Showing papers on "Bessel function published in 1977"

An investigation of temporal moments of stochastic waves

Abstract: It is proposed to describe the temporal characteristics of a wave propagating in a random medium in terms of its temporal moments. The first two moments are related to the mean arrival time and the mean pulse width. It is shown that the one-position two-frequency mutual coherence function enters in the formulation naturally. The form of the expression suggests expanding the mutual coherence function in a narrow-band expansion whose coefficients can be solved exactly from the parabolic equation that takes into account all multiple scattering effects except the backscattering. A brief survey of the literature shows that the irregularity spectrum, under various conditions, has a power-law dependence. In order to conform to this observation a Bessel function spectrum proposed by Shkarofsky is found convenient to use since it not only reduces to the desired power-law form in the proper range of wavenumber space, but also has all the finite moments. Exact expressions for the mean arrival time and mean square pulse width are obtained; some numerical examples are given. Finally, the effect of noise on these moments is discussed.

Scattering by periodic metal surfaces with sinusoidal height profiles--A theoretical approach

Abstract: A theory of scattering by periodic metal surfaces is presented that utilizes the physical optics approximation to determine the current distribution in the metal surface to first order, but modifies this approximate distribution by multiplication with a Fourier series whose fundamental period is that of the surface profile (Floquet's theorem). The coefficients of the Fourier series are determined from the condition that the field radiated by the current distribution into the lower (shielded) half-space must cancel the primary plane wave in this space range. The theory reduces the scatter problem to the familiar task of solving a linear system. For certain basic types of surface profiles, including the sinusoidal profile considered here, the coefficients of the linear system are obtained as closed form expressions in well-known functions (Bessel functions for sinusoidal profiles and exponential functions for piecewise linear profiles). The theory is thus amenable to efficient computer evaluation. Comparison of numerical results based on this theory with data obtained by recent numerical schemes shows that for depths of surface grooves less than a wavelength and for unrestricted groove widths, reliable and comparable, if not more accurate, data is obtained, in many cases at considerably cheaper computational cost.

Electrostatic potential and field in a cylindrical tank containing charged liquid

Abstract: If a metal tank is partly filled by charged liquid, such as hydrocarbons, the electrostatic potential can become very high and hazardous. In order to determine this potential, one must solve the Poisson and the Laplace equations simultaneously. With a cylindrical tank configuration, an analytical solution was obtained by using Bessel functions. Numerical computation of the model was performed and it can be shown that the convergence of the series solution is reasonably acceptable. The computed results are shown for a particular model. In order to estimate a maximum potential and field strength in the gas space for general configuration of a tank, more practical equations were derived.

CDC 6600 Subroutines IBESS and JBESS for Bessel Functions Iυ(x) and Jυ(x), x≥0,υ≥0

Abstract: Subroutines IBESS and JBESS for L(x) and J,(x), x >_ O, p > O, are presented which implement , in similar fashions, the power series, the asymptotic expansion for x ~ ~, and the uniform asymptotic expansion for ~-* ¢~ as basic elements. Where the asymptotic expansion for x-~ o0 is used, sequences are started by two evaluations of the expansion followed by backward recursion in IBESS and forward recursion in JBESS for p < x. Two evaluations of the series also start a backward reeursive sequence where x _< ½. Except for very large z and v, where it is more economical to perform two evaluations of a basic formula, other sequences are started by the Miller algorithm normalized by one evaluation of either the power series or the expansion for ~-~ ~o. Recursion is used for pairs (~, x) not covered by one of the basic formulas. A scaling option to remove the exponential growth of I,(x) is also provided together with appropriate diagnostics for underflow or overflow in both IBESS and JBESS.

Magnetized plasma kinetic theory. I. Derivation of the kinetic equation for a uniform magnetized plasma

Abstract: A kinetic equation for a multi-species plasma in an external uniform magnetic field is derived from the BBGKY hierarchy of equations. The equation generalizes the equation of Rostoker (1960), which assumes that the distribution function is independent of the azimuthal angle, and all previous results can be derived from it. An additional advantage is that the collision integral is obtained in a form which is free from infinite sums of Bessel functions, and this greatly facilitates calculations based on it.

A new (Laguerre) uniform approximation for vibrational transition probabilities

Abstract: A mapping between the exactly soluble forced oscillator and the general vibrationally inelastic scattering problem is shown to yield a new uniform approximation based on generalized Laguerre polynomials. Computations are reported for collinear He-H2 collisions in which H2 is represented by harmonic and Morse oscillators. The results show that the Laguerre approximation avoids the known failings of the existing Airy and Bessel uniform approximations.

Lower bounds for the zeros of Bessel functions

Abstract: Let jp „ denote the nth positive zero of J , p > 0. Then / ■■> 7\'/2 Jp.n > Oln + P) ■ We begin by considering the eigenvalue problem (1) -(•*/)' + x~y = X2x2p-Xy, X,p>0, (2) y(a) =y(\) = 0, 0 < a < 1. For simplicity of notation we will set q = p~x. It is easily verified that the general solution of (1) is y(x) = CxJq(Xqxx/q) + C2Yq(Xqxx'q) and that the eigenvalues are given by Jq(Xq)Yq(Xqax/q) Jq(Xqax/q)Yq(Xq) = 0. If zn(a, r) denotes the «th positive zero of Jr(z)Yr(zax/q) Jr(zax/q)Yr(z) = 0, then the «th eigenvalue, X2(a), of (1), (2) is given by (3) X2(a) = (z„(a, q)/qf. Let jrn denote the «th positive zero of Jr. On p. 38 of [4] it is shown that zn(a, r) —>jrn as a —> 0+ whenever r is a positive integer. The restriction on r is extrinsic so that (4) Mm zn(a,r)=jrn, r > 0. a—»0"1" Let R [p, y] denote the Rayleigh quotient R[p,y] = f\-(xy')' + x~xy)y dx / f\2p-xy2 dx. Ja Ja It is well known that the eigenvalues {X2(p)} of (1), (2) can be obtained from the Rayleigh quotient [5]. Let V denote the linear space of all functions in C2((a, 1)) which satisfy the boundary conditions (2). Then X2(p)= min R[p,y]y£ y,y=^o Let^,,^, . . . ,y„ be « functions in V, A denote the subspace of V spanned by yvy2, . . . ,yn and A x denote the orthogonal complement of A relative to V. Then Received by the editors January 5, 1976 and, in revised form, September 13, 1976. AMS (MOS) subject classifications (1970). Primary 33A40. © American Mathematical Society 1977 101 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

Algorithm 518: Incomplete Bessel Function I0. The Von Mises Distribution [S14]

Abstract: The incomplete modified Bessel function Io(0, ~) is formally equivalent to the cumulative distribution applied by yon Mises [9] to s tudy deviations of atomic weights from integer values, representable as points on the circumference of a circle or as circular directions. This distribution of points on a circle is analogous to the normal or Gaussian distribution of points on a line and has applications to the study of quantal or periodic data, directions of sedimentary bedding, surface fault lines, wildlife movements, etc. (cf. Batschelet [3] and Mardia [7]). The left tail area of this symmetrical distribution is evaluated by this For t ran F U N C T I O N of the angular deviation 0 and the concentration parameter K, where

Fraunhofer diffraction by semicircular apertures

Abstract: The far-field diffraction pattern of semicircular apertures is described. A general expression is given for the amplitude that is valid over the entire image plane and that reduces to Bessel and Struve functions on the axes. A simple approximation is also given that is valid for all angles at large distances from the origin.

The Computation of Real Fractional Order Bessel Functions of the Second Kind

Abstract: A new method is presented for the computation of sequences of real, fractional-order Bessel functions of the second kind. For x not too small the method involves evaluation of near-minimax approximations to Y/sub nu/(x) for fixed x and small variable ..nu.., followed by analytic continuation in x and then by recursion in ..nu... Direct expansions of Y/sub nu/(x) are used for x small and 0 < or = to nu < or = to 1. Finally, implementations of the method for the FUNPACK package are discussed.

Erratum: Algorithm 511: CDC, 6600 Subroutines IBESS and JBESS for Bessel Functions Iυ (x) and Jυ (x), x ≥ 0, υ ≥ 0 [S18]

Abstract: DESCRIPTION This algorithm is a complement to [1], where the theoretical background and development are described. ALGORITHM [Only those portions of the listings which are introductory comments explaining the subroutines IBESS and JBESS are printed here. The complete listing is available from the ACM Algorithms Distribution Service (see inside back cover for order form), or may be found in \"Collected Algorithms from ACM.\"] Gel~ral permission tO re-publish, but not for profit, all or part of this material is granted provided that ACM*~ copyright notice is given and that reference is made to the publication~ to lt~ d~ta of |~t~e, and to the fact that reprinting privileges were granted by perrhi~siot~ o[ the AasoQ|ation for (Join. puting Machinery.

Bessel function type automatic delay equalizer

Abstract: A Bessel function type automatic delay equalizer comprising a plurality of transversal filters each of which has an independent delay cosine equalization component having a predetermined period, and the superposition of those filters providing the desired characteristics, characterized in that, said equalizer further comprises a Bessel function generator for controlling the tap gain of said transversal filters

The Application of a Bessel Transform to the Determination of Stellar Rotational Velocities

Abstract: A method for analysing line profiles by means of a transform using Bessel functions is described. This yields the stellar rotational velocityv sini, to an accuracy of about ±1 km s−1 for rotational velocities greater than about 5 km s−1, provided that rotation is the major source of line broadening. The theory of the method is a special case of a general theory of linear transforms in data analysis, which is outlined in an appendix.

Strong differentiability properties of Bessel potentials

Abstract: This paper is concerned with the "strong" Lp differentiability properties of Bessel potentials of order a > 0 of Lp functions. Thus, for such a functionf, we investigate the size (in the sense of an appropriate capacity) of the set of points x for which there is a polynomial P1(y) of degree k < a such that lim sup (diam S)-k IISIl f If(Y) Px(y)lPdy/P = 0 diam(S)-.O where, for example, S is allowed to run through the family of all oriented rectangles containing the origin.

Wave propagation from a cylindrical cavity

Abstract: Here we study impact-initiated disturbances propagating from a cylindrical cavity in orthotropic cylindrically anisotropic inhomogeneous elastic materials. Employing a transform technique we ascertain conditions on the various parameters of the medium which, when satisfied, enable us to express the solutions for impact problems in terms of modified Bessel functions. From these we generate asymptotic wavefront expansions and exact closed-form solutions. A formal technique for directly generating such expansions is also employed to solve a variety of problems. Comparisons of results obtained by the two methods are included and an appendix contains a rigorous verification of the leading term in the formal expansion.

Asymptotic expansions of Hankel transforms of functions with logarithmic singularities

Abstract: Asymptotic expansions as λ → +∞ are obtained for the Hankel transform Ω V (λ)= ∫ 0 ∞ J V (λt)f(t)dt whereJv(t) is the Bessel function of the first kind and v is a fixed complex number. The function \tf(t) is allowed to have an asymptotic expansion near the origin of the form f(t)∼ ∑ n=0 ∞ C n t α n (–lnt) β n Here, Re αn ↑ +∞ and βn is an arbitrary complex number.

Integral Representations and Inequalities for Bessel Functions

Abstract: In a previous note the author showed how various inequalities, typified by Grunbaum’s inequality $1 + J_0 (a) \geqq J_0 (b) + J_0 (c)(a^2 = b^2 + c^2 )$, could easily be obtained from a certain integral representation for the Bessel functions. In the present note a method is described which provides more general representations of this kind, and examples of the resulting inequalities are given.

High-intensity single-mode laser theory

Abstract: A high-intensity single-mode gas laser is analyzed by several mathematical methods. Exact numerical results are best obtained by solving for the coefficients of a Fourier series solution by using a backward recurrence scheme or a successive convergents method for evaluating a continued fraction. Numerical integration techniques can also be used to convert the periodic boundary conditions to initial conditions and then solve the resulting initial-value problem. Direct analytic integration leads to exact solutions in terms of Bessel functions for the special case of zero detuning, equal decay rates, and neglecting of collision broadening. For general single-mode operation, an improved analytic approximation [the second recursion relation approximation (2RCRA)] is obtained by taking an additional term in the recursion relations for the Fourier coefficients. This approximation predicts the existence of one of the secondary resonances in graphs of the spatially averaged population inversion and quadrature polarization coefficient densities and matches the exact solution from the position of the first resonance and outward independent of the intensity. The 2RCRA also matches the exact detuning curve except at small detunings. The first-order ("rate equation") approximation, by comparison, does not predict any secondary resonances and does not match the exact solution at higher intensities. A physical interpretation for the appearance of the secondary resonances is presented in terms of multiphoton interactions and a prediction for the positions of the resonances for the special case discussed above is obtained. As predicted by Greenstein the exact detuning curves show quantitatively that the Lamb dip decreases and disappears as the intensity increases for high excitations.

The asymptotic distribution of torsional eigenfrequencies of a spherical shell. II.

Abstract: In the two previous papers of this series (SATO and LAPWOOD, 1977a, b) we examined approximate methods for calculating eigenfrequencies of radial overtones of torsional oscillations of spherically symmetrical shells. For shells composed of uniform layers we were able to obtain an exact frequency equation, in terms of spherical Bessel functions, for which roots could be computed with any desired precision. They thus supplied a standard for the measurement of the accuracy of approximate methods.In applications to shells of two and three uniform layers, which were simple representations of an Earth with inner surfaces of discontinuity, we noted the presence of the solotone effect, which is the existence of recurring patterns of eigenfrequencies owing to internal reflection.In this paper we take up the analysis of the solotone effect, showing how it may be predicted from knowledge of the Shell structure, and how it may be interpreted in terms of ray theory. Applications to the same Earth-models as used before show that for them the theory of the solotone gives an excellent fit to the precisely computed eigenfrequencies. The pattern of eigenfrequencies proves to be very sensitive to changes in layer thickness, and thus offers the possibility of future use in determining the positions of surfaces of discontinuity within the mantle of the Earth.