Showing papers on "Bessel function published in 1969"

A circular-harmonic computer analysis of rectangular dielectric waveguides

Abstract: This paper describes a computer analysis of the propagating modes of a rectangular dielectric waveguide. The analysis is based on an expansion of the electromagnetic field in terms of a series of circular harmonics, that is, Bessel and modified Bessel functions multiplied by trigonometric functions. The electric and magnetic fields inside the waveguide core are matched to those outside the core at appropriate points on the boundary to yield equations which are then solved on a computer for the propagation constants and field configurations of the various modes. The paper presents the results of the computations in the form of curves of the propagation constants and as computer generated mode patterns. The propagation curves are presented in a form which makes them refractive-index independent as long as the difference of the index of the core and the surrounding medium is small, the case which applies to integrated optics. In addition to those for small index difference, it also gives results for larger index differences such as might be encountered for microwave applications.

Statistical models for the vertical deflection from Gravity-Anomaly Models

Abstract: Statistical models for vertical deflections in the form of power spectral densities and autocorrelation functions are derived from theoretical gravity-anomaly models by means of the Vening Meinesz equations. Details are given for homogeneous and isotropic gravity-anomaly models described by exponential and Bessel autocorrelation functions. The use of these models for evaluation of mean-squared output errors in an inertial navigation system is described.

Uniform asymptotic expansions of integrals that arise in the analysis of precursors

Abstract: Asymptotic expansions for λ ≫1 of functions defined by integrals of the form $$I(\lambda ;\theta ) = \mathop \smallint \limits_\Gamma \exp \{ i\lambda \phi (k;\theta )\} g(k;\theta )dk$$ are considered in the case where there are two stationary points of φ which approach ±∞ as the second variable θ approaches some critical value, say θ 0. In this limit the results of the classical methods of stationary phase and steepest descents become invalid. This paper is devoted to the development of an asymptotic expansion of I that remains valid even for θ near and equal to θ 0. The motivating physical problem is the propagation of signals in dispersive media. Indeed, the results of the present paper can be used to study the behavior of that portion of a signal called the “precursor” in a neighborhood of its front of propagation. The technique used to obtain the uniform expansion is an adaptation of the method originally developed by Chester, Friedman and Ursell in their treatment of the problem of two nearby stationary points. Here, however, we find that a certain family of Bessel functions play the role of the Airy functions in that problem. We also obtain the interesting result that our expansion remains valid for λ merely bounded away from zero and θ → θ 0. In fact a theorem is proved which establishes the asymptotic nature of our results in the relevant limits. Finally, two examples are considered to illustrate the use of these results.

Analytical Evaluation of Multicenter Integrals of r12−1 with Slater‐Type Atomic Orbitals. V. Four‐Center Integrals by Fourier‐Transform Method

Abstract: The four‐center integral of r12−1 with Slater‐type atomic orbitals is evaluated analytically The Fourier‐transform convolution theorem is used to express the integral as an infinite sum in which the internuclear angles appear in spherical harmonics, and the internuclear distances in integrals over spherical Bessel functions and exponential‐type integrals These “radial” integrals are evaluated as convergent infinite expansions by contour integration techniques The formulas are valid for general values of the n, l, m, ζ parameters of the orbitals and for general nonzero values of the internuclear distance vectors

Reflection of a Plane Transient Electromagnetic Wave From a Cold Lossless Plasma Slab

Abstract: Reflection of a plane transient electromagnetic wave from a plasma half-space and a plasma slab is considered. The plasma is assumed to be cold and collisions are neglected. Under these assumptions, the Laplace transforms may be inverted exactly, yielding a solution in the form of Bessel function series. The numerical results, for the waveform of the reflected field, exhibit features that are easily related to the plasma frequency of the slab and its thickness.

Small-signal, high-frequency equivalent circuit for the metalߝoxideߝsemiconductor field-effect transistor

Abstract: The differential equations describing the small-signal sinusoidal operation of the 'intrinsic' m.o.s.f.e.t. structure are solved using modified Bessel functions of the first kind. Expressions for the small-signal short-circuit admittance parameters are obtained in series form. By retaining appropriate terms in the series, the elements of a convenient equivalent circuit are computed for both the nonpinchoff and the pinchoff cases. Results are compared with those presented by other authors, to show that previous calculations for the nonpinchoff case are incorrect.

Generalized Confluent Hypergeometric and Hypergeometric Transmission Lines

Abstract: A proliferation of exact closed-form solutions of the telegrapher's equation V_{xx} - Z_{x}Z^{-1}V_{x} - kZ Y V = 0 for the voltage V(x) in an RC or lossless transmission line, with distributed series impedance Z(x) and shunt admittance Y(x) , respectively, have emerged in recent years. Generalizations of known solutions have been constructed, sometimes using ad hoc methods. A systematic method is described for deriving exact solutions in terms of standard transcendental functions, which yields far more general profiles for Z(x), Y(x) , or Z(x)/ Y(x) than previously given. Examples of the procedure are given based upon Bessel's, Whittaker's, and the hypergeometric equation, and previously derived profiles emerge as special cases of the analysis.

Concerning the Zeros of Some Functions Related to Bessel Functions

Abstract: The real zeros of the Riccati‐Bessel functions (12πx)12Jv(x), (12πx)12Yv(x), of their derivatives d[(12πx)12Jv(x)]/dx, d[(12πx)12Yv(x)]/dx, and of their cross products 12πx[Jv(x)Yv(Kx) − Yv(x)Jv(Kx)], ddx[(πx2)12Jv(x)]ddx[(πx2)12Yv(Kx)] − ddx[(πx2)12Yv(x)]ddx[(πx2)12Jv(Kx)] are investigated. Expansions analogous to those provided by McMahon and Olver for the zeros of the Bessel functions are obtained for the zeros of the derivatives of the Riccati‐Bessel functions. The analysis of Kalahne for the zeros of the cross product of Bessel functions is considerably expanded and analogous results are obtained for the zeros of the cross product of the derivatives of the Riccati‐Bessel functions. Included are derivations of the expansions for large zeros at fixed v, of asymptotic expansions for large v at fixed number of the zero, and also asymptotic expansions for the zeros as K→1 and K→∞. Figures illustrating the behavior of the zeros are provided for v = l + ½, where l is an integer. These zeros correspond to th...

Theory of Bessel potentials. III : potentials on regular manifolds

Abstract: : The present part of the 'Theory of Bessel Potentials' contains Chapter 4 dealing with potentials on regular Riemannian manifolds. (Author)

Wave propagation in inhomogeneous elastic media, solution in terms of Bessel functions

Abstract: Wave propagation in an inhomogeneous elastic solid, when the elastic parameters of the solid depend on one space co-ordinate only, is considered. The stress and displacement components are assumed to depend on this same space co-ordinate and time alone. For the situations considered here the motion is governed by the wave equation with variable wave speed. The values of the elastic parameters which correspond to the solution of this equation in terms ofBessel functions are determined and some particular problems are considered.

Generalized transforms and their asymptotic behaviour

Abstract: Certain properties of generalized transforms of the type $\int\_{0}^{\infty}$ g(x)h($\alpha $, x) dx are derived when g is a generalized function in the terminology of Lighthill (1958) and Jones (1966 b). The kernel function h is assumed to be smooth and of sufficiently slow growth at infinity for the generalized transform to exist for any generalized function g. Nevertheless, the class of kernel functions is wide and includes functions such as e$^{\text{i}\alpha x^{2}}$ and the Bessel function J$\_{n}$($\alpha $x). Theorems concerning the derivative and the limit (in the generalized sense) of the generalized transform are established. The problem of the inversion of generalized transforms is also discussed. The analogue of the Riemann-Lebesgue lemma for generalized transforms is obtained when g is a conventional function and the restrictions on h are relaxed so that it need only be the derivative of a function with suitable properties. The asymptotic behaviour as $\alpha \rightarrow $ + $\infty $ of the generalized transform is examined under the condition that g is infinitely differentiable (in the ordinary sense) at all but a finite number of points. It is shown that the main contribution to the asymptotic development comes from intervals near these points and the point at infinity. Criteria are provided which demonstrate that in many important practical cases the contribution from the point at infinity is essentially exponentially small and therefore negligible. The contributions from the other critical points are determined under a variety of circumstances. In all cases the aim has been to consider conditions which are likely to be of practical value, to be capable of relatively straightforward verification and yet yield theorems of reasonable utility and wide applicability. Some illustrations of the applications of the theorems are given; they include Bessel functions, Laplace transforms and the Hankel transform.

Two Families of Apodization Problems

Abstract: Multidimensional Fourier transforms, Parseval’s theorem, and Schwarz’s inequality, used in concert, show how to apodize a lens to minimize the nth moment of the irradiance in the diffraction pattern for a given central irradiance. Replacing Schwarz’s inequality with the calculus of variations, we can find the pupils that minimize quotients of certain different moments. In particular, the mean-square miss distance of a photon gun is minimized if the waves it launches have amplitudes that decrease as the Jo Bessel function moves from the center of the projecting lens and reach zero at the rim.

Interaction Between an Electromagnetic Plane Wave and a Spherical Shell

Abstract: An efficient, exact solution for the complete field structure inside, in, and outside a spherical shell of arbitrary linear media, excited by a monochromatic plane wave is given. The fields in each region are expressed as vector multipole expansions whose coefficients are evaluated through the boundary conditions. The numerical reduction involves the use of modified spherical Bessel functions and recursion relations. The technique is illustrated by numerical results for the fields inside an imperfectly conducting shell. This analysis virtually exhausts the class of problems of spherical shells constructed of linear media.

Transient signal propagation through an inhomogeneous plasma

Abstract: The propagation of a transient electromagnetic field through a cold, isotropic, lossless, inhomogeneous plasma is considered. The electron density of the plasma decreases exponentially in the direction of propagation. For incident transient signals whose Laplace transforms are regular at infinity and whose only singularities are poles, the transient electric field propagated through the plasma can be expressed as a sum of exponential waves at discrete complex frequencies. The characteristic frequencies of the inhomogeneous plasma are related to the zeros of the modified Bessel function of the first kind of fixed argument and variable order. When the Laplace transform of the incident signal is not well-behaved at infinity, approximate methods may be used to evaluate the transient electric field. The propagation of a step sine wave and of a quasi-monochromatic Gaussian pulse are considered. It is found that for plasmas characteristic of reentry boundary layers, the effect of the plasma layer on dispersion is very small. The principal effect of the layer is to attenuate a signal propagated through it.

Distribution of product and quotient of Bessel function variates

Abstract: In this paper, the technique of Mellin transforms is employed to obtain the distribution of the product and quotient of two independent Bessel function random variables. Two different types of Bessel function variates are considered. The results are then specialized to yield a wide variety of classical distributions of importance in applications.

Steady rotation of a body of revolution in a conducting fluid

Abstract: We investigate the steady rotation of an insulating body of revolution in an unbounded electrically conducting fluid permeated by a uniform axial applied magnetic field. The assumptions of a small magnetic Reynolds number (Rm ≪ 1, i.e. the weakly conducting situation) and negligible inertia forces compared with the magnetic forces (R/M2 ≪ 1) permit us to suppress the inflow at the poles and outflow at the equator, which normally occurs for a non-conducting viscous fluid ((12), pp. 436–439). Thus in the case of the sphere, we find an exact solution of the reduced equations in terms of an infinite series of Legendre polynomials of order 1 with coefficients which are the ratios of modified spherical Bessel functions. This is the canonical problem by which results for arbitrary bodies of revolution are obtained.

On the evaluation of antenna quality factors

Abstract: Compact expressions are given for the reactive energies and modal quality factors in the literature. The formulas do not involve spherical Bessel functions and are in terms of, at most, two polynomials having positive coefficients which are readily calculable. Curves are presented for the exact forms for a number of modal quality factors.

Inverse Functions of the Products of Two Bessel Functions

Abstract: Special cases of the inverse function of the product of two spherical Bessel functions have been found recently by other writers as 1F2 hypergeometric functions. We give the general expression as the derivative of a product of spherical Bessel functions. These results have also been found in the classical literature.

Integral relations among Bessel functions

Abstract: This relation appears to be novel. Its derivation from similar integral relations in the literature requires a nontrivial effort. During examination of the manuscript, two referees derived Eq. (3b) independently. Their approach will now be sketched. The first is as follows : Expand Ii on the right-hand side of Eq. (3b) in powers of cos d. Integrate term by term using Eq. (2) on p. 374 of Watson's book [3]. Apply the definition of the Lommel functions sßtV(z) from p. 345 of Watson and carry out the summation, allowing for the identity Eq. (9) on p. 141 of Watson. This yields the left-hand side of the present Eq. (3). The second approach starts from Eq. (25) on p. 299 of the book by Luke [4]. The integral of this formula reduces to the present integral by the choice p = v = a = ß = \\, w = iz. Luke's formula gives the result as an infinite sum over generalized hypergeometric functions. In the present case the series expansion of these functions

Helicon-wave propagation in a laboratory plasma.

Abstract: Measurements have been made of the propagation of axially symmetric helicon waves in the afterglow of a pulsed plasma. The experimental values for the dispersion relation approach the theoretical values calculated for a cylindrical plasma column with zero under Bessel function radial density distribution, after time intervals several times longer than the radial diffusion time.

Axisymmetric deformation of a circular cylinder of piezoelectric cadmium selenide crystal

Abstract: This paper deals with the axisymmetric deformations in a circular cylinder of finite length made of Piezoelectric, crystalline cadmium selenide belonging to class 6mm. The displacements and the electric potential are expressed in terms of a single function. Displacements and electric potential prescribed on the lateral surfaces are expanded in Fourier series, and the stresses and the electric potential prescribed on upper and lower surfaces are expanded in series of Bessel functions of the first kind. The existence and uniqueness of the solution of the problem is discussed under certain restrictions on the coefficients that occur in the expansions. Numerical results are given.

Eigenvalues for higher-order conical-horn modes

Abstract: Macdonald's formula, which is used to relate the associated Legendre and Bessel functions, is employed to find the approximate eigenvalues for TE and TM modes in a conical horn Although the formula is restricted to small angles ?, it nevertheless provides a very simple and accurate method for calculating the eigenvalues of the fundamental and higher-order modes, even for large values of ?

An expansion formula for Fox's H-function

Abstract: The object of this paper is to establish an integral involving Fox's H-function and employ it to obtain an expansion formula for the H-function involving Bessel functions.