymptotics of the dispersion relation have been obtained when analyzing the stability of a transonic boundary layer with self-induced pressure. It is shown that the dependence of the perturbation frequency on the wave number (except for the range of small wave numbers), which was previously considered unambiguous, is an exceptional case.     

用去奇求积法计算等离子体色散函数及其它

提出一种处理奇异积分的数值方法。对于计算等离子体色散函数及其它具有奇异被积函数的特殊函数十分有效。对一些常用的奇异函数类型的积分,导出了相应的求积公式。对一些复杂奇异类型的奇异被积函数的处理,方法是十分方便的。