# §30.13 Wave Equation in Prolate Spheroidal Coordinates

## §30.13(i) Prolate Spheroidal Coordinates

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

 30.13.1 $\displaystyle x$ $\displaystyle=c\sqrt{(\xi^{2}-1)(1-\eta^{2})}\mathop{\cos\/}\nolimits\phi,$ $\displaystyle y$ $\displaystyle=c\sqrt{(\xi^{2}-1)(1-\eta^{2})}\mathop{\sin\/}\nolimits\phi,$ $\displaystyle z$ $\displaystyle=c\xi\eta,$ Defines: $\xi$: prolate spheroidal coordinate (locally), $\eta$: prolate spheroidal coordinate (locally) and $\phi$: prolate spheroidal coordinate (locally) Symbols: $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function, $\mathop{\sin\/}\nolimits\NVar{z}$: sine function, $z$: complex variable, $x$: real variable, $y$: real variable and $c$: positive constant A&S Ref: 21.2.2 (in different form) Referenced by: §30.13(i) Permalink: http://dlmf.nist.gov/30.13.E1 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for 30.13(i)

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 corresponds to

 30.13.2 $\displaystyle 1$ $\displaystyle<\xi<\infty,$ $\displaystyle-1$ $\displaystyle<\eta<1,$ $\displaystyle 0$ $\displaystyle\leq\phi<2\pi.$

The coordinate surfaces $\xi=\mbox{const}.$ are prolate ellipsoids of revolution with foci at $x=y=0$, $z=\pm c$. The coordinate surfaces $\eta=\mbox{const}.$ are sheets of two-sheeted hyperboloids of revolution with the same foci. The focal line is given by $\xi=1$, $-1\leq\eta\leq 1$, and the rays $\pm z\geq c$, $x=y=0$ are given by $\eta=\pm 1$, $\xi\geq 1$.

## §30.13(ii) Metric Coefficients

 30.13.3 $\displaystyle h_{\xi}^{2}$ $\displaystyle=\left(\frac{\partial x}{\partial\xi}\right)^{2}+\left(\frac{% \partial y}{\partial\xi}\right)^{2}+\left(\frac{\partial z}{\partial\xi}\right% )^{2}=\frac{c^{2}(\xi^{2}-\eta^{2})}{\xi^{2}-1},$ 30.13.4 $\displaystyle h_{\eta}^{2}$ $\displaystyle=\left(\frac{\partial x}{\partial\eta}\right)^{2}+\left(\frac{% \partial y}{\partial\eta}\right)^{2}+\left(\frac{\partial z}{\partial\eta}% \right)^{2}=\frac{c^{2}(\xi^{2}-\eta^{2})}{1-\eta^{2}},$ 30.13.5 $\displaystyle h_{\phi}^{2}$ $\displaystyle=\left(\frac{\partial x}{\partial\phi}\right)^{2}+\left(\frac{% \partial y}{\partial\phi}\right)^{2}+\left(\frac{\partial z}{\partial\phi}% \right)^{2}=c^{2}(\xi^{2}-1)(1-\eta^{2}).$

## §30.13(iii) Laplacian

 30.13.6 $\nabla^{2}=\frac{1}{h_{\xi}h_{\eta}h_{\phi}}\left(\frac{\partial}{\partial\xi}% \left(\frac{h_{\eta}h_{\phi}}{h_{\xi}}\frac{\partial}{\partial\xi}\right)+% \frac{\partial}{\partial\eta}\left(\frac{h_{\xi}h_{\phi}}{h_{\eta}}\frac{% \partial}{\partial\eta}\right)+\frac{\partial}{\partial\phi}\left(\frac{h_{\xi% }h_{\eta}}{h_{\phi}}\frac{\partial}{\partial\phi}\right)\right)=\frac{1}{c^{2}% (\xi^{2}-\eta^{2})}\left(\frac{\partial}{\partial\xi}\left((\xi^{2}-1)\frac{% \partial}{\partial\xi}\right)+\frac{\partial}{\partial\eta}\left((1-\eta^{2})% \frac{\partial}{\partial\eta}\right)+\frac{\xi^{2}-\eta^{2}}{(\xi^{2}-1)(1-% \eta^{2})}\frac{{\partial}^{2}}{{\partial\phi}^{2}}\right).$

## §30.13(iv) Separation of Variables

The wave equation

 30.13.7 $\nabla^{2}w+\kappa^{2}w=0,$ Defines: $\kappa$: parameter (locally) Symbols: $w_{j}$: solution to DE Referenced by: §30.13(iv), §30.13(v), §30.14(iv), §30.14(iv), §30.14(v) Permalink: http://dlmf.nist.gov/30.13.E7 Encodings: TeX, pMML, png See also: Annotations for 30.13(iv)

transformed to prolate spheroidal coordinates $(\xi,\eta,\phi)$, admits solutions

 30.13.8 $w(\xi,\eta,\phi)=w_{1}(\xi)w_{2}(\eta)w_{3}(\phi),$ Symbols: $\xi$: prolate spheroidal coordinate, $\eta$: prolate spheroidal coordinate, $\phi$: prolate spheroidal coordinate and $w_{j}$: solution to DE Referenced by: §30.13(iv), §30.13(v), §30.14(iv), §30.14(iv), §30.14(v) Permalink: http://dlmf.nist.gov/30.13.E8 Encodings: TeX, pMML, png See also: Annotations for 30.13(iv)

where $w_{1}$, $w_{2}$, $w_{3}$ satisfy the differential equations

 30.13.9 $\frac{\mathrm{d}}{\mathrm{d}\xi}\left((1-\xi^{2})\frac{\mathrm{d}w_{1}}{% \mathrm{d}\xi}\right)+\left(\lambda+\gamma^{2}(1-\xi^{2})-\frac{\mu^{2}}{1-\xi% ^{2}}\right)w_{1}=0,$
 30.13.10 $\frac{\mathrm{d}}{\mathrm{d}\eta}\left((1-\eta^{2})\frac{\mathrm{d}w_{2}}{% \mathrm{d}\eta}\right)+\left(\lambda+\gamma^{2}(1-\eta^{2})-\frac{\mu^{2}}{1-% \eta^{2}}\right)w_{2}=0,$
 30.13.11 $\frac{{\mathrm{d}}^{2}w_{3}}{{\mathrm{d}\phi}^{2}}+\mu^{2}w_{3}=0,$

with $\gamma^{2}=\kappa^{2}c^{2}\geq 0$ and separation constants $\lambda$ and $\mu^{2}$. Equations (30.13.9) and (30.13.10) agree with (30.2.1).

In most applications the solution $w$ has to be a single-valued function of $(x,y,z)$, which requires $\mu=m$ (a nonnegative integer) and

 30.13.12 $w_{3}(\phi)=a_{3}\mathop{\cos\/}\nolimits\!\left(m\phi\right)+b_{3}\mathop{% \sin\/}\nolimits\!\left(m\phi\right).$

Moreover, $w$ has to be bounded along the $z$-axis away from the focal line: this requires $w_{2}(\eta)$ to be bounded when $-1<\eta<1$. Then $\lambda=\mathop{\lambda^{m}_{n}\/}\nolimits\!\left(\gamma^{2}\right)$ for some $n=m,m+1,m+2,\dots$, and the general solution of (30.13.10) is

 30.13.13 $w_{2}(\eta)=a_{2}\mathop{\mathsf{Ps}^{m}_{n}\/}\nolimits\!\left(\eta,\gamma^{2% }\right)+b_{2}\mathop{\mathsf{Qs}^{m}_{n}\/}\nolimits\!\left(\eta,\gamma^{2}% \right).$

The solution of (30.13.9) with $\mu=m$ is

 30.13.14 $w_{1}(\xi)=a_{1}\mathop{S^{m(1)}_{n}\/}\nolimits\!\left(\xi,\gamma\right)+b_{1% }\mathop{S^{m(2)}_{n}\/}\nolimits\!\left(\xi,\gamma\right).$

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 line.

## §30.13(v) The Interior Dirichlet Problem for Prolate Ellipsoids

Equation (30.13.7) for $\xi\leq\xi_{0}$, and subject to 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 $\gamma$ is determined from the condition

 30.13.15 $\mathop{S^{m(1)}_{n}\/}\nolimits\!\left(\xi_{0},\gamma\right)=0.$

The corresponding eigenfunctions are given by (30.13.8), (30.13.14), (30.13.13), (30.13.12), with $b_{1}=b_{2}=0$. For the Dirichlet boundary-value problem of the region $\xi_{1}\leq\xi\leq\xi_{2}$ between two ellipsoids, the eigenvalues are determined from

 30.13.16 $w_{1}(\xi_{1})=w_{1}(\xi_{2})=0,$ Symbols: $\xi$: prolate spheroidal coordinate and $w_{j}$: solution to DE Permalink: http://dlmf.nist.gov/30.13.E16 Encodings: TeX, pMML, png See also: Annotations for 30.13(v)

with $w_{1}$ as in (30.13.14). The corresponding eigenfunctions are given as before with $b_{2}=0$.

For further applications see Meixner and Schäfke (1954), Meixner et al. (1980) and the references cited therein; also Ong (1986), Müller et al. (1994), and Xiao et al. (2001).