Radial Mathieu Functions Research Articles

Dynamic Green's functions for multiple elliptical inclusions with imperfect interfaces

The problem of an unbounded elastic solid with multiple elliptical inclusions subjected to a time-harmonic anti-plane concentrated force is semi-analytically solved by using the collocation multipole method. The displacement of matrix and inclusion are represented by angular and radial Mathieu functions. The imperfect interface between the matrix and the inclusion is characterized as a linear spring model with vanishing thickness. It is the derivative that the imperfect condition is involved. The addition theorem of Mathieu function is frequently used to solve multiply-connected domain problems in the traditional multipole method. An alternate here is a direct computation. The associated normal derivative with respect to a non-local elliptical coordinate system is developed by means of directional derivative. Besides simple computation, no truncation error is caused. The displacement field is determined by using the imperfect interface conditions through collocating points along the interface. Several numerical experiments are done to investigate the effects of the driving frequency of the concentrated force, imperfect interface and the convexity of elliptical inclusions on the dynamic Green's functions. © 2020 The Authors. Published by Elsevier Ltd.

Image Conditions for Elliptical-Coordinate Separation-of-Variables Acoustic Multiple Scattering Models with Perfectly Reflecting Flat Boundaries: Application to in Situ Tunable Noise Barriers

SummaryTwo-dimensional time-harmonic multiple scattering problems are addressed for a finite number of elliptical objects placed in wedge-shaped acoustic domains including half-plane and right-angled corners. The method of separation of variables in conjunction with the addition theorems is employed in the elliptical coordinates. The wavefunctions are represented in terms of radial and angular Mathieu functions. The method of images is applied to consider the effect of the infinitely long flat boundaries which are perfectly reflecting: either rigid or pressure release. The wedge angle is $\pi/n$ rad with integer $n$; image ellipses must be appropriately rotated to realise the mirror reflection. Then, the ‘image conditions’ are developed to reduce the number of unknowns by expressing the unknown expansion coefficients of image scattered fields in terms of real counterparts. Use of image conditions, therefore, leads to the $4n^2$-fold reduction in the size of a matrix for direct solvers and $2n$-times faster computation in building the system of linear equations than the approach without using them. Multiple scattering models using image conditions are formulated for rigid, pressure release and fluid ellipses under either plane- or cylindrical-wave incidence, and are numerically validated by the boundary element method. Furthermore, potential applications are presented: arrays of elliptically shaped scatterers make in situ tunable noise barriers by rotating scatterers. Finally, polar-coordinate image conditions (for circular objects) are also discussed when coordinates local to circles are also rotated. In Appendix, analytic formulae are provided, which permits the elliptical-coordinate addition theorems used in this article to be calculated by summation instead of numerical integration.

Mathieu Windows for Signal Processing

Two new windows for signal processing are introduced in this work. The windows are based on radial Mathieu functions and their performance is compared with well-known ones such as Kaiser, cosh, exponential, Hamming, prolate, and other windows. Most of the results are focused on 1D systems (or windows) and extensions to 2D and multidimensional cases are straightforward. These new Mathieu windows have three adjusting parameters and, by a proper choice of them, it is possible to have a wide span of window shapes that cover most of the existing ones and they might be more appropriate for some applications. An example dealing with the identification of three tones plus normal noise is studied, and the performance of these Mathieu windows is among the best ones. So, they are highly recommended for spectral or harmonic analysis applications. Moreover, due to the high flexibility in adjusting Mathieu windows shapes, these new proposed windows may be applicable for smoothing finite response filters, and related applications as well.

Dielectric Coated Slotted Elliptic Antenna Embedded Partly in Infinite Ground Plane

Radiation characteristics of an axially slotted elliptic antenna coated by conventional dielectric or metamaterial material and embedded partly in infinite ground plane are investigated analytically for TM case. The field in each region is written in terms of appropriate radial and angular Mathieu functions with expansion field coefficients. The boundary conditions at the conducting, dielectric coating and ground plane surfaces are enforced exactly to obtain the unknown field expansion coefficients. Results are obtained and plotted for the radiation pattern to show the effect of the presence of the infinite ground plane and parameters of the material dielectric coating.

Diffusion studies of anamorphic GRIN lenses

The present paper reports the diffusion study of cylindrical GRIN rod with elliptical cross section, developed by ion exchange process. The diffusion equation takes the form of Mathieu equations when transform into elliptic coordinate system and the solutions are derived in terms of angular and radial Mathieu functions. Computations of eigenvalues and expansion coefficients as well as angular and radial Mathieu functions are made which shows good agreement with the existing results. Simpler expression for ionic concentration is derived using asymptotic formulae of the functions which are used for final computation of ionic concentration of diffusing cations in elliptic GRIN. The plot of change in concentration versus diffusion depth along different directions approximately correlates with the results obtained by an earlier experimental study.

Reduction of Mutual Coupling Between Neighboring Strip Dipole Antennas Using Confocal Elliptical Metasurface Cloaks

In this paper, we propose a novel approach to reduce the mutual coupling between two closely spaced strip dipole antennas with the elliptical metasurfaces formed by conformal and confocal printed arrays of subwavelength periodic elements. We show that by covering each strip with the metasurface cloak, the antennas become invisible to each other and their radiation patterns are restored as if they were isolated. The electromagnetic scattering analysis pertained to the case of antennas with frequencies far from each other is shown to be as a good approximation of a 2-D metallic strip scattering cancellation problem solved by expressing the incident and scattered fields in terms of radial and angular Mathieu functions, with the use of sheet impedance boundary conditions at the metasurface. The results obtained by our analytical method are confirmed by full-wave numerical simulations.

Vertical mode expansion method for analyzing elliptic cylindrical objects in a layered background.

The vertical mode expansion method (VMEM) [J. Opt. Soc. Am. A31, 293 (2014)] is a frequency-domain numerical method for solving Maxwell's equations in structures that are layered separately in a cylindrical region and its exterior. Based on expanding the electromagnetic field in one-dimensional vertical modes, the VMEM reduces the original three-dimensional problem to a two-dimensional (2D) problem on the vertical boundary of the cylindrical region. However, the VMEM has so far only been implemented for structures with circular cylindrical regions. In this paper, we develop a VMEM for structures with an elliptic cylindrical region, based on the separation of variables in the elliptic coordinates. A key step in the VMEM is to calculate the so-called Dirichlet-to-Neumann (DtN) maps for 2D Helmholtz equations inside or outside the ellipse. For numerical stability reasons, we avoid the analytic solutions of the Helmholtz equations in terms of the angular and radial Mathieu functions, and construct the DtN maps by a fully numerical method. To illustrate the new VMEM, we analyze the transmission of light through an elliptic aperture in a metallic film, and the scattering of light by elliptic gold cylinders on a substrate.

AN ANALYTICAL SOLUTION FOR OSCILLATIONS WITHIN AN ELLIPTICAL HARBOR

A linear function is obtained by transforming the shallow-water wave equation from rectangular coordinates to elliptic coordinates, which gives the ordinary and the modified Mathieu equations respectively to describe oscillations in the polar and the radial directions by applying the method of separation of variables. Oscillations within an elliptical harbor can be described by appropriate products of radial and angular Mathieu functions. Eigenvalues are obtained by implementing the no-flux condition at the boundary. The oscillation is a two dimension distribution, and there are n nodes running in the polar direction, which is the same as the order of the angular Mathieu function; the nodes in radial direction are related with the boundary condition.

Exact solution for vibration analysis of an eccentric elliptical membrane

The classical method of separation of variables in elliptical coordinates in conjunction with the translational addition theorems for Mathieu functions are used to investigate free transverse vibrations of an elastic membrane of elliptical planform with an arbitrarily located elliptical perforation. Subsequently, the elaborated method of eigenfunction expansion is employed to obtain an exact time‐domain series solution, in terms of products of angular and radial Mathieu functions, for the forced transverse oscillations of the eccentric membrane. The analytical solution is illustrated through numerical examples including circular/elliptical membranes with a circular perforation or with an elliptical perforation of selected geometric, orientation, and location parameters. The first five natural frequencies are tabulated, and selected vibration mode shapes are presented in graphical form. Also, the displacement responses of representative membranes in a practical loading configuration (i.e., a uniformly distributed step load) are calculated. The accuracy of solutions is ensured through proper convergence studies, and the validity of results is demonstrated with the aid of a commercial finite element package as well as by comparison with the existing data. The set of data reported herein is believed to be the first rigorous attempt on the free/forced vibrational characteristics of eccentric elliptical membranes for a wide range of geometric parameters.

Acoustic eigenproblems of elliptical cylindrical cavities with multiple elliptical cylinders by using the collocation multipole method

Abstract Acoustic eigenproblems of 3-D elliptical cylindrical cavities having multiple elliptical cylinders are solved by means of a 3-D semi-analytical formulation based on a multipole expansion, directional derivative, collocation technique and singular value decomposition (SVD). The multipole expansion for the acoustic pressure is formulated in terms of angular and radial Mathieu functions due to elliptical boundaries considered here. The boundary conditions are satisfied by uniformly collocating points along the boundaries. When considering sound-hard or Neumann boundary conditions, the normal derivative of acoustic pressure with respect to non-local elliptic coordinates is derived using the directional derivative for multiply-connected domain problems. By truncating the multipole expansion, a finite linear algebraic system is acquired. The direct searching approach is applied to identify the natural frequencies by using the SVD. Several numerical examples are examined, including those of an elliptical cylindrical cavity, a confocal elliptical annulus cylindrical cavity and an elliptical cylindrical cavity with two elliptical cylinders. Convergence and comparison studies are done to demonstrate the validation and accuracy of the present method. No spurious eigensolutions are found in the proposed formulation. Excellent accuracy and fast rate of convergence are the main features of the present method thanks to its semi-analytical character.

Algorithm 934

Software to compute angular and radial Mathieu functions is provided in the case that the parameterqis a complex variable and the independent variablexis real. After an introduction on the notation and the definitions of Mathieu functions and their related properties, Fortran 90 subroutines to compute them are described and validated with some comparisons. A sample application is also provided.

Free vibration analysis of an elliptical plate with eccentric elliptical cut-outs

The elaborated collocation multipole method is employed to obtain a semi-analytical solution, involving proper products of angular and radial Mathieu functions, for the free flexural vibrations of a fully clamped thin elastic plate of elliptical planform containing multiple elliptical cutouts of arbitrary size, location, and orientation. The problem boundary conditions are satisfied by uniformly collocating points on the boundaries, and exactly calculating the normal derivative of plate displacement at the collocation points through use of appropriate directional derivative in each coordinate system. The multipole expansion is truncated to yield a coupled algebraic linear system of equations that is then solved for the nontrivial eigensolutions. Extensive numerical simulations present the first three calculated natural frequencies and the associated deformed mode shapes of an elliptical plate with elliptical/circular cutouts, for a wide range of plate/cutout aspect ratios, and cutout location/orientation parameters. The accuracy of solutions is checked through appropriate convergence studies, and the validity of results is established with the aid of a commercial finite element package as well as by comparison with the data available in the existing literature.

Free vibration and dynamic response of a fluid-coupled double elliptical plate system using Mathieu functions

A three-dimensional analytic model involving products of angular and radial Mathieu functions is developed for the exact free vibration analysis and transient acousto–structural response of two parallel elliptical plates, coupled with an internal bounded inviscid and compressible fluid medium, and under general external transverse loads of arbitrary temporal and spatial variations. Extensive numerical data are presented in an orderly fashion for the first ten symmetric/anti-symmetric system natural frequencies as a function of fluid layer thickness parameter for selected plate aspect ratios. Also, the occurrences of frequency veering phenomena between various modes of the same symmetry group and the interchange of associated mode shapes in the veering region are noted and discussed. Moreover, selected fluid-coupled structural deformation mode shapes are presented in vivid graphical form and the issue of mode localization is examined. The Laplace transform with respect to the time variable is subsequently invoked and a linear system of coupled algebraic equations is ultimately obtained, which is truncated and then solved by implementing Durbin's Laplace inversion algorithm accompanied with special solution convergence enhancement techniques for eradication of spurious oscillations (Gibbs' phenomenon). Numerical simulations are conducted for the displacement time histories of water-coupled double aluminum plates of selected aspect ratios and fluid depths, subjected to external loads of practical interest (i.e., an impulsive point load, a concentrated pulse load, and a uniformly distributed blast load). Validity of the results is established through computations made by using a commercial finite element package as well as by comparison with the data available in literature.

Flexural transient response of elastically supported elliptical plates under in-plane loads using Mathieu functions

The elaborated method of eigenfunction expansion in elliptic coordinates is employed to obtain an exact time-domain series solution, involving products of angular and radial Mathieu functions, for the forced flexural vibrations of a thin elastic plate of elliptical planform. The plate is supported by a constant moduli two-parameter foundation, while elastically restrained against translation and rotation at its edge, and subjected to the combined action of uniform in-plane static edge forces and general arbitrary time-dependent transverse loads with arbitrary initial conditions. Numerical calculations are carried out for the displacement response of clamped or simply supported elliptical plates of selected aspect ratios in various practical loading configurations (i.e., an impulsive point load, a point force in circular motion, a uniformly distributed harmonic load, and a blast load), with or without an elastic foundation, while taking the effects of initial tension or compression below the buckling load into consideration. Limiting cases are considered and good agreements with available results as well as with the computations made by using a commercial finite element package are obtained.

Cavity modes and their excitations in elliptical plasmonic patch nanoantennas

We present experimental and theoretical studies of two dimensional periodic arrays of elliptical plasmonic patch nanoantennas. Experimental and simulation results demonstrate that the azimuthal symmetry breaking of the metal patches leads to the occurrence of even and odd resonant cavity modes and the excitation geometries dependent on their modal symmetries. We show that the cavity modes can be described by the product of radial and angular Mathieu functions with excellent agreements with both simulations and experiments. The effects of the patch periodicity on the excitation of the surface plasmon and its coupling with the cavity modes are also discussed.

Acoustic scattering by multiple elliptical cylinders using collocation multipole method

This paper presents the collocation multipole method for the acoustic scattering induced by multiple elliptical cylinders subjected to an incident plane sound wave. To satisfy the Helmholtz equation in the elliptical coordinate system, the scattered acoustic field is formulated in terms of angular and radial Mathieu functions which also satisfy the radiation condition at infinity. The sound-soft or sound-hard boundary condition is satisfied by uniformly collocating points on the boundaries. For the sound-hard or Neumann conditions, the normal derivative of the acoustic pressure is determined by using the appropriate directional derivative without requiring the addition theorem of Mathieu functions. By truncating the multipole expansion, a finite linear algebraic system is derived and the scattered field can then be determined according to the given incident acoustic wave. Once the total field is calculated as the sum of the incident field and the scattered field, the near field acoustic pressure along the scatterers and the far field scattering pattern can be determined. For the acoustic scattering of one elliptical cylinder, the proposed results match well with the analytical solutions. The proposed scattered fields induced by two and three elliptical–cylindrical scatterers are critically compared with those provided by the boundary element method to validate the present method. Finally, the effects of the convexity of an elliptical scatterer, the separation between scatterers and the incident wave number and angle on the acoustic scattering are investigated.

Vibro-acoustic response of an elliptical plate-cavity coupled system to external shock loads

A general fully coupled three-dimensional vibro-acoustic model is developed to investigate the forced non-stationary acousto-structural response of a thin elastic plate of elliptical planform which is backed by a reverberant, rigid, and finite (closed) elliptic cylindrical acoustic enclosure, while under the action of general external transverse loads of arbitrary temporal and spatial variations. The Laplace transform with respect to the time coordinate is invoked, and the classical method of separation of variables in elliptic coordinates is used to obtain the transformed solutions as a linear combination of even and odd modes in terms of products of radial and angular Mathieu functions. A linear system of coupled algebraic equations is ultimately obtained, which is truncated and then solved numerically by implementing Durbin’s numerical Laplace transform inversion scheme. Detailed numerical simulations are conducted for the temporal histories of plate center-point displacement and on-axis cavity acoustic pressure for air-coupled elliptic aluminum plates of selected aspect ratios when subjected to external loadings of practical interest (i.e., an impulsive point load, a uniformly distributed pulse load, and a blast load). Also, acoustic radiation into the backing enclosure is examined by using appropriate 2D images of the internal sound field for selected cavity depths and plate eccentricities. The presented results confirm that the acousto-elastic characteristics of the coupled plate-cavity system are significantly influenced by the plate aspect ratio, cavity depth and the transverse loading configuration. Validity of the work is established through the computations made by using a commercial finite element package.

TE scattering by a perfect electromagnetic conducting semi-elliptic-cylindrical boss on a perfectly conducting plane

An exact analytic solution is presented to the problem of scattering of a plane wave by a perfect electromagnetic conducting semi-elliptic-cylindrical boss located on an infinite perfectly conducting ground plane. The problem is formulated by expressing the incident, reflected, as well as the scattered electromagnetic fields in terms of appropriate angular and radial Mathieu functions and three sets of expansion coefficients. The coefficients associated with the incident and reflected field expansions are known, but those associated with the scattered field expansions are unknown. The unknown expansion coefficients can be determined analytically, by imposing appropriate boundary conditions at the surface of the associated perfect electromagnetic conducting elliptic cylinder. Results are presented as normalized scattering widths for semi-elliptic-cylindrical bosses of different sizes and admittances, to show the effects of these on scattering.

Acoustical behavior of single inlet and multiple outlet elliptical cylindrical chamber muffler

Transmission loss (TL) of an elliptical cylindrical chamber muffler having a single side/end inlet and multiple side/end outlet is analyzed by means of the 3-D semi-analytical formulation based upon the modal expansion (in terms of the angular and radial Mathieu functions) and the Green's function. The acoustic pressure response obtained in terms of Green's function is integrated over surface area of the side/end ports (modeled as rigid pistons) and upon subsequent division by the port area, yields the acoustic pressure response or impedance Z] matrix parameters due to the uniform piston-driven model. The 3-D semi-analytical results are found to be in excellent agreement with the results obtained by means of 3-D FEA (SYSNOISE) simulations, thereby validating the semi-analytical procedure suggested in this work. Parametric studies such as the effect of chamber length (L), angular and axial locations of the ports, interchanging the locations of inlet and outlet ports as well as the addition of an outlet port for double outlet mufflers on the TL performance are reported, thereby leading to the formulation of design guidelines for obtaining muffler configurations exhibiting a broad-band TL spectrum. One such configuration is an axially long chamber having side-inlet and side-outlet ports such that one of the side ports is located at half the axial length on themajor/minor axis and the other side port is located at three-quarters (or one-quarter) of the axial length on the minor/major axis. (C) 2012 Institute of Noise Control Engineering.

Scattering from a coated perfect electromagnetic conducting elliptic cylinder

An exact analytic solution is presented to the problem of scattering from a coated perfect electromagnetic conducting (PEMC) elliptic cylinder, when it is illuminated by a plane wave as well as a line source. The problem is formulated by expanding the various fields in terms of appropriate angular and radial Mathieu functions and a set of expansion coefficients. The expansion coefficients associated with the incident fields are known, but those associated with the scattered fields and the fields within the coating are unknown. Imposing boundary conditions at the surface of the PEMC elliptic cylinder and the surface of the coating, enables the determination of the unknown expansion coefficients. Results are presented in the form of normalized scattering widths for elliptic cylinders of various sizes, PEMC admittances, and different coatings, to show the effects of these on scattering.