On the determination of the wave potential of a vibrating spherical particle in a half-infinite cylinder with liquid

1Kubenko, VD
1S.P. Timoshenko Institute of Mechanics of the NAS of Ukraine, Kyiv
Dopov. Nac. akad. nauk Ukr. 2017, 9:41-47
https://doi.org/10.15407/dopovidi2017.09.041
Section: Mechanics
Language: Russian
Abstract: 

The semiinfinite circular cylindrical cavity filled with ideal compressible fluid and containing a spherical body, whose surface is energized with the set frequency is observed. It is required to build a wave potential for the observed area for the purpose of definition of the influence of the end face of the cylinder on the fields of pressure and velocity in the system. The matching boundary problem for the equation of Helmgoltz is formulated. The general solution is built in the form of a superposition of the potentials written with the use of cylindrical and spheri cal wave functions. “The imaginary” sphere, with which the condition in the face cross-section of the cylinder is realized, is introduced. Redecomposition of spherical wave functions is applied to satisfy the boundary conditions on cylindrical ones and on the contrary, and the theorems of addition of spherical functions are used. As a result, the problem is reduced to the solution of an infinite system of algebraic equations, which can be solved by the truncation method.

Keywords: compressible liquid, semiinfinite cylinder, vibrating spherical particle
References: 
  1. Kubenko, V. D. (1987). Diffraction of Steady Waves at a Set of Spherical and Cylindrical Bodies in an Acoustic Medium. Internat. Appl. Mech., 23, No. 6, pp. 605-620. https://doi.org/10.1007/BF00887032
  2. Kubenko, V. D. & Dzyuba, V. V. (2000). A Pressure Field in an Acoustic Medium Containing a Cylindrical Solid and a Sphere Oscillating in a Predetermined Manner. Internat. Appl. Mech., 36, No. 12, pp. 1636-1650. https://doi.org/10.1023/A:1011348132172
  3. Kubenko, V. D. & Kuz'ma, A. V. (1999). Influence of the Boundary of a Column of Incompressible Liquid in Investigating Axisymmetric Oscillations of a Solid Sphere in a Cavity. Internat. Appl. Mech., 35, No. 12, pp. 1199-2007. https://doi.org/10.1007/BF02682392
  4. Morse, P. M., Feshbach, H. (1953). Methods of Theoretical Physics in 2 Vol. Part 1. NY: McGraw-Hill.
  5. Handbook of Mathematical Functions (Ed. M. Abramovitz and I.A. Stegun). (1964). NY: Nat. Bureau of Standards.
  6. Erofeenko, V. T. (1972). Relations between main solutions of Helmholtz and Laplace equations in spherical and cylindrical coordinates, Proc. Natl. Acad. Sci. Belorussian SSR 4, pp. 42-46 (in Russian).
  7. Ivanov, E. A. (1958). Diffraction of electro-magnetic waves at two bodies. Minsk: Nauka i technika (in Russian).
  8. Gelfand, I. M., Minlos, R. A., Cummins, G. (translater). (2012). Representations of the Rotation and Lorentz Groups and Their Applications. Martino Fine Books.