Showing papers on "Elliptic coordinate system published in 1976"
TL;DR: In this paper, a simple Cartesian to spheroidal coordinate system is presented, which eliminates the usual need for iteration and tests show that this formula is virtually rigorous for terrestrial stations.
Abstract: Formulae relating cartesian to spheroidal coordinate systems are examined. A simple equation eliminating the usual need for iteration is presented. Tests show that this formula is virtually rigorous for terrestrial stations.
99 citations
TL;DR: A list of orthogonal coordinate systems which permit R−separation of the wave equation ψtt−Δ2ψ=0 is presented in this article, where the basis operators associated with each coordinate system are also presented as symmetric second order operators in the enveloping algebra of the conformal group O(3,2).
Abstract: A list of orthogonal coordinate systems which permit R‐separation of the wave equation ψtt−Δ2ψ=0 is presented. All such coordinate systems whose coordinate curves are cyclides or their degenerate forms are given. In each case the coordinates and separation equations are computed. The two basis operators associated with each coordinate system are also presented as symmetric second order operators in the enveloping algebra of the conformal group O(3,2).
34 citations
01 Aug 1976
TL;DR: In this paper, a theoretical analysis for the membrane and bending stresses around an arbitrarily oriented crack in a long, thin, circular cylindrical shell subjected to uniform axial tension, internal pressure and torsional loadings is presented.
Abstract: A theoretical analysis is presented for the membrane and bending stresses around an arbitrarily oriented crack in a long, thin, circular cylindrical shell subjected to uniform axial tension, internal pressure and torsional loadings. The method of analysis involves perturbation in a curvature parameter defining the crack size with respect to dimensions of the shell and the results are valid for cracks of small length. The problem is first solved completely in elliptic coordinates and the stress intensity factors are then determined by carrying out a transformation to polar coordinates with crack tip as the origin. Closed-form expressions are obtained for the membrane and bending components of the elastic stress intensity factors and numerical results are presented over the complete rage of the crack angle. For the two symmetric orientations, the solutions agree with known solutions.
19 citations
01 Jan 1976
TL;DR: A coordinate grid has been imposed on the plates of Chap.
Abstract: A coordinate grid has been imposed on the plates of Chap. 6. Its three axes are; latero-lateral (X coordinate), posteroanterior (Y coordinate) and ventro-dorsal (Z coordinate).
11 citations
TL;DR: In this article, the authors proposed a conformal transformation that maps an infinite chevron in the complex Jacobi coordinate z=x+iy plane to an infinite strip in the w=u+iv plane, where u is the reaction coordinate; v is the vibrational coordinate.
Abstract: In order to build wavefunctions for colinear reactive scattering from solutions of a one‐dimensional Schrodinger equation, it is necessary to change the coordinate system from mass‐weighted Jacobi coordinates (x,y) to a system of reaction coordinates that retains the ∇2 character of the kinetic energy operator. Connor and Marcus [J. Chem. Phys. 53, 3188 (1970)] proposed a conformal transformation that maps an infinite chevron in the complex Jacobi coordinate z=x+iy plane to an infinite strip in the w=u+iv plane. u is the reaction coordinate; v is the (generalized) vibrational coordinate. We show that computational difficulties associated with this transformation can be partially removed by either of two methods of mapping part of the w‐plane strip into an infinte strip in a third plane, t=r+is. The first method introduces ’’dipole pairs’’ in the w plane. The second introduces a slit in the w‐plane strip. Relative advantages of the two methods are discussed. A Connor–Marcus‐like map from a semi‐infinite tr...
10 citations
01 Aug 1976
TL;DR: In this paper, the numerically generated body-fitted coordinate systems have been studied from a very basic point of view, which leads to a well structured method of coordinate system control in any region of interest and is also instrumental in obtaining the machine solutions of the field differential equations in general dependent variables on the generated coordinate systems.
Abstract: : This paper presents a study in depth on the numerically generated body-fitted coordinate systems All fundamental ideas have been developed from a very basic point of view which leads to a well structured method of coordinate system control in any region of interest It is also shown how the developed concepts are instrumental in obtaining the machine solutions of the field differential equations in general dependent variables on the generated coordinate systems (Author)
7 citations
TL;DR: In this article, the propagation of elastic waves in an oblate spheroidal model has been studied based on the ray theory, with axial symmetry starting from the general formulation of Hamilton's principle.
Abstract: The propagation of elastic waves in an oblate spheroidal model has been studied based on the ray theory, with axial symmetry starting from the general formulation of Hamilton's principle. Through certain transformations the wave motion in this curvilinear coordinate system may be treated as a flow problem in the rectangular coordinate system. Two parameters, orthogonal to each other, are introduced to describe the geometry of rays in the three-dimensional space. The inverse problem is formulated in terms of boundary values on the outer surface.
5 citations
TL;DR: In this article, the potential energy curves of the H2+ in the lowest even and odd π and δ states, that is in the 2pπu, 3dπg and 3dδg, 4fδu excited states, were determined with a simple analytical nonexponential wave function in elliptical coordinates using the variational method.
Abstract: The potential energy curves of the H2+ in the lowest even and odd π and δ states, that is in the 2pπu, 3dπg and 3dδg, 4fδu excited states are determined with a simple analytical non-exponential wave function in elliptical coordinates using the variational method. The calculated energy values agree well with the corresponding exact ones in the whole considered range of the internuclear distance (0.5≤R≤10a0). The 2pπu state has a shallow potential minimum.
2 citations
2 citations
TL;DR: In this article, the potential energy curves of the two-centre HeH2+, LiH3+ and HeLi4+ heteronuclear systems in the lowest lying π and δ states, that is in the 2pπ and 3dδ excited states, were determined with a simple analytical non-exponential wave function in elliptical coordinates using the variational method.
Abstract: The potential energy curves of the two-centre HeH2+, LiH3+ and HeLi4+ heteronuclear systems in the lowest lying π and δ states, that is in the 2pπ and 3dδ excited states are determined with a simple analytical non-exponential wave function in elliptical coordinates using the variational method. All considered states are repulsive. The energy values calculated in the appropriate approximation agree well with the exact ones, where they are available in the considered range of the internuclear distance (0.5a0 ≤R ≤ 6a0).
2 citations