Reflection and transmission of cylindrical sound wave through an elastic plate with an inhomogeneous coating
https://doi.org/10.22405/2226-8383-2022-23-1-312-327
Abstract
In paper the problem of harmonic cylindrical sound waves reflection and transmission through a homogeneous isotropic elastic plate with a continuously inhomogeneous in thickness elastic coating is considered. It is believed that the plate is placed in an infinite ideal fluid, the laws of heterogeneity of the coating material are described by continuous functions.
An analytical solution of the posed problem is obtained on the basis of the known solution of the problem about the passage of plane sound waves through plate with a continuously inhomogeneous coating and using integral representation of a cylindrical wave in the form of an expansion on flat waves.
Finding the displacement field in an inhomogeneous layer is reduced to solving boundary value problem for a system of ordinary differential equations of the second order.
The results of numerical calculations of frequency characteristics are presented for reflected and transmitted acoustic fields. Shown strong difference of frequency dependencies for different laws of inhomogeneity coating material.
Keywords
reflection and transmission of sound,
cylindrical sound wave,
homogeneous elastic plate,
inhomogeneous coating.
Supporting agencies: Supported by the Russian Science Foundation grant No. 18-11-00199, https://rscf.ru/project/18-11-00199/
About the Authors
Lev Alexeevich Tolokonnikov
Tula State University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences
Thi Sang Nguyen
Tula State University
Russian Federation
Russian Federation
postgraduate student
References
1. Brekhovskikh, L. M. 1973, “Waves in Layered Media”, Nauka, Moscow, 344 p., [in Russian].
2. Shenderov, E. L. 1972, “Wave problems of underwater acoustics”, Sudostroenie, Leningrad, 352 p., [in Russian].
3. Larin, N. V. 2015, “The transmission of sound through a uniform thermoelastic plane layer”, Izv. Tul. Gos. Univ., Ser. Estestv. Nauki, no. 3, pp. 145 – 153, [in Russian].
4. Larin, N. V. 2017, “Analysis of the resonance sound scattering by a thermoelastic plate”, Izv. Tul. Gos. Univ., Ser. Tekh. Nauki, no. 4, pp. 109 – 123, [in Russian].
5. Lonkevitch, M.P. 1971, “Transmission of sound through a finite-thickness layer of a transversal– isotropic materia”, Akust. Zhurnal, vol. 17, no. 1, pp. 85 – 92, [in Russian].
6. Shenderov, E.L. 1984, “Sound propagation through transversally-isotropic plate”, Akust. Zhurnal, vol. 30, no. 1, pp. 122 – 129, [in Russian].
7. Prikhod’ko, V.Yu. & Tyutekin, V. V. 1986, “Calculation of reflection coefficient of sound waves from solid layered media”, Akust. Zhurnal, vol. 32, no. 2, pp. 212 – 218, [in Russian].
8. Skobel’tsyn, S. A. & Tolokonnikov, L. A. 1990, “Transmission of sound waves through a transversely isotropic inhomogeneous plane layer”, Akust. Zhurnal, vol. 36, no. 4, pp. 740 – 744, [in Russian].
9. Rinkevich, A.B. & Smirnov, A. N. 2000, “Propagation of elastic waves in a non-uniform transversely isotropic plate”, Defektoskopiya, no. 8, pp. 78 – 83, [in Russian].
10. Tolokonnikov, L. A. 1999, “Reflection and refraction of a planar acoustic waves in an anisotropic inhomogeneous layer”, J. Appl. Mech. and Techn. Physics, vol. 40, no. 5, pp. 936 – 941.
11. Tolokonnikov, L. A. 1998, “The transmission of sound through an inhomogeneous anisotropic layer adjoining viscous liquids”, J. Appl. Mech. and Techn. Physics, vol. 62, no. 6, pp. 953 – 958.
12. Larin, N. V. & Tolokonnikov, L. A. 2006, “The transmission of a plane acoustic wave through a non-uniform thermoelastic layer”, J. Appl. Math. Mech., vol. 70, no. 4, pp. 590 – 598.
13. Tolokonnikov, L. A. & Larin, N. V. 2017, “Sound propagation through a discretely inhomogeneous thermoelastic plane layer adjacent to heat-conducting liquids”, J. Appl. Mech. and Techn. Physics, vol. 58, no. 1, pp. 95 – 102.
14. Huang C. & Nutt S. 2011, “An analytical study of sound transmission through unbounded panels of functionally graded materials”, J. of Sound and Vibration, vol. 330, no. 6, pp. 1153 — 1165.
15. Shamaev A. S. & Shumilova V. V. 2015, “Passage of a plane acoustic wave through a layered composite with components of elastic and viscoelastic materials”, Acoustical Physics, vol. 61, no. 1, pp. 8 – 18.
16. Larin, N. V., Skobel’tsyn, S. A. & Tolokonnikov, L. A. 2015, “Determination of the inhomogeneity laws for an elastic layer with preset sound-reflecting properties”, Acoustical Physics, vol. 61, no. 5, pp. 504 – 510.
17. Skobel’tsyn, S. A. 2016, “Determining the parameters of anisotropic elastic layer on the sound transmission”, Izv. Tul. Gos. Univ., Ser. Tekh. Nauki, no. 7-2, pp. 246 – 257, [in Russian].
18. Tolokonnikov, L. A. & Yudachev, V. V. 2015, “Reflection and refraction of a planar acoustic waves in an elastic planar layer with a non-uniform covering”, Izv. Tul. Gos. Univ., Ser. Estestv. Nauki, no. 3, pp. 219 – 226, [in Russian].
19. Larin, N. V., Skobel’tsyn, S. A. & Tolokonnikov, L. A. 2016, “Modeling the inhomogeneous coating of an elastic plate with optimum sound-reflecting properties”, J. Appl. Math. Mech., vol. 80, no. 4, pp. 339 – 344.
20. Larin, N. V. 2016, “Determination of the inhomogeneity laws for coating of the thermoelastic plate to obtain minimum sound reflection”, Izv. Tul. Gos. Univ., Ser. Tekh. Nauki, no. 11-2, pp. 216 – 234, [in Russian].
21. Tolokonnikov, L.A. & Nguyen, T. S. 2018, “About the influence of an non-uniform covering of the elastic plate on sound reflection and transmission”, Izv. Tul. Gos. Univ., Ser. Tekh. Nauki, no. 6, pp. 362 – 372, [in Russian].
22. Tolokonnikov, L. A. & Nguyen, T. S. 2019, "The transmission of sound through an elastic plate with an inhomogeneous coating adjoining viscous liquids Chebyshevskii sbornik, vol. 20, no. 2, pp. 311 -– 324, [in Russian].
23. Nguyen, T. S. 2021, “About reflection and transmission of a plane sound wave through an elastic plate with an inhomogeneous coating adjoining viscous liquids”, Izv. Tul. Gos. Univ., Ser. Tekh. Nauki, no. 5, pp. 404 – 414, [in Russian].
24. Skobel’tsyn, S. A. 2017, “Tstimation of the coating properties of a finite plate with a cavity providing the given parameters of the sound reflection”, Izv. Tul. Gos. Univ., Ser. Tekh. Nauki, no. 7, pp. 83 – 92, [in Russian].
25. Tolokonnikov, L. A. & Tolokonnikov, L. A. 2021, "Reflection and reflaction of a plane sound wave in an elastic plate with an inhomogeneous anisotropic coating Chebyshevskii sbornik, vol. 22, no. 3, pp. 423 – 437, [in Russian].
26. Piquette J. C. 1988, "Spherical-wave scattering by a finite-thickness solid plate of infinite lateral extent, with some implications for panel measurements J. Acoust. Soc. Am., vol. 83, No. 4, pp. 1284 – 1294.
27. Shushkevich G. Ch. & Kiselyova N. N. 2014, "Sound field shielding by flat elastic layer and thin unclosed spherical shell”, Informatika, no. 2, pp. 36 – 47, [in Russian].
28. Nowacki, W.1973, “Teoria sprezystosci”, PWN, Warszawa.
29. Tolokonnikov, L. A. & Nguyen, T. S. 2020, "Determination of the displacement field in an inhomogeneous coating of an elastic plate when passing through her plane sound wave Chebyshevskii sbornik, vol. 21, no. 1, pp. 310 -– 321, [in Russian].
30. Felsen L. B., Marcuvitz N. 1973, “Radiation and scattering of waves”, Prentice-Hall, Inc., Englewood Cliffs, New Jersey.
Review
For citations:
Tolokonnikov L.A., Nguyen T. Reflection and transmission of cylindrical sound wave through an elastic plate with an inhomogeneous coating. Chebyshevskii Sbornik. 2022;23(1):312-327. (In Russ.) https://doi.org/10.22405/2226-8383-2022-23-1-312-327
Views:
329
Email this article 