On the properties of the dispersion set for an inhomogeneous cylindrical waveguide

A. O. Vatulyan   ; V. O. Yurov

Vladikavkazskij matematičeskij žurnal, Tome 20 (2018) no. 1, pp. 50-60

Résumé

On the basis of the analysis of an operator spectral beam with two parameters, the dispersion relations for a cylindrical waveguide, inhomogeneous in the radial coordinate, with impedance boundary conditions on the external boundary are investigated. This boundary conditions permit to simulate free and clamped external boundary conditions as well as intermediate options. The stresses and displacements on the boundary are linearly related by means of two parameters. In the axisymmetric formulation, the spectral problem in the form of matrix differential operator of the 4th order with respect to the stress and displacement vectors components is formulated. A number of properties describing the general structure of the dispersion set are studied. Two spectral problems are formulated with two families of dispersion curves which are analytically continued from the points of the spectrum and differing by their eigenfunctions. Formulae reflecting the connection of the spectrum points with parameters entering the boundary conditions at the outer boundary are obtained. Based on the perturbation method, the structure of the curves of families considered is investigated. The property of solvability of the inhomogeneous problem proved in the article was used to construct an asymptotic approximation of the dispersion set components in the region of long waves. In the low-frequency range, in the particular case, the explicit dependence of the first dispersion curve slope angle on one of the parameters of the boundary conditions is constructed. At that, even a weak relationship between shear stresses and longitudinal displacements leads to changes for which the asymptotic behavior is not valid. On the basis of the shooting method, the schemes of constructing the dispersion curves components are stated. The results of the computational experiments for two kinds of radial inhomogeneity are presented. The dispersion set points that do not change their position depending on the boundary conditions parameters are revealed.

Bibliographie

[1] Pochhammer L., “Ueber die Fortpflanzungsgeschwindigkeiten kleiner Schwingungen in einem unbegrenzten isotropen Kreiscylinder”, J. Reine Angew. Math., 81 (1876), 324–336 | MR

[2] Chree C., “Longitudinal vibrations of a circular bar”, J. Quart. Pure Appl. Math., 21 (1886), 287–298

[3] Kostyuchenko A. G., Shkalikov A. A., “Self-adjoint quadratic operator pencils and elliptic problems”, Funct. Anal. Appl., 17 (1983), 109–128 | DOI | MR | Zbl

[4] Vorovich I. I., Babeshko V. A., Dynamic Mixed Elastic Problems for Nonclassical Regions, Nauka, M., 1979, 320 pp. (in Russian) | MR

[5] Getman I. P., Ustinov Yu. A., Mathematical Theory of Irregular Solid Waveguides, Izdat. RGU, Rostov-on-Don, 1993, 144 pp. (in Russian)

[6] Markus A. S., Introduction to the Spectral Theory of Polynomial Operator Sheaf, Shtiintsa, Kishinev, 1986, 260 pp. (in Russian) | MR

[7] Jia H., Jing M., Rose J. L., “Guided wave propagation in single and double layer hollow cylinders embedded in infinite media”, J. Acoust. Soc. Am., 129:2 (2011), 691–700 | DOI

[8] Castaings M., Lowe M., “Finite element model for waves guided along solid systems of arbitrary section coupled to infinite solid media”, J. Acoust. Soc. Am., 123:2 (2008), 696–708 | DOI

[9] Vatul'yan A. O., Morgunova A. V., “Study of the dispersion properties of cylindrical waveguides with variable properties”, Acoust. Phys., 61 (2015), 265 | DOI | DOI

[10] Vatul'yan A. O., Yurov V. O., “Wave processes in a hollow cylinder in an inhomogeneous prestress field”, J. Appl. Mech. Tech. Phys., 57 (2016), 731–739 | DOI | MR | MR | Zbl