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. 
@article{IVM_2023_8_a6,
     author = {M. Kh. Teshaev and I. M. Karimov and A. O. Umarov and Sh. I. Zhuraev},
     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},
     pages = {64--70},
     publisher = {mathdoc},
     number = {8},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2023_8_a6/}
}
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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/