Cylindrical Bessel Functions Research Articles

ACTION OF A SOLID BODY ON THE INNER SURFACE OF A THICK-WALLED BIMETAL CYLINDER

Purpose. The goal of the work is to obtain an exact solution to the problem of the stress-strain state of a long thick-walled bimetallic cylinder within the framework of the classical theory of elastic materials. Then, using the results obtained with this formulation, it is necessary to propose simpler engineering approaches and explore the possibilities of using asymptotic formulas for express analyzes at the design stage of such structural elements. Research methods. For the main body of the cylinder, the classical equations of the theory of elasticity in displacements are used. For the outer coating (sputtering), the shell theory equations are written based on the Kirchhoff-Love hypotheses. The complex integral Fourier transform and the Failon method are used to approximately find the original stresses and displacements. Asymptotic representations of cylindrical Bessel functions for large values of the argument and representations of improper integrals in the form of combinations of elementary and special tabulated functions are also used. Results. A mathematical model has been constructed to analyze the stress-strain state of a bimetallic cylinder with a thin outer layer of a material different from that of the inner layer. Various boundary conditions are recorded on the inner surface of the cylinder, describing the transmission from a rigid body of either specified forces or specified displacements. For all considered options, using the method of integral transformations, the results were obtained in the form of improper integrals, for the calculation of which a special method was used, aimed at calculating integrals with highly oscillating functions. Examples of specific graphs of changes in the components of the stress-strain state in the cylinder material are given. Depending on the conditions on the inner surface of the cylinder, simpler models are proposed to describe the main body, which are based, depending on the nature of the description of the interaction of the rigid body and the cylinder, on one equation of the theory of elasticity. With this approach, in some important cases it was possible to obtain improper inversion integrals using the asymptotic approach in closed form as a combination of elementary and special tabulated functions. Comparison with the exact approach allowed us to prove the possibility of using approximate models. Scientific novelty. A model of the behavior of a bimetallic cylinder as a body is constructed, the main layer of which is described by the equations of the theory of elasticity, and the theory of shells is used for the outer coating. Various methods of describing the transfer of forces and displacements from a rigid body to the inner surface of the cylinder are considered. The possibility of using the asymptotic approach to obtain relatively simple formulas for carrying out preliminary calculations at the design stage of such structural elements is shown.

Two-dimensional shielded vortices in a shear current

The interaction of shielded vortices, with a continuous vorticity distribution, and a shear current of weak vorticity amplitude but similar velocity compared to the vortex amplitude is numerically investigated in two-dimensional isochoric flows. Different types of axisymmetric shielded vortices, namely, a neutral unstable vortex, a neutral robust vortex, and a non-neutral vortex are considered. The vortices are linear combinations of vorticity layer-modes, which consist of conveniently normalized cylindrical Bessel functions of order 0, truncated by a zero of the Bessel function of order 1. The vortex–current interaction is investigated by superposing initially the vortices at different initial locations along the cross-flow axis in the shear current. The numerical results show that some shielded vortices, as well as the shear current, remain robust while the vortices cross the shear current and reach a stable equilibrium location, which is of the same sign vorticity as its amount of circulation. There exist two unstable equilibrium locations where most of the vortices persist during a relatively short time interval before heading to their stable equilibrium region in the shear current.

Effects of collar eccentricity on azimuthal response characteristics for acoustic reflection measurement while drilling

Acoustic reflection imaging has been successfully applied in postdrilling wireline logging, and the technique is potentially more valuable in acoustic logging while drilling (ALWD), particularly in geosteering while drilling. Because of the common occurrence of tool eccentering in drilling, the impact of drill collar eccentricity on the acoustic wavefield inside and outside the borehole must be considered. Therefore, we analyze the effect of an eccentric drill collar on acoustic reflection imaging while drilling using analytical solutions in a dual-cylindrical coordinate system. The virtual source analogy and the translational addition theorem for cylindrical Bessel functions are used to calculate the response of the fluid-filled borehole with an eccentric collar to the incident wavefield from a formation reflector. The analysis results are validated with those from 3D finite-difference simulations. We also have studied the azimuthal variation of the received wavefield and compared the results with their centralized counterparts, for which the effects of collar eccentricity, frequency of the source, and formation elastic property are analyzed. The results indicate that the asymmetric feature of the wavefield due to the eccentric collar allows for determining the azimuth of the reflector, providing a solution to the azimuth ambiguity problem of borehole acoustic reflection imaging. Our results, besides developing an effective method for eccentric collar ALWD modeling, can be used to provide a theoretical foundation for ALWD tool development and data interpretation.

Natural waves in a spatial viscoelastic cylinder with a radial crack

The investigation of the dispersion of waves in elastic bodies with different cutouts is of great interest in various scientific and technological fields. The objective of this research is to examine the propagation of free damped waves in elastic dissipative bodies with longitudinal notches. The dynamic behavior of cylindrical bodies with different cutouts is described by the equations of viscoelasticity. The solution of a system of differential equations is expressed by cylindrical Bessel and Hankel functions. The frequency equation is solved using the Muller, Gauss, and orthogonal run methods. It is found that with a decrease in the oscillation frequency in a cylinder of a given radius, the phase and group velocities of the longitudinal wave tend to the common limit - the phase velocity of the rod waves. It is found that as the oscillation frequency reduces in a cylinder of a given radius, the phase and group velocities of the longitudinal wave tend to converge towards the common limit. Additionally, it was found that for a specific value of the Poisson's ratio (0.2833), the frequencies of the shear and radial longitudinal resonances coincide in the cylinder.

Исследование математических подходов к определению частотно-зависимого внутреннего сопротивления провода воздушной линии электропередачи

Mathematical modeling of a power transmission line (PTL) is an important issue for a wide range of tasks in the electric power industry. Mathematical modeling is necessary when studying the transient processes in electric power systems. Analytical expressions that allow modeling power transmission lines in a wide frequency range, including low frequencies and direct current, make it possible to study the operation of relay protection and automation devices in various modes and improve the accuracy of devices of fault location. It is important not only to obtain analytical expressions to determine the frequency dependences of the resistance, but also to verify these expressions in comparison with more accurate methods based on the finite elements method (FEM). The study has been carried out using a mathematical tools based on cylindrical Bessel functions. To develop formulas, it is necessary to determine the constants of integration based on boundary conditions. To verify the obtained expressions, modeling has been performed in the COMSOL Multiphysics software package, which is based on the FEM. The article presents a study of the internal resistance of the wires of power transmission line using the example of AC 185/24 wire. An analytical expression has been obtained to determine the internal complex resistance of a bimetallic wire. The reliability of the obtained expressions is confirmed by the convergence of the simulation results in comparison with the results of simulation modeling in Comsol software and mathematical modeling using known analytical expressions. The proposed approach to determine the internal resistance of a wire makes it possible to get more accurate analytic definition of characteristics of an overhead power transmission line. And thus, to design more qualitative models to analyze transient processes in power transmission lines and investigate the operation of relay protection devices. The wire models developed in Comsol software can be considered as more accurate in a wide range of frequencies.

Illuminating finite-amplitude wave propagation: Hargrove’s use of recurrence relations

A classic problem in the theory of finite-amplitude wave propagation is that of the spectral evolution of a plane wave. An initial sinusoidal wave becomes triangular, with a vertical face at the shock distance. The wave in the pre-shock regime was represented by a Fourier series in the seminal work by Fubini, i.e., Fubini Ghiron [Alta Frequenza, 4, 530 (1935)]. Independently, Hargrove performed a similar derivation, also invoking recurrence relations for cylindrical Bessel functions, but presenting this with exceptional heuristic clarity and brevity [J. Acoust. Soc. Am. 32, 511 (1960)]. This was coincident with a third paper, by Keck and Beyer [Phys. Fluids 3, 346 (1960)], which knowingly followed Fubini. All of this work has facilitated understanding, providing cogent reminders of the power of recurrence relations when dealing with special functions.

Advances in Bessel image sharpness analysis

This paper introduces advances in the use of Bessel functions (cylinder functions) in achieving optimal sharpness of shapes delineated by collections of holes (dark valley regions) and bright regions in different areas of a video frame in terms of 4D voxels (grayscale video frame picture elements with coordinates (x, y, g, t) at location x, y with intensity g, and elapsed time t). An image hole is an island of low voxel intensities surrounded by varying voxel intensity peaks. The basic approach is to identify dominant collections of lights (high voxel intensity peaks) and darks (low voxel intensity clusters) in video frames. A main finding in this paper is that surface objects (recorded in images) are sharper wherever there are preponderant contrasting differences between image lights and darks. These contrasting differences lead to the highly accurate detection of shape holes (dark valleys) that delineate surface shapes recorded in sequences of video frames. With the use of cylindrical Bessel functions introduced by F. W. Bessel in 1824, contrasting differences between peaks and valleys can be measured in terms of voxel intensities and voxel indices.

Electronic States in a Doubly Eccentric Cylindrical Quantum Wire

Electronic states of a single electron in doubly eccentric cylindrical quantum wire are theoretically investigated in this paper. The motion of electron in quantum wire is free along axial direction in a cylindrical quantum wire and restricted in annular regions by three different parallel finite cylindrical barriers as soft wall confinement. The effective mass Schrödinger equation with effective mass boundary conditions is used to find energy eigenvalues and corresponding wavefunctions. Addition theorem for cylindrical Bessel functions is used to shift the origin for applying boundary conditions at different circular boundaries. Fourier expansion is applied after addition theorem to get wavefunctions in analytical form. A determinant equation is obtained as a result of applications of effective mass boundary conditions which roots gives energy of various electronic states. The lowest root gives ground state energy. The variation in ground state energy with eccentricity is obtained numerically and presented graphically. Electronic states in massive wall confinement and hard wall confinement is further obtained as limiting behavior of the states obtained in soft wall confinement. The knowledge of electronic states in such cylindrical hetrostructures semiconductor material can lead to improve the efficiency of many quantum devices.

Acoustic radiation force on an elastic cylinder in a Gaussian beam near an impedance boundary

Acoustic radiation force (ARF) is studied by considering an infinite elastic cylinder near an impedance boundary when the cylinder is illuminated by a Gaussian beam. The surrounding fluid is an ideal fluid. Using the method of images and the translation-addition theorem for the cylindrical Bessel function, the resulting sound field including the incident wave, its reflection from the boundary, the scattered wave from the elastic cylinder, and its image are expressed in terms of the cylindrical wave function. Then, we deduce the exact equations of the axial and transverse ARFs. The solutions depend on the cylinder position, cylinder material, beam waist, reflection coefficient, distance from the impedance boundary, and absorption in the cylinder. To analyze the effects of the various factors intuitively, we simulate the radiation force for non-absorbing elastic cylinders made of stainless steel, gold, and beryllium as well as for an absorbing elastic cylinder made of polyethylene, which is a well-known biomedical polymer. The results show that the impedance boundary, cylinder material, absorption in the cylinder, and cylinder position in the Gaussian beam significantly affect the magnitude and direction of the force. Both stable and unstable equilibrium regions are found. Moreover, a larger beam waist broadens the beam domain, corresponding to non-zero axial and transverse ARFs. More importantly, negative ARFs are produced depending on the choice of the various factors. These results are particularly important for designing acoustic manipulation devices operating with Gaussian beams.

Reversals of Acoustic Radiation Torque in Bessel Beams Using Theoretical and Numerical Implementations in Three Dimensions

Acoustic radiation torque (ART) plays an important role in the subject of acoustophoresis, which could induce the spinning rotation of particles on their own axes. The partial wave series method (PWSM) and T matrix method (TMM) are employed to investigate the three-dimensional radiation torques of both spherical and nonspherical objects in a single Bessel beam over broader frequencies (from Rayleigh scattering to geometric-optics regimes), with emphasis on the parametric conditions for torque reversal and the corresponding physical mechanisms. For elastic objects, the dipole, quadrupole, and even several higherscattering modes are dominant to induce the axial ARTs beyond the Rayleigh limit with a relatively large offset from the particle centroid to the beam axis in Bessel beams. The reversals appear with parametric conditions (including wave number, cone angle, and offset) for the extrema or null values of the corresponding cylindrical Bessel functions. The reversal of transverse ART from a rigid nonspherical particle is also investigated in an ordinary Bessel beam and the geometrical surface roughness is briefly studied for the effect on the radiation torques. This suggests the possibility of acoustic tweezers controlling the spinning motion of particles within and beyond the Rayleigh limit.

Theoretical study of acoustic radiation force and torque on a pair of polymer cylindrical particles in two Airy beams fields

In this work, we describe the acoustic radiation force (ARF) and torque acting on a pair of cylindrical particles induced by two Airy beams. The finite series expansion method and the addition theorem of the cylindrical Bessel function are used to analyze the acoustic scattered field by a pair of cylindrical particles in an effective incident acoustic field. The mathematical expressions for the ARF and the torque functions in multiple acoustic scattering by a pair of cylindrical particles are obtained. The influences of the phase difference, the beam distance, and the beam amplitude on the ARFs and torques are taken into consideration. The numerical examples illustrate that greater negative or positive forces on the two cylindrical particles emerge by adjusting the phase difference, the distance, or the amplitudes of the two Airy beams, which make it easier to separate the particles. The values of the torques will also increase or change between the positive and the negative, which increase the particles’ rotation velocity or change the particles’ rotation direction. This work will be conducive to the development of acoustic tweezers for polymer drugs separation (or cells separation) in medical domain.

Generalized coherent states related to the cylindrical Bessel functions and Legendre oscillator

The main objective of this paper is to discuss the Paley-Wiener space in the framework of a set of coherent states related to the cylindrical Bessel function and the Legendre oscillator. Thus, we show that the kernel of the Fourier transform of the L2-functions that are supported in form a set of coherent states. This also leads to the construction a coherent states integral transform.

Distribution Natural Waves on the Viscoelastic Cylindrical Body in Plane Strain State

In this paper the distribution of natural waves in the viscoelastic disc is discussed. The spectral problem is reduced to solving a system of ordinary differential equations of the first order with variable complex coefficients. The solution of differential equations is expressed by a special cylindrical Bessel functions and Hankel. The frequency equations are solved by numerical methods Muller and Gauss. The problem is solved numerically by the Godunov orthogonal sweep method and Mueller. Compares the numerical results.

Axial acoustic radiation force on a rigid cylinder near an impedance boundary for on-axis Gaussian beam

This work presents a theoretical model to calculate the acoustic radiation force on a rigid cylindrical particle immersed in an ideal fluid near a boundary for an on-axis Gaussian beam. An exact solution of the axial acoustic radiation force function is derived for a cylindrical particle by applying the translation addition theorem of cylindrical Bessel function. We analyzed the effects of the impedance boundary on acoustic radiation force of a rigid cylinder immersed in water near an impedance boundary with particular emphasis on the radius of the rigid cylinder and the distance from the cylinder center to impedance boundary. Simulation results reveal that the existence of particle trapping behavior depends on the choice of nondimensional frequency as well as the offset distance from the impedance boundary. The value of the radiation force function varies when the cylinder lies at the different position of the on-axis Gaussian beam. For the particle with different radius, the acoustic radiation force functions vary significantly with frequency. This study provides a theoretical basis for acoustic manipulation, which may benefit to the improvement and development of the acoustic control technology.

Multipole expansion of acoustical Bessel beams with arbitrary order and location

An exact solution of expansion coefficients for a T-matrix method interacting with acoustic scattering of arbitrary order Bessel beams from an obstacle of arbitrary location is derived analytically. Because of the failure of the addition theorem for spherical harmonics for expansion coefficients of helicoidal Bessel beams, an addition theorem for cylindrical Bessel functions is introduced. Meanwhile, an analytical expression for the integral of products including Bessel and associated Legendre functions is applied to eliminate the integration over the polar angle. Note that this multipole expansion may also benefit other scattering methods and expansions of incident waves, for instance, partial-wave series solutions.

An analytical solution for free liquid sloshing in a finite-length horizontal cylindrical container filled to an arbitrary depth

• Exact slosh analysis in an arbitrary-fill-depth finite horizontal cylindrical tank. • Full data for slosh frequencies as functions of fill depth and container length. • Extensive simulations exposing mode crossings among dissimilar frequency clusters. • Deep insights via benchmark general simulations and comparisons with other data. A general series-type theoretical formulation based on the linearized potential theory, the method of separation of variables, and the translational addition theorem for cylindrical Bessel functions is developed to study three-dimensional natural sloshing in a partially filled horizontally-mounted circular cylindrical tank of finite span. Assuming time-harmonic variations, the potential solutions associated with the Symmetric/Antisymmetric (S/A) modes of free liquid surface oscillations are first analytically expanded as series of bounded spatial functions with unknown modal coefficients. The impenetrability conditions of the rigid end-plates along with the free surface dynamic/kinematic boundary condition are then imposed. The zero-normal-velocity requirement of the lateral tank boundary is subsequently applied by innovative use of Graf's translational addition theorem for modified cylindrical Bessel functions. After truncation, four independent sets of homogeneous algebraic equations are obtained that are then numerically worked out for the natural sloshing eigen-frequencies and free surface oscillation mode shapes. Extensive numerical data include the first thirty six longitudinal/transverse Antisymmetric/Symmetric (AA, SA, AS, SS) dimensionless sloshing frequencies, for a wide range of liquid fill depths and container span to radius ratios. Also, the influence of fill depth on the free surface oscillation mode shapes is addressed through selected 2D images. Comprehensive numerical simulations illustrate the strong effects of container length and liquid fill depth on the calculated sloshing frequencies. It is revealed that the frequency branches with the same transverse mode number form a cluster that progressively merge together amid the tank fill-depth limits as the tank span ratio increases. On the other hand, when the tank length substantially decreases, the number of “frequency cross-overs” between various frequency clusters at certain liquid fill depths considerably increases. Moreover, primary advantages of proposed methodology in comparison to other approximate/numerical methods are explicitly pointed out, convergence of solution is tested, and accuracy/reliability of the results is demonstrated by comparisons with available data.

Evaluation of Heat Losses Behind the Front of the Detonation Moving Along the Metallic Porous Surface

The paper considers a computational technique of the heat flow from the hot products of detonation combustion into the porous coating and estimates the efficiency of the coating layer that results in slowing the flame front down with disregard the transverse displacement of the combustion products weight of a hydrogen-air mixture. Initial thermodynamic parameters of combustion products on the porous coating surface have been estimated. A drag (stagnation) temperature of flow was determined. The statement of task was to calculate the heat flow into the long cylindrical metal fiber with radius of 15 μm. The reference values of heat capacity and heat diffusivity were used to estimate a thermal diffusivity in a wide range of temperatures. An approximation of the parameters is given for a wide range of temperatures. The calculation algorithm using an explicit four-point scheme is presented. The convergence and accuracy of the results were confirmed. The theoretical estimation using cylindrical Bessel functions was made to prove the accuracy of the results. Total heat loss was estimated using the photos of moving detonation front and hot combustion gases. Comparison of the total heat loss and the amount of energy absorbed by a single fiber allowed us to find that the porous coating thickness, resulting in attenuation of detonation wave, is efficient.

An Efficient Rescaled Formulation for Tensor Green’s Function Computation in Cylindrical Multilayered Media

A robust formulation is presented to eliminate overflow, underflow, and convergence problems that arise the computation of tensor Green’s functions in cylindrical multilayered media under finite-precision arithmetic. This is done by first introducing a set of rescaled functions associated to the (canonical) cylindrical Bessel and Hankel functions representing standing waves and outgoing waves along the radial direction. The rescaled cylindrical functions are designed to avoid the poor scaling inherent to the canonical cylindrical functions for very small or very large arguments. In addition, a parabolic Sommerfeld integration path is constructed in the spectral complex plane to yield a numerical integration with good convergence properties for a wide range of parameter values.

Improvements for scattering from a large-sized chiral cylinder at an oblique incidence

Light scattering from a chiral cylinder is reconsidered based on Lorenz–Mie regime. The incident, internal, and scattered fields are expressed in terms of cylindrical vector wave functions. The scattering coefficients at an oblique incidence are obtained according to the boundary conditions and expressed in forms of ratios and logarithmic derivatives of cylindrical Bessel functions, which can be used for large-sized chiral cylinder scattering. Effects of the cylinder parameters such as chirality, loss and incident angle on scattering properties are numerically analyzed. Numerical results show that rainbow phenomenon similar to that of a chiral sphere also occurs for a large-sized chiral cylinder.

Vibroacoustic Response of an Annular Sandwich Electrorheological Disc

The steady-state sound radiation characteristics of a rigidly-baffled clamped-free sandwich annular disc with a tunable electrorheological fluid (ERF) core is studied. Hamilton's principle is employed to derive the governing equations of motion for the composite plate along with the associated boundary conditions in general form. The classical Galerkin procedure is subsequently utilized to reduce the structural problem into a linear system of algebraic equations for the unknown displacement coefficients. A modal summation approach based on Rayleigh integral formula along with the proper integral representation of the cylindrical Bessel functions are used to arrive at a useful single integral relation for the far-field modal radiated sound pressure and subsequently determine the radiation power and efficiency. Numerical results reveal the imperative influence of the applied electric field strength (0–3.5kV/mm) on controlling acoustic radiation from the adaptive panel in a wide frequency range. In particular, an effort is made to find the optimal electric field which yields improved sound radiation characteristics for each excitation frequency. Limiting cases are considered and good agreements with the calculations made by using a commercial finite element package as well as with the solutions available in the literature are obtained.