Elliptic Cylindrical Coordinates Research Articles

Elliptica topographic waves

Abstract Simple analytical solutions are presented for nondivergent topographic waves (rotational modes) in a certain type of elliptical basin. Under the assumption that the basin's depth contours form a family of confocal ellipses, the governing potential vorticity equation in elliptic cylindrical coordinates reduces to a Cartesian form, independent of the coordinate scale factors. As a consequence, for the exponential depth profile h=e −bξ, where ξ is the radial coordinate, the radial eigenfunctions for elliptically travelling waves in a basin with a partial vertical barrier along the centerline can be expressed in terms of elementary functions. For a lake without a barrier, approximate analytical solutions are obtained by the Rayleigh-Ritz (variational) method. The periods and streamline patterns of the first few modes of the variational solutions are compared with those due to Ball (1965) for an elliptic paraboloid. The gravest mode period of one of the variational solutions has also been computed for...

On the canonical equivalence of the Kepler problem in coordinate and momentum spaces

It is shown that the dynamics of the hydrogen atom in elliptic cylindrical coordinates on the Fock hypersphere S3 in momentum space is canonically equivalent to the R3 (usual position space) dynamics. The implication to the semiclassical quantisation procedure of the hydrogen atom in a weak magnetic field is briefly discussed.

A Vertically Averaged Circulation Model Using Boundary-Fitted Coordinates

Abstract A two-dimensional vertically averaged circulation model using boundary-fitted coordinates has been developed for predicting sea level and currants in estuarine and shelf waters. The basic idea of the approach is to use a set of coupled quasi-linear elliptic transformation equations to map the physical domain to a corresponding transformed plane such that all boundaries are coincident with coordinate lines and the transformed mesh is rectangular. The hydrodynamic equations are then solved by a multi-operation finite difference technique in the rectangular mesh transformed grid. Comparisons of the circulation model predictions for tidally forced flows in a wedge section with both flat and quadratic bottom topography, and in a flat channel with exponential variation in width, were in excellent agreement with corresponding analytic solutions. Simulation of steady-state wind-induced setup in a closed basin formed using elliptic cylindrical coordinates also was in excellent agreement with the analytic ...

Initiation of mode I degradation in sodium-beta alumina electrolytes

The extension of initial surface cracks by the focusing of the ionic current in beta alumina electrolytes (Mode I degradation) is discussed in terms of existing models. Focusing for an ion current impinging on an elliptic-cylindrical flaw is calculated by solving for the electric potential with suitable boundary conditions. The current density distribution along the crack is used to calculate the sodium flow velocity and Poiseuille pressure inside the flaw. Calculated critical current densities using aKlc criterion are several orders of magnitude higher than measured average critical current densities. This implies a lower effectiveKlc for electrolytic degradation than for mechanical testing. Current density enhancement around insulating barriers, such as non-wetted surface areas, is also calculated using elliptic-cylindrical coordinates. Significant current density enhancements are found, but they are localized in very small regions. Crack growth would occur within these regions, but should be arrested once the flaw extends past the high current density zone. A plausible mechanism for decreasingKlc in the electrolytic case is discussed.

On the Unified General Solutions of Linear Wave Motions of Thermoelastodynamics and Hydrodynamics With Practical Examples

In analogy to the development of the potential equations of motion of linear elastodynamics, the governing potential equations for linear wave motions of hydrodynamics and thermoelastodynamics are systematically exploited. As a result of these developments, problems of linear wave motions of homogeneous, isotropic, thermally conducting multi-layered elastic solids and viscous fluids can be systematically solved for the media whose boundaries are described by rectangular, circular, parabolic, and elliptic cylindrical coordinates, as well as by spherical and conical coordinates. Five practical examples are analytically solved to illustrate how the use of the potential equations of motion leads to more systematic solution procedures; these examples can be used for the modeling studies of aero and hydrospace vehicles, geological wave problems, and macroscopic biomechanics.