Geodesic
Diffraction of a plane sound wave by an elastic sphere with an non-uniform coating located near a plane
Čebyševskij sbornik, Tome 19 (2018) no. 2, pp. 199-216

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In paper the problem of diffraction of a plane sound wave by a homogeneous elastic sphere with radially non-uniform elastic coating located near a plane. It is necessary that the body is placed in an ideal fluid, the spreading flat surface is absolutely rigid and absolutely soft, heterogeneity laws of a coating material are described by continuous functions.The problem is replaced by a problem of diffraction on two bodies. According to a method of imaginary radiants the dividing boundary of mediums is substituted by with mirrorly mapped imaginary sphere which is situated in the field of two plane waves. The analytical solution of the problem of diffraction of a plane sound wave by two identical homogeneous elastic spheres with radially non-uniform coatings situated in an ideal unlimited fluid is received. For solution of the problem the addition theorem for spherical wave functions is used. Analytic expressions In the form of decomposition on spherical functions are obtained which describe the wave fields in the containing medium and the homogeneous elastic bodies. The boundary-value problem for the system of ordinary differential equations of the second order is constructed for determination of the displacement fields in non-uniform coatings. On the basis of solution of problem of diffraction a plane wave by two bodies the diffraction problem for case of scattering of second plane wave is received. By summation of results of solutions of two diffraction problems the analytical solution of the problem of diffraction of a plane sound wave by a elastic sphere with coating located near a plane is received.By means of an continuous-non-uniform elastic coatings it is possible to change effectively scattering performances of bodies in determinate directions if to pick up corresponding the inhomogeneity laws for mechanical parametres of a coating.
Keywords: diffraction, sound waves, uniform elastic sphere, non-uniform coating. 
@article{CHEB_2018_19_2_a15,
     author = {L. A. Tolokonnikov},
     title = {Diffraction of a plane sound wave by an elastic sphere with an non-uniform coating located near a plane},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {199--216},
     publisher = {mathdoc},
     volume = {19},
     number = {2},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2018_19_2_a15/}
}
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L. A. Tolokonnikov. Diffraction of a plane sound wave by an elastic sphere with an non-uniform coating located near a plane. Čebyševskij sbornik, Tome 19 (2018) no. 2, pp. 199-216. http://geodesic.mathdoc.fr/item/CHEB_2018_19_2_a15/