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Elliptic coordinate system
About: Elliptic coordinate system is a research topic. Over the lifetime, 670 publications have been published within this topic receiving 11135 citations. The topic is also known as: elliptical coordinate system & elliptic coordinates.
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TL;DR: In this paper, the elliptic coordinates are used to build a new families of 2D coordinate systems which are orthogonal and admit the separation of variables, and charts of characteristic curves are constructed for these systems and compared with Mathieu's.
Abstract: The elliptic coordinates are used to build a new families of 2D coordinate systems which are orthogonal and admits the separation of variables. The charts of characteristic curves are constructed for these systems and compared with Mathieu's. The possible applications are in quantum mechanics, diffraction theory, vibrations and classical dynamics.
TL;DR: In this paper, the potential of a homogeneous cylinder in a cylindrical coordinate system has been analyzed. But, because of the involved mathematical operations, the analytical formula of the potential for a homogenous cylinder at an arbitrary point has not been seen from others.
Abstract: Some special cases of the potential for a homogeneous cylinder in a cylindrical coordinate system may be treated by virtue of simple integrals, for example, the potential for a straight rod or wire segment and that for a homogeneous cylinder at the point on its axis. However, because of the involved mathematical operations, the analytical formula of the potential for a homogeneous cylinder at an arbitrary point has not been seen from others. In order to solve the problem, the author has taken the following steps: (1) expanding Green's function elk],x -r'/|r'-r| in the cylindrical coordinate system; (2) transforming Green's function elk'r _r|/|r' r| into Green's function l/|r' r| by setting the wave number k to be zero and integrating the separated azimuthal function cos\"('-); (3) using the integral recursion relation for the function r'2m+1/[(z' z)2 + r'2 + r2](4m+I)/2 with respect to r and those for the functions l/[(z'-z)2 + r2](2m_1)/2 and l/[(z/-z)2-i-r2-(-a2](4m_2/_1)/'2 with respect to z , then we can complete the integrals for the function 1 /1 r' — r| and obtain the analytical expression of the potential for the cylinder in the cylindrical coordinate system. For numerical comparison, we have calculated the potentials for the cylinder and the prolate or oblate spheroid with equivalent volume and same high aspect ratio at far field point. The results are satisfactory.
01 Jan 1996
TL;DR: In this paper, the authors developed a procedure to identify elasticity coefficients for materials that have no more symmetry planes, which is necessary to characterize a medium by ultrasonic techniques, but this procedure requires the assumption of orthorhombic symmetry and knowledge of the material symmetry axes.
Abstract: The hypothesis of an orthorhombic symmetry and the knowledge of the material symmetry axes is usually necessary to characterize a medium by ultrasonic techniques [1–5]. However a wrong setting up of the sample or the strata’s stacking defects in industrial composite material lead to the non-superposition between the material symmetry coordinate system and the observation coordinate system. The development of a procedure to identify elasticity coefficients for materials that have no more symmetry planes, is necessary.
01 Jan 2007
TL;DR: In this article, the authors developed and presented techniques, using analytic Green's functions in elliptic cylindrical coordinates and in prolate spheroidal coordinates, for formulating integral equations for physically practicable structures and sources.
Abstract: It is desired to develop and present techniques, using analytic Green’s functions in elliptic cylindrical coordinates and in prolate spheroidal coordinates, for formulating integral equations for physically practicable structures and sources. Equivalent models for two structures are introduced which serve as guides in the derivation of analytic Greens functions, comprising special functions and satisfying required boundary conditions. Expressions for needed field components are represented in terms of integrals of Greens functions times unknowns, and integral equations follow from the proper relationships among sources and field components needed to satisfy Maxwell’s equations.
TL;DR: In this paper, conditions for the confinement of ions in the trapping field were determined and the finite character of ion motion under the proposed confinement conditions was independently checked by numerical integration of the Lagrange equations.
Abstract: Electrostatic traps with ideal spatiotemporal focusing of ions in one direction of their motion and the motion in transverse directions integrable in the elliptic coordinates are considered. Conditions for the confinement of ions in the trapping field are determined. The finite character of ion motion under the proposed confinement conditions is independently checked by numerical integration of the Lagrange equations.