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 $O$.
See for example, G. N. Watson, Bessel functions, Cambridge, 1948, pp. 359-361, or Stratton, Electromagnetic theory, McGraw Hill, 1941, pp. 372-374.
- Victor Twersky, On the nonspecular reflection of sound from planes with absorbent bosses, J. Acoust. Soc. Amer. 23 (1951), 336–338. MR 43718, DOI https://doi.org/10.1121/1.1906768
J. A. Stratton; Electromagnetic theory, McGraw Hill, N. Y., 1941, p. 409.
Infeld and Hull, Reviews Modern Physics, 23, 54 (1951).
J. A. Stratton, Electromagnetic theory, McGraw Hill, 1941, p. 410.
Ibid., p. 578.
Ibid., p. 414.
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.
J. A. Stratton, Electromagnetic theory, McGraw Hill, 1941, p. 414.
Magnus and Oberhettinger, Special functions of mathematical physics, Chelsea, 1949, p. 64-66.
See for example, G. N. Watson, Bessel functions, Cambridge, 1948, pp. 359-361, or Stratton, Electromagnetic theory, McGraw Hill, 1941, pp. 372-374.
V. Twersky, J. Acoust. Soc. Am. 24, 42 (1952); J. Appl. Phys. 23, 407, 1099 (1952).
J. A. Stratton; Electromagnetic theory, McGraw Hill, N. Y., 1941, p. 409.
Infeld and Hull, Reviews Modern Physics, 23, 54 (1951).
J. A. Stratton, Electromagnetic theory, McGraw Hill, 1941, p. 410.
Ibid., p. 578.
Ibid., p. 414.
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.
J. A. Stratton, Electromagnetic theory, McGraw Hill, 1941, p. 414.
Magnus and Oberhettinger, Special functions of mathematical physics, Chelsea, 1949, p. 64-66.
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Article copyright:
© Copyright 1954
American Mathematical Society
