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 two-surface Cartesian coordinate system and cylindrical coordinate system measurement techniques are applied to obtain the scattered sound fields of a general shape, based on the principle that any waveform can be decomposed using a twodimensional spatial Fourier transform into wave components that propagate in a known manner.
Abstract: Two‐surface Cartesian coordinate system and cylindrical coordinate system measurement techniques are applied to obtain the scattered sound fields of a general shape. This decomposition method is based on the principle that any waveform can be decomposed using a two‐dimensional spatial Fourier transform into wave components that propagate in a known manner. The Cartesian method was developed by Tamara [J. Acoust. Soc. Am. 88, 2259–2264 (1990)] to measure the reflection coefficients of waves incident at oblique angles onto planar surfaces, so our paper focuses on the optimization of this method when applied to scattering investigations. A two‐surface technique to separate the incident and scattered field is also developed in cylindrical coordinates. The separation process is carried out in the wave‐number domain in a manner similar to what has been developed for Cartesian coordinates. However, because the incident and scattered fields are not directly separable in cylindrical coordinates, knowing general pr...
25 citations
TL;DR: In this article, the authors considered the sub-Riemannian length minimization problem on the group of motions of pseudo-Euclidean plane that form the special hyperbolic group SH(2), which comprises of left invariant vector fields with 2-dimensional linear control input and energy cost functional.
Abstract: We consider the sub-Riemannian length minimization problem on the group of motions of pseudo-Euclidean plane that form the special hyperbolic group SH(2). The system comprises of left invariant vector fields with 2-dimensional linear control input and energy cost functional. We apply the Pontryagin maximum principle to obtain the extremal control input and the sub-Riemannian geodesics. A change of coordinates transforms the vertical subsystem of the normal Hamiltonian system into the mathematical pendulum. In suitable elliptic coordinates, the vertical and the horizontal subsystems are integrated such that the resulting extremal trajectories are parametrized by the Jacobi elliptic functions. Qualitative analysis reveals that the projections of normal extremal trajectories on the xy-plane have cusps and inflection points. The vertical subsystem being a generalized pendulum admits reflection symmetries that are used to obtain a characterization of the Maxwell strata.
25 citations
TL;DR: In this paper, the direct transformation of the Cartesian coordinate system into a system of geodetic coordinates is discussed, and relations for direct transformation from Cartesian coordinates to geodetectors are established.
Abstract: Relations for the direct transformation of the Cartesian coordinate system into a system of geodetic coordinates.
25 citations
TL;DR: Oscillatory properties of linear matrix differential system, applying generalization of polar coordinate transformation to certain systems of second order as discussed by the authors, applied generalization to linear matrices of the second order.
Abstract: Oscillatory properties of linear matrix differential system, applying generalization of polar coordinate transformation to certain systems of second order
24 citations
TL;DR: In this article, a study of separability conditions for the Helmholtz and Laplace equations is made, and necessary and sufficient conditions are tabulated, by which one can easily determine if a proposed coordinate system will allow solutions by the method of separation of variables.
Abstract: A study is made of separability conditions for the Helmholtz and Laplace equations. Attention is confined to the most useful case for physical applications; namely, 1. (a) The coordinate system is either cylindrical or it has rotational symmetry, and 2. (b) The potential is independent of the third space variable. Necessary and sufficient conditions are tabulated, by which one can easily determine if a proposed coordinate system will allow solutions by the method of separation of variables.
24 citations