# §30.14 Wave Equation in Oblate Spheroidal Coordinates

## §30.14(i) Oblate Spheroidal Coordinates

Oblate spheroidal coordinates $\xi,\eta,\phi$ are related to Cartesian coordinates $x,y,z$ by

 30.14.1 $\displaystyle x$ $\displaystyle=c\sqrt{(\xi^{2}+1)(1-\eta^{2})}\cos\phi,$ $\displaystyle y$ $\displaystyle=c\sqrt{(\xi^{2}+1)(1-\eta^{2})}\sin\phi,$ $\displaystyle z$ $\displaystyle=c\xi\eta,$ ⓘ Defines: $\xi$: oblate spheroidal coordinate (locally), $\eta$: oblate spheroidal coordinate (locally) and $\phi$: oblate spheroidal coordinate (locally) Symbols: $\cos\NVar{z}$: cosine function, $\sin\NVar{z}$: sine function, $z$: complex variable, $x$: real variable, $y$: real variable and $c$: positive constant A&S Ref: 21.3.2 (in different form) Referenced by: §30.14(i) Permalink: http://dlmf.nist.gov/30.14.E1 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for 30.14(i), 30.14 and 30

where $c$ is a positive constant. (On the use of the symbol $\theta$ in place of $\phi$ see §1.5(ii).) The $(x,y,z)$-space without the $z$-axis and the disk $z=0$, $x^{2}+y^{2}\leq c^{2}$ corresponds to

 30.14.2 $\displaystyle 0$ $\displaystyle<\xi<\infty,$ $\displaystyle-1$ $\displaystyle<\eta<1,$ $\displaystyle 0$ $\displaystyle\leq\phi<2\pi.$

The coordinate surfaces $\xi=\mbox{const}.$ are oblate ellipsoids of revolution with focal circle $z=0$, $x^{2}+y^{2}=c^{2}$. The coordinate surfaces $\eta=\mbox{const}.$ are halves of one-sheeted hyperboloids of revolution with the same focal circle. The disk $z=0$, $x^{2}+y^{2}\leq c^{2}$ is given by $\xi=0$, $-1\leq\eta\leq 1$, and the rays $\pm z\geq 0$, $x=y=0$ are given by $\eta=\pm 1$, $\xi\geq 0$.

## §30.14(ii) Metric Coefficients

 30.14.3 $\displaystyle h_{\xi}^{2}$ $\displaystyle=\frac{c^{2}(\xi^{2}+\eta^{2})}{1+\xi^{2}},$ ⓘ Symbols: $h_{\xi},h_{\eta},h_{\phi}$: metric coefficients, $\xi$: oblate spheroidal coordinate, $\eta$: oblate spheroidal coordinate and $c$: positive constant A&S Ref: 21.4.3 (in different form) Permalink: http://dlmf.nist.gov/30.14.E3 Encodings: TeX, pMML, png See also: Annotations for 30.14(ii), 30.14 and 30 30.14.4 $\displaystyle h_{\eta}^{2}$ $\displaystyle=\frac{c^{2}(\xi^{2}+\eta^{2})}{1-\eta^{2}},$ 30.14.5 $\displaystyle h_{\phi}^{2}$ $\displaystyle=c^{2}(\xi^{2}+1)(1-\eta^{2}).$

## §30.14(iii) Laplacian

 30.14.6 $\nabla^{2}=\frac{1}{c^{2}(\xi^{2}+\eta^{2})}\left(\frac{\partial}{\partial\xi}% \left((\xi^{2}+1)\frac{\partial}{\partial\xi}\right)+\frac{\partial}{\partial% \eta}\left((1-\eta^{2})\frac{\partial}{\partial\eta}\right)+\frac{\xi^{2}+\eta% ^{2}}{(\xi^{2}+1)(1-\eta^{2})}\frac{{\partial}^{2}}{{\partial\phi}^{2}}\right).$

## §30.14(iv) Separation of Variables

The wave equation (30.13.7), transformed to oblate spheroidal coordinates $(\xi,\eta,\phi)$, admits solutions of the form (30.13.8), where $w_{1}$ satisfies the differential equation

 30.14.7 $\frac{\mathrm{d}}{\mathrm{d}\xi}\left((1+\xi^{2})\frac{\mathrm{d}w_{1}}{% \mathrm{d}\xi}\right)-\left(\lambda+\gamma^{2}(1+\xi^{2})-\frac{\mu^{2}}{1+\xi% ^{2}}\right)w_{1}=0,$

and $w_{2}$, $w_{3}$ satisfy (30.13.10) and (30.13.11), respectively, with $\gamma^{2}=-\kappa^{2}c^{2}\leq 0$ and separation constants $\lambda$ and $\mu^{2}$. Equation (30.14.7) can be transformed to equation (30.2.1) by the substitution $z=\pm i\xi$.

In most applications the solution $w$ has to be a single-valued function of $(x,y,z)$, which requires $\mu=m$ (a nonnegative integer). Moreover, the solution $w$ has to be bounded along the $z$-axis: this requires $w_{2}(\eta)$ to be bounded when $-1<\eta<1$. Then $\lambda=\lambda^{m}_{n}\left(\gamma^{2}\right)$ for some $n=m,m+1,m+2,\dots$, and the solution of (30.13.10) is given by (30.13.13). The solution of (30.14.7) is given by

 30.14.8 $w_{1}(\xi)=a_{1}S^{m(1)}_{n}\left(i\xi,\gamma\right)+b_{1}S^{m(2)}_{n}\left(i% \xi,\gamma\right).$

If $b_{1}=b_{2}=0$, then the function (30.13.8) is a twice-continuously differentiable solution of (30.13.7) in the entire $(x,y,z)$-space. If $b_{2}=0$, then this property holds outside the focal disk.

## §30.14(v) The Interior Dirichlet Problem for Oblate Ellipsoids

Equation (30.13.7) for $\xi\leq\xi_{0}$ together with the boundary condition $w=0$ on the ellipsoid given by $\xi=\xi_{0}$, poses an eigenvalue problem with $\kappa^{2}$ as spectral parameter. The eigenvalues are given by $c^{2}\kappa^{2}=-\gamma^{2}$, where $\gamma^{2}$ is determined from the condition

 30.14.9 $S^{m(1)}_{n}\left(\mathrm{i}\xi_{0},\gamma\right)=0.$

The corresponding eigenfunctions are then given by (30.13.8), (30.14.8), (30.13.13), (30.13.12), with $b_{1}=b_{2}=0$.

For further applications see Meixner and Schäfke (1954), Meixner et al. (1980) and the references cited therein; also Kokkorakis and Roumeliotis (1998) and Li et al. (1998b).