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

$$c\equiv {\textstyle{1\over 2}}ak.$$ (2)

Substitute in a trial solution

$$\Phi=R_{mn}(c,\xi_1)S_{mn}(c,\xi_2){\cos \atop \sin}(m\phi).$$ (3)

The radial differential equation is

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

and the angular differential equation is

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