Lommel Functions Research Articles

Green’s function for external excited Bohm–Gross waves

Analytical expressions for the impulsive causal Green’s function based on the resolution of the differential equation describing the external excitation of the Bohm and Gross electrostatic wave are derived. The exact solution corresponding to the longitudinal plasma wave is put in the compact form of a two-real-variable Lommel function.

Laplace transform pairs of N-dimensions

The main theme of the present paper is to establish a formula for calculating the Inverse Laplace transformation of functions of the form p 1(⦶| s 1 2 )/ p n (⦶| s 1 2 ) F[( p 1(⦶| s 1 2 )) 2. This result is used to obtain the Inverse Laplace transforms of generalized hypergeometric and Lommel functions of n variables.

On an integral of Lommel and Bessel functions

AbstractIn this paper we have evaluated an infinite integral of product of the Lommel and Bessel functions and powers. Some special cases of the result are discussed.

Diffraction of light by an opaque sphere 1: Description and properties of the diffraction pattern

In this paper we discuss the diffraction pattern resulting from the propagation of light past an opaque obstacle with a circular cross section. A mathematical description of the diffraction pattern is obtained in the Fresnel region using scalar diffraction theory and is presented in terms of the Lommel functions. This description is shown experimentally to be quite accurate, not only for near axis points within the shadow region but also well past the shadow's edge into the directly illuminated region. The mathematical description is derived for spherical wave illumination and an isomorphic relation is developed relating it to plane wave illumination. The size of the central bright spot (as well as the subsequent diffraction rings), the axial intensity, and the intensity along the geometric shadow are characterized in terms of point source location and the distance of propagation past the circular obstacle.

Fresnel diffraction by spherical obstacles

Lommel functions were used to solve the Fresnel–Kirchhoff diffraction integral for the case of a spherical obstacle. Comparisons were made between Fresnel diffraction theory and Mie scattering theory. Fresnel theory is then compared to experimental data. Experiment and theory typically deviated from one another by less than 10%. A unique experimental setup using mercury spheres suspended in a viscous fluid significantly reduced optical noise. The major source of error was due to the Gaussian-shaped laser beam.

Exact solutions for buckling of some divergence-type nonconservative systems in terms of bessel and lommel functions

Closed-form solutions for a divergence-type nonconservative system, that of the column under follower distributed forces, is given for three types of boundary conditions. The solution generalizes the previous classical studies by Pflüger, who found the solution for the column simply supported at its ends in terms of Bessel functions, as well as by Leipholz and Madan, who formulated the series solutions for the column clamped at one end, and simply supported or clamped at the other. In the present work the solution is formulated in terms of Bessel and Lommel functions, yielding exact characteristics equations, with attendant buckling loads found within any desired numerical accuracy.

Simulation of Fresnel diffraction of output beams of unstable resonators

The solution is given of the problem of diffraction of a spherical wave by an aperture in the form of a doubly connected region with a rectangular or circular contour, followed by the propagation of this wave across an optical system characterized by an arbitrary ray matrix. The exact expression is obtained for the Fresnel-Kirchhoff integral in terms of Fresnel integrals or Lommel functions of two variables. An investigation is made of the evolution of the effective beam width when its effective length and the shape of the aperture altered. A simple method is proposed for applying this solution in approximate simulation of the distribution of the field in laser output beams generated in unstable optical resonators.

Lens effect on synthetic image generation based on light particle theory

The synthetic image generation with the lens effect consists of two consecutive processors: the hidden-surface processor, and the focus processor. The normal ray-tracing algorithms are used in the first processor. The second processor computes Lommel's function, which is an infinite series of Bessel functions. In order to avoid the complicated calculation and the huge memory consumption, an approximation method based on the light particle theory is developed. No noticeable differences can be detected between the results from the approximation method and those from the exact calculation.

Green’s functions for nonlinear Klein–Gordon kink perturbation theory

A Green’s function is defined for nonlinear Klein–Gordon theories in terms of the solutions to the eigenvalue equation obtained by linearizing the nonlinear wave equation about a static kink waveform. Analytic forms in terms of ‘‘modified’’ Lommel functions of two variables are derived for the sine–Gordon, phi-4, and double quadratic potentials. Asymptotic forms for the Green’s functions are obtained by investigating the asymptotic behavior of the modified Lommel functions. Methods for calculating the Lommel functions are also outlined.

Peak accelerations from subshear and supershear radiation of kinematic dislocation models.

The field of accelerations of a kinematic fault-model are evaluated in terms of fundamental interference-integrals that depend upon the source-observer geometry and the various spatial and 'temporal source elements. The dependence of the accelerations at various distances r from the fault's center, on the fault's major dimension L and the radiation's wavelength λ are scaled to the three dimensionless 'regionalization-indices': 2πr/λ, 2L2/λr and 0.62 × (L/r) √L/λ. These determine the limits of the far-field, the outer Fresnel-zone and the inner Fresnel-zone respectively. The interference integrals are then evaluated through the stationary-phase approximation and the Fresnel approximation both in the time and frequency domains. In the Fresnel zone they are expressed in terms of the Lommel functions of two variables. The ensuing acceleration field is shown to depend strongly on the shear Mach-number. In subshear-rupture, the acceleration in the near-fault and Fresnel zones decreases exponentially in a direction normal to the fault. In supershear-rupture the acceleration is a Mach-wave that propagates without attenuation along the Mach-lines. The theoretical scaling law for the acceleration in each region is determined. We assert: (1) peak accelerations in the near-fault zone are essentially independent of the earthquake's magnitude; (2) peak accelerations in the near-fault and Fresnel zones are proportional to the particle velocity on the fault; (3) accelerations in the near-fault and Fresnel zones are determined by the radiation from the nearest fault-segment. This explains the elliptical shape of isoseismals, which are locii of equidistant points from the fault.

Particular solutions of forced generalized Airy equations

Differential equations with delayed forcing terms arise naturally in the explicit solution of differential-delay equations [ 11. In our present studies of singular perturbation analyses of boundary-value problems for secondorder differential-difference equations with a turning point (see [2] for earlier results), we have had to deal with forcing terms consisting of shifted Airy and Lommel functions and their derivatives. One example of the type of equation which arises is the inhomogeneous Airy equation (see the Appendix for a derivation) y”-xy=c,Ai(x-a)+c,Bi(x-a)+c,L(x-a) + d, Ai’(x a) + d, Bi’(x a) + d3 L’(x a), (1) where a is a constant (0 < a < co); ci, di, i = 1, 2, 3, are constants; Ai, Bi are Airy functions; and L is a Lommel function (see [ 31) satisfying L”-xL= 1. (2)

Optimizing photodetection in a focused field

In a consideration of the effect of focal shift, light energy which falls inside a coaxial circular detector in a focused field will achieve its absolute maximum in an ascertainable place between the aperture and the geometrical focal plane. In this paper the optimum position of the detector is determined by solving a pair of transcendental equations involving Lommel functions. The numerical results obtained are displayed by curves with the Fresnel numbers sampled between 0.1 and 100.

Stress distribution in rotating aeolotropic laminated heterogeneous disc under action of a time-dependent loading

The problem of stress distribution in rotating aeolotropic laminated heterogeneous disc under action of a time-dependent loading is studied. A closed type solution in terms of modified Bessel and Lommel functions is obtained. The general procedure presented herein is valid for symmetrically laminated discs as well as for discs composed of materials with proportional properties. Numerous examples are shown indicating the effect of layers heterogeneity and layers thickness on the stress distribution.

Three-dimensional intensity distributions in the region of focus of two superposed converging spherical waves

The three-dimensional distribution of intensity in the neighborhood of focus of two converging coaxial spherical waves has been calculated for various focal separations and phase differences between the waves. Lommel functions were used to solve the diffraction integral, and computer techniques were used to generate the complex-field amplitude arrays. Subsequent superposition of the arrays provided contour maps of the intensity distributions.

Transients on lossy transmission lines with arbitrary boundary conditions

In this work the problem of transients on a lossy transmission line terminated by an arbitrary, including nonlinear, load is formulated. The tranmission line parameters are the constants <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R, L, G</tex> , and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</tex> . The exact relation between the input and output voltages and currents in the form of two coupled integral equations is derived by the Laplace transform method. It is shown that the kernels of the integral equations may be represented in terms of either Lommel functions or integrals involving zeroth order modified Bessel functions. Simultaneous (numerical) solutions of these integral equations fulfilling the boundary conditions at the input and output of the line yields the input and output voltages and currents on the line. Finally the exact analytical relations in time domain of the voltage and current at an arbitrary point on the line (and the voltages and currents at the input and output terminals) are derived. In all parts, the problem has been formulated in such a way as to impose the causality condition explicitly.

A Numerical Method of Hankel Transform for Axisymmetric Problems of Elasticity

In numerical calculations of the Hankel Transform to solve axisymmetric elastic problems of semi-infinite bodies or infinite thick plates, Simpson's rule is usually adopted. But it is difficult to obtain accurate results by Simpson's rule because the Bessel function in the kernel of the Hankel transform oscillates. In this paper, f(x), a function to be transformed, xf(x) is approximated piecewise by the quadratic and the result of the Hankel transform is expressed by Lommel's functions. In this way accurate Hankel transforms are obtained. Two examples are illustrated to show the availability of this method.

Effect of refined shape of input atomic beam on plasma density distribution in a double-ended Q machine

The density distributions of equilibrium plasma in a double-ended Q machine are analyzed by means of fluid theory in connection with various shapes of input atomic beam. An inhomogeneous ordinary differential equation for plasma density is analytically solved in terms of modified Bessel functions, and modified Lommel function exhibiting the effect of refined atomic beam impingement. The radial density profiles are discussed in various cases of the atomic beam to find the optimum input-flux shape which results in the improved plasma confinement. As the slope of input distribution becomes steeper, it turns out to have better confinement effect on plasma distribution.

Hoff’s Problem in a Probabilistic Setting

Hoff’s problem—that of investigation of the maximum load supported by an elastic column in a compression test—is considered in a probabilistic setting. The initial imperfections are assumed to be Gaussian random fields with given mean and autocorrelation functions, and the problem is solved by the Monte Carlo Method. The Fourier coefficients of the expansion for the initial imperfection function are simulated numerically; for each realization of the initial imperfection function, the maximum load supported by an elastic column in a compression test is found by the solution of a set of coupled nonlinear differential equations. For slightly imperfect columns, the closed solution is given in terms of Bessel and Lommel functions and turns out to compare well with the result of numerical integration. Results of the Monte Carlo solution are used in constructing the reliability function at a specified load. Reliability functions for different manufacturing processes (represented by different autocorrelation functions with equal variance) are calculated; design requirement suggests then that, other conditions being equal, the preference should be given to the manufacturing process resulting higher reliability.

Transient stresses and voltage developed in a spinning disc of radially inhomogeneous piezoelectric material

The stress distributions and the voltage developed in an annular disc of inhomogeneous piezoelectric material spinning with a variable angular velocity have been investigated. The stiffness coefficients and piezoelectric stress coefficients as well as the permittivity of the material of the disc may vary as any power of the radial distance. The solution of the problem is presented in terms of Bessel and Lommel functions, in which both the boundaries are kept free from radial stress. It deals with the determination of the hoop stress, and others, for a prescribed value of the applied shearing stress and voltage at the boundaries. The corresponding results for a homogeneous spinning disc can be deduced. Graphical representation of the results may be helpful to designers.

Excitation of1S(1s,3s) state of helium by electron impact in the Glauber approximation

The authors report Glauber cross sections for the electron-impact optically-forbidden transition 1S(1s,3s) from 1S(1s)2 in helium in the incident energy range 29.2 to 500 eV. The Glauber amplitude is calculated using an alternative procedure for the evaluation of the modified Lommel function instead of the usual series presentation which requires very high precision arithmetic for the intermediate values of the argument. The present results for both the differential and integral cross sections in the single-particle Glauber approximation (SPGA) and the full Glauber approximation (GA) are compared with experimental findings and with the predictions of recent theoretical calculations. They find that the GA predicts the shape of differential cross sections even at a few electron volts beyond threshold. At higher incident energies the GA yields cross sections in good agreement with experiment. At low energies the SPGA deviates strongly from the GA and fails to predict the angular distributions of scattered electrons.