Abnormality of the longitudinal Pochhammer–Chree waves in the vicinity of C2 phase speed

2018 ◽  
Vol 24 (23) ◽  
pp. 5642-5649 ◽  
Author(s):  
Sergey V Kuznetsov
Keyword(s):  
Spectral Analysis ◽  
Phase Speed ◽  
Harmonic Waves ◽  
Wave Polarization ◽  
Axially Symmetric ◽  
Displacement Fields ◽  
The Matrix ◽  
Circular Frequency ◽  
Shear Speed ◽  
Analytical Expressions

The exact solutions of the linear Pochhammer–Chree equation for propagating harmonic waves in a cylindrical rod are analyzed. Spectral analysis of the matrix dispersion equation for longitudinal axially symmetric modes is performed and analytical expressions for displacement fields are obtained. The variation of wave polarization on the free surface due to the variation of Poisson’s ratio and circular frequency is analyzed. It is observed that at the phase speed coinciding with the bulk shear speed all the components of the displacement field vanish, meaning that no longitudinal axisymmetric Pochhammer–Chree waves can propagate at this phase speed.

Polarization of the Longitudinal Pochhammer–Chree Waves

2020 ◽  
Vol 22 (4) ◽  
pp. 1329-1336
Author(s):  
Alla V. Ilyashenko ◽  
Sergey V. Kuznetsov
Keyword(s):  
Wave Speed ◽  
Phase Speed ◽  
Harmonic Waves ◽  
Wave Polarization ◽  
Axially Symmetric ◽  
Displacement Fields ◽  
Shear Wave Speed ◽  
The Matrix ◽  
Circular Frequency ◽  
Analytical Expressions

AbstractThe exact solutions of the linear Pochhammer – Chree equation for propagating harmonic waves in a cylindrical rod, are analyzed. Spectral analysis of the matrix dispersion equation for longitudinal axially symmetric modes is performed. Analytical expressions for displacement fields are obtained. Variation of wave polarization on the free surface due to variation of Poisson’s ratio and circular frequency is analyzed. It is observed that at the phase speed coinciding with the bulk shear wave speed all the components of the displacement field vanish, meaning that no longitudinal axisymmetric Pochhammer – Chree wave can propagate at this phase speed.

Longitudinal Pochhammer — Chree Waves in Mild Auxetics and Non-Auxetics

2018 ◽  
Vol 35 (3) ◽  
pp. 327-334 ◽  
Author(s):  
A. V. Ilyashenko ◽  
S. V. Kuznetsov
Keyword(s):  
Poisson’S Ratio ◽  
Wave Speed ◽  
Poisson's Ratio ◽  
Abnormal Behavior ◽  
Axially Symmetric ◽  
Displacement Fields ◽  
Shear Wave Speed ◽  
The Matrix ◽  
The Waves ◽  
Analytical Expressions

ABSTRACTThe exact solutions of Pochhammer — Chree equation for propagating harmonic waves in isotropic elastic cylindrical rods, are analyzed. Spectral analysis of the matrix dispersion equation for the longitudinal axially symmetric modes is performed. Analytical expressions for displacement fields are obtained. Variation of the wave polarization due to variation of Poisson’s ratio for mild auxetics (Poisson’s ratio is greater than -0.5) is analyzed and compared with the non-auxetics. It is observed that polarization of the waves for both considered cases (auxetics and non-auxetics) exhibits abnormal behavior in the vicinity of the bulk shear wave speed.

Image Contrast of Dislocation Loops Near a Free Surface

1977 ◽  
Vol 35 ◽  
pp. 52-53
Author(s):  
S. M. Ohr
Keyword(s):  
Free Surface ◽  
Dislocation Loop ◽  
Image Contrast ◽  
Stress System ◽  
Shear Stresses ◽  
Dislocation Loops ◽  
Axially Symmetric ◽  
Displacement Fields ◽  
Elastic Fields ◽  
Analytical Expressions

The image contrast of dislocation loops computed in the past has made use of the displacement fields which do not take into account the presence of stress- free foil surfaces. The free surface modifies the elastic fields around a dislocation loop and hence can influence the image contrast observed in the electron microscope. The effect can be significant particularly when the loops lie close to one of the foil surfaces. In general, the elasticity problem of dislocation loops that takes the free surface into account is difficult to handle mathematically. In the present paper, the method of Bastecka1 was extended to obtain explicitly the analytical expressions for the displacement fields around a pure edge circular dislocation loop lying parallel to the foil surface. In this method, the stress fields of an image dislocation loop and another axially symmetric stress system were added in order to eliminate the normal as well as shear stresses at the surface.

scholarly journals Analytical angular solutions for the atom–diatom interaction potential in a basis set of products of two spherical harmonics: two approaches

2021 ◽  
Author(s):  
Mariusz Pawlak ◽  
Marcin Stachowiak
Keyword(s):  
Spherical Harmonics ◽  
Interaction Potential ◽  
Chem Phys ◽  
Basis Set ◽  
Matrix Elements ◽  
Trigonometric Functions ◽  
Variational Theory ◽  
Molecular Collision ◽  
The Matrix ◽  
Analytical Expressions

AbstractWe present general analytical expressions for the matrix elements of the atom–diatom interaction potential, expanded in terms of Legendre polynomials, in a basis set of products of two spherical harmonics, especially significant to the recently developed adiabatic variational theory for cold molecular collision experiments [J. Chem. Phys. 143, 074114 (2015); J. Phys. Chem. A 121, 2194 (2017)]. We used two approaches in our studies. The first involves the evaluation of the integral containing trigonometric functions with arbitrary powers. The second approach is based on the theorem of addition of spherical harmonics.

Field Computation for Two-Dimensional Array Transducers with Limited Diffraction Array Beams

2005 ◽  
Vol 27 (4) ◽  
pp. 237-255 ◽  
Author(s):  
Jian-Yu Lu ◽  
Jiqi Cheng
Keyword(s):  
Continuous Wave ◽  
Weighting Function ◽  
Near Field ◽  
Bessel Beam ◽  
Two Dimensional ◽  
Axially Symmetric ◽  
Computationally Efficient ◽  
Annular Array ◽  
Array Transducer ◽  
Analytical Expressions

A method is developed for calculating fields produced with a two-dimensional (2D) array transducer. This method decomposes an arbitrary 2D aperture weighting function into a set of limited diffraction array beams. Using the analytical expressions of limited diffraction beams, arbitrary continuous wave (cw) or pulse wave (pw) fields of 2D arrays can be obtained with a simple superposition of these beams. In addition, this method can be simplified and applied to a 1D array transducer of a finite or infinite elevation height. For beams produced with axially symmetric aperture weighting functions, this method can be reduced to the Fourier-Bessel method studied previously where an annular array transducer can be used. The advantage of the method is that it is accurate and computationally efficient, especially in regions that are not far from the surface of the transducer (near field), where it is important for medical imaging. Both computer simulations and a synthetic array experiment are carried out to verify the method. Results (Bessel beam, focused Gaussian beam, X wave and asymmetric array beams) show that the method is accurate as compared to that using the Rayleigh-Sommerfeld diffraction formula and agrees well with the experiment.

The matrix M/M/∞ system: retrial models and Markov Modulated sources

1993 ◽  
Vol 25 (2) ◽  
pp. 453-471 ◽  
Author(s):  
J. Keilson ◽  
L. D. Servi
Keyword(s):  
Numerical Evaluation ◽  
Stability Conditions ◽  
Kummer Functions ◽  
The Matrix ◽  
Service Rates ◽  
System Properties ◽  
Markov Modulated ◽  
The Stability ◽  
Analytical Expressions

The matrix-geometric work of Neuts could be viewed as a matrix variant of M/M/1. A 2 × 2 matrix counterpart of Neuts for M/M/∞ is introduced, the stability conditions are identified, and the ergodic solution is solved analytically in terms of the ten parameters that define it. For several cases of interest, system properties can be found from simple analytical expressions or after easy numerical evaluation of Kummer functions. When the matrix of service rates is singular, a qualitatively different solution is derived. Applications to telecommunications include some retrial models and an M/M/∞ queue with Markov-modulated input.

scholarly journals Case studies of the propagation characteristics of auroral TIDS with EISCAT CP2 data using maximum entropy cross-spectral analysis

1998 ◽  
Vol 16 (2) ◽  
pp. 161-167 ◽  
Author(s):  
S. Y. Ma ◽  
K. Schlegel ◽  
J. S. Xu
Keyword(s):  
Spectral Analysis ◽  
Phase Velocity ◽  
Maximum Entropy ◽  
Case Studies ◽  
Large Scale ◽  
Phase Speed ◽  
Wave Number ◽  
Propagation Characteristics ◽  
Dominant Period ◽  
Horizontal Wavelength

Abstract. In this paper case studies of propagation characteristics of two TIDs are presented which are induced by atmospheric gravity waves in the auroral F-region on a magnetic quiet day. By means of maximum entropy cross-spectral analysis of EISCAT CP2 data, apparent full wave-number vectors of the TIDs are obtained as a function of height. The analysis results show that the two events considered can be classified as moderately large-scale TID and medium-scale TID, respectively. One exhibits a dominant period of about 72 min, a mean horizontal phase speed of about 180 m/s (corresponding to a horizontal wavelength of about 780 km) directed south-eastwards and a vertical phase speed of 55 m/s for a height of about 300 km. The other example shows a dominant period of 44 min, a mean horizontal phase velocity of about 160 m/s (corresponding to a horizontal wavelength of about 420 km) directed southwestwards, and a vertical phase velocity of about 50 m/s at 250 km altitude.Key words. Ionosphere · Auroral ionosphere · Ionosphere-atmosphere interactions · Wave propagation)

ALGEBRAIC APPROACH TO THE PSEUDOHARMONIC OSCILLATOR IN 2D

2002 ◽  
Vol 11 (02) ◽  
pp. 155-160 ◽  
Author(s):  
SHI-HAI DONG ◽  
ZHONG-QI MA
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Algebraic Approach ◽  
Matrix Elements ◽  
Ladder Operators ◽  
Commutation Relations ◽  
The Matrix ◽  
Pseudoharmonic Oscillator ◽  
Analytical Expressions

A realization of the ladder operators for the solutions to the Schrödinger equation with a pseudoharmonic oscillator in 2D is presented. It is shown that those operators satisfy the commutation relations of an SU(1, 1) algebra. Closed analytical expressions are evaluated for the matrix elements of some operators r2 and r∂/∂ r

VELOCITY AND ATTENUATION OF SEISMIC WAVES IN TWO‐PHASE MEDIA: PART I. THEORETICAL FORMULATIONS

Geophysics ◽  
1974 ◽  
Vol 39 (5) ◽  
pp. 587-606 ◽  
Author(s):  
Guy T. Kuster ◽  
M. Nafi Toksöz
Keyword(s):  
Elastic Moduli ◽  
Seismic Waves ◽  
Matrix Material ◽  
Composite Medium ◽  
Solid Matrix ◽  
Seismic Velocities ◽  
Displacement Fields ◽  
Two Phase ◽  
The Matrix ◽  
In Series

The propagation of seismic waves in two‐phase media is treated theoretically to determine the elastic moduli of the composite medium given the properties, concentrations, and shapes of the inclusions and the matrix material. For long wavelengths the problem is formulated in terms of scattering phenomena in an approach similar to that of Ament (1959). The displacement fields, expanded in series, for waves scattered by an “effective” composite medium and individual inclusions are equated. The coefficients of the series expansions of the displacement fields provide a relationship between the elastic moduli of the effective medium and those of the matrix and inclusions. The expressions are derived for both solid and liquid inclusions in a solid matrix as well as for solid suspensions in a fluid matrix. Both spherical and oblate spheroidal inclusions are considered. Some numerical calculations are carried out to demonstrate the effects of fluid inclusions of various shapes on the seismic velocities in rocks. It is found that the concentration, shapes, and properties of the inclusions are important parameters. A concentration of a fraction of one percent of thin (small aspect ratio) inclusions could affect the compressional and shear velocities by more than ten percent. For both sedimentary and igneous rock models, the calculations for “dry” (i.e.,air‐saturated) and water‐saturated states indicate that the compressional velocities change significantly while the shear velocities change much less upon saturation with water.

Continuum Theory for a Laminated Medium

1968 ◽  
Vol 35 (3) ◽  
pp. 467-475 ◽  
Author(s):  
C.-T. Sun ◽  
J. D. Achenbach ◽  
George Herrmann
Keyword(s):  
Strain Energy ◽  
Equations Of Motion ◽  
Continuum Theory ◽  
Laminated Composite ◽  
Harmonic Waves ◽  
Approximate Theory ◽  
Lagrangian Multipliers ◽  
Phase Velocities ◽  
Wave Numbers ◽  
The Matrix

A system of displacement equations of motion is presented, pertaining to a continuum theory to describe the dynamic behavior of a laminated composite. In deriving the equations, the displacements of the reinforcing layers and the matrix layers are expressed as two-term expansions about the mid-planes of the layers. Dynamic interaction of the layers is included through continuity relations at the interfaces. By means of a smoothing operation, representative kinetic and strain energy densities for the laminated medium are obtained. Subsequent application of Hamilton’s principle, where the continuity relations are included through the use of Lagrangian multipliers, yields the displacement equations of motion. The distinctive trails of the system of equations are uncovered by considering the propagation of plane harmonic waves. Dispersion curves for harmonic waves propagating parallel to and normal to the layering are presented, and compared with exact curves. The limiting phase velocities at vanishing wave numbers agree with the exact, limits. The lowest antisymmetric mode for waves propagating in the direction of the layering shows the strongest dispersion, which is very well described by the approximate theory over a substantial range of wave numbers.

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