ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Usmonov B. S., Mukhitdinov R. T., Eliboyev N. R., Akhmedov N. B. Nonstationary scattering of elastic waves by a spherical inclusion. Izvestiya VUZ. Applied Nonlinear Dynamics, 2026, vol. 34, iss. 1, pp. 84-97. DOI: 10.18500/0869-6632-003200, EDN: MSPDVQ

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Language: 
Russian
Heading: 
Applied Problems of Nonlinear Oscillation and Wave Theory
Article type: 
Article
UDC: 
534.1
DOI: 
10.18500/0869-6632-003200
EDN: 
MSPDVQ

Nonstationary scattering of elastic waves by a spherical inclusion

Autors: 
Usmonov Botir Shukurillaevich, Tashkent chemical - technological Institute
Mukhitdinov Ramazon Tukhtaevich, Tashkent chemical - technological Institute
Eliboyev Nurali Rajabaliyevich, Tashkent chemical - technological Institute
Akhmedov Nasriddin Bakhodirovich,
Abstract: 

Problems of elastic wave scattering by various types of inhomogeneities rank among the most complex and relevant topics in the field of deformable solid dynamics. From an applied perspective, this is due to the fact that information about the dynamic stress–strain state in the vicinity of such inhomogeneities is of significant interest for various engineering and physical applications.

The objective of this study is to investigate the nonstationary scattering of elastic waves by a spherical inclusion embedded in an infinite elastic medium. The analytical approach to the solution involves the application of Fourier integral transforms with respect to time. It
is established that the eigenfunctions of the considered problem cannot be treated as vectors in a Hilbert space, since they are not square-integrable due to their exponential growth with distance. This necessitates the use of generalized functions and specialized methods from scattering theory.
 

Key words: 
spherical shell
wave scattering
wave amplitude
eigenfunctions
eigenfrequencies
Reference: 
  1. Formalev V, Kartashov E, Kolesnik S. Wave heat transfer in anisotropic half-space under the action of a point exponential-type heat source based on the wave parabolic-type equation. J. Eng. Phys. Thermophys. 2022;95:366-373. DOI: 10.1007/s10891-022-02490-2.
  2. Formalev VF, Garibyan BA, Kolesnik SA. Mathematical simulation of thermal shock wave dynamics in nonlinear local non-equilibrium media. Herald of the Bauman Moscow State Technical University, Series Natural Sciences. 2022;4:80-94 (in Russian). DOI: 10.18698/1812-3368-2022-4-80-94.
  3. Formalev VF, Kolesnik SA. Wave heat transfer in heat-shielding materials with an exponential-like nonlinear dependence of thermal conductivity on temperature. High Temp. 2022;60:731-734. DOI: 10.1134/S0018151X22050030.
  4. Tushavina OV, Kriven GI, Hein TZ. Study of thermophysical properties of polymer materials enhanced by nanosized particles. Int. J. Circuits, Syst. Signal Process. 2021;15:1436-1442. DOI: 10.46300/9106.2021.15.155.
  5. Sha M, Utkin YA, Tushavina OV, Pronina PF. Experimental studies of heat and mass transfer from tip models made of carbon-carbon composite material (CCCM) under conditions of high-intensity thermal load. Per. Tche Quim. 2020;17(35):988-997. DOI: 10.52571/ptq.v17.n35.2020.81_sha_pgs_988_997.pdf.
  6. Garnier B, Boudenne A. Use of hollow metallic particles for the thermal conductivity enhancement and lightening of filled polymer. Polym. Degrad. Stab. 2016;127:113-118. DOI: 10.1016/j.polymdegradstab.2015.11.026.
  7. Mohammad SMH, Merkel M, Ochsner A. Influence of the joint shape on the uniaxial mechanical properties of non-homogeneous bonded perforated hollow sphere structures. Comput. Mater. Sci. 2012;58:183-187. DOI: 10.1016/j.commatsci.2012.01.024.
  8. Safarov II, Teshayev MH, Juraev ShI, Khomidov FF. Vibrations of viscoelastic plates with attached concentrated masses. Lobachevskii J. Math. 2024;45:1729-1737. DOI: 10.1134/S1995080224601474.
  9. Usmonov BSh, Safarov II, Teshaev МKh. Nonlinear flutter of the transient process of hereditarily deformable systems in supersonic flight mode. Tomsk State University Journal of Mathematics and Mechanics. 2024;(88):124-137 (in Russian). DOI: 10.17223/19988621/88/10.
  10. Safarov I, Nuriddinov B, Nuriddinov Z. Propagation of own waves in a viscoelastic cylindrical panel of variable thickness. Lobachevskii J. Math. 2024;45:1246-1253. DOI: 10.1134/S1995080224600663.
  11. Safarov I, Teshaev M. Control of resonant oscillations of viscoelastic systems. Theoretical and Applied Mechanics. 2024;51(1):1-12. DOI: 10.2298/TAM220510007S.
  12. Safarov II, Teshaev МKh. Unsteady movements of spherical shells in a viscoelastic medium. Tomsk State University Journal of Mathematics and Mechanics. 2023;(83):166-179. DOI: 10.17223/19988621/83/14.
  13. Teshaev MK, Safarov II, Kuldashov NU, Ishmamatov MR, Ruziev TR. On the distribution of free waves on the surface of a viscoelastic cylindrical cavity. J. Vib. Eng. Technol. 2020;8:579-585. DOI: 10.1007/s42417-019-00160-x.
  14. Safarov II, Teshaev MK. Dynamic damping of vibrations of a solid body mounted on viscoelastic supports. Izvestiya VUZ. Applied Nonlinear Dynamics. 2023;31(1):63-74 (in Russian). DOI: 10.18500/0869-6632-003021.
  15. Adamov AA, Matveenko VP, Trufanov NA, Shardakov IN. Methods of Applied Viscoelasticity. Ekaterinburg: UrO RAN; 2003. 411 p.
  16. Fedorov AYu, Matveenko VP, Shardakov IN. Numerical analysis of stresses in the vicinity of internal singular points in polymer composite materials. IJCIET. 2018;9(8):1062-1075.
  17. Bykov AA, Matveenko VP, Shardakov IN, Shestakov AP. Shock wave method for monitoring crack repair processes in reinforced concrete structures. Mech. Solids. 2017;52:378-383. DOI: 10.3103/S0025654417040033.
  18. Karimov K, Akhmedov A, Karimova A. Development of mathematical model, classification, and structures of controlled friction and vibration mechanisms. AIP Conf. Proc. 2023;2612:030014. DOI: 10.1063/5.0116891.
  19. Safarov II. Numerical modeled static stress-deformed state of parallel pipes in deformable environment. Adv. Sci. Technol. Res. J. 2018;12(3):114-125. DOI: 10.12913/22998624/92177.
  20. Teshaev МKh, Karimov IM, Umarov AO, Zhuraev SI. Diffraction of harmonic shear waves on an elliptical cavity located in a viscoelastic medium. Russ Math. 2023;67:44-48. DOI: 10.3103/S1066369X23080108.
  21. Popova TV, Mayer AE, Khishchenko KV. Evolution of shock compression pulses in polymethylme-thacrylate and aluminum. Journal of Applied Physics. 2018;123:235902. DOI: 10.1063/1.5029418.
  22. Popova TV, Mayer AE, Khishchenko KV. Numerical investigations of shock wave propagation in polymethylmethacrylate. J. Phys.: Conf. Ser. 2015;653:012045. DOI: 10.1088/1742-6596/653/1/012045.
  23. Bouak F, Lemay J. Use of the wake of a small cylinder to control unsteady loads on a circular cylinder. J. Visualization. 2001;4:61-72. DOI: 10.1007/BF03182456.
  24. Bhagyashekar M, Rao K, Rao RMVGK. Studies on rheological and physical properties of metallic and non-metallic particulate filled epoxy composites. J. Reinforced Plastics and Composites. 2009;28(23):2869-2878. DOI: 10.1177/0731684408093976.
  25. Dubrovsky VA, Morozhnik VS. Scattering of elastic waves on a large-scale low-contrast spherical inclusion. Izvestiya of the Academy of Sciences of the USSR. Physics of the Solid Earth. 1986;(4):32-41 (in Russian).
Received: 
28.07.2025
Accepted: 
27.10.2025
Available online: 
13.11.2025
Published: 
30.01.2026
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