Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Addition theorems for spherical waves


Authors: Bernard Friedman and Joy Russek
Journal: Quart. Appl. Math. 12 (1954), 13-23
MSC: Primary 33.0X
DOI: https://doi.org/10.1090/qam/60649
MathSciNet review: 60649
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Abstract: Expansions or ``addition theorems'' for the spherical wave functions $ {j_n}\left( {kR} \right)P_n^m\left( {\cos \theta } \right)\exp \left( {im\phi } \right)$, $ h_n^{\left( 1 \right)}\left( {kR} \right)P_n^m\left( {\cos \theta } \right)\exp \left( {im\phi } \right)$, and $ h_n^{\left( 2 \right)}\left( {kR} \right)P_n^m\left( {\cos \theta } \right)\exp \left( {im\phi } \right)$, with reference to the origin $ O$, have been obtained in terms of spherical wave functions with reference to the origin $ O'$, where $ O'$, has the coordinates $ \left( {{r_0},{\theta _0},{\phi _0}} \right)$ with respect to $ O$.


References [Enhancements On Off] (What's this?)

  • [1] See for example, G. N. Watson, Bessel functions, Cambridge, 1948, pp. 359-361, or Stratton, Electromagnetic theory, McGraw Hill, 1941, pp. 372-374.
  • [2] Victor Twersky, On the nonspecular reflection of sound from planes with absorbent bosses, J. Acoust. Soc. Amer. 23 (1951), 336–338. MR 0043718, https://doi.org/10.1121/1.1906768
  • [3] J. A. Stratton; Electromagnetic theory, McGraw Hill, N. Y., 1941, p. 409.
  • [4] Infeld and Hull, Reviews Modern Physics, 23, 54 (1951).
  • [5] J. A. Stratton, Electromagnetic theory, McGraw Hill, 1941, p. 410.
  • [6] Ibid., p. 578.
  • [7] Ibid., p. 414.
  • [8] A similar integral representation for $ h_n^{\left( 2 \right)}\left( {kR} \right)P_n^m\left( {\cos \theta } \right)\exp \left( {im\phi } \right)$ has been obtained by Satô, Yasuo, Bull. Earthquake Res. Institute, Tokyo, 28, 1-22 and 175-217 (1950). In this paper Yasuo obtains an addition theorem for $ h_n^{(2)}\left( {kR} \right)P_n^m\left( {\cos \theta } \right)\exp \left( {im\phi } \right)$ also; however, his expansion is valid only for a translation of the origin along the $ z$-axis. The addition theorem he obtains contains powers of $ kr$ instead of the Bessel functions $ {j^v}\left( {kr} \right)$, and the coefficients of his expansion are not given by a general formula but must be evaluated by a recurrence relation.
  • [9] J. A. Stratton, Electromagnetic theory, McGraw Hill, 1941, p. 414.
  • [10] Magnus and Oberhettinger, Special functions of mathematical physics, Chelsea, 1949, p. 64-66.

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DOI: https://doi.org/10.1090/qam/60649
Article copyright: © Copyright 1954 American Mathematical Society


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