# Simulation of elastic wave diffraction by a sphere in semibounded region

 1Khimich, AN2Selezov, ITSydoruk, VA1V.M. Glushkov Institute of Cybernetics of the NAS of Ukraine, Kyiv2Institute of Hydromechanics of the NAS of Ukraine, Kyiv Dopov. Nac. akad. nauk Ukr. 2020, 10:22-27 https://doi.org/10.15407/dopovidi2020.10.022 Section: Information Science and Cybernetics Language: English Abstract:  The problem of scattering of plane elastic waves by a rigid sphere located near a plane rigid boundary is considered, which leads to the generation of multiply re-reflected dilatation and shear waves. The formulation of the problem is given when slippage conditions are specified on a flat boundary (equality of tangential stresses to zero). The problem is reduced to the definition of scalar functions. General solutions are written down, and approximate solutions are constructed for the field in the far zone characterized by the fact that the distance from the plane boundary to the obstacle is much greater than the radius of the sphere. In addition, the Rayleigh approximation is used, when the wave number is much lesser than the radius of the sphere. The method of images is used to construct multiply reflected waves. Approximate formulas are given for the field in the far zone and in the case of the long-wave Rayleigh approximation. The calculations of scattered wave fields, presented in the form of scattering diagrams, are carried out, from which a strongly oscillating wave field can be seen. Keywords: elastic waves, image method, oscillating field, semibounded region, sphere, wave diffraction, wavelength
References:

1. Selezov, I. T., Kryvonos, Yu. G. & Gandzha, I. S. (2018). Wave propagation and diffraction. Mathematical methods and applications. Springer. In series Foundations of Engineering Mechanics, DOI 10.1007/978-981-10-4923-1.
2. Selezov, I. T. (1993). Diffraction of waves by radially inhomogeneous inclusions. Physical Express, March. 1(2), pp. 104-115.
3. Morse, Ph. M. & Feshbach, H. (1953). Methods of theoretical physics. Part I, New York. Mc Gray-Hill Book Company.
4. Seismic diffraction. (2016). SEG Geophysics reprint series N.30. Society of Exploration Geopgycists. 8801 S. Yale. Tulsa, USA.
5. Jackson, J. D. (1965). Classic electrodynamics. John Wiley & Sons.
6. Friedman, B. &Russek, J. (1954). Addition theorem for spherical waves. Quart. Appl. Math., 12, No. 1, pp. 13-23.