# §30.13 Wave Equation in Prolate Spheroidal Coordinates

## §30.13(i) Prolate Spheroidal Coordinates

Prolate spheroidal coordinates are related to Cartesian coordinates by

30.13.1

where is a positive constant. (On the use of the symbol in place of see §1.5(ii).) The -space without the -axis corresponds to

The coordinate surfaces are prolate ellipsoids of revolution with foci at , . The coordinate surfaces are sheets of two-sheeted hyperboloids of revolution with the same foci. The focal line is given by , , and the rays , are given by , .

## §30.13(iv) Separation of Variables

The wave equation

30.13.7

transformed to prolate spheroidal coordinates , admits solutions

where , , satisfy the differential equations

with and separation constants and . Equations (30.13.9) and (30.13.10) agree with (30.2.1).

In most applications the solution has to be a single-valued function of , which requires (a nonnegative integer) and

Moreover, has to be bounded along the -axis away from the focal line: this requires to be bounded when . Then for some , and the general solution of (30.13.10) is

The solution of (30.13.9) with is

If , then the function (30.13.8) is a twice-continuously differentiable solution of (30.13.7) in the entire -space. If , then this property holds outside the focal line.

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

Equation (30.13.7) for , and subject to the boundary condition on the ellipsoid given by , poses an eigenvalue problem with as spectral parameter. The eigenvalues are given by , where is determined from the condition

The corresponding eigenfunctions are given by (30.13.8), (30.13.14), (30.13.13), (30.13.12), with . For the Dirichlet boundary-value problem of the region between two ellipsoids, the eigenvalues are determined from

with as in (30.13.14). The corresponding eigenfunctions are given as before with .

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).