Oblate Spheroidal Coordinates

A system of Curvilinear Coordinates in which two sets of coordinate surfaces are obtained by revolving the curves of the Elliptic Cylindrical Coordinates about the y-Axis which is relabeled the z-Axis. The third set of coordinates consists of planes passing through this axis.

$$x = a\cosh\xi\cos\eta\cos\phi$$ (1)

$$y = a\cosh\xi\cos\eta\sin\phi$$ (2)

$$z = a\sinh\xi\sin\eta,$$ (3)

where $\xi \in [0,\infty)$, $\eta \in [-\pi/2, \pi/2]$, and $\phi \in [0,2\pi)$. Arfken (1970) uses $(u, v, \varphi)$ instead of $(\xi, \eta, \phi)$. The Scale Factors are

$$h_\xi = a\sqrt{\sinh^2\xi+\sin^2\eta}$$ (4)

$$h_\eta = a\sqrt{\sinh^2\xi+\sin^2\eta}$$ (5)

$$h_\phi = a \cosh \xi \cos\eta.$$ (6)

The Laplacian is

$\nabla^2 f = {1\over a^3(\sinh^2\xi+\sin^2\eta)\cosh\xi\cos\eta}$
$\times \left[{{\partial f\over\partial\xi}\left({a\cosh\xi\cos\eta{\partial f\... ...i+\sin^2\eta)\over a\cosh\xi\cos\eta}{\partial^2 f\over\partial\phi^2 }}\right]$
$={1\over a^3(\sinh^2\xi+\sin^2\eta)\cosh\xi\cos\eta} \left[{a\sinh\xi\cos\eta{\partial f\over\partial\xi}}\right.$
$+a\cosh\xi\cos\eta{\partial^2 f\over\partial\xi^2}+a\sinh\xi\cos\eta{\partial f\over\partial\eta}$
$\left.{+a\cosh\xi\cos\eta{\partial^2 f\over\partial\eta^2}}\right]+{1\over a^2(\sinh^2\xi+\sin^2\eta)}{\partial^2 f\over\partial\phi^2}$
$={1\over a^2 (\sinh^2\xi+\sin^2\eta)}\left[{{1\over\cosh\xi}{\partial\over\part... ...}\right]+ {1\over a^2(\cosh^2\xi +\cos^2\eta)}{\partial^2 f\over\partial\phi^2}$	(7)
$={1\over\sin^2\eta+\sinh^2\xi}\left[{(\mathop{\rm sech}\nolimits ^2\xi\tan^2\et... ...ver\partial\xi^2} -\tan\eta{\partial\over\eta}+{\partial^2\over\eta^2}}\right].$	(8)

An alternate form useful for ``two-center'' problems is defined by

$$\xi_1 = \sinh\xi$$ (9)

$$\xi_1' = \cosh\xi$$ (10)

$$\xi_2 = \cos\eta$$ (11)

$$\xi_3 = \phi,$$ (12)

where $\xi_1 \in [1,\infty]$, $\xi_2\in [-1,1]$, and $\xi_3\in [0,2\pi)$. In these coordinates,

$$y = a\xi_1'\xi_2\sin\xi_3$$ (13)

$$z = a\sqrt{({\xi_1'}^2-1)(1-{\xi_2}^2)}$$ (14)

$$x = a\xi_1'\xi_2\cos\xi_3$$ (15)

(Abramowitz and Stegun 1972). The Scale Factors are

$$h_{\xi_1} = a\sqrt{{\xi_1}^2-{\xi_2}^2\over{\xi_1}^2-1}$$ (16)

$$h_{\xi_2} = a\sqrt{{\xi_1}^2-{\xi_2}^2\over 1-{\xi_2}^2}$$ (17)

$$h_{\xi_3} = a\xi\eta,$$ (18)

and the Laplacian is

$$\nabla^2 f= {1\over a^2}\left\{{{1\over{\xi_1}^2+{\xi_2}^2}{... ...(1-{\xi_2}^2)} {\partial^2 f\over\partial {\xi_3}^2}}\right\}.$$ (19)

The Helmholtz Differential Equation is separable.

See also Helmholtz Differential Equation--Oblate Spheroidal Coordinates, Latitude, Longitude, Prolate Spheroidal Coordinates, Spherical Coordinates

References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Definition of Oblate Spheroidal Coordinates.'' §21.2 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 752, 1972.

Arfken, G. ``Prolate Spheroidal Coordinates ($u$, $v$, $\phi$).'' §2.11 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 107-109, 1970.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 663, 1953.