Eigen Functions Research Articles

Propagation characteristics of an elliptical chirowaveguide

The coupled Helmhotz-type differential equations for a general cylindrical structure containing chiral material were transformed into a pair of decoupled equations, and this new set was solved to find expressions for Ez and Hz with the expansion of the complete eigen functions in elliptical chiralwaveguide. There are no pure TE （TM） or even-type （odd-type） modes existing individually in the elliptical structure if Ez and Hz are coupled with each other. The influences of eccentricity and chiral admittance on dispersion of modes are discussed in this paper.

The two-dimensional periodic structure in a bifrequency magnetically insulated transmission line oscillator

The eigen function of a bifrequency magnetically insulated transmission line oscillator （BFMILO） is deduced, and the dispersion characteristics and the field distribution are analyzed in detail. By dividing a traditional symmetrical coaxial disk loaded slow wave structures （SWS） into two different azimuthal partitions, the SWS of BFMILO becomes a two-dimensional SWS. In this structure, different electromagnetic modes are distributed in different azimuthal partitions, so that two different electromagnetic modes, which are competition modes in a traditional MILO, can interact with electrons separately and stably. This special SWS is the key point in designing a BFMILO. The proposed method can also be used in analyzing a mlti-frequency MILO.

Structure of Near-Tip Stress Field and Variation of Stress Intensity Factor for a Crack in a Transversely Graded Material

The existing studies on the behavior of cracks in continuously graded materials assume the elastic properties to vary in the plane of the crack. In the case of a plate graded along the thickness and having a crack in its plane, the elastic properties will vary along the crack front. The present study aims at investigating the effect of elastic gradients along the crack front on the structure of the near-tip stress fields in such transversely graded materials. The first four terms in the expansion of the stress field are obtained by the eigenfunction expansion approach (Hartranft and Sih, 1969, “The Use of Eigen Function Expansion in the General Solution of Three Dimensional Crack Problems,” J. Math. Mech., 19(2), pp. 123–138) assuming an exponential variation of the elastic modulus. The results of this part of the study indicated that for an opening mode crack, the angular structure of the first three terms in the stress field expansion corresponding to r(−1∕2), r0, and r1∕2 are identical to that given by Williams’s solution for homogeneous material (Williams, 1957, “On the Stress Distribution at the Base of a Stationary Crack,” ASME J. Appl. Mech., 24, pp. 109–114). Transversely graded plates having exponential gradation of elastic modulus were prepared, and the stress intensity factor (SIF) on the compliant and stiffer face of the material was determined using strain gauges for an edge crack subjected to pure bending. The experimental results indicated that the SIF can vary as much as two times across the thickness for the gradation and loading considered in this study.

Shape Invariant Potential and Semi-Unitary Transformations (SUT) for Supersymmetry Harmonic Oscillator in T 4-Space

Constructing the Semi - Unitary Transformation (SUT) to obtain the supersymmetric partner Hamiltonians for a one dimensional harmonic oscillator, it has been shown that under this transformation the supersymmetric partner loses its ground state in T^{4}- space while its eigen functions constitute a complete orthonormal basis in a subspace of full Hilbert space. Keywords: Supersymmetry, Superluminal Transformations, Semi Unitary Transformations. PACS No: 14.80Lv

EXACT BOUND STATES OF THE D-DIMENSIONAL KLEIN–GORDON EQUATION WITH EQUAL SCALAR AND VECTOR RING-SHAPED PSEUDOHARMONIC POTENTIAL

We present the exact solution of the Klein–Gordon equation in D-dimensions in the presence of the equal scalar and vector pseudoharmonic potential plus the ring-shaped potential using the Nikiforov–Uvarov method. We obtain the exact bound state energy levels and the corresponding eigen functions for a spin-zero particles. We also find that the solution for this ring-shaped pseudoharmonic potential can be reduced to the three-dimensional (3D) pseudoharmonic solution once the coupling constant of the angular part of the potential becomes zero.

Study of eigen function and eigen values for energy by muller s method and COGI method by use Causs potential

This study concern here on we shall compare the Eigenvalues and eigenfunction calculated numerically with those obtained by the asymptotic series (Muller,s method) and COGI method where large and moderately large values of angular momentum L and coupling constants g2 are involved.

Determination of vibrational energy levels and transition dipole moments of CO 2 molecules by density functional theory

An efficient method is presented to calculate the intra-molecular potential energies and electrical dipole moments of CO 2 molecules at the electronic ground state by solving the Kohn–Sham (KS) equation for a total of 101 992 nuclear configurations. The projector-augmented wave (PAW) exchange-correlation potential functionals and plane wave (PW) basis functions were used in solving the KS equation. The calculated intra-molecular potential function was then included in the pure vibrational Schrödinger equation to determine the vibrational energy eigen values and eigen functions. The vibrational wave functions combined with the calculated dipole moment function were used to determine the transition dipole moments. The calculated results were compared with the experimental data.

On the numerical solution of Cauchy singular integral equations in neutron transport

The use of Case eigen functions in the solution of Boltzmann equation for neutron transport results in an integral equation with a singular kernel. We show a solution prescription that combines a polynomial expansion for the unknown, a collocation procedure for fixing the expansion coefficients and a Double Exponential quadrature for the Cauchy principal value integral. For a set of test problems, we demonstrate the advantages of this method over the same procedure based on the TANH quadrature.

Existence and basisness of eigen and adjoined elements of linear operators

In this paper we continue the study described earlier in No. 5, 2006, of Russian Mathematics (Izv. Vyssh. Uchebn. Zaved., Matematika). We establish conditions, providing the asymptotics mentioned in the cited paper. We prove the basis property of eigen functions and adjoined ones in linear problems for differential equations with deviating arguments.

Computation and scaling of inductance gradients in electromagnetic launch systems

When a pulsed electric potential is applied to the terminals of a sliding electrical contact system, the armature translates under the influence of Lorentz forces caused by diffusing currents. The inductance gradient is conventionally used as a parameter to quantify the total Lorentz force acting on the armature and size the systems. The determination of the variation of the inductance gradient with system parameters such as the dimensions of the rails and armature, and the ramp rate of the applied potential is of interest in the design of sliding electrical contact systems with specified objectives. It is also useful in scaling results from laboratory to fieldable systems. The inductance gradient can be calculated with finite-element analyses using codes such as EMAP3D; this will be a time-consuming process. Semi-analytical formulations are possible for systems with simple and regular geometries such as rectangular sections. Solutions of the field diffusion equation can be expressed as a weighted sum of eigen functions characteristic of the geometry with imposed boundary conditions. A semi-analytical formulation based on eigen functions is presented here for computing and scaling inductance gradients of a sliding electrical contact system with rectangular rails and armatures. The effects of the dimensions of the rails and armatures, and the effect of the external ramp rate of the applied current on the inductance gradients are studied. The results are compared with available computational and experimental data.

波浪中非定常運動する浮体に作用する非線形波浪流体力 第一報

There exist nonlinear wave loads such as wave drift force, wave drift damping and wave drift added mass, when the floating body is slowly oscillating in waves. In the present study, wave drift added mass acting on an arbitrarily shaped body is evaluated based on the potential theory. A moving coordinate frame following the low frequency motion of the body is adopted to express this interaction model. Two small parameters which express wave slope(ε) and the frequency of slow oscillations(σ) are used for the perturbation expansion. The boundary value problem of each order is solved by the hybrid method. The fluid domain is divided into two regions. The potentials are expressed by the eigen function expansion in the outer region while the inner region is solved by the boundary element method (BEM) with the Rankine source as the Green's function. Element functions with higher order are utilized to improve the accuracy. Wave drift added mass is contributed from five components of potentials. The following potentials are solved in this work. First is the wave potential (Ο(ε)). Second is the disturbance potential for slowly oscillation of the body (Ο(σ)). The last one is the linear potential referring to the incident wave but affected by disturbance of the body (Ο(ε σ)). In the case of the Ο(ε σ) problem, it is important to treat the inhomogeneous free surface boundary condition. In the present paper, the Emmerhoff solution and a new approach about integral over the free surface are expanded for it. The wave drift added mass is calculated. Result is compared with the semi-analytical one. Good agreement is observed.

Study on stress fields in a V-shape notched disc under distributed load

A new method is developed for determining the stress and displacement fields around a sharp V-shape notched disc, which is symmetrically loaded on the circumferential edge. Complex eigen function expansion is used to satisfy the stress free condition of the sharp V-shape notch. Boundary condition of the external load applied on the circumferential edge is satisfied with the aid of the Schmidt method. An example of numerical calculation on the stress field is presented and examined. Approximation expressions based on the eigen function expansion are proposed and its validity is confirmed. Finally, the numerical results are compared to photoelastic experiment.

Estimation of Diffraction Forces on Moored Ships in Harbors through Wave Field Calculations with Boussinesq Model

On ship motions in harbors, a method for estimating diffraction forces from wave field calculations with the Boussinesq model is considered based on potential theory. Since the potential and its normal derivative on any arbitrary boundary enclosing the ship is obtained from the Boussinesq model as time histories of the water surface oscillation and the depthaveraged velocities at each grid point, the incident wave potential along the ship is calculated through Green's Identity Formula. Then a boundary value problem on the diffraction potential is formulated using the method of matched eigen function expansions. By solving the problem, the diffraction potentials are determined, and the ship motions are estimated through the retardation function method using thus obtained diffraction forces.

Unsteady Couette flows in a second grade fluid with variable material properties

Fluid solid mixtures are generally considered as second grade fluids and are modeled as fluids with variable physical parameters. Thus, an analysis is performed for a second grade fluid with space dependent viscosity, elasticity and density. Two types of time-dependent flows are investigated. An eigen function expansion method is used to find the velocity distribution. The obtained solutions satisfy the boundary and initial conditions and the governing equation. Remarkably some exact analytic solutions are possible for flows involving second grade fluid with variable material properties in terms of trigonometric and Chebyshev functions.

Double plate system with a discontinuity in the elastic bonding layer

The double plate system with a discontinuity in the elastic bonding layer of Winker type is studied in this paper. When the discontinuity is small, it can be taken as an interface crack between the bi-materials or two bodies (plates or beams). By comparison between the number of multifrequencies of analytical solutions of the double plate system free transversal vibrations for the case when the system is with and without discontinuity in elastic layer we obtain a theory for experimental vibration method for identification of the presence of an interface crack in the double plate system. The analytical analysis of free transversal vibrations of an elastically connected double plate systems with discontinuity in the elastic layer of Winkler type is presented. The analytical solutions of the coupled partial differential equations for dynamical free and forced vibration processes are obtained by using method of Bernoulli’s particular integral and Lagrange’s method of variation constants. It is shown that one mode vibration corresponds an infinite or finite multi-frequency regime for free and forced vibrations induced by initial conditions and one-frequency or corresponding number of multi-frequency regime depending on external excitations. It is shown for every shape of vibrations. The analytical solutions show that the discontinuity affects the appearance of multi-frequency regime of time function corresponding to one eigen amplitude function of one mode, and also that time functions of different vibration basic modes are coupled. From final expression we can separate the new generalized eigen amplitude functions with corresponding time eigen functions of one frequency and multi-frequency regime of vibrations.

Response and Calibration of A New Logging‐Whiledrilling Resistivity Tool

AbstractThe distribution of electromagnetic fields and the feature of receiver coil's response for a new loggingwhile‐drilling resistivity tool in homogeneous earth are analyzed, and the calibration method of the tool in vacuum is studied. The vector eigen function expansion formulae of the dyadic Green's functions with magnetic current source in a radial layered medium are derived in the cylindrical coordinate system, and hence the needed analytical formulae are obtained to study the distribution of electromagnetic fields, the receiver coil's response and the calibration method of the tool. It is shown through computation that the directional exploring capacity of the tool is greatly enhanced because of the existence of the drill collar. It is also shown that different sizes of the drill collar and different distances between the coils and the drill collar have obviously different influence on the response of the coils in vacuum. The real‐part signal produced by the drill collar in the earth can be approximately regarded as constant, and the imaginary part can also be approximately regarded as constant when the conductivity of the earth is greater than 0.01S·m−1. Numerical computation has shown that the bigger the tool's eccentric level and the drill collar's radius, the smaller the apparent conductivity of the calibration ring's resistance

The wave forces and moments on a floating rectangular box in a two-layer fluid

A comprehensive set of computational formulations is presented for the hydrodynamic forces on a floating rectangular box in a two-layer fluid by matching eigen functions of the interior and exterior problems. Some examples are given and the results indicate that density stratification can have a relative large effect on the hydrodynamic forces on the floating rectangular box over a wide range of frequencies.

The Hellmann–Feynman theorem and its relation with the universal density functional

The integrated Hellmann–Feynman theorem provides an expression for the energy as a functional of the density. However, this result is completely defined only for the densities corresponding to the eigen functions of the Schrödinger equation. Since this density functional is obtained by integrating the Hellmann–Feynman theorem, it is not possible to obtain bounds to the true energy by using the variation method together an approximate density. This point is discussed through some elementary examples. The relation between the integrated Hellmann–Feynman theorem and the Hohenberg and Kohn theorem, is also stressed in this Letter.

Theoretic solution of rectangular thin plate on foundation with four edges free by symplectic geometry method

The theoretic solution for rectangular thin plate on foundation with four edges free is derived by symplectic geometry method. In the analysis proceeding, the elastic foundation is presented by the Winkler model. Firstly, the basic equations for elastic thin plate are transferred into Hamilton canonical equations. The symplectic geometry method is used to separate the whole variables and eigenvalues are obtained simultaneously. Finally, according to the method of eigen function expansion, the explicit solution for rectangular thin plate on foundation with the boundary conditions of four edges frees are developed. Since the basic elasticity equations of thin plate are only used and it is not need to select the deformation function arbitrarily. Therefore, the solution is theoretical and reasonable. In order to show the correction of formulations derived, a numerical example is given to demonstrate the accuracy and convergence of the current solution.

Sd-model with strong exchange coupling and a metal-insulator phase transition

Sd-exchange model (Kondo lattice model) is formulated for strong sd-coupling within the framework of the Xoperators technique and the generating functional approach. The X-operators are constructed based on the exact eigen functions of a single-site sd-exchange Hamiltonian. Such representation enables us to develop a perturbation theory near the atomic level. A locator-type representation was derived for the electron Green’s function. The electron self-energy includes interaction of electrons and spin fluctuations. An integral equation for the self-energy was obtained in the limit of infinite loca lized spins. A solution of this equation in the static approximation for spin fluctuations leads to a structure of electron Green’s function showing a metal-insulator phase transition. This transition is similar to that in the Hubbard model at half filling.