Topic
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, a computational method to study three-body rearrangement processes for nonzero total angular momentum is presented, which is applied to muon exchange between muonic hydrogen and a heavier nucleus.
Abstract: A computational method to study three-body rearrangement processes for nonzero total angular momentum is presented. The method is applied to muon exchange between muonic hydrogen and a heavier nucleus X{sup Z+}:p{mu}+X{sup Z+}{yields}p+({mu}X){sup (Z-1)+}. Muon-transfer rates are calculated for oxygen and neon including partial waves up to J=4 for collision energies in the range 10{sup -3}-10 eV. An excellent agreement is obtained with available experimental results.
16 citations
TL;DR: In this paper, the authors give a complete proof of the existence of an infinite set of eigenmodes for a vibrating elliptic membrane in one to one correspondence with the well-known eigen mappings for a circular membrane.
Abstract: We give a complete proof of the existence of an infinite set of eigenmodes for a vibrating elliptic membrane in one to one correspondence with the well-known eigenmodes for a circular membrane. More exactly, we show that for each pair $(m,n) \in \{0,1,2, \cdots\}^2$ there exists a unique even eigenmode with $m$ ellipses and $n$ hyperbola branches as nodal curves and, similarly, for each $(m,n) \in \{0,1,2, \cdots\}\times \{1,2, \cdots\}$ there exists a unique odd eigenmode with $m$ ellipses and $n$ hyperbola branches as nodal curves. Our result is based on directly using the separation of variables method for the Helmholtz equation in elliptic coordinates and in proving that certain pairs of curves in the plane of parameters $a$ and $q$ cross each other at a single point. As side effects of our proof, a new and precise method for numerically calculating the eigenfrequencies of these modes is presented and also approximate formulae which explain rather well the qualitative asymptotic behavior of the eigenfrequencies for large eccentricities.
16 citations
TL;DR: A theoretical approach is developed for the elliptic shear wave pattern observed in transverse isotropic materials subjected to axisymmetric excitation creating radially converging shear waves normal to the fiber axis that could aid in analysis of other anisotropic tissue structures.
Abstract: Dynamic elastography methods—based on optical, ultrasonic, or magnetic resonance imaging—are being developed for quantitatively mapping the shear viscoelastic properties of biological tissues, which are often altered by disease and injury. These diagnostic imaging methods involve analysis of shear wave motion in order to estimate or reconstruct the tissue's shear viscoelastic properties. Most reconstruction methods to date have assumed isotropic tissue properties. However, application to tissues like skeletal muscle and brain white matter with aligned fibrous structure resulting in local transverse isotropic mechanical properties would benefit from analysis that takes into consideration anisotropy. A theoretical approach is developed for the elliptic shear wave pattern observed in transverse isotropic materials subjected to axisymmetric excitation creating radially converging shear waves normal to the fiber axis. This approach, utilizing Mathieu functions, is enabled via a transformation to an elliptic coordinate system with isotropic properties and a ratio of minor and major axes matching the ratio of shear wavelengths perpendicular and parallel to the plane of isotropy in the transverse isotropic material. The approach is validated via numerical finite element analysis case studies. This strategy of coordinate transformation to equivalent isotropic systems could aid in analysis of other anisotropic tissue structures.
16 citations
TL;DR: In this article, the authors analyzed the properties of a hydrogen atom at the focus of a confining, prolate ellipsoid and obtained the energies for several states, for different positions of the foci and different values of the semi-major axis, and also developed simple wavefunctions for some states, based on the cusp and boundary conditions.
Abstract: We have analysed the properties of a hydrogen atom at the focus of a confining, prolate ellipsoid. This system is separable in terms of elliptic coordinates, and has the unusual property that the solution is a product of two identical functions of elliptic coordinates. We have obtained the energies for several states, for different positions of the foci and different values of the semi-major axis, and also developed simple model wavefunctions for some states, based on the cusp and boundary conditions. The model may provide a good representation of the distortions in the confining surface when the confined atom moves away from the centre.
16 citations
TL;DR: In this paper, two sets of balance equations for a compressible and non-Newtonian fluid in helical coordinate systems are developed for describing flow in a screw extruder, in which the flow path is characterised by two inherent geometrical features: curvature and helicity.
Abstract: Two sets of the balance equations for a compressible and non-Newtonian fluid in helical coordinate systems are developed. They are associated with two different ways of decomposing a velocity vector; one with three non-orthogonal velocity components in the normal, tangential and helical directions and the other with three orthogonal velocity components in the radial, cross-channel and down-channel directions. It is shown that that compared with classical rectangular and cylindrical coordinate systems, a helical coordinate system is most suitable for describing flow in a screw extruder, in which the flow path is characterised by two inherent geometrical features: curvature and helicity.
16 citations