Geodesic
Diffraction of harmonic shear waves on an elliptical cavity located in a viscoelastic medium
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 8 (2023), pp. 64-70.

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The problem of diffraction of harmonic shear waves on an elliptical cylindrical cavity located in a viscoelastic medium is considered. The relationship between stresses and deformations is taken into account using the hereditary integral Boltzmann-Voltaire relation. The problem of a dynamic stress-strain state around an elliptical cavity in an unlimited viscoelastic medium under the action of harmonic shear waves is reduced to a plane problem (plane deformable state) of viscoelasticity. The Lame equation reduces to the solution of the Mathieu equation with complex arguments. Its solution is expressed in terms of Mathieu functions. Numerical results are obtained for different frequencies of incident waves, angles of incidence of the transverse wave and the ratio of the axes of the elliptical cavity.
Keywords: shear waves, elliptical cylinder
Mots-clés : Mathieu equation, Boltzmann–Voltaire relations, complex argument. 
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     title = {Diffraction of harmonic shear waves on an elliptical cavity located in a viscoelastic medium},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
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M. Kh. Teshaev; I. M. Karimov; A. O. Umarov; Sh. I. Zhuraev. Diffraction of harmonic shear waves on an elliptical cavity located in a viscoelastic medium. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 8 (2023), pp. 64-70. http://geodesic.mathdoc.fr/item/IVM_2023_8_a6/

[1] Mow C.C., McCabe W.L., “Dynamic Stresses in an Elastic Cylinder”, J. Engineering Mech. Div., ASCE, 89:3 (1963), 21–41 | DOI

[2] Mow C.C., Mente L.J., “Dynamic Stresses and Displacements Around Cylindrical Discontinuities Due to Plane Harmonic Shear Waves”, J. Appl. Mech., 30:4 (1963), 598–604 | DOI

[3] Das D., Mandal B.N., “Wave radiation by a sphere submerged in a two-layer ocean with an ice-cove”, Appl. Ocean Res., 32:3 (2010), 358–366 | DOI

[4] Mohapatra S., Bora S.N., “Radiation of Water Wave by a Sphere in an Ice-Covered Two-Layer Fluid of Finite Depth”, J. Adv. Res. Appl. Math., 2:1 (2010), 46–63 | DOI | MR

[5] Safarov I.I., Teshaev M., Toshmatov E., Boltaev Z., Homidov F.F., “Torsional vibrations of a cylindrical shell in a linear viscoelastic medium”, IOP Conf. Ser. Mater. Sci. and Eng., 883:1 (2020), 12–19 | DOI

[6] Mirzaev I.M., Nikiforovskii V.S., “Rasprostranenie ploskoi volny i razrushenie v uprugikh i nesovershenno elastichnykh sochlenennykh konstruktsiyakh”, Sovetsk. gornaya nauka, 9 (1973), 161–165

[7] Safarov I.I., Homidov F.F., Rakhmonov B.S., Almuratov Sh.N., “Seismic vibrations of complex relief of the surface of the naryn canyon $($on the Norin river in Kyrgyzstan$)$ during large-scale underground explosions”, J. Phys. Conf. Ser., 1706:1 (2020), 012125 | DOI

[8] Safarov I.I., Teshaev M.Kh., Boltaev Z.I., “Propagation of linear waves in multilayered structural - inhomogeneous cylindrical shells”, J. Critical Reviews, 7:12 (2020), 893–904

[9] Mirzaev Ya.M., “Vzaimodeistvie mezhdu dolotom i kamnem”, Sovetsk. dobycha poleznykh iskopaemykh, 11 (1975), 70–73

[10] Boltaev Z., Safarov I., Razokov T., “Natural vibrations of spherical inhomogeneity in a viscoelastic medium”, Internat. J. Sci. and Tech. Research, 9:1 (2020), 3674–3680

[11] Safarov I., Teshaev M.Kh., Axmedov S., Rayimov D., Homidov F., “Manometric Tubular Springs Oscillatory Processes Modeling with Consideration of its Viscoelastic Properties”, E3S Web of Conf., 2021, 264

[12] Safarov I.I., Teshaev M.Kh., Marasulov A.M., Nuriddinov B.Z., “Propagation of Own Non-Axisymmetric Waves in Viscoelastic Three-Layered Cylindrical Shells”, Engineering J., 25:7 (2021), 97–107 | DOI

[13] Lependin L.F., Akustika, Vyssh. shk., M., 1978

[14] Viktorov I.A., Fizicheskie osnovy primeneniya ultrazvukovykh voln Releya i Lemba v tekhnike, Nauka, M., 1966

[15] Salieva O., Safarov I., Abidov K., “Investigation of the propagation of free waves in a rod in the design of building structures”, IOP Conf. Ser. Earth and Environmental Sci., 839:5 (2021), 052035 | DOI

[16] Arshidinova M., Begaliyeva K., Kudaykulov A., Tashev A., “Numerical Modeling of Nonlinear Thermomechanical Processes in a Rod of Variable Cross Section in the Presence of Heat Flow”, 2018 5th Internat. Conf. on Information Sci. and Control Engineering (Zhengzhou, China, 2018), 351–354

[17] Kudaykulov A., Arinov E., Ispulov N., Abdul Qadir, Begaliyeva K., “Numerical Study of a Thermally Stressed State of a Rod”, Hindawi. Adv. Math. Phys., 2019, 9 | MR | Zbl

[18] Teshaev M.Kh., “Ob osuschestvlenii servosvyazei elektromekhanicheskoi sledyaschei sistemoi”, Izv. vuzov. Matem., 2010, no. 12, 44–51 | MR | Zbl

[19] Timergaliev S.N., “K probleme razreshimosti nelineinykh kraevykh zadach dlya proizvolnykh izotropnykh pologikh obolochek tipa Timoshenko so svobodnymi krayami”, Izv. vuzov. Matem., 2021, no. 4, 90–107 | MR | Zbl