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Oblate Spheroidal Wave Function

The wave equation in Oblate Spheroidal Coordinates is

$\nabla^2\Phi+k^2\Phi={\partial\over\partial\xi_1}\left[{({\xi_1}^2+1){\partial\...
...}^2)}{\partial^2\Phi\over\partial\phi^2} + c^2({\xi_1}^2+{\xi_2}^2)\Phi=0,\quad$ (1)
where

\begin{displaymath}
c\equiv {\textstyle{1\over 2}}ak.
\end{displaymath} (2)

Substitute in a trial solution
\begin{displaymath}
\Phi=R_{mn}(c,\xi_1)S_{mn}(c,\xi_2){\cos \atop \sin}(m\phi).
\end{displaymath} (3)

The radial differential equation is


\begin{displaymath}
{d\over d\xi_2} \left[{(1+{\xi_2}^2){d\over d\xi_2} S_{mn}(c...
...-c^2{\xi_2}^2+{m^2\over 1+{\xi_2}^2}}\right)R_{mn}(c,\xi_2)=0,
\end{displaymath} (4)

and the angular differential equation is


\begin{displaymath}
{d\over d\xi_2} \left[{(1-{\xi_2}^2){d\over d\xi_2} S_{mn}(c...
...}-c^2{\xi_2}^2+{m^2\over 1-{\xi_2}^2}}\right)R_{mn}(c,\xi_2)=0
\end{displaymath} (5)

(Abramowitz and Stegun 1972, pp. 753-755).

See also Prolate Spheroidal Wave Function


References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Spheroidal Wave Functions.'' Ch. 21 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 751-759, 1972.




© 1996-9 Eric W. Weisstein
1999-05-26