Modified Mathieu Functions Research Articles

Wave interaction with an elliptic disc submerged in a two-layer fluid

Wave interaction with a horizontal elliptic disc submerged in a two-layer fluid is studied here assuming linear theory. An elliptic coordinate system is used to solve the three-dimensional problem analytically. The fluid region is divided into a number of subregions and by means of separation of variables, velocity potential in each region is expressed in terms of series of Mathieu and modified Mathieu functions of real argument. Matching conditions along with the orthogonal properties of eigenfunctions provide solutions for the fluid velocity potentials. Numerical results are presented for the wave-induced forces and moments. The effect of variations in the depth of submergence of the disc, angle of incidence of the incoming wave, density ratio of the fluid and aspect ratio of the disc on these physical quantities have been analyzed. In particular, sudden amplification of forces and moments have been observed at certain frequencies when the disc is situated at close proximity to the interface. Various other behaviors of the solution are investigated in some detail.

A Theoretical Model for Analyzing the Thickness-Shear Vibration of a Circular Quartz Crystal Plate With Multiple Concentric Ring Electrodes.

A dynamic model to analyze the thickness-shear vibration of a circular quartz crystal plate with multiple concentric ring electrodes on its upper and bottom surfaces is established with the aid of coordinate transformation. The theoretical solution is obtained, which can be written in a superposition form of Mathieu functions and modified Mathieu functions. The convergence of the solution is demonstrated, and the correctness is numerically validated via results from the finite element method (FEM). Subsequently, a systematic investigation is carried out to quantify the effect of the electrode size on the energy trapping phenomenon, i.e., the resonant frequency and mode shape, which reveals that the ring electrode has a great influence on the work performance of resonators. With the increase of the electrode inertia, i.e., the radius and mass ratio, new trapped modes emergence with the vibration mainly focused on the plate with partial electrodes. Besides, owing to the anisotropy, degenerated trapped modes have different resonant frequencies and the frequency discrepancy between them will become smaller for higher modes. Finally, the influence of multiple ring electrodes is investigated, and the qualitative analysis and quantitative results demonstrate that multiple ring electrodes will lead to a more uniform mass sensitivity compared with a single ring electrode. The outcome is widely applicable, which can provide theoretical guidance for the structural design and manufacturing of quartz resonators, as well as a thorough interpretation about the underlying physical mechanism.

Transient Pressure-Driven Electroosmotic Flow through Elliptic Cross-Sectional Microchannels with Various Eccentricities

The semi-analytical solution for transient electroosmotic flow through elliptic cylindrical microchannels is derived from the Navier-Stokes equations using the Laplace transform. The electroosmotic force expressed by the linearized Poisson-Boltzmann equation is considered the external force in the Navier-Stokes equations. The velocity field solution is obtained in the form of the Mathieu and modified Mathieu functions and it is capable of describing the flow behavior in the system when the boundary condition is either constant or varied. The fluid velocity is calculated numerically using the inverse Laplace transform in order to describe the transient behavior. Moreover, the flow rates and the relative errors on the flow rates are presented to investigate the effect of eccentricity of the elliptic cross-section. The investigation shows that, when the area of the channel cross-sections is fixed, the relative errors are less than 1% if the eccentricity is not greater than 0.5. As a result, an elliptic channel with the eccentricity not greater than 0.5 can be assumed to be circular when the solution is written in the form of trigonometric functions in order to avoid the difficulty in computing the Mathieu and modified Mathieu functions.

Computation and Applications of Mathieu Functions: A Historical Perspective

Mathieu functions of period $\pi$ or $2\pi$, also called elliptic cylinder functions, were introduced in 1868 by Emile Mathieu together with so-called modified Mathieu functions, in order to help u...

A method to derive thermoelastic free vibration in a simply supported annulus elliptic plate

The present article investigates the thermoelastic free vibration for a simply supported elliptic plate subjected to a thermal load. The realistic solution involving the Mathieu functions and also their derivatives for the heat conduction differential equation subjected to sinusoidal sectional heating on the upper face with the lower face and the curved inner surface is kept at zero temperature, and the outer curve is kept thermally insulated and is derived using the classical method. The strain energy due to bending of the middle surface of the plate undergoing large deflection was well-thought-out by neglecting the second strain invariant terms for the analysis of large amplitude (nonlinear)-free vibrations of a simply supported plate. Furthermore, nonlinear free vibration equation of elliptic structure is developed with the aid of Berger assumption and Hamilton’s principle and obtained its solution using a new integral transform involving Mathieu and modified Mathieu functions. A closed-form bending stress function obtained has been equated with those obtained by Berger’s methodology. The thermal stress components are obtained in terms of resultant bending moments and resultant forces for numerical analysis. The free-vibration mode of the corresponding nonlinear problem, the Jacobi elliptic function, is obtained from the exact resolution of the natural frequency of the simply supported elliptic plate. For the particular case by applying limiting conditions, the elliptic region can degenerate into the problem of the circular zone, and some numerical results have also been plotted in a few instances.

Production performance analysis of fractured vertical wells with SRV in triple media gas reservoirs using elliptical flow

The stimulated reservoir volume (SRV) exists near hydraulic fractures in gas reservoirs after hydraulically fracturing stimulation, which creates one region with distinct gas reservoir properties from initial formation. Due to strong anisotropy, compared with radial flow model, the elliptical flow model is important for the optimal exploitation of triple media gas reservoirs (TMGR) consisting of matrix, vugs and fractures. The paper presents a composite elliptical flow model to analyse the fractured vertical well transient pressure and rate performance in TMGR with SRV. The inter-porosity flow from matrix to fractures is transient flow. Modified Mathieu functions is employed to solve the mathematical model. There are nine flow regimes based on dimensionless pseudo-pressure curves' characteristics and the comparison of transient inter-porosity flow with steady inter-porosity flow is illustrated. Sensitivity analysis for relevant parameters are conducted, such as well radius, SRV radius, mobility ratio, storativity ratio and inter-porosity flow coefficient.

The Dependence of Resonance Frequency to Landing Angle in Reciprocal Scattering Phenomena of the Waves From an Elliptical Plasma Dielectric Antenna

Scattering of an electromagnetic plane wave in transverse electric mode which is obliquely incident to an elliptical plasma antenna with a dielectric rod is comprehensively investigated. The field solutions of the incident, transmitted, and the scattered waves will be derived in terms of Mathieu and modified Mathieu functions. Using asymptotic expansions of Mathieu functions, the scattering cross section per unit length is obtained. It will be shown that in such an elliptical plasma antenna, the backscattering cross section depends on the incident angle and the bistatic scattering cross section varies with scattering angle significantly. Such effects are not seen in antennas with circular cross section due to the circular symmetry in them. The validation of the presented formulations and numerical computation will be examined with the circular case that is an asymptotic well-known case. The graphs of backscattering cross section, scattering pattern, and their corresponding to the polarized charge densities are presented. The dependence of the backscattering cross section and the scattering pattern upon the density and geometrical dimension of the plasma and dielectric are investigated. In addition, it will be shown that the resonance frequencies, under which the plasma antenna response will be maximum, essentially depend on the incident angle of the incident wave.

Analytical solution for the short-crested wave diffraction by an elliptical cylinder

The diffraction of short-crested waves around an elliptical cylinder is solved using the elliptical coordinates, and an analytical solution is obtained in terms of Mathieu and modified Mathieu functions. Analytical expressions for the pressure, the water run-up and the total wave forces on an elliptical cylinder are also derived in this study. The wave run-up of short-crested waves on an elliptical cylinder is quite different from that of a plane incident wave. The maxima and location of the wave run-up as well as the total wave forces are affected significantly by the incident angle and wave phase. Moreover, the total wave forces induced by the short-crested wave on a circular cylinder is overall smaller than that induced by the plane wave with the same wave number, but in some cases the total wave forces induced by the short-crested waves on an elliptical cylinder may be larger than that induced by the plane wave. Thus, it is necessary to take the short-crested wave forces in to account in the design of an offshore structure with elliptical cross-section.

An analysis of the thickness vibration of an unelectroded doubly-rotated quartz circular plate.

The scalar differential equation in the thickness eigendisplacement for the doubly-rotated quartz plates is applied to analyze thickness vibrations of an unelectroded circular plate with free edges at the neighborhood of the pure thickness vibration mode. The scalar differential equation is transformed into an elliptical coordinate system. With the boundary conditions of free edges, the frequencies and the modes are solved in terms of the Mathieu function and the Modified Mathieu function. The results of frequencies of the fundamental harmonic and its third overtone of an AT-cut quartz circular plate by the present approach agree well with the existing theoretical results and the experiment results. The frequencies and modes of an SC-cut quartz circular plate are investigated by the present approach. The frequencies are close to each other when the order of the harmonics is the same. A rotation angle of the symmetric axes of the vibration modes are observed that is dependent on the anisotropic material constants and the order of the harmonics. This approach has potential applications in the design of the doubly-rotated quartz circular resonators.

A Theoretical Description for Elliptical Plasma Antenna Response in Presence of Terahertz Electromagnetic Plane Wave Based on Surface Plasmon Concept

Excitation of THz surface plasmons due to the scattering of transverse magnetic (TM) wave by an elliptical plasma antenna consisting of a metallic rod with dielectric coating is investigated. The incident, transmitted, and scattered electromagnetic waves are expressed in terms of Mathieu and modified Mathieu functions. The geometrical structure and its effect on the scattering cross-section and resulting resonance frequencies are studied and analyzed based on theory of Fabry-Perot resonator. It is shown that the resonance frequency can be varied by changing the geometrical structure and material used between the surfaces of this structure.

Integral transform approach for solving dynamic thermal vibrations in the elliptical disk

ABSTRACTThe present article is concerned with the determination of thermally induced transverse vibration of a uniform thin elliptical disk with elastic supports at both radial boundaries. The temperature field on the disk is considered to be a ramp-type prescribed on the top face while the bottom face is kept at zero temperature. In this study, integral transform technique is used to investigate the conductivity equation in an exact manner by the use of Mathieu functions and modified Mathieu functions. The thermal moment is derived on the basis of temperature distribution, and its stresses are derived based on the resultant bending moments per unit width. The numerical calculations of the distributions of the transient temperature and its associated thermoelastic results are performed, and the numerical results are shown in the figures.

Mathieu functions and its useful approximation for elliptical waveguides

The standard form of the Mathieu differential equation is d2ydη2+(a−2qcos2η)y=0 where a and q are real parameters and q > 0. In this paper we obtain closed formula for the generic term of expansions of modified Mathieu functions in terms of Bessel and modified Bessel functions in the following cases: (i)Ce1'(ξi,γ12)Ce1(ξi,γ12)(ii)Fey1'(ξi,γ12)Fey1(ξi,γ12)(iii)Gey1'(ξi,γ12)Gey1(ξi,γ12)(iv)Ce1'(ξi,γ22)Ce1(ξi,γ22)(iv)Se1'(ξi,γ22)Se1(ξi,γ22). Let ξ 0 = ξ 0 , where i can take the values 1 and 2 corresponding to the first and the second boundary. These approximations also provide alternative methods for numerical evaluation of Mathieu functions.

Effect of Chirality on Dispersion Characteristics of Uniaxial Anisotropic Chiral Elliptical Waveguide

AbstractIn this paper, the dispersion characteristic of elliptical waveguide consisting of uniaxial anisotropic chiral medium is investigated. The eigenvalue equation for the waveguide that is to be studied is deduced by solving the Mathieu and modified Mathieu functions. The cutoff frequencies for several lower order even modes have been calculated and their dispersion characteristics are correspondingly plotted. The effects of elliptical eccentricity and chirality on the mode cutoff frequencies and mode transmission are examined numerically.

Modal analysis and dispersion curves of an elliptical W-type single mode fiber

In this paper, the propagation of electromagnetic waves in W-type elliptical dielectric optical fiber having various cladding layers is presented. The presented fiber has concentric core and cladding which have elliptical cross section and the refractive index of one of the inner cladding (the cladding between first clad and last clad) maximum. Using elliptic cylindrical coordinates, boundary conditions are derived and longitudinal field components for the even and odd modes are obtained. The characteristic equation for the fiber to be studied is determined by solving the Mathieu (q > 0) and the modified Mathieu functions (q < 0). In order to study the fundamental mode, the modal index m is put as m = 1. Finally, the cutoff frequencies for several lower order modes have been calculated and their dispersion characteristics are plotted. The effects of elliptical eccentricity e on the mode cutoff frequencies and mode transmissions are also addressed. The analysis shows that one can control the propagation property of optical fiber by increasing the number of inner claddings. These claddings provide additional degree of freedom to control the modes.

VIBRATION OF THE CIRCULAR ANNULAR PLATE WITH AN INTERNAL FREE EDGE

Whole and with opening elliptic membranes meet as elements of machines of devices: diaphragms, winders. Results of calculations of frequency parameter for the symmetric vibrations of such membranes at the different values of excentricity are presented in background literature and based on one of works. In scientific literature there are not experimental information on determination of frequencies of vibrations of such plates, which would enable to estimate exactness of the presented results. Absent calculations of frequency parameter of asymmetrical vibrations, and also results of calculation frequency the parameter of elliptic membrane with opening. A whole elliptic membrane is stretched by equipartition forces. Worked out an equation of eigentones in the rectangular system of coordinates. The formulas of transformation of rectangular co-ordinates are entered to elliptic and equalization of eigentones is determined in the set elliptic system of coordinates. The decision of this system is given through the ordinary and modified Mathieu function of the first family. Worked out a frequency equation and his decision is found for symmetric vibrations and asymmetrical vibrations at different. For an elliptic membrane with opening to the decision the Mathieu function of the second family is added on the basis of Floquet theorem. Resulted decisions of frequency equalization at the different values of excentricity of internal and external ellipse it is Necessary to conduct experimental researches of tasks on determination of frequency parameter of symmetric and asymmetrical whole with opening of membranes. .

Computing Eigen value equation and modal dispersion characteristics of an elliptical Bragg waveguide

Using the method of separation of variables in the elliptical coordinate system, a recursive formula for the electromagnetic fields in a new type of Bragg waveguide having elliptical core cross section with multilayered cladding is derived. The eigen equation is written in the form of Mathieu and the modified Mathieu functions and a dispersion relation is obtained for various modes supported by the proposed Bragg waveguide. The cutoff frequencies for several lower order even-odd modes have been calculated and their propagation characteristics are plotted. The results show that the dispersion curves are discontinuous and modes can exist only in particular wavelength bands. The effects of elliptical eccentricity on the mode cutoff values and mode transmission are addressed. Finally, the modal birefringence in the said waveguide is also estimated.

True and Spurious Eigensolutions of an Elliptical Membrane by Using the Nondimensional Dynamic Influence Function Method

In this paper, we employ the nondimensional dynamic influence function (NDIF) method to solve the free vibration problem of an elliptical membrane. It is found that the spurious eigensolutions appear in the Dirichlet problem by using the double-layer potential approach. Besides, the spurious eigensolutions also occur in the Neumann problem if the single-layer potential approach is utilized. Owing to the appearance of spurious eigensolutions accompanied with true eigensolutions, singular value decomposition (SVD) updating techniques are employed to extract out true and spurious eigenvalues. Since the circulant property in the discrete system is broken, the analytical prediction for the spurious solution is achieved by using the indirect boundary integral formulation. To analytically study the eigenproblems containing the elliptical boundaries, the fundamental solution is expanded into a degenerate kernel by using the elliptical coordinates and the unknown coefficients are expanded by using the eigenfunction expansion. True and spurious eigenvalues are simultaneously found to be the zeros of the modified Mathieu functions of the first kind for the Dirichlet problem when using the single-layer potential formulation, while both true and spurious eigenvalues appear to be the zeros of the derivative of modified Mathieu function for the Neumann problem by using the double-layer potential formulation. By choosing only the imaginary-part kernel in the indirect boundary integral equation method (BIEM) to solve the eigenproblem of an elliptical membrane, spurious eigensolutions also appear at the same position with those of NDIF since boundary distribution can be lumped. The NDIF method can be seen as a special case of the indirect BIEM by lumping the boundary distribution. Both the analytical study and the numerical experiments match well with the same true and spurious solutions.

A semi-analytical method for near-trapped mode and fictitious frequencies of multiple scattering by an array of elliptical cylinders in water waves

In this paper, we focus on the water wave scattering by an array of four elliptical cylinders. The null-field boundary integral equation method (BIEM) is used in conjunction with degenerate kernels and eigenfunctions expansion. The closed-form fundamental solution is expressed in terms of the degenerate kernel containing the Mathieu and the modified Mathieu functions in the elliptical coordinates. Boundary densities are represented by using the eigenfunction expansion. To avoid using the addition theorem to translate the Mathieu functions, the present approach can solve the water wave problem containing multiple elliptical cylinders in a semi-analytical manner by introducing the adaptive observer system. Regarding water wave problems, the phenomena of numerical instability of fictitious frequencies may appear when the BIEM/boundary element method (BEM) is used. Besides, the near-trapped mode for an array of four identical elliptical cylinders is observed in a special layout. Both physical (near-trapped mode) and mathematical (fictitious frequency) resonances simultaneously appear in the present paper for a water wave problem by an array of four identical elliptical cylinders. Two regularization techniques, the combined Helmholtz interior integral equation formulation (CHIEF) method and the Burton and Miller approach, are adopted to alleviate the numerical resonance due to fictitious frequency.

Free vibration analysis of a finite-length isotropic solid elliptic cylinder using exact three dimensional elasticity theory

A three-dimensional analytical model based on Navier’s displacement equation of motion is developed to describe the free vibrations of a simply supported elastic isotropic solid elliptical cylinder of finite length. Helmholtz decomposition theorem is invoked and the classical technique of separation of variables in elliptical coordinates is employed to solve the subsequent uncoupled governing differential equations as products of ordinary and modified Mathieu functions. Inclusive numerical data are provided in a systematic manner for the first several symmetric/anti-symmetric eigen-frequencies as a function of cross sectional eccentricity for selected cylinder length ratios. The incidences of “mode crossing” and “frequency veering” among different modes of the same symmetry group as well as swapping of the corresponding mode shapes in the veering region are examined. Also, selected three dimensional deformation mode shapes are illustrated. The precision of calculations is checked by performing proper convergence studies, and the correctness of results is demonstrated using a commercial finite element software as well as by comparison with the information in the accessible literature. The collection of presented data is believed to be the first meticulous attempt to determine the free vibration frequencies of finite-length simply-supported elastic elliptical cylinders in an exact manner.

A Semianalytical Approach for a Nonconfocal Suspended Strip in an Elliptical Waveguide

A problem of a nonconfocal suspended strip in an elliptical waveguide is analyzed by using a semianalytical approach, which is the so-called null-field boundary integral-equation method (BIEM). The null-field BIEM is proposed by introducing the idea of null field, degenerate kernels, and eigenfunction expansion to improve the conventional dual boundary-element method (BEM). A closed-form fundamental solution can be expressed in terms of the degenerate kernel containing the Mathieu and modified Mathieu functions in the elliptical coordinates. Boundary densities are represented by using the eigenfunction expansion. By this way, the efficiency is promoted in three aspects: analytical boundary integral without numerical error, natural bases for boundary densities, and exact description of boundary geometry. Due to the semianalytical formulation, the null-field BIEM can fully capture the property of geometry and the error only occurs from the truncation of the number of the eigenfunction expansion terms in the real implementation. The present method is also a kind of meshless method since only boundary nodes are needed to construct influence matrices instead of using boundary elements. Both TE and TM cases are considered in this paper. To verify the validity of the present method, the dual BEM and finite-element method are also utilized to provide cutoff wavenumbers. Besides, the analytical solution of a confocal elliptical waveguide can be derived by using the present method. After comparing with published data, good agreement is made.