Showing papers on "Bessel function published in 1972"

Note on backward recurrence algorithms

Abstract: An algorithm is given for the computation of the recessive solution of a second- order linear difference equation, based upon a combination of algorithms due to J.C.P. Miller and F.W.J. Olver. A special feature is automatic and rigorous control of trunca- tion error. The method is illustrated by application to the well-used example of the Bessel func- tions Jr(x). 1. Introduction and Summary. Let (1) aryr-1 - brYr + CrYr+l = 0 (r = 1, 2, be a given difference equation in which the coefficients a. and cr do not vanish. Suppose that the equation has a pair of solutions fr and gr such that fr/gr -> 0 as r -*> . Then fr is said to be a recessive (or subdominant or distinguished) solution of the difference equation at r = o, and g. is said to be dominant. The recessive solution is unique, apart from a constant factor. The dominant solution is not unique, however, since any constant multiple of fr may be added to gr without affecting the asymptotic form of gr. Computation of fr from (1) by forward recurrence is usually impractical owing to strong instability. On the other hand, backward application of (1) provides a stable way of computing fr (but not g9), since rounding errors grow no faster than the wanted solution, as a rule.* In the next section, we describe briefly two published algorithms which enable fr to be computed without the need for accurate starting values at high values of r. In Section 3, certain difficulties in the implementation of the algorithms are described, and in the next section, it is shown how these difficulties can be overcome by combining the algorithms. In Section 5, the well-used Bessel function example is considered. A computing routine is described in which the truncation error is bounded rigorously, without loss of efficiency. The method is compared with methods of earlier writers. The concluding section, Section 6, gives proofs of certain results used in earlier sections.

Electromagnetic scattering by arbitrarily oriented ice cylinders.

Abstract: The scattering of electromagnetic waves by arbitrarily oriented, infinitely long circular cylinders is solved by following the procedures outlined by van de Hulst. The far-field intensities for two cases of a linearly polarized incident wave are derived. The scattering coefficients involve the Bessel functions of the first kind, the Hankel functions of the second kind, and their first derivatives. Calculations are made for ice cylinders at three wavelengths: 0.7, 3, and 10 microns. The numerical results of intensity coefficients are presented as functions of the observation angle. A significant cross-polarized component for the scattered field, which vanishes only at normal incidence, is obtained. It is also shown that the numerous interference maxima and minima of the intensity coefficients due to single-particle effects depend on the size parameter as well as on the oblique incident angle.

Diffraction of Radio Waves Around the Earth's Surface

Abstract: : ;Contents: Statement of the problem and its solution in series form; Summation formula; Calculation of the Hertz function for the illuminated regions; Asymptotic expressions for the Hankel function; Expression of the Hertz function applicable to the twilight regions(penumbrae); Study of the expression for the Hertz function. Summary of results.

On propagation of long waves in curved ducts

Abstract: A two‐dimensional detailed study of the behavior of long waves in curved ducts and in junctions between straight and curved ducts will be given. The mathematical treatment of the problem utilizes the method of separation of variables. Solutions and expressions for principal mode of the wave are obtained by using the linearized equation of motion solved for its characteristic values. The unavoidable approximations in the numerical solutions of the cylindrical functions are due to use of series expansion of Bessel functions and from restrictions necessary to solve infinite matrices.

Integrals of Products of Bessel Functions

Abstract: Simple expressions for a variety of integrals involving the product of three cylindrical or three spherical Bessel functions are obtained in terms of the angular functions arising in the decomposition of a plane wave in two or three dimensions.

Approximations for the non-central Wishart distribution

Abstract: The non-central Wishart distribution was first derived by Anderson and Girshick (1944) and Anderson (1946) for the linear and planer cases. Herz (1955) expressed the distribution as a Bessel function and James (1961) as a series of zonal polynomials. This non-central distribution plays an important part in deriving a number of test-statistics in multivariate analysis. The distribution as well as the distributions derived from it, are difficult to handle. In this paper it is shown how the non-central distribution can be approximated by a central distribution or by a central distribution together with the generalised Laguerre polynomials.

An Interpolation Series Associated with the Bessel–Hankel Transform

Abstract: [5]; they form a complete orthonormal set in the Hilbert function space known as the Paley-Wiener functions, and wn(m) = 5nm (Kronecker's Symbol) for all integers n and m. This means that the cardinal series is not only an orthogonal expansion for the Paley-Wiener functions, but it also provides a process for interpolation at the integers since the series reduces formally to am when x is an integer m. In the present note we shall consider further sets of this type and in particular a set involving Bessel functions. Cardinal series interpolation has important applications in information theory, where it was introduced by C. E. Shannon [11]. It is, for example, important for the electrical engineer to know that a certain type of transmitted signal, a function of time, lies in a subspace (the Paley-Wiener functions) of L(-oo, oo), and that this subspace possesses an orthogonal basis with respect to which the \" coordinates \" of the signal are actually values taken by the signal at certain instants of time. It was with this application in mind that H. P. Kramer introduced a generalisation of the cardinal series in a lemma which we adopt as the starting point for the present discussion. LEMMA 1 (Kramer [7]). Let (a, b) be a finite interval of U (the real numbers). Let K(x, t) e L(a, b)for each xeU and suppose that the sequence of real numbers {xn} (where n runs over some indexing set of integers) is such that {K(xn, t)} forms a complete orthogonal set (COS) in L(a, b). If

The theory of X-ray crystal diffraction for finite polyhedral crystals. III. The Bragg–(Bragg)m cases

Abstract: The plane-wave and spherical-wave theories are described for the Bragg-(Bragg)m cases. The treatment is similar to that of Parts I and II [Saka, Katagawa & Kato (1972). Acta Cryst. A28, 102-113, 113-120] for the Laue-(Bragg)m cases. In the plane-wave theory of the Bragg case, a few aspects which up to now have not been well understood, are described to clarify the mathematical structures of the wave field. In particular, emphasis is put on a method for specifying the plane-wave solution by using a Riemann sheet instead of the dispersion surface. In the spherical-wave theory, the reflected vacuum wave and the transmitted crystal wave at the entrance surface can be represented by two Bessel functions. The crystal wave of the Bragg-(Bragg)m case reflected at the exit surface is represented by a combination of two Bessel functions. The transmitted vacuum wave, however, is given by a combination of four Bessel functions. It is shown that the solutions are compatible with those of the Laue-(Bragg)m cases. The solution for finite polyhedral crystals can be constructed by superposing the solutions for individual cases such as of Laue, Laue-(Bragg)m [Kato (1968). J. Appl. Phys. 39, 2225-2230; Parts I and II] and Bragg-(Bragg)m obtained in the present paper. A comparison is made with Uragami's results obtained by another approach [J. Phys. Soc. Japan (1969), 27, 147-154; (1970), 28, 1508-1527].

An Extension of the Bessel‐Fubini Series for a Multiple‐Frequency CW Acoustic Source of Finite Amplitude

Abstract: A fundamental operational solution of the lossless Burgers' equation is used to derive the spectrum of a two‐frequency CW source of finite amplitude as a function of range. The resulting expression is then generalized to the case of a multiple N‐frequency CW source of finite amplitude. The region of validity of these solutions is determined by specifying the critical ranges at which the wavefronts become discontinuous. Finally, a modification of these omnidirectional solutions which makes them applicable to directive sources is briefly discussed.

A method for computing Bessel function integrals

Abstract: Infinite integrals involving Bessel functions are recast, by means of an Abel transform, in terms of Fourier integrals. As there are many efficient numerical methods for computing Fourier integrals, this leads to a convenient way of approximating Bessel func- tion integrals.

Expansions for the Density of the Absolute Value of a Strictly Stable Vector

Abstract: : Let q be the density function of the absolute value of a strictly stable random vector in R sup N, N-dimensional Euclidean space. Asymptotic expressions for q(r) for large r and for small r are found. The proofs use the Fourier inversion formula and contour integration. Bessel functions play a role occupied by the exponential and trigonometric functions when N = 1. (Author)

Computation of bessel functions in light scattering studies.

Abstract: Computations of light scattering require finding Bessel functions of a series of orders. These are found most easily by recurrence, but excessive rounding errors may accumulate. Satisfactory procedures for cylinder and sphere functions are described. If argument z is real, find Yn(z) by recurrence to high orders. From two high orders of Yn(z) estimate Jn(z). Use backward recurrence to maximum Jn(z). Correct by forward recurrence to maximum. If z is complex, estimate high orders of Jn(z) without Yn(z) and use backward recurrence.

The asymptotic expansions of Hankel transforms and related integrals

Abstract: In this paper, the asymptotic expansion of integrals of the form f F(kr)f(k) 4kis considered, as r tends to infinity, and where F(kr) are Bessel functions of the first and secQnd kind, or functions closely related to these. Asymptotic expansions for several functions of this type are presented under suitable restrictions on f(k). The expansion given by Willis for Hankel transforms is seen to be valid under conditions of f(k) less restrictive thtn those imposed by that author.

Cross-spectral functions based on von Karman's spectral equation

Abstract: Cross-spectral functions for the vertical and longitudinal components of turbulence of a two-dimensional gust field are derived from the point correlation function for turbulence due to von Karman. Closed form solutions in terms of Bessel functions of order 5/6 and 11/6 are found. An asymptotic expression for large values of the frequency argument, and series results for small values of frequency, are also given. These results now form the base for studying the effect of spanwise variations in turbulence for a turbulence environment which is characterized by the von Karman isotropic spectral relations. Previous studies were based mainly on the Dryden-type spectral representation.

Shoaling of finite amplitude long waves on a beach of constant slope

Abstract: A solution of finite amplitude long waves on constant sloping beaches is obtained by solving the equations of the shallow water theory of the lowest order. Non-linearity of this theory is taken into account, using the perturbation method. Bessel functions involved in the solution are approximated with trigonometric functions. The applicable range of this theory is determined from the two limit conditions caused by the hydrostatic pressure assumption and the trigonometric function approximation of Bessel functions. The shoaling of this finite amplitude long waves on constant sloping beaches is discussed. Especially, the effects of the beach slope on the wave height change and the asymmetric wave profile near the breaking point are examined, which can not be explained by the concept of constancy of wave energy flux based on the theory of progressive waves in uniform depth. These theoretical results are presented graphically, and compared with curves of wave shoaling based on finite amplitude wave theories. On the other hand, the experiments are conducted with respect to the transformation of waves progressing on beaches of three kinds of slopes ( 1/30, 1/2.0 and 1/10 ) . The experimental results are compared with the theoretical curves to confirm the validity of the theory.

Clebsch-Gordan Coefficients and Special Function Identities. I. The Harmonic Oscillator Group

Abstract: It is shown that by constructing explicit realizations of the Clebsch‐Gordan decomposition for tensor products of irreducible representations of a group G, one can derive a wide variety of special function identities with physical interest. In this paper, the representation theory of the harmonic oscillator group is used to give elegant derivations of identities involving Hermite, Laguerre, Bessel, and hypergeometric functions.

Continuity of Bessel potentials

Abstract: It is shown that certain capacities associated with potentials of functions in Lebesgue classes are non-increasing under orthogonal projection of sets. This inequality is then used to discuss continuity of traces of potentials on subspaces of possibly low dimension. The case of principal interest is the Bessel potential.

Harmonics on Stiefel manifolds and generalized Hankel transforms

Abstract: In this note we announce an extension of these theorems to the setting of Stiefel manifolds and matrix space. Our work makes it possible to construct holomorphic discrete series representations for the real symplectic group by decomposing a tensor product of certain projective representations introduced earlier by Shale and Weil. (See Weil [11] and also Shalika [10].) Proofs of the results announced here and their application to the construction of discrete series will appear elsewhere. We let Mnm denote then x m real matrix space, S w,m the Stiefel manifold of matrices VeMnm such that VV = Im9 and Pm the cone of m x m positive-definite symmetric matrices. The rotation group SO(n) acts on S\"' and Mnttn by left matrix multiplication so that S\"' m s SO(n)/SO(n m). Corresponding to the decomposition Mnm = S w,m x Pm we have the integral formula

Effect of Spanwise Variation of Gust Velocity on Airplane Response to Turbulence

Abstract: When predicting the vertical response of a flexible aircraft to atmospheric turbulence, the designer may use either the assumption of flight through cylindrical turbulent waves, or through an isotropic field of random vertical flow, which is no longer uniform in span. In this first part, the paper is devoted to the demonstation of a very simple formula which makes it possible to determine the range of validity of both methods. It appears that the assumption of turbulence uniform in span gives a good approximation in the range of frequency associated with rigid-airplane motion. It is no longer valid, in general, at flexible modes natural frequencies. The method proposed by ONERA to compute transfer functions to isotropic turbulence, using lifting surface theory, is briefly described. However, this second part of the paper is mainly concerned with discussions of the power spectral densities of the response at different locations of the structure of the Concorde SST. These evaluations have been obtained both for isotropic turbulence and for turbulence uniform in span. The first method gives loads about 14 % lower. In the third part, comparison is extended to fatigue damage, and consequence on fatigue life is emphasized. Nomenclature b^ = spanwise dimension of the aircraft F(t) = inverse Fourier Transform of F(a>) g(r) = transverse correlation function of turbulence G(a>) = turbulence spectral model j\(x) = bessel function of first kind and order A K(M,M,'a>) = kernel of the integral equation of the lifting surface theory K»(x) = modified Bessel function of second kind and order v I = transverse coherence length of turbulence L = scale of turbulence N0 = mean number of zero crossings of turbulence per second Nos = mean number of zero crossings of output per second Nr(s) — number of cycles to failure under stress s p(M,t) = pressure field on the wing qk(t) = generalized coordinate associated with mode k r = distance between two points Rw(M,M',r) = cross correlation of the vertical component of

A basic theorem in the computation of ellipsoidal error bounds

Abstract: In thispaper we formulate and prove the basic principles of a procedure for computing a bound on the error in the numerical solution of a system of linear differential equations. The bound on the error at each integration step is expressed in terms of an ellipsoid whose size and orientation is determined by the computations. To illustrate the procedure, Bessel's equation (of order zero) is integrated over the interval 2?x?3 at steps of length 0.1 and bounds on the error are given for each step.

Complex zeros of the modified Bessel function _

Abstract: The complex zeros of Kn(Z) are computed for integer orders n = 2(1)10, to 9D figures, using an iterative interpolation scheme. 1. Introduction. The investigation of wave propagation and scattering in elastic media is often performed by means of integral transform methods. The analysis of such problems in cylindrical coordinates often leads to waves whose transformed potential functions are expressed in terms of modified Bessel functions. In particular, the potentials for outgoing radiating waves which decay with increasing distance from the source are expressed in terms of modified Bessel functions of the second kind, Kn(Z). In the course of a recent study using the Laplace transform, it was necessary to determine complex zeros of Kn(Z) in order to locate poles of the solution required for the inversion. It is believed that the tabulated zeros given below will permit the evaluation of several significant scattering problems. Several methods for the evaluation of zeros of Bessel functions, notably by means of asymptotic expansions, have been given by Olver (1) and Luke (2). The methods developed by Olver, however, are not entirely applicable in the present case, since the convergence only improves with large orders of n. On the other hand, the rational approximations given by Luke have been proved, under appropriate restrictions of the parameters, to converge in the first quadrant; convergence for

On a Nonlinear Bessel Equation

Abstract: In the theory of type II superconductivity A. A. Abrikosov discovered in 1957 that the so-called Abrikosov's mixed state can be described as a special solution of Ginzburg-Landau equation the basic equation of the theory of superconductivity (C2H). Suppose that there exists a cylindrical superconductor of type II at temperature below its critical value Tc and there exists external magnetic field parallel to its axis of cylinder the strength of which is lower than upper critical field HcZ. Abrikosov's mixed state is the phenomenon that the magnetic flux penetrates the superconductor forming triangular lattices of flux lines and the fluxoid of each flux line is quantized. To describe one flux line Abrikosov derived from Ginzburg-Landau equation the singular boundary value problem for a nonlinear Bessel equation.