The Asymptotic Nature of Zeros of Cross-Product Bessel Functions

Cochran, James Alan

doi:10.1093/qjmam/19.4.511

Cited by 11 publications

(14 citation statements)

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“…The reason why the roots in Table II for are not shown is because they are the same as those ones of Table I for as it has been pointed out before. It can be seen that the values of the normalized roots tend to multiples of , as expected when since this limit corresponds to the solution of a parallel-plate waveguide [18]. The results certify that the interlacing properties of the roots of both functions hold for all of them.…”

Section: Resultssupporting

confidence: 70%

“…The calculations of the roots have been carried out on a common desktop computer (Intel 4-cores 2.4 GHz), the algorithm showing a speed of around 81 000 and around 16 000 roots per second for and , respectively. The previously developed algorithms that analyzed the calculation of the roots of (1) were based either on the approximation by a series [14], [15], sometimes only valid for asymptotic values of or [16], [18], or on the solution of an eigenvalue problem in a fast and efficient way as most recent works have done [3], [4], [30], [31]. These last studies showed very efficient calculation of the eigenvalues of more general structures, though the accuracy presented for coaxial cables was lower than achieved with the algorithm presented in this paper.…”

Section: Resultsmentioning

confidence: 99%

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Algorithm to Calculate a Large Number of Roots of the Cross-Product of Bessel Functions

Sorolla

Mosig

Mattes

2013

IEEE Trans. Antennas Propagat.

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Abstract-This paper describes an algorithm to calculate a large number of roots of the cross-product of Bessel functions and of their first derivatives. The algorithm initially finds the roots of the zeroth order using an auxiliary function that exhibits the same roots as the original cross-products but with better behavior for numerical root search with the Newton-Raphson algorithm. In order to find the roots for higher orders, the algorithm follows a pyramidal scheme using the interlacing property of the cross-product of Bessel functions. The algorithm shows globally convergent behavior for a large range of values of the argument and of the order of the Bessel functions. The roots can be computed to any precision, limited only by the computer implementation, and the convergence is attained in six iterations per root in average, showing a much better performance than previous works for the calculation of these roots.

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Section: Resultssupporting

confidence: 70%

Section: Resultsmentioning

confidence: 99%

Algorithm to Calculate a Large Number of Roots of the Cross-Product of Bessel Functions

Sorolla

Mosig

Mattes

2013

IEEE Trans. Antennas Propagat.

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“…Using the uniform expansions of Olver [1954Olver [ , 1962, the following asymptotic expression has been obtained [Cochran, 1966]: (8)…”

Section: "High" Frequencymentioning

confidence: 99%

“…In his original paper Buchholz (1939], with the help of several auxiliary integral representations, carefully deduced approximations to the angular propagation constants for this situation (a minor error observed by Rice [1948] is corrected in a subsequent paper [Buchholz, 1949D. These same approximations have been rederived by Rice [1948] and Lewin [1955], for the dominant modes, and by Marshak [1943], de Hoop [1953, Kislyuk [1961], and Cochran [1966] (and probably others) in the general case. The form of the asymptotic expression is The gradual bend approximation has also been the case of interest for the analogous problem in radio "waveguide" studies.…”

Section: Gradual Bendsmentioning

confidence: 99%

Mode Propagation in Continuously Curved Waveguides

Cochran¹,

Pecina²

1966

Radio Science

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This paper presents the results of precise numerical computations and extensive analytical investigations of the angular propagation constants for the various electromagnetic modes which may exist in waveguide bends of rectangular cross section. Although the values of the several independent parameters have been chosen to be commensurate with microwave applications, the results are displayed and discussed in a manner which should be convenient for both the analysis and the design of such components in analogous problems in other and diverse practical situations. Moreover, special consideration is given to the relationship of this work to various efforts in the study of radio wave propagation. Two important approximations, one for “high” frequencies and the other for gradual bends, are also considered as a part of the present inquiry. These approximations, especially the latter, turn out to be remarkably good, and rather accurate values of the propagation constants can be obtained with their use for wide ranges of parameter values. The following are typical of the conclusions which can be drawn from this research: (1) cutoff frequencies for modes existing in continuously curved waveguides are curvature dependent; (2) there exist waveguide bends such that circularly polarized waves are undistorted by transmission along the curved section; (3) the angular propagation constants are generally monotonically increasing functions of the operating frequency f (for fixed mean radius) and of the mean radius R (for fixed frequency). Moreover, they vary linearly with f for frequencies far above cutoff and with R for small curvatures.

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“…Cross product zeros were first analyzed by McMahon ( 1894 ) who derived an expression for large zeros. In addition, Cochran examined the asymptotic nature and analyticity of cross product Bessel functions (Cochran 1964 , 1966a , b ). Recurrence relations for the Bessel function cross products are given by Goodwin ( 1949 ).…”

Section: Introductionmentioning

confidence: 99%

Orthogonality, Lommel integrals and cross product zeros of linear combinations of Bessel functions

et al. 2015

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The cylindrical Bessel differential equation and the spherical Bessel differential equation in the interval with Neumann boundary conditions are considered. The eigenfunctions are linear combinations of the Bessel function or linear combinations of the spherical Bessel functions . The orthogonality relations with analytical expressions for the normalization constant are given. Explicit expressions for the Lommel integrals in terms of Lommel functions are derived. The cross product zeros and are considered in the complex plane for real as well as complex values of the index and approximations for the exceptional zero are obtained. A numerical scheme based on the discretization of the two-dimensional and three-dimensional Laplace operator with Neumann boundary conditions is presented. Explicit representations of the radial part of the Laplace operator in form of a tridiagonal matrix allow the simple computation of the cross product zeros.

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The Asymptotic Nature of Zeros of Cross-Product Bessel Functions

Cited by 11 publications

References 0 publications

Algorithm to Calculate a Large Number of Roots of the Cross-Product of Bessel Functions

Algorithm to Calculate a Large Number of Roots of the Cross-Product of Bessel Functions

Mode Propagation in Continuously Curved Waveguides

Orthogonality, Lommel integrals and cross product zeros of linear combinations of Bessel functions

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