Scattering of sound waves by an cylinder with an radial non-uniform elastic coating in a~planar waveguide

L. A. Tolokonnikov

Čebyševskij sbornik, Tome 20 (2019) no. 1, pp. 272-283.

Résumé

In paper the problem of sound wave scattering by absolutely rigid cylinder with radially inhomogeneous isotropic elastic coating in a planar waveguide is considered. It is believed that a waveguide filled with a homogeneous ideal fluid, one of its borders is absolutely rigid and the other — acoustically soft, heterogeneity laws of a coating material are described by differentiable functions, harmonic sound wave excited by a given distribution of sources in the section waveguide. In the case of steady state oscillations the propagation of small perturbations in ideal fluid is described by the Helmholtz's equation. The oscillations of an inhomogeneous isotropic elastic cylindrical layer described by general motion equations of the continuous medium. The boundary-value problem for the system of ordinary second order differential equations is constructed for determination of the displacement field in inhomogeneous coating. The primary field of disturbances is represented by a set of its own waveguide waves. The pressure of the field scattered by the cylindrical body is sought as potential of a simple layer. The Green function for the Helmholtz equation that satisfies the given boundary conditions on the waveguide walls and conditions of radiation at infinity is constructed. The function of distribution density of sources are sought as a Fourier series expansion. The infinite linear system of equations is obtained for determination of the coefficients of this decomposition. The solution of truncated infinite system is found by the inverse matrix method. Analytical expressions for the scattered acoustic field in different areas of the waveguide are obtained.

Détail

Keywords: scattering, sound waves, cylinder, non-uniform elastic coating, planar waveguide.

Bibliographie

[1] R. H. Hackman, G. S. Sammelman, “Acoustic scattering in an inhomogeneous waveguide: Theory”, J. Acoust. Soc. Amer., 80:5 (1986), 1447–1458

[2] R. H. Hackman, G. S. Sammelman, “Multiple-scattering analysis for a target in oceanic waveguide”, J. Acoust. Soc. Amer., 84:5 (1988), 1813–1825

[3] Kuz'kin V. M., “Sound scattering by a body in a planar layered waveguide”, Acoustical Physics, 49:1 (2003), 68–74 | MR

[4] Sharfarets B. P., “Method for calculating the field of an opaque source and the field scattered by an inhomogeneous inclusion in a planar layered waveguide”, Acoustical Physics, 50:1 (2004), 107–111

[5] Belov V. E., Gorskij S. M., Zinov'ev A. Y., Khil'ko A. I., “Application of the method of integral equations to diffraction of acoustic waves at elastic bodies in a layer of liquid”, Acoustical Physics, 40:4 (1994), 548–560

[6] Andrianova T. A., Belov V. E., Zalezsky A. A., Sharonov G. A., “An analysis of the mode composition and spatial distribution of acobstic fields for different localizations of inhomogeneities in an liquid layer”, Acoustical Physics, 42:4 (1996), 407–419

[7] Belov V. E., Gorskij S. M., Zalezsky A. A., Zinov'ev A. Y., Khil'ko A. I., “Scattering of acoustic waves by elastic cylinders in multimode layered waveguides”, Formation of acoustic fields in oceanic waveguides. Reconstruction of inhomogeneities, Inst. Prikladnoj Fiziki RAN, Nizhnij Novgorod, 1994, 63–88 (in Russian)

[8] V. E. Belov, S. M. Gorsky, A. A. Zalezsky, A. Yu. Zinovyev, “Application of the integral equation method to acoustic wave diffraction from elastic bodies in a fluid layer”, J. Acoust. Soc. Amer., 103:3 (1994), 1288–1295

[9] Romanov A. G., Tolokonnikov L. A., “The scattering of acoustic waves by a cylinder with a non-uniform elastic coating”, J. Appl. Math. Mech., 75:5 (2011), 595–600 | MR | Zbl

[10] Tolokonnikov L. A., “Scattering of an obliquely incident plane sound wave by an elastic cylinder with a non-uniform covering”, Izv. Tul. Gos. Univ., Ser. Estestv. Nauki, 2013, no. 2–2, 265–274 (in Russian)

[11] Shenderov E. L., Wave problems of underwater acoustics, Sudostroenie, L., 1972, 352 pp. (in Russian)

[12] Nowacki W., Teoria sprezystosci, Mir, 1975, 872 pp. (in Russian)

[13] Skudrzyk E., The Foundations of Acoustics, v. 2, Mir, 1976, 544 pp. (in Russian)

[14] Tikhonov A. N., Samarskiy A. A., Equations of Mathematical Physics, Nauka, 1972, 736 pp. (in Russian) | MR

[15] Ivanov E. A., Diffraction of electromagnetic waves by two bodies, Nauka i tekhnika, Minsk, 1968, 584 pp. (in Russian)