Showing papers on "Bessel function published in 1971"

Wave forces on a circular dock

Abstract: The scattering of surface gravity waves by a circular dock is considered in order to determine the horizontal and vertical forces and torque on the dock. An incident plane wave is expanded in Bessel functions, and for each mode the problem is formulated in terms of the potential on the cylindrical surface containing the dock and extending to the bottom. The solution is shown to have phase independent of depth and so may be obtained from an infinite set of real equations, which are solved numerically by Galerkin's method. The convergence of the solution is discussed, and some numerical results are presented.This problem has been investigated previously by Miles & Gilbert (1968) by a different method, but their work contained errors.

Axisymmetric Vibrations of Isotropic Composite Circular Plates

Abstract: A Kirchhoff‐type theory is established for axisymmetric motions of heterogeneous isotropic circular plates. It is shown that a coupled extensional‐flexural inertia term exists, in addition to the classical extensional and rotatory inertia terms. An analogy is found between the composite plate problem and the vibrations of homogeneous shallow spherical shells. The obtained sixth‐order system of equations is solved in closed form in terms of Bessel functions, with an argument determined from a characteristic cubic equation. A transcendental frequency equation is then derived for a circular composite plate with clamped edge conditions. Numerous examples are studied, showing the significant effect of plate heterogeneity on its vibrational response. Possibility of composite systems to transcend the frequencies of the individual constituents is clearly indicated by the theoretical results and checked experimentally.

High-Frequency Backscattering From a Finite Cone

Abstract: The high-frequency backscattered field produced by a plane electromagnetic wave at oblique incidence on a perfectly conducting, right circular cone with a flat base is considered. The first two terms of the asymptotic expansion are obtained by applying the geometrical theory of diffraction; these terms reduce to results derived earlier in the particular case of a circular disk. Owing to the axial caustic, functions must be introduced to match the wide-angle formulas to the known results for nose-on incidence. This matching is effected by employing Bessel functions and Fresnel integrals for the first- and second-order terms, respectively. The resulting expressions are valid for all cone angles and for a wide range of aspect angles about nose-on; they are also found to be in good agreement with experimental data.

Impulse Reflection with Arbitrary Angle of Incidence and Polarization From Isotropic Plasma Slabs

Abstract: The nature of the reflected signal is considered for an electromagnetic impulse impinging on a semi-infinite plasma and a finite plasma slab terminated by a perfectly conducting surface at arbitrary angles of incidence and for both polarizations. The inverse Fourier transform technique is used and leads to a statement for the reflected signal in terms of Bessel functions of the first kind. The results are shown graphically for both the TE and TM case for all angles of incidence between normal and π/2.

Propagation of harmonic waves in composite circular cylindrical shells. II - Numerical analysis

Abstract: The frequency equation for harmonic waves with an arbitrary number of circumferential nodes traveling in composite traction-free circular cylindrical shells established in the first part of this investigation has been programmed for numerical evaluation on an IBM 7094 digital computer. The numerical results obtained are employed in evaluating the effect of the changes of the shell parameters on the frequency and shape of the first few modes of wave propagation. Moreover, the asymptotic limits of the phase velocities for waves having short axial wavelengths are established analytically and verified by the numerical results.

Minima in Generalized Oscillator Strengths of Hydrogen-Like Atoms

Abstract: Algebraic expressions are obtained for the generalized oscillator strengths of hydrogen-like atoms for the transitions involving s-, p-, and d-states with the principal quantum number n of 2, 3, and 4 Zero minima are found for all the transitions involving s-states, and their positions are shown to be proportional to the effective charge, the coefficients being tabulated The Transitions from the 1s-state sometimes show zero minima if different effective charges are used for the initial and the final states Even nodeless wave functions can produce zeros owing to the nodes of the spherical Bessel function Some excitation processes of Ne, Na, Mg, and Ar caused by high-energy charged particles are investigated in the first Born hydrogen-like approximation, and the possibility of finding zero minima in the differential cross sections is discussed; the 3s→4p transitions of Na and Mg have the most conspicuous zeros of all that are studied

High-frequency backscattering from a metallic disc

Abstract: The paper treats the backscattered field produced by a plane electromagnetic wave at oblique incidence on a perfectly conducting circular disc. The geometric theory of diffraction is used to obtain the first two terms of the asymptotic expansion at high frequencies. The first term is attributable to direct scatter from the edge of the disc and the second term arises from rays that cross the disc once. It has been found that the ray geometry is fixed for one polarisation but is aspect-dependent for the other. Owing to the axial caustic, functions must be introduced to match the wide-angle formulas to the known results for axial incidence. Bessel functions are used for the 1st-order terms and Fresnel integrals for the 2nd-order terms and evidence is presented to support this choice of function. The resulting 1st-order expression is in reasonable agreement with experimental data, and may be adequate for many purposes. However, the expressions incorporating the 2nd-order terms reveal the polarisation dependence of the scattering, and are in excellent agreement with measured data over a widt range of aspect angles.

Integral‐Transform‐Generated Basis Sets

Abstract: Two alternative versions of a procedure for generating new basis sets via the integral‐transform (IT) method are outlined. The first version (fixed shape function, set of primitive functions) is tested on 3‐electron (Z=2, 3, 4, 5) and 4‐electron (Z=3, 4, 5, 6) atomic systems (ground state) and the 21P excited state of He. The set of primitive functions are either hydrogenic or Slater orbitals and the shape function is the square pulse that generates the Hulthen transforms. In these calculations, a conjecture about the dependence of the parameters of higher principal quantum number (n) IT orbitals on the parameters of lower n IT orbitals is also tested and substantiated. The second version (single primitive function, set of shape functions) is tested on the 21P state of He. The primitive function is a 1s Gaussian function, the set of shape functions is related to the set of generalized Laguerre polynomials Lnv‐1(x). The first member of this set gives kv(qr), the reduced modified Bessel function of the seco...

New Tool for Scattering Studies

Abstract: In potential scattering, from the transformation kernel, a function K(ρ) is constructed whose Bessel transforms are the Jost functions. K(ρ) contains the whole information on the potential V(ρ). There is a one‐to‐one correspondence between the two functions, and they are related, in both senses of this correspondence, through integral equations. Since the relation between K(ρ) and the phase shift is very direct, it is a useful tool for all analyses of the relations between the information contained in the dynamics of the problem (viz., the potential) and the measurable information (viz., the phase shifts). This tool will be applied to the inverse problem in forthcoming publications. Besides, the derivation of K(ρ) makes clear that a similar study can be done in all cases to which the Gel'fand‐Levitan scheme applies and therefore in most scattering problems in physics.

Generation of Bessel functions of complex order and argument

Abstract: Recurrence techniques are described for the generation of Bessel functions of the first and second kinds where both the argument and order may be complex. The method is shown to be accurate for several well known forms of functions, including Kelvin and spherical Bessel functions. The accuracy of the general case of complex order and argument is determined by computing the Wronskians and by verifying some addition theorems of the Bessel functions over wide ranges of order and argument. Procedures for the accurate generation of complex-argument gamma functions are also described.

Modification of the Bessel‐Fubini Solution That Includes Attenuation

Abstract: Burgers' equation is a nonlinear acoustical wave equation which approximately governs the propagation of finite‐amplitude waves with dissipation and spreading losses. A new solution is derived for a spherically diverging sinusoidal waveform. By means of this solution, the amplitudes of the harmonic levels above the third can be more readily estimated than by successive approximation.

Generalized Lipschitz conditions and Riesz derivatives on the space of Bessel potentials

Abstract: In this paper, the first of a series, the space of n-dimensional Bessel potentials Lρ α, 0 < α ≦2, is considered with the aim of describing smoothness properties of its elements. This is achieved by forming norms involving the existence of derivatives or the order of Lipschitz conditions of f or its Riesz transform, and by showing these to be equivalent to the Lα ρ- The method of proof, inspired by Sunouchi and Shapiro, consists in interpreting the characterization itself as a saturation problem with Favard class Lα ρ; thus, the characterizations have only to satisfy the conditions of a general saturation theorem, established in Lρ,1≦ρ≦∞ To obtain more specific results in case 1 < ρ < ∞ the Marcinkiewicz–Mikhlin multiplier theorem is applied. Our general results contain particular ones due to Berens–Nessel, Butzer, Butzer–Trebels, Calderon, Cooper, Gorlich, Nessel–Trebels, and Trebels.

A New Approximation Method for Wave Theories

Abstract: The scalar field due to a bounded source and obeying the wave equation is analyzed. As a result of this, a set of rules is derived for solving a class of ``wave theories,'' including electrodynamics and general relativity, by expanding the field in a power series of c−1 in null‐spherical coordinates. The method is applied for the Maxwell equations to give all the well‐known results without the use of Fourier analysis and Bessel functions.

The near field of a locally illuminated diffracting edge

Abstract: It is shown that when both the source and field point are sufficiently close to a diffracting edge to require a closer approximation to the field than is provided by the usual Fresnel integrals, the rigorous solution can be put in terms of the incomplete Hankel function, whose asymptotic properties are known and which is well tabulated. The solution remains valid without change of form in the vicinity of the shadow boundary, and it reduces, respectively, to the known Hankel or Fresnel formulation at the shadow boundary or when either the source or field point is distant from the edge.

Ferrite-Loaded Helical Line with Coaxial Inner Conductor (Correspondence)

Abstract: A transcendental characteristic equation of a thin-wire cylindrical helix with coaxial inner conductor uniformly filled with circumferentially magnetized gyromagnetic medium is derived in terms of modified Bessel functions and confluent hypergeometric functions. Two special cases of the structure are considered.

Bessel function expansions of incomplete Lipschitz-Hankel integrals

Abstract: TWO methods are described for constructing Neumann series for incomplete Lipschitz-Hankel integrals of Bessel functions of the first kind.

Root and delay parameters for normalized Bessel and Butterworth low-pass transfer functions

Abstract: Tables are given for the roots of the normalized low-pass Bessel and Butterworth transfer functions in both Cartesian and polar form. The latter form is especially useful when designing transitional (Butterworth-Thomson) functions. The tables cover the orders 1 through 30 inclusive. In addition, the group delay characteristics for each of the above orders are tabulated for both the Bessel and Butterworth functions.

Diffraction by a conducting wedge in the near field

Abstract: The purpose of this paper is to derive a near-field asymptotic solution for the diffraction of waves by a perfectly conducting wedge. Thus, starting from the series solution for cylindrical-wave incidence and using an integral expression for products of the Bessel functions involved, the total field is represented as a geometrical-optics term plus a diffraction integral. Using an integral form for the field of a spherical wave, the solution for the cylindrical-wave excitation is extended to the spherical-wave case. The method of steepest descent is used to obtain an asymptotic expression for the diffraction integral in terms of Fresnel integrals. The accuracy of our expressions is established by comparison with previous results as well as the exact solutions.

Two-dimensional analysis of Bessel RC lines

Abstract: A two-dimensional analysis of Bessel RC lines is numerically carried out by using a successive over-relaxation method, and the results are compared with characteristics conventionally obtained by one-dimensional analysis.

Wavefunction expansion in terms of spherical bessel functions.

Abstract: A formalism is derived for expanding the solutions of the Schrodinger equation in terms of spherical Bessel functions. The regular and the irregular solutions are treated. A relation between the expansion coefficients and the phase shifts is derived. As an application, the expansion coefficients of both the irregular and regular Coulomb wavefunctions are given in a form of a simple recurrence relation. The expansions have been checked numerically and found to be very suitable for calculating the regular Coulomb wavefunction in a very large region of the coordinate and the Coulomb parameter.

Extraction of accurate scattering information from apparently divergent Fredholm series: the centrifugal barrier

Abstract: By treating the centrifugal barrier l(l+1)/2r2 as part of the physical potential it is shown that partial wave scattering information may be calculated, using the Fredholm method, for arbitrary values of l without using the usual distorted basis of spherical Bessel functions. This is true in spite of the fact that the Fredholm series formally diverges for a potential for which limr to 0 r2V(r) not=0. A general method is outlined which is capable of handling both single and multichannel scattering problems. Examples of its use are given.