The Theoretical Solutions of Several Integrations in Relation with the Plane Wave Functions

2012 ◽  
Vol 166-169 ◽  
pp. 2009-2015
Author(s):  
Wen Pu Shi ◽  
Cui Hua Li
Keyword(s):  
Wave Function ◽  
Plane Wave ◽  
Analytical Solutions ◽  
Elastic Waves ◽  
Plane Waves ◽  
Stationary Wave ◽  
Wave Functions ◽  
Analysis Method ◽  
Principle Of Superposition ◽  
Scattering Problems

Several integrations of the plane wave functions frequently appearing in the scattering problems of elastic waves are considered in this study. In conjunction with the principle of superposition and the stationary wave function expansion, the exact solutions are obtained. As in the scattering problems of elastic waves and plane waves, the theoretical solutions can improve the convergence speed of the arithmetic and the computation precision. The analysis method and the analytical solutions can also be very helpful to solving other complex integrations in relation with the cylinder functions and plane functions.

FLUCTUATIONS OF ELASTIC WAVES DUE TO RANDOM SCATTERING FROM INCLUSIONS

2001 ◽  
Vol 09 (03) ◽  
pp. 973-991 ◽  
Author(s):  
V. A. KORNEEV ◽  
L. R. JOHNSON
Keyword(s):  
Exact Solutions ◽  
Analytical Solutions ◽  
Material Properties ◽  
Elastic Waves ◽  
Plane Waves ◽  
Spatial Averaging ◽  
Scattering Medium ◽  
Homogeneous Sphere ◽  
Mean And Variance ◽  
The Mean

Exact solutions for elastic compressional and shear waves scattered from a homogeneous sphere are used to obtain formulas for fluctuations of velocity and attenuation of plane waves propagating through a layer of randomly distributed inclusions over a broad range of frequencies. The size and contrast of the inclusions are arbitrary, but interactions between scatterers are not considered and the concentration of scatterers is assumed to be small. The analytical solutions are also compared with numerical simulations and it is demonstrated that they satisfactorily explain the effects of scattering on both the mean and variance of the phase and the mean and variance of the attenuation. The need for spatial averaging of observational data and methods of interpreting such averaged data in terms of the material properties of the scattering medium are discussed.

The application of variational methods to atomic scattering problems II. Impact excitation of the 2s level of atomic hydrogen-distorted wave treatment

1952 ◽  
Vol 212 (1111) ◽  
pp. 521-530 ◽  
Keyword(s):  
Cross Section ◽  
Atomic Hydrogen ◽  
Plane Waves ◽  
Wave Functions ◽  
Impact Excitation ◽  
Wave Method ◽  
Distorted Wave ◽  
Scattering Problems ◽  
The Cross ◽  
Distorted Wave Method

The cross-section for excitation of the 2 S level of atomic hydrogen by electrons is calculated using the distorted wave method with full allowance for exchange. The distorted wave functions used in the calculations are determined by Hulthèn’s variational method. The initial wave functions, representing the motion of an electron in the field of a normal atom with allowance for exchange, are taken to be those calculated by Massey & Moiseiwitsch (1950). The final wave functions, representing the motion of an electron in the field of a hydrogen atom in the 2 S state, have been obtained by a modification of the same method. Exchange effects are found to be less important in determining the forms of these wave functions. The cross-sections obtained are considerably smaller than those calculated by the Born-Oppenheimer method, in which the electron wave functions are undistorted plane waves. This is largely because the symmetrical cross-section, which has the greater weight in determining the mean cross-section, is much greater than the antisymmetrical according to the Born-Oppenheimer method, but the reverse is true if distortion is allowed for. In no case does the distorted wave method give results exceeding the theoretical upper limit, whereas with plane waves this limit is exceeded at certain electron energies by the symmetrical cross-section.

scholarly journals Massless Composite Bosons Formed by the Coupled Electron-Positron Pairs and Two-Photon Angular Correlations in the Colliding Beam Reaction e-e+→Bγγ with Emission of the Massless Boson

2018 ◽  
Vol 2018 ◽  
pp. 1-18 ◽  
Author(s):  
A. I. Agafonov
Keyword(s):  
Wave Function ◽  
Free Particle ◽  
Plane Waves ◽  
Wave Functions ◽  
Two Photon ◽  
Electron Positron ◽  
Correlation Spectrum ◽  
Composite Bosons ◽  
Massless Boson ◽  
Colliding Beams

The approach in which the electron and positron are treated as ordinary, different particles, each being characterized by the complete set of the Dirac plane waves, is examined. This completely symmetric representation that is beyond the standard QED makes it necessary to choose another solution of the Dirac equation for the free particle propagator as compared to that used currently. The Bethe-Salpeter equation with these particle propagators is solved in the ladder approximation. A new solution has been found represented by the massless composite bosons formed by the coupled electron-positron pairs with the coupling equal to the fine structure constant. It has been demonstrated that (1) the massless boson states have normalizable complex wave functions which are transversely compressed plane waves; (2) the transverse radius of the wave functions diverges as the boson energy goes to zero; that is, the composite bosons cannot be at rest; (3) increasing the boson energy results in an extension of the transverse wave function in the momentum space and a corresponding contraction of the real space coordinate wave function. The new reaction e-e+→Bγγ is investigated with the products composed of the massless composite boson and two photons. The cross-section of this reaction is derived for nonrelativistic colliding beams of spin-polarized electrons and positrons. In this case the 2γ angular correlation spectrum is characterized by a narrow peak with the full-width-at-half-maximum not exceeding 0.2 mrad. It is shown that in order to distinguish between the conventional annihilation of the singlet electron-positron pair with the two-photon emission and the new examined reaction yielding the three particles, experiments are proposed with the extremely nonrelativistic colliding beams.

The electronic properties of tetrahedral intermetallic compounds II. Conduction band in diamond

1962 ◽  
Vol 270 (1342) ◽  
pp. 373-382 ◽  
Keyword(s):  
Plane Wave ◽  
Intermetallic Compounds ◽  
Valence Band ◽  
Conduction Band ◽  
Plane Waves ◽  
Wave Functions ◽  
Wave Method ◽  
Linear Combinations ◽  
Plane Wave Method ◽  
Hybrid Orbitals

A set of plane waves, made orthogonal to the 1 s band and to the approximate valence band, is used to calculate the conduction band in diamond. This is a modification of the usual orthogonalized plane-wave method where the plane waves are made orthogonal only to the 1 s band. The approximate valence-band wave functions are obtained by assuming that the equivalent orbitals, employed by Hall to discuss the valence band in diamond, can be written as linear combinations of atomic hybrid orbitals. The following results are obtained: The lowest conduction b and level at k = 0 is antibonding and threefold degenerate. The minimum of the conduction band is not at k = 0 but is about half-way along the (1, 0, 0) direction.

Gravity as the curvature of the wave function of the universe.

2019 ◽  
Author(s):  
Vitaly Kuyukov
Keyword(s):  
Wave Function ◽  
Wave Functions ◽  
Main Task ◽  
Reference Systems ◽  
Theory Of Relativity ◽  
General Theory Of Relativity ◽  
Principle Of Equivalence ◽  
Relative Wave Function ◽  
New Equation ◽  
The Universe

Modern general theory of relativity considers gravity as the curvature of space-time. The theory is based on the principle of equivalence. All bodies fall with the same acceleration in the gravitational field, which is equivalent to locally accelerated reference systems. In this article, we will affirm the concept of gravity as the curvature of the relative wave function of the Universe. That is, a change in the phase of the universal wave function of the Universe near a massive body leads to a change in all other wave functions of bodies. The main task is to find the form of the relative wave function of the Universe, as well as a new equation of gravity for connecting the curvature of the wave function and the density of matter.

Electromagnetic waves

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Plane Wave ◽  
Electromagnetic Waves ◽  
Maxwell Equations ◽  
Plane Waves ◽  
Geometrical Optics ◽  
Particle Detection ◽  
Spatial Coordinates ◽  
A Charge

This chapter examines solutions to the Maxwell equations in a vacuum: monochromatic plane waves and their polarizations, plane waves, and the motion of a charge in the field of a wave (which is the principle upon which particle detection is based). A plane wave is a solution of the vacuum Maxwell equations which depends on only one of the Cartesian spatial coordinates. The monochromatic plane waves form a basis (in the sense of distributions, because they are not square-integrable) in which any solution of the vacuum Maxwell equations can be expanded. The chapter concludes by giving the conditions for the geometrical optics limit. It also establishes the connection between electromagnetic waves and the kinematic description of light discussed in Book 1.

scholarly journals Ultrasonic Scattered Field Distribution of One and Two Cylindrical Solids with Phased Array Technique

2019 ◽  
Vol 32 (1) ◽  
Author(s):  
Xiaozhou Liu ◽  
Jian Ma ◽  
Haibin Wang ◽  
Sha Gao ◽  
Yifeng Li ◽  
...  
Keyword(s):  
Plane Wave ◽  
Field Distribution ◽  
Scattered Field ◽  
Phased Array ◽  
Simulation Analysis ◽  
Plane Waves ◽  
Theoretical Solution ◽  
Steel Matrix ◽  
Field Distributions ◽  
Scattered Fields

AbstractThe scattered fields of plane waves in a solid from a cylinder or sphere are critical in determining its acoustic characteristics as well as in engineering applications. This paper investigates the scattered field distributions of different incident waves created by elastic cylinders embedded in an elastic isotropic medium. Scattered waves, including longitudinal and transverse waves both inside and outside the cylinder, are described with specific modalities under an incident plane wave. A model with a scatterer embedded in a structural steel matrix and filled with aluminum is developed for comparison with the theoretical solution. The frequency of the plane wave ranged from 235 kHz to 2348 kHz, which corresponds to scaling factors from 0.5 to 5. Scattered field distributions in matrix materials blocked by an elastic cylindrical solid have been obtained by simulation or calculated using existing parameters. The simulation results are in good agreement with the theoretical solution, which supports the correctness of the simulation analysis. Furthermore, ultrasonic phased arrays are used to study scattered fields by changing the characteristics of the incident wave. On this foundation, a partial preliminary study of the scattered field distribution of double cylinders in a solid has been carried out, and the scattered field distribution at a given distance has been found to exhibit particular behaviors at different moments. Further studies on directivities and scattered fields are expected to improve the quantification of scattered images in isotropic solid materials by the phased array technique.

Surface and interfacial anti-plane waves in micropolar solids with surface energy

2020 ◽  
pp. 108128652096564
Author(s):  
Mriganka Shekhar Chaki ◽  
Victor A Eremeyev ◽  
Abhishek K Singh
Keyword(s):  
Surface Wave ◽  
Plane Wave ◽  
Numerical Study ◽  
Plane Waves ◽  
Angular Frequency ◽  
Elastic Half Space ◽  
Surface Strain ◽  
Theoretical Frameworks ◽  
Algebraic Form ◽  
New Type

In this work, the propagation behaviour of a surface wave in a micropolar elastic half-space with surface strain and kinetic energies localized at the surface and the propagation behaviour of an interfacial anti-plane wave between two micropolar elastic half-spaces with interfacial strain and kinetic energies localized at the interface have been studied. The Gurtin–Murdoch model has been adopted for surface and interfacial elasticity. Dispersion equations for both models have been obtained in algebraic form for two types of anti-plane wave, i.e. a Love-type wave and a new type of surface wave (due to micropolarity). The angular frequency and phase velocity of anti-plane waves have been analysed through a numerical study within cut-off frequencies. The obtained results may find suitable applications in thin film technology, non-destructive analysis or biomechanics, where the models discussed here may serve as theoretical frameworks for similar types of phenomena.

scholarly journals Reducing a Class of Two-Dimensional Integrals to One-Dimension with an Application to Gaussian Transforms

Atoms ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 53
Author(s):  
Jack C. Straton
Keyword(s):  
Quantum Theory ◽  
Arbitrary Function ◽  
Plane Waves ◽  
Wave Functions ◽  
One Dimension ◽  
Two Dimensional ◽  
Transition Amplitudes ◽  
Multidimensional Integrals ◽  
Coulomb Potentials

Quantum theory is awash in multidimensional integrals that contain exponentials in the integration variables, their inverses, and inverse polynomials of those variables. The present paper introduces a means to reduce pairs of such integrals to one dimension when the integrand contains powers multiplied by an arbitrary function of xy/(x+y) multiplying various combinations of exponentials. In some cases these exponentials arise directly from transition-amplitudes involving products of plane waves, hydrogenic wave functions, and Yukawa and/or Coulomb potentials. In other cases these exponentials arise from Gaussian transforms of such functions.

Study of wave attenuation in concrete

1993 ◽  
Vol 8 (9) ◽  
pp. 2344-2353 ◽  
Author(s):  
J-M. Berthelot ◽  
Souda M. Ben ◽  
J.L. Robert
Keyword(s):  
Experimental Study ◽  
Plane Wave ◽  
Wave Attenuation ◽  
Plane Waves ◽  
Propagation Distance ◽  
Experimental Results ◽  
The Other ◽  
Wave Frequency ◽  
Concrete Composition ◽  
Analytical Expression

The experimental study of wave attenuation in concrete has been achieved in the case of the propagation of plane waves in concrete rods. Different mortars and concretes have been investigated. A transmitter transducer coupled to one of the ends of the concrete rod generates the propagation of a plane wave in the rod. The receiver transducer, similar to the previous one, is coupled to the other end of the rod. The experimental results lead to an analytical expression for wave attenuation as function of the concrete composition, the propagation distance, and the wave frequency.

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