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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: To extend the applicability to nonflat boundaries, this paper formulates one-way wave equations and corresponding reciprocity theorems in terms of curvilinear coordinates of the semiorthogonal (SO) type.
Abstract: One-way wave equations conveniently describe wave propagation in media with discontinuous and/or rapid variations in one direction, but with smooth and slow variations in the complementary transverse directions. In the past, reciprocity theorems have been developed in terms of one-way wave fields. The boundaries of the integration volumes and the variations of the medium parameters must adhere to strict conditions. The variations must have the smoothness required by pseudodifferential operators, while the boundaries have to be flat. To extend the applicability to nonflat boundaries, this paper formulates one-way wave equations and corresponding reciprocity theorems in terms of curvilinear coordinates of the semiorthogonal (SO) type. In SO coordinate systems, one of the covariant basis vectors is orthogonal to the others, which can be nonorthogonal among each other. The same applies to the contravariant basis vectors. Furthermore, the orthogonal directions coincide; that is, the orthogonal co- and contravariant basis vectors coincide. SO coordinates are characterized by a local property of the basis vectors. An extra specification is necessary to make them conform in any way to nonflat boundaries. This can be done in terms of so-called lateral Cartesian (LC) coordinates. Cartesian coordinates are mapped to LC coordinates by applying an invertible transformation to one coordinate while keeping the others the same. LC coordinates are a straightforward means to describe or conform to nonflat boundaries. Applications of the extended reciprocity theorems include removal of multiple reflections, removal of complex propagation effects, wave field extrapolation, and synthesis of unrecorded data.

17 citations

Journal ArticleDOI
TL;DR: In this paper, an analytic solution to the problem of a plane electromagnetic wave scattering by two parallel elliptic conducting cylinders is presented using an iterative procedure to account for the multiple scattered field between the cylinders.
Abstract: An analytic solution to the problem of a plane electromagnetic wave scattering by two parallel elliptic conducting cylinders is presented using an iterative procedure to account for the multiple scattered field between the cylinders. To compute the higher order terms of the scattered fields, the translation addition theorem for Mathieu functions is implemented to express the field scattered by one cylinder in terms of the elliptic coordinate system of the other cylinder in order to impose the boundary conditions. Scattered field coefficients of various orders are obtained and written in matrix form. Numerical results are obtained for the scattered field in the far zone for different axial ratios, electrical separations and angles of incidence.

16 citations

Journal ArticleDOI
TL;DR: In this paper, the authors used integral transforms to obtain rapidly convergent series for one-dimensional elliptical lattice sums and use these lattice sum to determine the transport properties of composites constructed from multiple layers of elliptical cylinders.
Abstract: The Rayleigh method has been used in previous work to determine the effective transport properties of a rectangular array of elliptical cylinders for all cases where the ellipses are non‐intersecting. However, the calculation of the elliptic lattice sums that naturally arose was numerically inefficient and analytically cumbersome. In this present work we use integral transforms to obtain rapidly convergent series for one‐dimensional elliptical lattice sums and use these lattice sums to determine the transport properties of composites constructed from multiple layers of elliptical cylinders. We study the convergence of effective transport properties as a function of the number of layers and show that this convergence is extremely rapid. Further, we use the integral transform representation to exhibit the simplest form of the interior addition formula for harmonic functions in elliptical coordinates.

16 citations

Journal ArticleDOI
TL;DR: In this paper, a numerical procedure for the orthogonalization of a discrete non-orthogonal coordinate system in two dimensions is described, based on an initial-value formulation.

16 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20237
202211
202111
202010
201913
201810