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A linear combination of modified Bessel functions is defined, discussed briefly, and tabulated. This combination was found to recur in the analysis of various heat transfer problems and in the analysis of the thermal behavior of living tissue when modeled by cylindrical shells.
Calculated roots of cross product of bessel functions over an extended range of index
We consider an oscillator model with nonpolynomial interaction. The model admits exact solutions for two situations: for energy eigenvalues in terms of zeros of Bessel functions, that were considered as functions of the continuous index; and for the corresponding eigenstates in terms of Lommel polynomials.
This paper investigates a problem on the steady-state, conduction-convection heat transfer process in cylindrical porous heat exchangers. The governing partial differential equations for the system are obtained using the energy conservation law. Solution of these equations and the concept of enthalpy lead to a new approach to prove a theorem that the sum of inverse squares of all the positive roots of the zero order Bessel function of the first kind equals to one-forth. As a corollary, it is shown that the sum of one over pth power (p greater than or equal to 2) of the roots converges to some constant.
Application of theory to axial compressor noise
Interpolation between values given in table of zeros is permitted provided curve is traced between at least three values from table. Zeros have been obtained on digital computer and results were rounded off to the fourth decimal point.
The Expressions Module is a software module that has been incorporated into the Satellite Orbit Analysis Program (SOAP). The module includes an expressions- parser submodule built on top of an analytical system, enabling the user to define logical and numerical variables and constants. The variables can capture output from SOAP orbital-prediction and geometric-engine computations. The module can combine variables and constants with built-in logical operators (such as Boolean AND, OR, and NOT), relational operators (such as >, <, or =), and mathematical operators (such as addition, subtraction, multiplication, division, modulus, exponentiation, differentiation, and integration). Parentheses can be used to specify precedence of operations. The module contains a library of mathematical functions and operations, including logarithms, trigonometric functions, Bessel functions, minimum/ maximum operations, and floating- point-to-integer conversions. The module supports combinations of time, distance, and angular units and has a dimensional- analysis component that checks for correct usage of units. A parser based on the Flex language and the Bison program looks for and indicates errors in syntax. SOAP expressions can be built using other expressions as arguments, thus enabling the user to build analytical trees. A graphical user interface facilitates use.
Error in induced electric field modes under Stix coil resulting from assumption of orthogonality of natural plasma mode - Bessel function
Attempts to systematically measure and fit a large sample of flare spectra from IMP 7 and 8 satellite data to determine the spectral forms that characterize the data and the range of spectral slopes are reported. Proton spectra are constructed covering the range 0.9 to over 400 MeV, and alpha fluxes from 1.4-80 MeV/nucleon. The event spectra are composed of fluxes measured at different times at each energy, and fitting procedures were employed such as a power law in kinetic energy, an exponential in rigidity, an exponential in energy, and a Bessel function of momentum/nucleon. Best fits were obtained with the exponential in rigidity and the Bessel function, with the Bessel function having a better numerical accuracy for all 21 proton spectra considered.
A particle size spectrometer having a fixed field of view within the forward light scattering cone at an angle .theta..sub.s between approximately 100 and 200 minutes of arc (preferably at 150 minutes), a spectral range extending approximately from 0.2 to 4.0 inverse micrometers (.mu.m.sup.-1), and a spectral resolution between about 0.1 and 0.2 .mu.m.sup.-1 (preferably toward the lower end of this range of spectral resolution), is employed to determine the distribution of particle sizes, independently of the chemical composition of the particles, from measurements of incident light, I.sub.o, at each frequency, .sigma. (=1/.lambda.), and scattered light, I(.sigma.), according to the equation: ##EQU1## where l=2.pi.sin.theta., .theta. being the fixed viewing angle .theta..sub.s at which scattered light is measured, r is particle size, .sigma. is the reciprocal of wavelength, J.sub.1 is a Bessel function of first kind and order unity, Y.sub.1 is a Bessel function of second kind and order unity. The quantity, I.sub..sigma., is the ratio of scattered light to incident light at each frequency interval. The apparatus is a passive remote sensor that can be used in laboratories, field stations, flying aircrafts and airships, and on board an orbiting satellite.
Spherical wave functions, Legendre and Bessel functions, and addition theorems
Expansions of integrals of bessel functions of large order
A dynamic analysis is presented for a spin-stabilized spacecraft employing four radial wire antennas with tip masses. The satellite is modelled as a 14-degree-of-freedom system, and the linearized equations of motion are found. The lowest-order vibrational modes and natural frequencies of the gyroscopically coupled system are then determined. Because the satellite spin rate is decreased by antenna deployment, a spin-up maneuver is needed. The response of the time-varying mode equations during spin-up is found for the planar modes in terms of Bessel functions and a Struve function of order -0.25. The particular solution is expressed in various forms, including an infinite series of Bessel functions and a particularly useful asymptotic expansion. An error formula for the latter is derived, showing that it gives good accuracy. A simple approximation to the complementary function is obtained using the Wenzel-Kramers-Brillouin method, and the phase error in the approximation is shown to be small.
Three methods for performing Hankel transforms with optical or digital processors are described. The first method is applicable when the input data is available in Cartesian (x-y) format and uses the close connection between generalized Hankel transform and the two dimensional Fourier transform in Cartesian coordinates. The second method is useful when the input data is in polar (r - theta) format and uses change of variables to perform the nth order Hankel transform as a correlation integral. The third method utilizes the von Neumann addition theorem for Bessel functions to extract the Hankel coefficients from a correlation between the radial part of the input and a Bessel function. Initial experimental results obtained for optical implementation of the first two methods are presented.
Series expansion in powers of bessel function
A dynamic analysis is presented for a spin stabilized spacecraft employing four radial wire antennas with tip masses, a configuration first employed in the IMP-J spacecraft. The use of wires in place of the usual booms represents the ultimate in weight reduction at the expanse of flexibility. The satellite is modelled as a 14 degree of freedom system, and the linearized equations of motion are found. The lowest order vibrational modes and natural frequencies of the gyroscopically coupled system are then determined. Because the satellite spin rate is decreased by antenna deployment, a spin-up maneuver is needed. The response of the time varying mode equations during spin-up is found, for the planar modes, in terms of Bessel functions and a Struve function of order -1/4. Because tables of the latter are not readily available, the particular solution is expressed in various forms including an infinite series of Bessel functions and a particularly useful asymptotic expansion.
We investigate, in detail, the production of solar flare gamma ray emission above 100 MeV via the interaction of high energy protons with the ambient solar atmosphere. We restrict our considerations to the broadband gamma ray spectrum resulting from the decay of neutral pions produced in p-H reactions. Thick-target calculations are performed to determine the photon fluences. However, proton transport is not considered. Inferences about the form of the proton spectrum at 10-100 MeV have already been drawn from de-excitation gamma ray lines. Our aim is to constrain the proton spectrum at higher energies. Thus, the injected proton spectrum is assumed to have the form of a Bessel Function, characteristics of stochastic energy at higher energies. The detailed shape of the gamma ray spectra around 100 MeV is found to have a strong dependence on the spectral index of the power law and on the turnover energy (from Bessel function to power law). As would be expected, the harder the photon spectrum the wider the 100 MeV feature. The photon spectra are to be compared with observations and used to place limits upon the number of particles accelerated and to constrain acceleration models.
Closed form asymptotic expressions for computing high frequency noise generated by an annular cascade in an infinite duct containing a uniform flow are presented. There are two new elements in this work. First, the annular duct mode representation does not rely on the often-used Bessel function expansion resulting in simpler expressions for both the radial eigenvalues and eigenfunctions of the duct. In particular, the new representation provides an explicit approximate formula for the radial eigenvalues obviating the need for solutions of the transcendental annular duct eigenvalue equation. Also, the radial eigenfunctions are represented in terms of exponentials eliminating the numerical problems associated with generating the Bessel functions on a computer. The second new element is the construction of an unsteady response model for an annular cascade. The new construction satisfies the boundary conditions on both the cascade and duct walls simultaneously adding a new level of realism to the noise calculations. Preliminary results which demonstrate the effectiveness of the new elements are presented. A discussion of the utility of the asymptotic formulas for calculating cascade discrete tone as well as broadband noise is also included.