§30.13 Wave Equation in Prolate Spheroidal Coordinates

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

where $c$ is a positive constant. (On the use of the symbol $\theta$ in place of $\varphi$ see §1.5(ii).) The $(x,y,z)$-space without the $z$-axis corresponds to

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

The wave equation

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

where $w_1$, $w_2$, $w_3$ satisfy the differential equations

with ${\gamma }^2={\kappa }^2{c}^2\ge 0$ 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

Moreover, $w$ has to be bounded along the $z$-axis away from the focal line: this requires $w_2(\eta )$ to be bounded when . Then $\lambda ={\lambda }_n^m\left({\gamma }^2\right)$ for some $n=m,m+1,m+2,\mathrm{\dots }$, and the general solution of (30.13.10) is

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

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.

Equation (30.13.7) for $\xi \le {\xi }_0$, 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

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 ${\xi }_1\le \xi \le {\xi }_2$ between two ellipsoids, the eigenvalues are determined from

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).