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On Green's function for the reduced wave equation in a spherical annular domain with Dirichlet's boundary conditions

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Summary

In this study Green's function for the reduced wave equation (Helmholtz equation) in a spherical annular domain with Dirichlet's boundary conditions is derived. The convergence of the series solution representing Green's function is then established. Finally it is shown that Green's function for the Dirichlet problem reduces to Green's function for the exterior of a sphere as given by Franz and Etiènne, when the outer radius is moved towards infinity, and when a special position of the coordinate system is chosen.

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References

  1. 1)

    Franz, W., Z. Naturforschung9a (1954) 705.

  2. 2)

    Etiènne, J., Bull. Soc. Roy. des Sciences de Liège,30 (1961) 416.

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    Jeffreys, H. and E. R. Lapwood, Proc. Roy. Soc. London00A (1957) 455.

  4. 4)

    Martinek, J. and H. P. Thielman, Laurent Type of Expansion, General Radiation Conditions related to Solutions of the Reduced Wave equation. Report No. 107, United Electro Dynamics, Inc., also in print in “Acta Mechanica”.

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    Sommerfeld, A., Partial Differential Equations in Physics, Academic Press, New York, 1949. p. 199.

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    Gray, Andrew and G. B. Mathew, A treatise on Bessel Functions, Macmillan, London, 1922.

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Martinek, J., Thielman, H.P. On Green's function for the reduced wave equation in a spherical annular domain with Dirichlet's boundary conditions. Appl. Sci. Res. 12, 315–324 (1965). https://doi.org/10.1007/BF00382130

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Keywords

  • Boundary Condition
  • Coordinate System
  • Wave Equation
  • Dirichlet Problem
  • Special Position