Showing papers on "Bessel function published in 1995"
TL;DR: In this article, the non-diffractive vector Bessel beams of an arbitrary order are examined as both the solution to the vector Helmholtz wave equation and the superposition of vector components of the angular spectrum.
Abstract: The non-diffractive vector Bessel beams of an arbitrary order are examined as both the solution to the vector Helmholtz wave equation and the superposition of vector components of the angular spectrum. The transverse and longitudinal intensity components of the vector Bessel beams are analysed for the radial, azimuthal, circular and linear polarizations. The radially and azimuthally polarized beams are assumed to be formed by the axicon polarizers used with the initially unpolarized or linearly polarized light. Conditions in which the linearly polarized Bessel beams can be approximated by the scalar solutions to the wave equation are also discussed.
244 citations
TL;DR: In this paper, two simple but elegant solutions for the flow of an Oldroyd-B fluid are presented for flow past an infinite porous plate and find that the problem admits an asymptotically decaying solution in the case of suction at the plate, and that in case of blowing it admits no such solution.
Abstract: We present two simple but elegant solutions for the flow of an Oldroyd-B fluid. First, we consider the flow past an infinite porous plate and find that the problem admits an asymptotically decaying solution in the case of suction at the plate, and that in the case of blowing it admits no such solution. Second, we study the longitudinal and torsional oscillations of an infinitely long rod of finite radius. The solutions are found in terms of Bessel functions.
223 citations
TL;DR: In this article, an approach is presented for obtaining exact analytical solutions for sound propagation in ducts with an axial mean temperature gradient, and the results obtained using the developed analytical solution are in excellent agreement with experimental as well as numerical results.
97 citations
TL;DR: In this article, a method for evaluating infinite integrals involving products of Bessel functions is proposed, which makes use of extrapolation on a sequence of partial sums, and requires rewriting the product of Belsen functions as the sum of two more well-behaved functions.
90 citations
TL;DR: In this paper, a series transformation technique for integrals involving Bessel functions and sinusoidal functions is presented. But the main aim is to critically examine the popular series transformation (acceleration) methods which are used in electromagnetic problems and to introduce a new acceleration technique.
Abstract: Most electromagnetic problems can be reduced to either integrating oscillatory integrals or summing up complex series. However, limits of the integrals and the series usually extend to infinity, and, in addition, they may be slowly convergent. Therefore numerically efficient techniques for evaluating the integrals or for calculating the sum of an infinite series have to be used to make the numerical solution feasible and attractive. In the literature there are a wide range of applications of such methods to various EM problems. In this paper our main aim is to critically examine the popular series transformation (acceleration) methods which are used in electromagnetic problems and to introduce a new acceleration technique for integrals involving Bessel functions and sinusoidal functions.
82 citations
TL;DR: Algorithm 644 computes all major Bessel functions of a complex argument and of nonnegative order and modified to reduce the computation time for the Y Bessel function by approximately 25% over a wide range of arguments and orders.
Abstract: Algorithm 644 computes all major Bessel functions of a complex argument and of nonnegative order. Single-precision routine CBESY and double-precision routine ZBESY are modified to reduce the computation time for the Y Bessel function by approximately 25% over a wide range of arguments and orders. Quick check (driver) programs that exercise the package are also modified to make tests more meaningful over larger regions of the complex plane.
77 citations
TL;DR: In this paper, two distinct approaches to finding bounds, as functions of the order ν, for the smallest real or purely imaginary zero of Bessel and some related functions are presented.
Abstract: We give two distinct approaches to finding bounds, as functions of the order ν, for the smallest real or purely imaginary zero of Bessel and some related functions. One approach is based on an old method due to Euler, Rayleigh, and others for evaluating the real zeros of the Bessel function Jν(x) when ν > −1. Here, among other things, we extend this method to get bounds for the two purely imaginary zeros which arise in the case −2 < ν < −1. If we use the notation jν1 for the smallest positive zero, which approaches 0 as ν → −1, we can think of j ν1 as continued to −2 < ν < −1, where it has negative values. We find an infinite sequence of successively improving upper and lower bounds for j ν1 in this interval. Some of the weakest, but simplest, lower bounds in this sequence are given by 4(ν + 1) and 25/3(ν + 1)[(ν + 2)(ν + 3)]1/3 while a simple upper bound is 4(ν + 1)(ν +2)1/2. The second method is based on the representation of Bessel functions as limits of Lommel polynomials. In this case, the bounds for the zeros are roots of polynomials whose coefficients are functions of ν. The earliest bounds found by this method already are quite sharp. Some are known in the literature though they are usually found by ad hoc methods. The same ideas are applied to get bounds for purely imaginary zeros of other functions such as J ′ ν(x), J ′′ ν (x), and αJν(x) + xJ ′ ν(x).
67 citations
TL;DR: In this paper, the authors compare various extrapolation techniques as well as choices of endpoints in dividing the integral, and establish the most efficient method for evaluating infinite integrals involving Bessel functions of any order n, not just zero or one
66 citations
TL;DR: The propagation features of several apertured Bessel beams are calculated and show that the relations of axial intensity versus propagation distance are similar to the radial distribution of the aperture functions, which may be helpful in choosing the proper aperture functions in experiments.
Abstract: The propagation features of several apertured Bessel beams are numerically calculated. The calculations show that the relations of axial intensity versus propagation distance are similar to the radial distribution of the aperture functions, which may be helpful in choosing the proper aperture functions in experiments.
63 citations
TL;DR: In this paper, the exact solutions for stability analysis of bars with varying cross sections subjected to simple or complicated loads, including concentrated and variably distributed axial loads, are presented in terms of Bessel functions and super geometric series.
56 citations
TL;DR: In this paper, two methods are presented to detect the impact-force history on elastic plates, one involves construction of a Green's function to relate the strain responses to an impact force acting axisymmetrically on a circular plate.
Abstract: Two methods are presented to detect the impact-force history on elastic plates. The first method involves construction of a Green’s function to relate the strain responses to an impact force acting axisymmetrically on a circular plate. The classical plate theory and a series of Bessel functions were used to obtain the Green’s functions in the time domain. The gradient projection method was employed to search for the optimal force history. Examples using circular plates with free and fixed boundary conditions are demonstrated. On the other hand, the second method is purely experiment-based and involves no Green’s function. Thus, this method can be applied to structures of various geometries, boundary conditions and material properties. The very satisfactory agreement between the measured and the detected force histories using both methods shows that they are reliable for impact-force determination purposes.
Journal Article•
TL;DR: In this article, a class of pseudo-differential operators (p.d.o.), generalizing Bessel differential operator d2/dx2 + (1 − 4μ2)/(4x2), is defined.
TL;DR: In this article, a q-difference analog of the Painleve III equation is presented, which admits the solutions expressed by the Casorati determinant whose entries are given by the discrete Bessel functions.
Abstract: The discrete Painleve III equation is investigated based on the bilinear formalism. It is shown that it admits the solutions expressed by the Casorati determinant whose entries are given by the discrete Bessel functions. Moreover, based on the observation that these discrete Bessel functions are transformed to the q‐Bessel functions by a simple variable transformation, we present a q‐difference analog of the Painleve III equation.
TL;DR: In this paper, a comparison between the wave transmission characteristics of cylinder/plate junctions and their equivalent plate/plate junction is made and numerical examples are presented and a comparison is made.
TL;DR: Bessel series expansions for the incomplete Lipschitz-Hankel integral are derived by using contour integration techniques to evaluate the inverse Laplace transform representation for Je0(a, z) as mentioned in this paper.
Abstract: Bessel series expansions are derived for the incomplete Lipschitz-Hankel integralJe0(a, z). These expansions are obtained by using contour integration techniques to evaluate the inverse Laplace transform representation for Je0(a, z). It is shown that one of the expansions can be used as a convergent series expansion for one definition of the branch cut and as an asymptotic expansion if the branch cut is chosen differently. The effects of the branch cuts are demonstrated by plotting the terms in the series for interesting special cases. The Laplace transform technique used in this paper simplifies the derivation of the series expansions, provides information about the resulting branch cuts, yields integral representations for Je0(a, z), and allows the series expansions to be extended to complex values of z. These series expansions can be used together with the expansions for Ye0(a, z), which are obtained in a separate paper, to compute numerous other special functions, encountered in electromagnetic applications. These include: incomplete Lipschitz-Hankel integrals of the Hankel and modified Bessel form, incomplete cylindrical functions of Poisson form (incomplete Bessel, Struve, Hankel, and Macdonald functions), and incomplete Weber integrals (Lommel functions of two variables).
TL;DR: In this paper, an iterative algorithm was developed to calculate phase optical elements serving to transform incident coherent light into non-diffracting beams characterized by an unchanging transverse intensity distribution.
Abstract: The iterative algorithm developed allows calculation of phase optical elements serving to transform incident coherent light into non-diffracting beams characterized by an unchanging transverse intensity distribution. Such non-diffracting light beams represent Bessel modes and are described as a superposition of a small number of Bessel functions with equal arguments. The results of numerical calculations are discussed.
TL;DR: In this paper, a random rough surfaces under illumination of coherent light with a circular ring slit produces the speckles with a zero-order Bessel function for the intensity correlation when observed in the Fraunhofer diffraction region.
TL;DR: In this paper, a method for evaluating analytical long-range contributions to scattering lengths for some potentials used in atomic physics was described, and exact analytical formulae for the scattering lengths were derived.
Abstract: We describe a method for evaluating analytical long-range contributions to scattering lengths for some potentials used in atomic physics. We assume that an interaction potential between colliding particles consists of two parts. The form of a short-range component, vanishing beyond some distance from the origin (a core radius), need not be given. Instead, we assume that a set of short-range scattering lengths due to that part of the interaction is known. A long-range tail of the potential is chosen to be an inverse power potential, a superposition of two inverse power potentials with suitably chosen exponents or the Lent potential. For these three classes of long-range interactions a radial Schrodinger equation at zero energy may be solved analytically with solutions expressed in terms of the Bessel, Whittaker and Legendre functions, respectively. We utilize this fact and derive exact analytical formulae for the scattering lengths. The expressions depend on the short-range scattering lengths, the core radius and parameters characterizing the long-range part of the interaction. Cases when the long-range potential (or its part) may be treated as a perturbation are also discussed and formulae for scattering lengths linear in strengths of the perturbing potentials are given. It is shown that for some combination of the orbital angular momentum quantum number and an exponent of the leading term of the potential the derived formulae, exact or approximate, take very simple forms and contain only polynomial and trigonometric functions. The expressions obtained in this paper are applicable to scattering of charged particles by neutral targets and to collisions between neutrals. The results are illustrated by accelerating convergence of scattering lengths computed for e--Xe and Cs-Cs systems.
TL;DR: The construction of such diagonal forms for translation operators for the Helmholtz equation in two dimensions is rigorous and straightforward, and numerical experiments indicate that it is within a factor of two of being optimal, in terms of the number of nodes required to discretize the translation operator to a specified precision ϵ.
TL;DR: In this paper, a few transformations are presented for reducing certain cases of Meijer's G-function to a G function of lower order and their applications to the integration of a product of Bessel functions are given.
TL;DR: In this article, the higher order 3D fundamental solutions to the modified Helmholtz and the modified spherical Bessel functions have been derived and each degree of these solutions has a singularity of order (1 r ).
Abstract: The higher order 3-D fundamental solutions to the Helmholtz and the modified Helmholtz equations have been derived. The Lth order fundamental solution for the 3-D Helmholtz equation has the form of a spherical Bessel function multiplied by a distance to the power L. In the case of the 3-D modified Helmholtz equation a modified spherical Bessel function is required instead of the spherical Bessel function. Each degree of these solutions in the 3-D cases has a singularity of order ( 1 r ). These solutions can be used for applying the multiple reciprocity boundary element method to the 3-D Helmholtz or modified Helmholtz problems.
31 Mar 1995
TL;DR: In this paper, the authors propose a perturbation method based on the Laplace transform and initial value problems, which is the state-of-the-art in computer algebra.
Abstract: Preface Achnowledgments 1. Formulation of physical problems 2. Classification of equations 3. One-dimensional waves 4. Finite domains and separation of variables 5. Elements of Fourier series 6. Introduction to Green's functions 7. Unbounded domains and Fourier transforms 8. Bessel functions and circular domains 9. Complex variables 10. Laplace transform and initial value problems 11. Conformal mapping and hydrodynamics 12. Riemann-Hilbert problems in hydrodynamics and elasticity 13. Perturbation methods - the art of approximation 14. Computer algebra for perturbation analysis Appendices Bibliography Index.
TL;DR: A selection of uniform asymptotic methods for integrals, in particular on uniform approximations, can be found in this article, together with references to recent papers on the Stokes phenomenon.
TL;DR: A quadrature formula for the Bessel function of the first kind was recently obtained by C. Frappier and P. Olivier as mentioned in this paper, where the nodes are the zeros of Bessel functions.
Abstract: A quadrature formula for entire functions of exponential type wherein the nodes are the zeros of the Bessel function of the first kind was recently obtained by C. Frappier and P. Olivier. Here the condition imposed on the function is relaxed. Some applications of the formula are also given
TL;DR: In this article, a convolution of the Hankel transform is constructed for the calculation of an integral containing Bessel functions of the first kind, and the convolution is used to calculate an integral of the form
Abstract: In the paper a convolution of the Hankel transform is constructed. The convolution is used to the calculation of an integral containing Bessel functions of the first kind.
TL;DR: A new algorithm for cylindrical Bessel functions that is similar to the one for spherical Bessel function allows us to compute scattering functions for infinitely long cylinders covering sizes ka = 2πa/λ up to 8000 through the use of only an eight-digit single-precision machine computation.
Abstract: A new algorithm for cylindrical Bessel functions that is similar to the one for spherical Bessel functions allows us to compute scattering functions for infinitely long cylinders covering sizes ka = 2πa/λ up to 8000 through the use of only an eight-digit single-precision machine computation. The scattering function and complex extinction coefficient of a finite cylinder that is seen near perpendicular incidence are derived from those of an infinitely long cylinder by the use of Huygens's principle. The result, which contains no arbitrary normalization factor, agrees quite well with analog microwave measurements of both extinction and scattering for such cylinders, even for an aspect ratio p = l/(2a) as low as 2. Rainbows produced by cylinders are similar to those for spherical drops but are brighter and have a lower contrast.
TL;DR: In this paper, the authors derived new approximate solutions to the 0th-and 1st-order Bessel functions of the first kind based on using a new integral with no previously known solution.
Abstract: New approximate solutions to the 0th- and 1st order Bessel functions of the first kind are derived. The formulations are based upon using a new integral with no previously known solution. The new integral in the limiting case is identical to the 0th-order Bessel function integral. It is solved in closed form, and the solution is expressed as a simple even order polynomial with integer coefficients. The polynomial coefficients are all of integer value. The 1st-order Bessel function approximation can then be found through a simple derivative. Comparisons are made between the exact solution, classic solutions, and the new approximation. The new approximation proves to be much more accurate than the classic small argument approximation. It is also sufficiently accurate to bridge the gap between the classic large and small argument approximations and has potential applications in allowing one to analytically evaluate integrals containing Bessel functions. >
TL;DR: In this article, a portable software package, named RFSFNS, is presented for the localization and computation of the simple real zeros of the Bessel functions of first and second kind, J,, (z), Y~(z), respectively, and their derivatives, where v > 0 and z > 0.