 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
Archive of Issues
| Total articles in the database: | | 12854 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
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| In English (Mech. Solids): | | 4810 |
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| V.A. Kovalev and Yu.N. Radaev, "Estimates of Azimuthal Numbers Associated with Elementary Elliptic Cylinder Wave Functions," Mech. Solids. 49 (3), 253-269 (2014) |
| Year |
2014 |
Volume |
49 |
Number |
3 |
Pages |
253-269 |
| DOI |
10.3103/S0025654414030029 |
| Title |
Estimates of Azimuthal Numbers Associated with Elementary Elliptic Cylinder Wave Functions |
| Author(s) |
V.A. Kovalev (Moscow City Government University of Management, ul. Sretenka 28, Moscow, 107045 Russia, vlad_koval@mail.ru)
Yu.N. Radaev (Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, pr. Vernadskogo 101, str. 1, Moscow, 119526 Russia, radayev@ipmnet.ru) |
| Abstract |
The paper deals with issues related to the construction of
solutions, 2π-periodic in the angular variable, of the Mathieu differential equation for the circular elliptic cylinder harmonics, the associated characteristic values, and the azimuthal numbers needed to form the elementary elliptic cylinder wave functions. A superposition of the latter is one possible form for representing the analytic solution of the thermoelastic wave propagation problem in long waveguides with elliptic cross-section contour. The classical Sturm-Liouville problem for the Mathieu equation is reduced to a spectral problem for a linear self-adjoint operator in the Hilbert space of infinite square summable two-sided sequences. An approach is proposed that permits one to derive rather simple algorithms for computing the characteristic values of the angular Mathieu equation with real parameters and the corresponding eigenfunctions. Priority is given to the application of the most symmetric forms and equations that have not yet been used in the theory of the Mathieu equation. These algorithms amount to constructing a matrix diagonalizing an infinite symmetric pentadiagonal matrix. The problem of generalizing the notion of azimuthal number of a wave propagating in a cylindrical waveguide to the case of elliptic geometry is considered. Two-sided mutually refining estimates are constructed for the spectral values of the Mathieu differential operator with periodic and half-periodic (antiperiodic) boundary conditions. |
| Keywords |
elliptic cylinder, thermoelastic field, Mathieu equation, eigenvalue, azimuthal number, spectral problem, wave number, wave function, diagonalization, Gershgorin circle, Cassinian oval |
| References |
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[in Russian]. |
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1 (9), 183-196 (2011). |
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|
| Received |
31 July 2012 |
| Link to Fulltext |
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