Diffraction of a plane sound wave by an elastic sphere with an inhomogeneous transversal isotropic layer

L. A. Tolokonnikov ; S. L. Tolokonnikov

Čebyševskij sbornik, Tome 22 (2021) no. 4, pp. 332-343.

Résumé

In paper the problem of diffraction of a harmonic plane sound wave by a homogeneous isotropic elastic sphere with a continuously inhomogeneous anisotropic elastic layer is considered. It is believed that the body is placed in an infinite ideal fluid, the laws of heterogeneity of the coating material are described by continuous functions. An analytical solution to the diffraction problem is obtained for the case when the material of the sphere layer is radially inhomogeneous and transversally isotropic. Wave field in a containing medium and a homogeneous isotropic sphere are described by expansions in spherical wave functions. A boundary value problem is constructed for a system of ordinary differential equations of the second order for finding displacement fields in an inhomogeneous anisotropic layer of sphere. The results of numerical calculations of directional patterns for scattered acoustic field in the far zone are presented. It is shown that anisotropy of continuously inhomogeneous elastic layer one can substantially change the scattering characteristics of spherical bodies.

Détail

Keywords: diffraction, sound waves, uniform elastic sphere, inhomogeneous anisotropic layer.

Bibliographie

[1] Faran J. J., “Sound scattering by solid cylinders and spheres”, J. Acoust. Soc. Amer., 23:4 (1951), 405–418 | DOI | MR

[2] Junger M. C., “Sound scattering by thin elastic shells”, J. Acoust. Soc. Amer., 24:4 (1952), 366–373 | DOI | MR

[3] Flax L., Dragonette L. R., Uberall H., “Theory of elastic resonance excitation by sound scattering”, J. Acoust. Soc. Amer., 63:3 (1978), 723–731 | DOI | Zbl

[4] Goodman R. D., Stern R., “Reflection and transmission of sound by elastic spherical shells”, J. Acoust. Soc. Amer., 34:3 (1962), 338–344 | DOI | MR

[5] Prikhodko V. Yu., Tyutekin V. V., “O sobstvennykh chastotakh i formakh kolebanii radialno-sloistykh uprugikh tel”, Prikl. mekhanika, 23:6 (1987), 9–14 | Zbl

[6] Shenderov E. L., Izluchenie i rasseyanie zvuka, Sudostroenie, L., 1989, 304 pp.

[7] Skobeltsyn S. A., Tolokonnikov L. A., “Rasseyanie zvuka neodnorodnym transversalno-izotropnym sfericheskim sloem”, Akusticheskii zhurn., 41:6 (1995), 917–923

[8] Larin N. V., Tolokonnikov L. A., “Rasseyanie zvuka neodnorodnym termouprugim sfericheskim sloem”, Prikladnaya matematika i mekhanika, 74:4 (2010), 645–654 | Zbl

[9] Brigadirova T. E., Medvedskii A. L., “Difraktsiya nestatsionarnoi akusticheskoi volny na neodnorodnoi transversalno-izotropnoi poloi sfere”, Mekhanika kompoz. mater. i konstruktsii, 13:1 (2007), 119–130

[10] Tolokonnikov L. A., “Rasseyanie ploskoi zvukovoi volny uprugim sharom s neodnorodnym pokrytiem”, Prikladnaya matematika i mekhanika, 78:4 (2014), 519–526 | Zbl

[11] Tolokonnikov L. A., Rodionova G. A., “Difraktsiya sfericheskoi zvukovoi volny na uprugom share s neodnorodnym pokrytiem”, Izvestiya Tulskogo gos. un-ta. Estestvennye nauki, 2014, no. 3, 131–137

[12] Tolokonnikov L. A., “Difraktsiya tsilindricheskikh zvukovykh voln na uprugoi sfere s neodnorodnym pokrytiem”, Prikladnaya matematika i mekhanika, 79:5 (2015), 663–673 | Zbl

[13] Tolokonnikov L. A., “Difraktsiya ploskoi zvukovoi volny na uprugom share s neodnorodnym pokrytiem i proizvolno raspolozhennoi sfericheskoi polostyu”, Izvestiya Tulskogo gos. un-ta. Estestvennye nauki, 2014, no. 2, 181–193

[14] Tolokonnikov L. A., Larin N. V., Skobeltsyn S. A., “Modelirovanie neodnorodnogo pokrytiya uprugogo shara s trebuemymi zvukootrazhayuschimi svoistvami”, Matematicheskoe modelirovanie, 29:11 (2017), 89–98 | Zbl

[15] Larin N. V., Tolokonnikov L. A., “Rasseyanie zvuka termouprugim sharom s nepreryvno-neodnorodnym pokrytiem v teploprovodnoi zhidkosti”, Matematicheskoe modelirovanie, 31:5 (2019), 20–38 | DOI | Zbl

[16] Tolokonnikov L.A., “Modelirovanie nepreryvno-neodnorodnogo pokrytiya uprugogo shara sistemoi odnorodnykh uprugikh sloev v zadache rasseyaniya zvuka”, Prikladnaya matematika i mekhanika, 81:6 (2017), 699–707 | Zbl

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

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

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

[20] Lebedev N.N., Spetsialnye funktsii i ikh prilozheniya, Fizmatgiz, M., 1963, 358 pp.