Showing papers on "Antisymmetric relation published in 1968"

The Eigenfunctions of Laplace's Tidal Equations over a Sphere

Abstract: Numerical calculations are presented for the eigenvalues of Laplace’s tidal equations governing a thin layer of fluid on a rotating sphere, for a complete range of the parameter e ( omega = rate of rotation, R = radius, g = gravity, h — depth of fluid layer). The corresponding eigenfunctions or ‘Hough functions’ are shown graphically for the lower modes of oscillation. Negative values of e, which have application in problems involving forced motions, are also considered The calculations reveal many asymptotic forms of the solution for various limiting values of e. The corresponding analytical expressions are derived in the present paper. Thus, as e0 through positive values we have the well-known waves of the first and second class respectively, which were found by Margules and Hough. These can be represented in terms of spherical harmonics. As e-> + oo there are three distinct asymptotic forms. In each of these the energy is concentrated near the equator. In the first type, the kinetic energy is three times the potential energy. In the other two types the kinetic and potential energies are equal. The waves of the second type are all propagated towards the west. The waves of the third type are Kelvin waves propagated eastwards along the equator. All three types are described in terms of Hermite polynomials. As e 0 through negative values there is only one asymptotic form of solution, representing motions which are analytically continuous with Hough’s ‘waves of the second class’. As e —> — oc there are three different asymptotic forms, in each of which the energy tends to be concentrated near the poles of rotation. In the first two types the energy is mainly kinetic and the motion is in inertial circles. In the third type the energy is mainly potential. The modes tend to occur in pairs of almost the same frequency, one being symmetric and the other antisymmetric about the equator. The analytical forms of the solutions involve generalized Laguerre polynomials. In the special case of zonal oscillations, the first two limiting forms as E -> — oo go over into a different form in which the frequency tends to zero as e tends to a finite negative value. In this case the third type does not occur. The way in which the various asymptotic solutions are connected can be traced in figures 1 to 6 (e > 0) and figures 16 to 21 (e 4 4 are tabulated in tables 1 to 10. The eigenfunctions for the lower modes are presented graphically.

Two‐Dimensional Double‐Minimum Model of Hydrogen Bonding: The Symmetric Case

Abstract: A linear AH···A model, representing hydrogen bonding, which incorporates a double minimum in the potential energy, and interaction between the symmetric and antisymmetric stretching motion, is examined. Using harmonic‐oscillator functions as a basis for expansion, solutions of the two‐dimensional wave equation are obtained. Three tests for the convergence of the matrix expansion are satisfied. Using ranges of parameters appropriate to hydrogen‐bonded systems, the combined effects of tunneling and vibrational interaction on the tunneling transition νt, the far‐infrared transition νσ, the normal‐infrared transition νs, and the combination bands νs ± νσ are determined. The effect of replacement of H by D is also examined. The validity of applying the Stepanov approximation to these systems is tested.

Asymmetry in Magnetic Second‐Rank Tensor Quantities

Abstract: The possibility of asymmetry in a variety of second‐rank tensor quantities of importance in magnetic resonance phenomena is discussed. A classical model demonstrating this effect in the case of shielding is described. The present methods for the measurement of anisotropy of tensor quantities in crystals are discussed, and the contributions of the antisymmetric part of such a tensor are shown to occur in terms quadratic in the shielding (or g) tensor components.

On two arbitrarily located identical parallel antennas

Abstract: The conventional problem of two arbitrarily located parallel antennas is solved by using an integral equation technique. The two simultaneous integral equations for the two antennas are first decoupled into two independent integral equations and then solved by an approximate method with currents represented by five trigonometric functions, three for the symmetric and two for the antisymmetric parts. Typical current distributions and input admittances are obtained for half-wave and full-wave antennas in nonstaggered, in 45\deg echelon, and in collinear arrangements. For the nonstaggered case, the results agree with experimental data. For the other two arrangements, no experimental data are yet available. However, the current distribution is also obtained by a numerical method. The two theoretical results agree favorably for all three cases. The five-term method can be extended to a general array of N -parallel elements. This is reserved for a further report.

The effect of macroscopic rotation on anistotropic bilinear spin interactions in solids

Abstract: An examination is made of the effect of rapid specimen rotation on the N.M.R. spectra of solid specimens, with particular reference to the anisotropic electron-coupled spin-spin interaction. It is shown that when the specimen is rotated rapidly about an axis inclined at the special angle of 54° 44′ to the direction of the external magnetic field the spectral broadening arising from any anisotropy of the electron-coupled spin-spin interactions is removed provided the coupling tensors are symmetric. The dipolar interactions and the anisotropy of the chemical shift tensors are of course removed also, and the spectra are in such cases determined by the scalar shifts σ and coupling constants J as in isotropic fluids. It is however found that when a spin-coupling tensor has an antisymmetric component, this component is not in general removed by the rapid rotation. The spectrum is then determined by antisymmetric coupling constants A in addition to σ and J. A discussion is given of the role played by the constan...

Uniform asymptotic solutions for potential flow around a thin airfoil and the electrostatic potential about a thin conductor

Abstract: The two-dimensional irrotational flow of an incompressible inviscid fluid past a thin cylindrical airfoil and the two-dimensional electrostatic field in the exterior of a thin cylindrical conductor are considered. The complete uniform asymptotic expansions of the two potentials are obtained with respect to $\epsilon $, the slenderness ratio of the profile curve C of the cylinder. Only curves C which have axes of symmetry are treated. To find the symmetric part of the potential due to the presence of the body, it is represented as a superposition of potentials of point sources distributed along a segment of the axis inside C. The source strength distribution satisfies a linear integral equation. The uniform asymptotic expansion of the solution of this equation is obtained by adapting the method of Handelsman and Keller [5], [6].The antisymmetric part is obtained by using the harmonic conjugate of an appropriate symmetric potential. Complete expansions of the virtual mass, the polarizability, and the lift o...

Convective Spherical Shell: II. With Rotation

Abstract: The problem of a convective, rotating spherical shell is considered in Herring's approximation. The temperature is expanded in spherical harmonics, YLm(θ,Φ), and the velocity field in basic poloidal pLm(r,t) and toroidal tLm(r,t) vectors. The scalar pLm(r,t) [tLm(r,t)], together with YLm(θ,Φ), defines a basic poloidal [toroidal] vector. The equations for pLm(r,t) and tLm(r,t) with different L's are coupled by the Taylor number and two types of solutions are possible: symmetric or antisymmetric about the equator. For the case of axial symmetry and for a Rayleigh number equal to 1500, we calculate the convective steady-state solution with rotation by successively increasing the Taylor number from zero, its value for no rotation. Using free surface boundary conditions, the relevant equations determine the radial and time-dependent parts of the temperature and velocity field, with the exception of t10(r), the lowest toroidal component of the axisymmetric solution having equatorial symmetry. The conse...

Antisymmetric Projection in the Approximation of No Spin‐Orbit Coupling

Abstract: The explicit form of the projection operator for constructing antisymmetric wavefunctions for N fermions in the approximation of no spin‐orbit coupling is developed. Projection is applied within the one particle approximation. It is shown that if the orbitals associated with the minority spin can be completely expanded in terms of the orbitals associated with the majority spin, then the projected Hartree‐Fock scheme is completely equivalent to unprojected Hartree‐Fock theory. In the unrestricted case, deviations from this condition are not expected to be large, and integral properties such as energies calculated in the projected scheme should not be significantly different from unprojected results. However, for such properties as spin density at the nucleus in atoms or ions with nominally closed s shell, there may be significant differences between projected and unprojected schemes.

Many-particle systems VI. N-fermion systems: derivation of a lower-bound shell model

Abstract: The programme of reducing N-particle problems to one-particle problems is extended to include the Pauli principle. To calculate the ground-state energy of a system of N fermions interacting by pair forces, a rigorous lower-bound shell model is derived. This shell model follows a building-up principle, and tends to improve with increasing number of particles as well as with increasing strength of interaction. For twenty particles antisymmetric in ordinary space interacting by square-well interaction of strength V 0a2 = 200 hslash 2/2m the shell model differs by less than 8% from the calculated upper bound, and hence a fortiori from the exact energy. In order to test the quality of the approximation for various interactions, it was necessary to calculate upper bounds.

Matrix analysis of non-proportionally damped dynamical systems employing the A-P algorithm

Abstract: The A-P algorithm was originally formulated for the eigenreduction of real symmetric matrices but has been extended in recent years to handle normalizable real and complex matrices. Considerable experience with the method for the determination of normal modes of free vibration and buckling modes in complex structures has indicated it to be an efficient computer method, especially in conjunction with the matrix structural methods developed by Argyris [1, 2, 3]. Recent research involving vibration problems having non-proportional damping indicates that the method is equally valuable in this area. The extension of this problem to include aerodynamic damping and stiffness is discussed with reference to a dynamic stability analysis based on the free-free antisymmetric modes of a whole aircraft.

Unsymmetrical Yield Point Loads of Spherical Domes

Abstract: Bounds for the yield-point load of thin, spherical shells exposed to antisymmetric hydraulic loading on surfaces parallel to the middle surface are given and investigated experimentally The bounds are in agreement with the test results obtained for one steel dome and one aluminum dome each approximately 6 in in radius and subjected to a load equipollent to the antisymmetric hydraulic load