Addition Theorem For Bessel Functions Research Articles

Semi-analytical study of the first order horizontal force induced by oblique waves on the array of circular columns of offshore wind turbines equipped with permeable cylinders

This paper studied the incident of linear oblique waves to circular columns of offshore wind turbines in conjunction with concentric and non-concentric thin permeable cylinders. The unique research objective of this work is to assess the possibility of using permeable vertical cylinders to reduce wave forces on the column of offshore wind turbines. The developed semi-analytical method benefits from the mathematical relation between Bessel functions to express the BEM-related integrals in the Fourier modes of the Hankel function and to calculate singular integrals indirectly. Moreover, Graf's addition theorem for Bessel functions was used for shifting between different coordinate systems to avoid calculating related integrals numerically. The basic geometries parameters such as permeability parameter, cylinders arrangement and the radius of the permeable cylinders, and the wavenumber on the horizontal wave-induced force were studied. The results are presented in graphs. The results indicate that covering the columns of DeepCwind and OC3 wind turbines with the permeable cylinders or using the permeable cylinders upstream helps reduce the horizontal wave-induced force at a particular permeability parameter. The results of this research are beneficial in the design of columns of offshore wind turbines and other structures with vertical piles, such as semi-submersibles.

A dynamical study on sea waves interaction with an array of circular elastic floaters of aquaculture fish cages

In this paper, linear oblique wave incident to circular elastic floaters of aquaculture fish cages was studied semi-analytically. This work's unique research objectives include considering the array of elastic floaters where each floater consists of two tori with minor differences in their main radii. Hydrodynamic interaction between the inner and outer of each floater body and hydro-elastic interaction between floaters in the array are the particular features of the problem. Using slender body theory and matched asymptotic expansions, each 3D problem was decomposed into two separate, more straightforward problems named near-field and far-field problems. This approach helps derive fundamental solutions for symmetric and antisymmetric motions. Also, using linear wave theory, the diffracted and radiated waves due to the presence of floaters were expressed in Fourier modes of symmetric and antisymmetric motions. Moreover, Graf's and Gegenbauer's addition theorems for Bessel functions are used to shift between different coordinate systems. This approach is helpful to bridge on calculating related integrals numerically. The developed model verification by BEM-based software demonstrates that the model solves the mathematical formulas with enough accuracy. The effect of various parameters such as floater dimensions, wavenumber, the distances between floaters, and the bending stiffness of floater on the floater's motions was studied. The results demonstrate that as the wavenumber increases, the floater's hydro-elastic interaction becomes essential. Furthermore, the upstream floaters do encounter larger wave loads than a single floater in waves.

A Novel Type of Wave Energy Converter with Five Degrees of Freedom and Preliminary Investigations on Power-Generating Capacity

In order to further improve the power-generating capacity of the wave energy converter (WEC) of oscillating buoy type, this paper puts forward a novel type where the WEC can move and extract power in five degrees of freedom. We make a detailed hydrodynamic analysis of such WECs. Each buoy is modeled as a floating truncated cylinder with five degrees of freedom: surge, sway, heave, roll, and pitch, and there are relative motions among buoys in the array. Linear power take-off (PTO) characteristics are considered for simplicity. Under the linear wave theory, a semi-analytical method based on the eigenfunction expansion and Graf’s addition theorem for Bessel functions is proposed to analyze the hydrodynamic interactions among the WEC array under the action of incident waves, and the amplitude response and power extraction of the WEC array are then solved. After verifying the accuracy of hydrodynamic analysis and calculation, we make preliminary case studies, successively investigating the power-generating capacity of a single WEC, an array of two WECs, and an array of five WEC; then, we compare their results with the conventional heaving WECs. The results show that the WEC with five degrees of freedom can significantly improve the power extraction performance.

Wave diffraction from an array of porous cylinders with porous plates fixed inside

The interaction between water wave and an array of porous cylinders with porous plates fixed inside is investigated theoretically in this study. The porous cylinders are bottom-mounted and surface-piercing, which contain porous plates fixed below the free water surface. The analytical solution of velocity potential in the flow field is derived using eigenfunction expansion and Graf’s addition theorem for Bessel functions under the assumptions of potential flow theory and linear wave theory. This paper focus on investigating the influence of the porous plates on the diffraction process of the multiple porous cylinders. The results show that the existence of the porous plates has little effect on the surge force of the porous cylinders, but it can reduce the wave run up inside the cylinders. The behavior of the porous plates and porous cylinders in combination is similar to that of a wave absorber, which may cause a setdown of the free surface elevation on the array’s downstream side.

A novel mathematical method to analyze the free vibration of eccentric annular plates

Vibration analysis of nominally axisymmetric plate structures with either imperfections or geometric asymmetries due to practical motivations is of interest in designing and developing some mechanical structures. Semi-analytical methods to model such structures suffer from either choosing inappropriate admissible functions or both plausible convergence issues and additional computations owing to employing the addition theorem of Bessel functions. Therefore, the present study aims at developing a new mathematical method to analyze the vibrational behavior of circular plates with geometric asymmetries. The suggested approach makes use of the separation of variables to determine general solutions of the partial differential equation of the plate transverse displacement while defining multiple polar coordinate systems each of which offers a formulation of the plate deformation. Moreover, closed-form geometric equations and the chain rule for determining derivatives are implemented to move from one coordinate system to the other to satisfy boundary conditions without any need for the cumbersome transformation involved in using the addition theorem. A finite element model is also constructed to evaluate the validity of the proposed method before studying the effects of the cutout location and size on natural frequencies and mode shapes of eccentric annular plates.

Bessel beam superposition for analyzing off-axial scattering, forces, and torques

Illumination of an arbitrary-order Bessel beam on an off-axial object is expanded as illuminations of a series of Bessel beam components of different orders on an axially centered object [L. Zhang, J. Acoust. Soc. Am. 143(5), 2796–2800 (2018)]. The expansion follows from a parallel-axis relation that is derived as a generization of Graf's addition theorem of Bessel functions. The off-axial Bessel beam expansion enables to understand the off-axial scattering, forces, and torques in terms of superposing that in the on-axial illuminations by the series of beam components. The superposition reveal negative axial radiation force and negative radiation torque in terms of contributions of different beam components in the on-axial configuration. The exact locations of the object for torque direction reversal are analytically identified for a small particle. The superposition also reveals that the off-axial radiation torque on a small particle exists even for higher-order vortex beams or vortices.

Invited Article: Refined analysis of synchrotron radiation for NIST’s SURF III facility

We have developed a new method for the exact calculation of synchrotron radiation for the National Institute of Standards and Technology Synchrotron Ultraviolet Radiation Facility, SURF III. Instead of using the Schwinger formula, which is only an approximation, we develop formulae based on Graf's addition theorem for Bessel functions and accurate asymptotic expansions for Hankel functions and Bessel functions. By measuring the radiation intensity profile at two distances from the storage ring, we also confirm an apparent vertical emittance that is consistent with the vertical betatron oscillations that are intentionally introduced to extend beam lifetime by spreading the electron beam spatially. Finally, we determine how much diffraction by beamline apertures enhances the spectral irradiance at an integrating sphere entrance port at the end station. This should eliminate small but treatable components of the uncertainty budget that one should consider when using SURF III or similar synchrotrons as standard, calculable sources of ultraviolet and other radiation.

Solutions of vibration problems for thin infinite plates subjected to harmonic loads

New closed form solutions for harmonic vibrations of infinite Kirchhoff plates subjected to a constant harmonic ring load, a constant harmonic circular load and an alternating harmonic circular load are derived. Two different approaches are used to define the closed form solutions. The first approach uses the integration of the harmonic point force and the addition theorem for Bessel functions, while the second approach applies the Hankel transform to solve the inhomogeneous partial differential equation of the Kirchhoff plate theory. The new closed form particular solutions can especially be used in Trefftz like methods and extend their field of application.

Electromagnetic Scattering by 2-D PEC Cylinder of General Cross Section Under <inline-formula> <tex-math notation="LaTeX">$\text {TM}^{z}$ </tex-math> </inline-formula> Illumination

A semianalytical method is presented for the evaluation of scattered fields from arbitrarily shaped 2-D Perfect Electric Conductor (PEC) scatterers under $\text {TM}^{z}$ wave incidence. Surface equivalence principle for scattered fields is used in conjunction with modal expansion of fields in terms of cylindrical eigenfunctions. It is assumed that no field variation exists in axial ( $z$ ) direction. Implementation of boundary conditions on scatterer surface leads to a system of linearly independent equations to solve for equivalent surface currents. Scattered fields are calculated by equivalent surface currents leading to evaluation of scattering widths (Radar Cross-Section of 2-D objects) for different observation angles. $\text {TM}^{z}$ polarized incident wave does not excite any $\text {TE}^{z}$ component. The novelty of proposed method is use of addition theorem of Bessel functions for representation of scattered fields in terms of equivalent surface currents, giving an integral equation with an analytic integrand and a well-behaved impedance matrix. Orthogonality of cylindrical modes results in excitation matrix to be a discrete function of mode numbers. Therefore, contrary to method of moments, there is no need for weighting functions. Four examples are considered, namely, line source and plane wave incidence on a 2-D PEC circular cylinder and also plane wave incidence on 2-D elliptical and equilateral triangular PEC cylinders.

Hydrodynamic interactions of water waves with a group of independently oscillating truncated circular cylinders

In this study, we examine the water wave radiation by arrays of truncated circular cylinders. Each cylinder can oscillate independently in any rigid oscillation mode with a prescribed amplitude, including translational and rotational modes such as surge, sway, heave, pitch, roll, and their combinations. Based on the eigenfunction expansion and Graf’s addition theorem for Bessel functions, we developed an analytical method that includes the effects of evanescent modes in order to analyze such arrays of cylinders. To investigate the effects of several influential factors on convergence, our objective is to dramatically reduce the number of tests required and determine the influencing relationships between truncation number and convergence behavior for different factor combinations. We use the orthogonal test method to fulfill the objective. Lastly, we present our results regarding the effects of evanescent modes on hydrodynamic coefficients.

Bounds for truncation errors of Graf’s and Neumann’s addition theorems

Graf's and Neumann's addition theorems for Bessel functions have been widely used in acoustic and electromagnetic scattering problems, especially the fast multipole method for 2-D scattering problems. This paper studies the truncation errors of Graf's and Neumann's addition theorems and their linear combinations. Explicit bounds and convergence rates of the truncation errors are derived, and convergence calculated. The conclusions are tested by numerical experiments and show that the derived bounds for the truncation errors of the addition theorems are valid.

Effects of eccentric circular perforation on thermal vibration of circular graphene sheets using translational addition theorem

In this article, nonlocal thin plate theory of Eringen is employed to investigate effects of thermal environment on behavior of freely vibrating circular single layer graphene sheet containing a circular perforation of arbitrary size and location. In order to analytically solve equation of motion, the separation of variables method in conjunction with the translational addition theorem for Bessel functions is used. Accuracy and stability of results are verified by the literature. The behavior of sheets in low and high temperature conditions is considered. The influences of temperature change and various geometrical parameters on the natural frequencies are investigated by considering size-dependent material properties. In some cases, thermal buckling phenomenon was observed. Results show that the temperature changes play an important role in free vibration behavior of eccentric circular graphene sheets.

Cutoff Properties of a Circular Waveguide Loaded with Two Eccentric Dielectric Cylinders

An exact closed-form technique is employed to calculate the cutoff wavenumbers of a conducting circular waveguide loaded with two circular dielectric cylinders. An analytical expression of the hybrid modes is derived in terms of cylindrical coordinates of the geometry. The addition theorem of Bessel functions is used to transfer the electromagnetic fields from one coordinate system to another to apply the boundary conditions. A MATLAB (The MathWorks, Natick, Massachusetts, USA) code is developed to solve the system of equations and produce the cutoff wavenumbers for different geometrical parameters. The present problem can be extended to model a circular waveguide loaded with multiple dielectric cylinders that resembles a petroleum-carrying pipeline. A professional High Frequency Structure Simulator (HFSS; ANSYS, Inc., Canonsburg, PA, USA) is used to solve this model and to determine the ratio of the multi-phase fluid.

An approximate analytical solution of the integral equations of non-axisymmetric contact problems for a ring-shaped domain

An algorithm is developed for solving the integral equations of the first and third kind to which non-axisymmetric mixed problems in continuum mechanics and mathematical physics reduce on replacing the boundary conditions in a ring-shaped domain. The use of the Bubnov–Galerkin procedure in conjunction with addition theorems for Bessel functions is the basis of this method. In the final stage of solving integral equations corresponding to arbitrary harmonics of a mixed problem for a ring-shaped domain, the method enables the coefficients of the linear algebraic systems to be represented in the form of simple quadratures that are convenient for numerical implementation. The discussion is carried out using the example of a contact problem in the theory of elasticity for a linearly deformable base of a general type strengthened by a thin coating along its boundary. The effect of the relative thickness of the coating, its stiffness and the shape of the bottom of a ring-shaped punch on the basic contact characteristics is investigated. Publications on the known results are available in the special case of axisymmetric problems and the problem of an inclined ring-shaped punch.

Dynamic effective elastic modulus of polymer matrix composites with dense piezoelectric nano-fibers considering surface/interface effect

Based on effective field method, the dynamic effective elastic modulus of polymer matrix composites embedded with dense piezoelectric nano-fibers is obtained, and the interacting effect of piezoelectric surfaces/interfaces around the nano-fibers is considered. The multiple scattering effects of harmonic anti-plane shear waves between the piezoelectric nano-fibers with surface/interface are averaged by effective field method. To analyze the interacting results among the random nano-fibers, the problem of two typical piezoelectric nano-fibers is introduced by employing the addition theorem of Bessel functions. Through numerical calculations, the influence of the distance between the randomly distributed piezoelectric nano-fibers under different surface/interface parameters is analyzed. The effect of piezoelectric property of surface/interface on the effective shear modulus under different volume fractions is also examined. Comparison with the simplified cases is given to validate this dynamic electro-elastic model.

Symbolic methods for the evaluation of sum rules of Bessel functions

The use of the umbral formalism allows a significant simplification of the derivation of sum rules involving products of special functions and polynomials. We rederive in this way known sum rules and addition theorems for Bessel functions. Furthermore, we obtain a set of new closed form sum rules involving various special polynomials and Bessel functions. The examples we consider are relevant for applications ranging from plasma physics to quantum optics.

The hydrodynamic interactions of an array of truncated circular cylinders as each cylinder oscillates independently with different prescribed modes

An analytical method based on the eigenfunction expansion and the Graf's addition theorem for Bessel functions is developed to study the hydrodynamic interactions of an array of truncated circular cylinders with each cylinder oscillating independently in different prescribed modes. The hydrodynamic radiation and diffraction of linear waves by such an array of cylinders are investigated and the analytical solutions of the velocity potentials are obtained. After comparisons for degenerated cases and program verifications, several cases for an array of truncated cylinders with each cylinder oscillating independently in surge, sway, heave, roll, and pitch modes with different prescribed amplitudes, are studied and the hydrodynamic forces and moments acting on the cylinders are obtained.

Radiations from an Eccentric Coated Cylinder with N Slots

The transverse magnetic (TM) radiation characteristics are investigated for a cylinder with N infinite axial slots of arbitrary opening size and location. The cylinder is a thin circular conductor and coated by an eccentric material. Fields are found by applying the boundary conditions to the cylindrical wave functions. The addition theorem of Bessel functions is used to obtain an infinite series solution in Fourier–Bessel series form. Results are computed by shrinking the generated infinite series to a finite number of terms and compared to other available data. Numerical results in graphical forms for different values are also developed and discussed for small eccentricities.

Impedance Calculation of a Bobbin Coil in a Conductive Tube With Eccentric Walls

The complex impedance of an air-cored coil in a conductive tube with eccentric inner and outer cylindrical surfaces is calculated. The analytic expressions for the induced fields and the impedance variation due to the eddy-current flow inside the tube wall are derived using a second-order potential approach. The addition theorem of Bessel functions is employed to perform the transition between the local coordinate systems that conform to the boundaries of the structure. Although the model can be used for any coil shape and orientation, we focus our study on the configuration of a bobbin coil with axis parallel to the axes of the tube surfaces, but not necessarily coinciding with either of them. The results of the presented analysis are verified by a finite-element-method (FEM) solution.

Multiple scattering from two cylindrical inclusions in a semi-infinite thin plate under flexural waves

In this study, a theoretical method is applied to investigate the multiple scattering of flexural waves from two cylindrical inclusions and dynamic stress in a semi-infinite thin plate, and the roller-supported boundary at the semi-infinite edge is considered. A computationally efficient approach utilizing the wave expansion method together with the image method is employed to formulate the problem. The addition theorem for Bessel functions is employed to accomplish the translation of wave fields between different local coordinate systems. The scattered far field amplitudes of the two inclusions are presented. Through numerical examples and analyses, it is found that the angular distribution of the dynamic stress around the inclusions shows a great difference when the positions of two inclusions vary. The effects of the elastic modulus, density, Poisson’s ratio of the inclusions on the dynamic stress are closely related to the distance between the inclusions and the semi-infinite edge and the distance between the two inclusions. The thickness and Poisson’s ratio of the inclusions show little effect on the dynamic stress. Comparisons with other existing models are also discussed.