Mathieu Progressive Waves

A new family of exact solutions to the wave equation representing relatively undistorted progressive waves is constructed using separation of variables in the elliptic cylindrical coordinates and one of the Bateman transforms. The general form of this Bateman transform in an orthogonal eurvilinear cylindrical coordinate system is discussed and a specific problem of physical feasibility of the obtained solutions, connected with their dependence on the cyclic coordinate, is addressed. The limiting case of zero eccentricity, in which the elliptic cylindrical coordinates turn into their circular cylindrical counterparts, is shown to correspond to the focused wave modes of the Bessel-Gauss type.

Planar Retro-Reflector for Grazing Incident Beam of High-Energy X-Ray Photons

The grazing-incidence hard X-ray retro-reflector (GIRR) is a novel optical element, which completely suppresses the conventional mirror beam in the retro-reflection mode, and, at the same time, totally reflects the primary grazing incident X-radiation backwards to its source in strictly anti-parallel direction with minimal scattering. The GIRR is a good alternative to hard X-ray normal-incidence Bragg mirror, and has a high potential for applications in various X-ray optical devices and techniques, such as, hard X-ray optical noise (or background) filters, high-quality hard X-ray waveguides, low-gain hard X-ray free electron laser resonators (XFELRs), X-ray holography, coherent X-ray diffraction imaging, phase-contrast imaging, as well as in hard X-ray optical data storage devices and deep space hard X-ray communications. The proposed optical element consists of single-crystal wafer covered by a thin, non-diffracting layer of low-absorbing material.

Surface motion of a semi-elliptical hill for incident plane SH waves

A closed-form analytical solution of surface motion of a semi-elliptical cylindrical hill for incident plane SH waves is presented. Although some previous analytical work had already dealt with hill topography of semi-circular and shallow circular, our work aims at calculating surface motion of very prolate hill for high incident frequency, and explaining the special vibrating is checked by boundary conditions, numerical results for and some conclusions are obtained. properties of very prolate hill. Accuracy of the solution surface motion of oblate and prolate hills are calculated,

Propagating Characteristics of Confocal Elliptical Waveguide Filled with Multilayered Dielectrics

Using the method of separation of variables in the elliptical coordinate system, a recursive formula for the electromagnetic fields in a confocal elliptical waveguide filled with multi-layered homogeneous isotropic media is derived; then the eigenequation for it is given. When an elliptical waveguide becomes a circular waveguide, the electromagnetic fields and the eigenequation of the circular waveguide can be obtained from the eigenequation of the elliptical waveguide using the asymptotic formulae of Mathieu and modified Mathieu functions for a large radial coordinate in the elliptical coordinate system, and the eigenequation of a circular waveguide filled with multilayered dielectrics can be treated as a special case of an elliptical waveguide. In addition, some numerical examples are presented to analyze the propagating characteristics influenced by the permittivity, permeability of dielectrics filled in the elliptical waveguide, etc. The results show that changing the permittivity or permeability of the dielectrics filled in the waveguide and the major semiaxis value of the i-th layer can change the propagating characteristics of an elliptical waveguide.

Dynamic stress concentration of an elliptical cavity in a semi-elliptical hill under SH-waves

The method of wave-function expansion in elliptical coordinates,elliptical cosine half-range expansion and Mathieu function were applied to obtain an exact analytical solution of the dynamic stress concentration factor(DSCF)around an elliptical cavity in a shallow,semi-elliptical hill.An infinite system of simultaneous linear equations for solving this problem was established by substituting the wave expression obtained by the Mathieu function including the standing wave expression of elliptical lining given herein into the boundary condition obtained by the region-matching method.The finite equations system with unknown coefficients obtained by truncation were solved numerically,and the results in the case of an ellipse degenerating into a circle were compared with previous results to verify the accuracy of the method.The effects of different aspect ratios,incident wave angles and aperture ratios on the dynamic stress concentration around the elliptical cavity were described.Some numerical results,when the elliptical hill was changed into a circular one,were analyzed and compared in detail.In engineering,this model can be regarded as a semi-cylindrical hill with an elliptical cylindrical unlined tunnel under the action of SH waves,and the results are significant in aseismic design.

Coupled hydroelastic vibrations of an elliptical cylindrical tank with an elastic bottom

An exact three-dimensional analysis based on the linear potential theory and the elaborated method of eigenfunction expansion in elliptic coordinates are presented to study the free coupled elasto-hyrodynamic characteristics of an upright non-deformable cylindrical container of elliptical planform with a flexible bottom plate, filled to an arbitrary depth with an inviscid incompressible liquid. Extensive numerical data are presented in an orderly fashion for the first few symmetric/anti-symmetric coupled hydroelastic natural frequencies as a function of fluid depth parameter for two plate aspect ratios. Also, selected hydrodynamic and structural deformation modes shapes are presented in graphical form. The effects of liquid level, bottom plate elasticity, and cross sectional aspect ratio on the sloshing frequencies and hydrodynamic pressure modes are examined. The validity of the results is examined through computations using a commercial finite element package as well as by comparison with the data available in literature.