Showing papers on "Bessel function published in 1980"

A numerical study of P-, SV-, and SH-wave generation in a plane layered medium

Abstract: A solution for the surface displacements due to buried dislocation sources in a multi-layered elastic medium is found using the Haskell (1964) paper as a starting point and more importantly, for notation. Through the introduction of some simple matrix operations, the Haskell (1964) solution is made simultaneously more compact and computationally stable. Time histories are computed for a perfectly elastic medium by performing classical contour integration in the complex wavenumber plane. A new aspect in the evaluation of those contours is introduced because of the recognition of nonzero singularity contributions of the Hankel and modified Bessel functions at k = 0. Theoretical ground motion time histories are presented to show the usefulness of the formulation. The overall objective of this paper is to incorporate the modifications made since 1964 to the Haskell (1964) paper in an easily understandable, step-by-step development.

Shaped reflector antenna analysis using the Jacobi-Bessel series

Abstract: The physical optics approximation is employed to derive the radiation integral for a doubly curved offset reflector antenna illuminated by an arbitrary source. A novel procedure is presented for expressing the radiation integral in terms of a summation of Fourier transforms of an "effective" aperture distribution which includes the effect of the curvature of the surface. The Jacobi-Bessel series is used to evaluate the Fourier transforms. The vector nature of the far-field pattern is studied by evaluating its three Cartesian components in a unified fashion. The rapid numerical evaluations of the expressions obtained are demonstrated via extensive test cases. In particular, the scattering characteristics of symmetric and offset parabolic, spherical, and shaped reflectors are studied in detail, and comparisons are made with other available data.

A new method of calculating the generalized Q function (Corresp.)

Abstract: A recursion relationship for the modified Bessel function is used to develop recursive formulas for functions which are expressed as a Neumann series expansion The method is applied to devise an algorithm for calculating the generalized Q function Comparisons with some alternative schemes are discussed and an error analysis is presented

Self‐modulation of ion Bernstein waves

Abstract: A self‐modulation equation for coherent ion Bernstein waves is derived by a multiple‐time‐scale, perturbative expansion of the Vlasov equation. The dominant nonlinearity acting on the waves is shown to be the nonlinear ion gyrofrequency shift. The waves obey a version of the nonlinear Schrodinger equation with the dispersive terms in the three spatial directions having, in general, differing signs. If the modulation is independent of one spatial variable, the elliptic or hyperbolic two‐dimensional nonlinear Schrodinger equation results. In contrast to the elliptic case which exhibits collapse, wave energy always disperses if initially localized in the hyperbolic case.

A boundary integral formulation of the eddy current problem

Abstract: The eddy current problem for slowly varying fields is formulated using Fredholm integral equations. The problem is expressed in terms of the electric vector potential, T. The development here is for two dimensions so that in rectangular coordinates only J x and J y are permitted. The formulation to be presented uses a modified Bessel function as the kernel. In this way only the boundary of the conductors need be represented. The result is in general far fewer equations than a differential formulation such as finite difference or finite elements.

The inverse electromagnetic scattering problem for a perfectly conducting cylinder

Abstract: The problem of determining the shape of the cross section of a simply connected perfectly conducting infinite cylinder from a knowledge of the far-field pattern for all angles of observation and small values of the wavenumber is considered. The method proposed relies heavily on conformal mapping techniques. In particular it is shown that if the transfinite diameter is known each Fourier coefficient of the far-field pattern of the electric field determines a Laurent coefficient of the conformal mapping taking the exterior of the unit disk onto the exterior of the unknown cross section. The transfinite diameter is determined by changing the polarization of the incoming wave and measuring the far-field pattern of the resulting magnetic field. Of particular interest is the case when only a finite number of the Fourier coefficients of the far-field pattern are known. In this situation error estimates are obtained by using results on coefficient estimates for univalent functions.

On Temme's Algorithm for the Modified Bessel Function of the Third Kind

Abstract: Some modlficatmns to Temme's algorithm for the evaluation of the modified Bessel functmn of the third kind have been made Temme evaluates K,(x) and K,+dx) for x > 1 and ]~1 --1⁄2 from auxdmry functmns k(,(x) and kdx), which he determines by Miller's backward recurrence algorithm In this paper the starting order for the backward recurrence algorithm for ko(x) and k~(x) is given by a pmcewlse linear functmn of 1/x Also, we have found that ~t is more efficmnt, for small values of x, to calculate only kdx)/ko(x), whmh can be used with the values L(x) and L÷dx) and the Wronskian to obtam K,.(x) and K,+dx)

Demonstration for observing J0(x) on a resonant rotating vertical chain

Abstract: A simple example of mechanical resonance using a light chain rotating about a vertical axis is described. The equation of motion for the chain can be reduced to Bessel’s function of the first kind of order zero [J0(x)]. Stable lobe patterns are formed with eigenfrequencies and internode spacing which agree well with the zeros of J0(x).

Long Atmospheric Waves and the Polar-Plane Approximation to the Earth’s Spherical Geometry

Abstract: The spherical geometry of the earth is replaced by polar cylindrical geometry, with a plane tangential to the earth at the pole. The resulting frequency and structure of free motions in an isothermal, adiabatic atmosphere with a resting basic state is studied in both geometries. The solutions for ν (meridional wind) may be written as a single Bessel function if certain approximations are made. For positive equivalent depths, the geometrical approximation is best when the Lamb parameter ϵ≳ 10, so that Rossby waves are well modeled, while fast moving gravity waves are not well approximated. The impact of setting f to a constant value when undifferentiated, as in the usual midlatitude beta-plane approximation, is examined. It is found that the value of f is as important in determining how well the model behaves as are the geometrical and other approximations.

Asymptotic approximations for modified Bessel functions

Abstract: The behavior of a qν(x) =Iν(x)/Iνa(x), where Iν is a modified Bessel function with integral or half‐integral index ν and Iνa the leading term of its asymptotic series, is investigated for x≫1. It is shown that qν(x) may be approximated by eν(x) =exp(−ν2/2x), the difference rν(x) =qν(x)−eν(x) being of order x−1/4. Bounds for rν(x) depending only on x are derived for each of the two classes of ν’s and an application of these results in scattering theory is indicated.

Sugli zeri delle funzioni di bessel

Abstract: In this paper we give some inequalities for the zerosc ν,n of the cilinder functionJ ν (x)cosα−Y ν (x)sindα, 0≤α<π, for the zerosc′ν,n ofC′ν(x) and we make an application to negative zerosa n ,b n of Airy functions Ai(x), Bi(x) of first and second kind respectively. We also make a comparison between inequalities related to the first zeroj ν,1 of the Bessel functionJ ν (x) of first kind with ν>0 and other known inequalities. Some results are new, other not, but in this case they are derived in a simpler way than usual ones.

The interaction of acoustic waves with flux tubes

Abstract: The interaction between a magnetic flux tube and an acoustic wave front propagating in the nonmagnetic region in which it is embedded is investigated by expressing the incident wave as a Fourier Bessel series. Using the velocity and pressure balance conditions at the interface, one may determine the amplitudes of the reflected and transmitted waves for each Bessel component.

Low-frequency scattering by a slit in a conducting plane

Abstract: The problem of diffraction of an H polarized plane wave by a narrow slit in a perfectly conducting screen can be formulated as an integral equation with a Hankel function difference kernel. For narrow slits, we replace the Hankel function by the first three terms of its series expansion and solve exactly the resultant integral equation. The results are superior to those obtained by solving a succession of integral equations, each with a static kernel, which is the traditional approach. Also solved is the slightly more difficult problem of E polarization. We indicate that the method can be generalized to include more terms in the kernel expansion, to diffraction by multiple slits, with different media and with impedance boundary conditions on the screen.

Fourier transform of two-centre charge distribution

Abstract: At medium and high collision energies travelling orbitals must be employed in charge-transfer calculations. This results in two-centre matrix elements of the form of a Fourier transform. Similar Fourier transform matrix elements arise in the Bethe theory of the scattering of a structureless charged particle by a diatomic molecule as well as in the computation of the coherent and incoherent intensity for X-rays scattered by molecules. No simple general expression exists for Slater-type orbitals (STO). The author derives such a general expression which contains only small sums and a single numerical integral over an integrand which contains a spherical Bessel function.

Uniformly Valid Expansions for Laplace Integrals

Abstract: Uniformly valid asymptotic expansions for a class of parameter-dependent Laplace integrals involving coalescing saddle points are obtained by a direct method based on matched asymptotic expansion theory. Bessel functions of large order are treated as an example.

The reflected impulse response of a Lorentz medium

Abstract: The wave reflected from a Lorentz medium when illuminated by a transverse electric polarized plane wave with impulsive time dependence is expressed as an infinite sum of spherical Bessel functions. This solution is useful in numerical calculation for small and intermediate values of time. For small values of time the Lorentz medium solution simplifies to the result for the impulse response of a cold plasma.

Generalized Bessel functions and the representation theory of U(2) σ C2×2

Abstract: We construct the matrix elements of both finite transformations and infinitesimal generators in irreducible representations of the motion group U(2) σ C2×2 with the aid of the contraction limit of the analogous structures of U(4). The matrix elements of finite transformations are found to have a structure similar to that of the classical Bessel function in that they contain two inverse gamma matrices which couple Wigner D functions. An integral representation is established and related to the matrix‐valued Bessel functions of Gross and Kunze. By means of the representation property of the matrix elements we obtain a new sum rule for classical Bessel functions and an analog of the binomial theorem for the sum of two 2×2 matrices which involves the U(2) gamma matrix instead of the classical gamma function.

On Enveloping Series for Some of the Special Functions, and on Integral Inequalities Involving Them

Abstract: The work of the present paper deals with enveloping series for certain of the special functions of mathematical physics. In many (but not all) cases, the Maclaurin-series expansion of the function concerned envelops the function itself and can be regarded as an asymptotic expansion of the function about the origin.

Analysis of the Adsorption Potential in the Hydrodynamic Model of Metals

Abstract: We analyse the perturbatively calculated adsorption potential U{/i = 2covzf/3) of a neutral molecule a distance z from a flat metal surface, adopting for the metal the simplest non-trivial spatially dispersive model, nam ely a continuous fluid confined by impenetrable barriers and with assigned plasm a frequency wp and sound velocity /?. Then U emerges as a weighted sum , over molecular states with virtual excitations En0 = of a function G(en0,y) which is the Laplace transform of a function g{en0, y), where k = (vpy /3is the tangential component of a plasmon wavenum ber. For large y, Watson’s lemma applies because the Taylor series for g converges for small y, the lemma yields an asymptotic series for G{e,y) which in general has a leading term of order For positive e, this expansion was previously obtained by Mahanty & Paranjape (1977). It must be modified if e is negative, and in particular if e is close to the surface-plasmon branch point at — when is large b u t; + A l is n o t> the effective leading term is of order If e is close to the bulk-plasmon branch point at —1, similar modifications are needed in some next-toleading term s. G is singular at y = 0, where its behaviour is governed by the fact that, for large y, g has a convergent expansion in powers of y~x. For the Laplace transform G of such a function we prove a theorem which is a natural complement to W atson’s lemma but which, surprisingly, seems to be new. I t allows G to be expressed in term s of power series in y converging everywhere, with all singularities displayed explicitly: G = A n{e) y n + lnX ”=0 Bn(e)yn.The coefficients and B n are determined by the coefficients in the expansion of g(e, y); rem arkably, they are entire functions of e, whence those parts of G which are singular in y are entire in e and vice versa. Sum rules and closure relations allow the sum over molecular states to be found in closed form for the parts of G proportional to y~x and In y. In the special case where all en0 are negligible (wp > | En0), U(y) reduces to the far simpler yet still nontrivial classical limit Ue{y), expressible in closed form in term s of Bessel and Struve functions. The non-dispersive limit (/? = 0) is obtainable trivially, but is so special a case that it can shed no light on the small-distance behaviour of U, nor on the large-distance behaviour when e is close to the branch point at —

Further comments on the computation of modified Bessel function ratios

Abstract: A theorem by W. J. Thron is used to enhance and further substantiate empirical results by Gautschi and Slavik on the convergence of continued fractions used to compute ratios of modified Bessel functions. 1. Thron's Theorem Applied to Continued Fractions for Ratios of Modified Bessel Functions. A theorem by Thron* [2] offers analytic confirmation of recent empirical results [1] on the convergence of continued fractions, cfs, for computing ratios of modified Bessel functions. Defining Thron's Theorem 6.1 states: The continued fraction K(an/1) converges to a value u which satisfies R(u) > -Vi provided that J2(an) < R(an) + % and \\an\\ 1. Here M is an arbitrary large positive quantity. As a consequence of Thron's theorem we can obtain a priori bounds on the truncation error, 2Rn, of Gauss's and Perron's cfs for modified Bessel function ratios. In the case of Thron's Theorem 6.1, Rn is the radius of the value region (nested circles within which the value of the cf lies) of the cfs. From Thron's theorem we thus have *7,<1 Üil + Ui4Mk)). I *=i Relatively larger values of M are associated with overall slower convergence. Both Gauss's and Perron's cfs for Iv(x)/Iv_x(x) are in the form required by the theorem [1]. For positive x the assumptions of Thron's theorem are trivially satisfied for Gauss's cf and are easily verified for Perron's cf if v > 1. Thus, to apply the theorem, note that when x » v, the an of Perron's cf will be dominated by l/a0{x) = x/ix + 2v) and the an of Gauss's cf will be dominated by axix) = (l/4)x2/v(v + 1), which we will refer to asMp and MG, respectively. Clearly, as* » v, Mp< 1 and MG » 1. In this case, Rn, or 2Rn (the truncation error), of Gauss's cf is much larger than Perron's, indicating slower convergence of the Gauss cf. These results are given in Table 1.1 which reproduces Table 3.2 of [1] supplemented by values of Mp and MG Received March 20, 1978; revised September 5, 1979. 1980 Mathematics Subject Classification. Primary 33A40; Secondary 40A15, 65D15. * Thron's theorem referred to here is actually a special case of an earlier result due to Scott and Wall (Trans. Amer. Math. Soc, v. 47, 1940, pp. 155-172). However, the results of this paper depend on the proof as given by Thron. © 1980 American Mathematical Society 0025-5718/80/0000-0119/$01.75 937 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 938 MARIETTA J. TRETTER AND G. W. WALSTER (k in Table 1.1 represents the number of terms required in the Gauss and Perron cfs to obtain relative accuracy of tëlO-8, as originally given in Table 3.2 of [1]). One interesting observation in Table 1.1 is the fact that for x » v, Mp is increasing slightly as x increases, even though the table indicates, for 8 decimal digit accuracy, that Perron's cf is converging faster with increasing x. This phenomenon is explained by looking at Figure 3.3 in [1]. The increase in Mp reflects the fact that, although initial convergence is fast, at extreme accuracies the convergence slows down before regaining speed. The phenomenon of slowing down also becomes more pronounced as x increases. Thus, one must be careful to interpret M as an overall measure of convergence. Table 1.1 Number of terms, k, required in the Gauss and Perron cfs to obtain relative accuracy of &10~ 8 (as given in Table 3.2 of [1 ] ), and M as defined above.

Asymptotic Solutions Without Special Functions for Steep and Shallow Shells

Abstract: The exponential “boundary-layer” solutions for “steep” shells are well known as are the uniformly valid solutions for domes and toroids, which have a “shallow” region, i.e. a transition point in the differential equation. The uniformly valid solution requires, however, a special function, usually a Bessel function, but in some cases a generalized hypergeometric function. In this work the advantage of the exact eigenvalue of the coefficient matrix is explored. The first term of the formal expansion is essentially that of the boundary-layer solution, but with some extra elements which give the proper limit of a very shallow shell, i.e. a flat plate. From numerical calculations of edge flexibility coefficients, it is evident that the one term does not give a uniformly valid approximation; however, the error in the zone of transition from steep to shallow is bounded. Indeed the two-term approximation has at worst a 10% error. Thus the approach which requires only elementary functions seems more flexible and more readily adoptable for an automated calculation procedure.

The Jost solutions for Yukawa potentials

Abstract: Using a suitable Laplace transformation of the regular solution and the expansion of its Laplace transform in partial fractions we found new explicit expressions for the Jost solutions of the radial Schrodinger equation with complex wave numbers and angular momenta, in general, and for potentials of the exponential and, especially, Yukawa type, which were described by means of the Stiltjes integral with respect to a function μ(t) of bounded variation. Many other relations including those for the Jost functions were derived or rederived, which have their counterpart among relations for the Bessel functions.

Implementation of a novel procedure to optimize electron optical systems through the use of orthogonal functions

Abstract: It is shown how Fourier‐Bessel series solutions published recently, representing the electric potential in various electrostatic electron optical configurations with rotational symmetry, can be used in a constrained optimization procedure through which local minima of an aberration function A=A (B1, B2, ...) can be found by direct search or by descent methods. The variables Bi are Fourier coefficients. As a simple illustration, a Bessel Box with a coaxial annular aperture is obtained, which produces a tangential ray image of low field curvature and distortion. Optimization with respect to both paraxial and zonal focal properties can be carried out, because the method is not restricted by the use of the paraxial ray equation. The procedure can be applied to one‐foil, and open systems.

Modulus and phase of the reduced logarithmic derivative of the cylindrical Bessel function

Abstract: The modulus and phase of the reduced logarithmic derivative of the cylindrical Bessel function zJ,(Z)/JV(Z), for complex variable z and real order v, are investigated. Special attention is paid to the location of saddle points and their trajectories as v varies.

On Electromagnetic Environmental Fidelity in Damped Cylindrical Cavities

Abstract: A study is made of the electromagnetic cavity mode structure of a cylindrical tank containing a cylindrical damper membrane. Maxwell's equations for the most general mode of a cylindrical tank is considered and reduced to the problem of solving a single transcendental equation written in terms of the Bessel function of the first and second kind. This equation is solved numerically for some representative damped cavities yielding modal frequencies and damping rates.