Prolate spheroidal coordinates are related to Cartesian coordinates by
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 , .
The wave equation
transformed to prolate spheroidal coordinates , admits solutions
where , , satisfy the differential equations
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
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 .