Rotational-translational addition theorems for scalar spheroidal wave functions
Authors:
R. H. MacPhie, J. Dalmas and R. Deleuil
Journal:
Quart. Appl. Math. 44 (1987), 737-749
MSC:
Primary 33A55
DOI:
https://doi.org/10.1090/qam/872824
MathSciNet review:
872824
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: Rotational-translational addition theorems for the scalar spheroidal wave function $\psi _{mn}^{\left ( i \right )}\left ( {h;\eta ,\xi ,\phi } \right )$, with $i = 1,3,4$, are deduced. This permits one to represent the $m{n^{th}}$ scalar spheroidal wave function, associated with one spheroidal coordinate system $\left ( {{h_q};{\eta _q},{\xi _q},{\phi _q}} \right )$ centered at its local origin ${O_q}$, by an addition series of spheroidal wave functions associated with a second rotated and translated system $\left ( {{h_r};{\eta _r},{\xi _r},{\phi _r}} \right )$, centered at ${O_r}$. Such theorems are necessary in the rigorous analysis of radiation and scattering by spheroids with arbitrary spacings and orientations.
- Bernard Friedman and Joy Russek, Addition theorems for spherical waves, Quart. Appl. Math. 12 (1954), 13–23. MR 60649, DOI https://doi.org/10.1090/S0033-569X-1954-60649-8
- Seymour Stein, Addition theorems for spherical wave functions, Quart. Appl. Math. 19 (1961), 15–24. MR 120407, DOI https://doi.org/10.1090/S0033-569X-1961-0120407-5
- Orval R. Cruzan, Translational addition theorems for spherical vector wave functions, Quart. Appl. Math. 20 (1962/63), 33–40. MR 132851, DOI https://doi.org/10.1090/S0033-569X-1962-0132851-2
J. Bruning and Y. T. Lo, Multiple scattering of EM waves by spheres, part I and part II, IEEE Trans. AP-19, 378–400 (1971)
- Bateshwar P. Sinha and Robert H. Macphie, Translational addition theorems for spheroidal scalar and vector wave functions, Quart. Appl. Math. 38 (1980/81), no. 2, 143–158. MR 580875, DOI https://doi.org/10.1090/S0033-569X-1980-0580875-9
J. Dalmas and R. Deleuil, Diffusion multiple des ondes électromagnétiques par des ellipsoïdes de révolution allongés. Opt. Acta 29, 1117–1131 (1982)
- Jeannine Dalmas and Roger Deleuil, Translational addition theorems for prolate spheroidal vector wave functions ${\bf M}^r$ and ${\bf N}^r$, Quart. Appl. Math. 44 (1986), no. 2, 213–222. MR 856176, DOI https://doi.org/10.1090/S0033-569X-1986-0856176-1
B. P. Sinha and R. H. MacPhie, Electromagnetic plane wave scattering by a system of two parallel conducting prolate spheroids, IEEE Trans. AP-31, 294–304 (1983)
J. Dalmas and R. Deleuil, Multiple scattering of electromagnetic waves from two infinitely conducting prolate spheroids which are centered in a plane perpendicular to their axes of revolution, Radio Sci. 20, 575–581 (1985)
B. P. Sinha and R. H. MacPhie, Mutual admittance characteristics for two-element parallel prolate spheroidal antenna systems, IEEE Trans. AP-33, 1255–1263 (1985)
- Carson Flammer, Spheroidal wave functions, Stanford University Press, Stanford, California, 1957. MR 0089520
- A. R. Edmonds, Angular momentum in quantum mechanics, Investigations in Physics, Vol. 4, Princeton University Press, Princeton, N.J., 1957. MR 0095700
J. A. Stratton, Electromagnetic theory, McGraw-Hill, New York (1941)
M. E. Rose, Elementary theory of angular momentum, Wiley, New York (1967)
D. M. Brink and G. R. Satchler, Angular momentum, Clarendon, Oxford (1962)
A. Erdelyi, Higher transcendental functions, McGraw-Hill, New York (1953)
M. Abramowitz and I. Stegun, Handbook of mathematical functions, Dover, New York (1965)
- I. M. Gel′fand and Z. Ya. Šapiro, Representations of the group of rotations of $3$-dimensional space and their applications, Amer. Math. Soc. Transl. (2) 2 (1956), 207–316. MR 0076290
- S. L. Altmann and C. J. Bradley, A note on the calculation of the matrix elements of the rotation group, Philos. Trans. Roy. Soc. London Ser. A 255 (1963), 193–198. MR 148215, DOI https://doi.org/10.1098/rsta.1963.0001
B. Friedman and J. Russek, Addition theorems for spherical waves, Quart. Appl. Math. 12, 13–23 (1954)
S. Stein, Addition theorems for spherical wave functions, Quart. Appl. Math. 19, 15–24 (1961)
O. Cruzan, Translational addition theorems for spherical vector wave functions, Quart. Appl. Math. 20, 33–40 (1962)
J. Bruning and Y. T. Lo, Multiple scattering of EM waves by spheres, part I and part II, IEEE Trans. AP-19, 378–400 (1971)
B. P. Sinha and R. H. MacPhie, Translational addition theorems for spheroidal scalar and vector wave functions, Quart. Appl. Math. 38, 143–158 (1980)
J. Dalmas and R. Deleuil, Diffusion multiple des ondes électromagnétiques par des ellipsoïdes de révolution allongés. Opt. Acta 29, 1117–1131 (1982)
J. Dalmas and R. Deleuil, Translational addition theorems for prolate spheroidal vector wave functions $M^{r}$ and $N^{1}$, Quart. Appl. Math. 44, 213–222 (1986)
B. P. Sinha and R. H. MacPhie, Electromagnetic plane wave scattering by a system of two parallel conducting prolate spheroids, IEEE Trans. AP-31, 294–304 (1983)
J. Dalmas and R. Deleuil, Multiple scattering of electromagnetic waves from two infinitely conducting prolate spheroids which are centered in a plane perpendicular to their axes of revolution, Radio Sci. 20, 575–581 (1985)
B. P. Sinha and R. H. MacPhie, Mutual admittance characteristics for two-element parallel prolate spheroidal antenna systems, IEEE Trans. AP-33, 1255–1263 (1985)
C. Flammer, Spheroidal wave functions, Stanford Univ. Press, Stanford, Calif. (1957)
A. R. Edmonds, Angular momentum in quantum mechanics, Princeton Univ. Press, Princeton, N. J. (1957)
J. A. Stratton, Electromagnetic theory, McGraw-Hill, New York (1941)
M. E. Rose, Elementary theory of angular momentum, Wiley, New York (1967)
D. M. Brink and G. R. Satchler, Angular momentum, Clarendon, Oxford (1962)
A. Erdelyi, Higher transcendental functions, McGraw-Hill, New York (1953)
M. Abramowitz and I. Stegun, Handbook of mathematical functions, Dover, New York (1965)
I. M. Gel’Fand and Z. Ya. S̃apiro, Representations of the group theory of rotations of a 3-dimensional space and their applications, Ann. Math. Soc. Transl., Ser. 2, 2, 207–316 (1956)
S. L. Altmann and C. J. Bradley, A note on the calculation of the matrix elements of the rotation group, Philos. Trans. Roy. Soc. London, Ser. A, 255, 193–198 (1963)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
33A55
Retrieve articles in all journals
with MSC:
33A55
Additional Information
Article copyright:
© Copyright 1987
American Mathematical Society