Sound waves diffraction by an elastic cylinder with inhomogeneous anisotropic outer layer near the plane

S. A. Skobel'tsyn   ; N. Y. Peshkov

Čebyševskij sbornik, Tome 22 (2021) no. 4, pp. 225-240

Résumé

The problem of diffraction of a plane monochromatic sound wave by an elastic cylinder with a layered inhomogeneous transversely isotropic outer layer is considered. It is assumed that the cylinder is located near a plane with an ideal surface (absolutely hard or acoustically soft). In order to get rid of the boundary conditions in the plane, in accordance with the so-called imaginary scatterer method, an additional obstacle is introduced in the form of a second elastic cylinder, which is mirror-like with respect to the initial one on the other side of the plane. The plane itself is excluded from consideration, and the fulfillment of the boundary conditions on it is ensured by introducing a second incident plane wave with the same amplitude as that of the first. The direction of propagation of the second wave is mirrored to the direction of the original wave relative to the plane. The phase shift in the second wave is equal to the phase shift in the first if the plane is absolutely rigid. If the plane is absolutely soft, the phase shift in the second wave is shifted relative to the phase shift in the first one by $\pi$. Thus, the problem is reduced to the problem of scattering of two plane waves by two identical elastic cylinders with parallel axes. Assuming that the incident wave propagates along the normal to the cylinder axis, a two-dimensional problem is solved. The solution of the problem in a modified formulation is carried out using the finite element method. Numerical simulation of the solution in the near zone of a scattered acoustic field is carried out. The calculation results show that in a number of cases of combinations of the parameters of the cylinder and the incident wave, the anisotropy and inhomogeneity of the material properties of the outer layer of the cylinder have a significant effect on the scattered field.

Détail

Mots-clés : diffraction
Keywords: plane sound wave, elastic cylinder, inhomogeneous anisotropic layer, underlying surface, imaginary scatterer method, finite element method.

Bibliographie

[1] Skuchik E., Osnovy akustiki, v. 1, Mir, M., 1976, 520 pp.

[2] Koshlyakov N.S., Gliner E.B., Smirnov M.M., Uravneniya v chastnykh proizvodnykh matematicheskoi fiziki, Vysshaya shkola, M., 1970, 712 pp.

[3] Novatskii V., Teoriya uprugosti, Mir, M., 1975, 872 pp.

[4] Fedorov F.I., Teoriya uprugikh voln v kristallakh, Nauka, M., 1965, 388 pp.

[5] Lekhnitskii S.G., Teoriya uprugosti anizotropnogo tela, Nauka, M., 1977, 415 pp.

[6] Isakovich M.A., Obschaya akustika, Nauka, M., 1973, 496 pp.

[7] Ivanov E.A., Difraktsiya elektromagnitnykh voln na dvukh telakh, Nauka i tekhnika, Minsk, 1968, 584 pp.

[8] Skobeltsyn S.A., Tolokonnikov L.A., “Difraktsiya ploskoi zvukovoi volny na uprugom sferoide s neodnorodnym pokrytiem v prisutstvii podstilayuschei poverkhnosti”, Izv. TulGU. Estestvennye nauki, 2015, no. 2, 64–75

[9] Skobeltsyn S.A., Peshkov N.Yu., “Opredelenie geometricheskikh parametrov konechnogo tsilindra, raspolozhennogo u granitsy poluprostranstva, po rasseyannomu zvuku”, Aktualnye problemy prikladnoi matematiki, informatiki i mekhaniki, Sb. trudov Mezhdunar. nauchno-tekhnicheskoi konf., Nauchno-issledovatelskie publikatsii, Voronezh, 2018, 1263–1269

[10] Skobeltsyn S.A., “Rasseyanie zvukovykh voln uprugim ellipsoidom s neodnorodnym pokrytiem v poluprostranstve s idealnoi poverkhnostyu”, Chebyshevskii sbornik, 19:1 (2018), 220–237 | DOI | MR | Zbl

[11] Skobeltsyn S.A., Fedotov I.S., Titova A.S., “Difraktsiya zvuka na uprugom share s neodnorodnym pokrytiem i polostyu v poluprostranstve”, Chebyshevskii sbornik, 19:4 (2018), 177–193 | DOI | MR | Zbl

[12] Skobeltsyn S.A., Peshkov N.Yu., “Rasseyanie zvuka neodnorodnym uprugim ellipticheskim tsilindrom v akusticheskom poluprostranstve”, Izv. TulGU. Tekhnicheskie nauki, 2018, no. 7, 183–200

[13] Skobelt'syn S.A., Peshkov N.Y., “Finding, by means of a scattered sound, the geometric parameters of a finite elastic cylinder located near the half-space border”, J. Phys.: Conf. Ser., 1203 (2019), 1–10

[14] Ihlenburg F., Finite element analysis of acoustic scattering, Springer Publishing Company, Inc, New York, 2013, 226 pp. | MR

[15] Ivanov V.I., Skobeltsyn S.A., “Modelirovanie reshenii zadach akustiki s ispolzovaniem MKE”, Izv. TulGU. Estestvennye nauki, 2008, no. 2, 132–145

[16] Skobeltsyn S.A., “O poryadke resheniya zadachi difraktsii zvuka uprugim telom s polostyu s ispolzovaniem MKE”, Vestn. TulGU. Seriya: Differentsialnye uravneniya i prikladnye zadachi, 2012, no. 1, 51–58

[17] Shenderov E.L., Volnovye zadachi gidroakustiki, Sudostroenie, L., 1972, 352 pp.