Prolate 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 x-Axis, which is relabeled the z-Axis. The third set of coordinates consists of planes passing through this axis.

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

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

$$z = a\cosh\xi\cos\eta,$$ (3)

where $\xi \in [0, \infty)$, $\eta \in [0, \pi)$, and $\phi \in [0, 2\pi)$. Arfken (1970) uses $(u, v, \varphi)$ instead of $(\xi, \eta, z)$. 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)

$$h_\phi = a \sinh\xi\sin\eta.$$ (6)

The Laplacian is

$\nabla^2 f={1\over\sin\eta\sinh\xi(\sin^2\eta+\sinh^2\xi)}$
$\times\left\{{{\partial\over\partial\xi}\left({\sin\eta\sinh\xi{\partial f\ove... ...i\sin\eta+\csc\eta\sinh\xi){\partial f\over\partial\phi}}\right]}\right\}.\quad$	(7)
$={1\over\sin^2\eta+\sinh^2\xi}\left[{(\csc^2\eta+\mathop{\rm csch}\nolimits ^2\... ...2}+\coth\xi{\partial f\over\partial\xi}+{\partial^2f\over\partial\xi^2}}\right]$	(8)

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

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

$$\xi_2 = \cos\eta$$ (10)

$$\xi_3 = \phi,$$ (11)

where $\xi_1 \in [1,\infty]$, $\xi_2\in [-1,1]$, and $\xi_3\in [0,2\pi)$ (Abramowitz and Stegun 1972). In these coordinates,

$$z = a\xi_1\xi_2$$ (12)

$$x = a\sqrt{({\xi_1}^2-1)(1-{\xi_2}^2)}\,\cos\xi_3$$ (13)

$$y = a\sqrt{({\xi_1}^2-1)(1-{\xi_2}^2)}\,\sin\xi_3.$$ (14)

In terms of the distances from the two Foci,

$$\xi_1 = {r_1+r_2\over 2a}$$ (15)

$$\xi_2 = {r_1-r_2\over 2a}$$ (16)

$$2a = r_{12}.$$ (17)

The Scale Factors are

$$h_{\xi_1} = a\sqrt{{\xi_1}^2-{\xi_2}^2\over{\xi_1}^2-1}$$ (18)

$$h_{\xi_2} = a\sqrt{{\xi_1}^2-{\xi_2}^2\over 1-{\xi_2}^2}$$ (19)

$$h_{\xi_3} = a\sqrt{({\xi_1}^2-1)(1-{\xi_2}^2)},$$ (20)

and the Laplacian is

$\nabla^2 f = {1\over a^2}\left\{{{1\over{\xi_1}^2-{\xi_2}^2} {\partial \over \partial \xi_1}\left[{({\xi_1}^2-1) {\partial f\over \partial \xi_1}}\right]}\right.$
$\mathop{+} \left.{{1\over{\xi_1}^2-{\xi_2}^2}{\partial\over\partial\xi_2}\left... ...({\xi_1}^2-1)(1-{\xi_2}^2)} {\partial^2 f\over\partial{\xi_3}^2}}\right\}.\quad$	(21)

The Helmholtz Differential Equation is separable in prolate spheroidal coordinates.

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

References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Definition of Prolate 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 ($u$, $v$, $\phi$).'' §2.10 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 103-107, 1970.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 661, 1953.