About the Project
30 Spheroidal Wave FunctionsApplications

§30.14 Wave Equation in Oblate Spheroidal Coordinates

  1. §30.14(i) Oblate Spheroidal Coordinates
  2. §30.14(ii) Metric Coefficients
  3. §30.14(iii) Laplacian
  4. §30.14(iv) Separation of Variables
  5. §30.14(v) The Interior Dirichlet Problem for Oblate Ellipsoids

§30.14(i) Oblate Spheroidal Coordinates

Oblate spheroidal coordinates ξ,η,ϕ are related to Cartesian coordinates x,y,z by

30.14.1 x =c(ξ2+1)(1η2)cosϕ,
y =c(ξ2+1)(1η2)sinϕ,
z =cξη,

where c is a positive constant. (On the use of the symbol θ in place of ϕ see §1.5(ii).) The (x,y,z)-space without the z-axis and the disk z=0, x2+y2c2 corresponds to

30.14.2 0 <ξ<,
1 <η<1,
0 ϕ<2π.

The coordinate surfaces ξ=const. are oblate ellipsoids of revolution with focal circle z=0, x2+y2=c2. The coordinate surfaces η=const. are halves of one-sheeted hyperboloids of revolution with the same focal circle. The disk z=0, x2+y2c2 is given by ξ=0, 1η1, and the rays ±z0, x=y=0 are given by η=±1, ξ0.

§30.14(ii) Metric Coefficients

30.14.3 hξ2 =c2(ξ2+η2)1+ξ2,
30.14.4 hη2 =c2(ξ2+η2)1η2,
30.14.5 hϕ2 =c2(ξ2+1)(1η2).

§30.14(iii) Laplacian

30.14.6 2=1c2(ξ2+η2)(ξ((ξ2+1)ξ)+η((1η2)η)+ξ2+η2(ξ2+1)(1η2)2ϕ2).

§30.14(iv) Separation of Variables

The wave equation (30.13.7), transformed to oblate spheroidal coordinates (ξ,η,ϕ), admits solutions of the form (30.13.8), where w1 satisfies the differential equation

30.14.7 ddξ((1+ξ2)dw1dξ)(λ+γ2(1+ξ2)μ21+ξ2)w1=0,

and w2, w3 satisfy (30.13.10) and (30.13.11), respectively, with γ2=κ2c20 and separation constants λ and μ2. Equation (30.14.7) can be transformed to equation (30.2.1) by the substitution z=±iξ.

In most applications the solution w has to be a single-valued function of (x,y,z), which requires μ=m (a nonnegative integer). Moreover, the solution w has to be bounded along the z-axis: this requires w2(η) to be bounded when 1<η<1. Then λ=λnm(γ2) for some n=m,m+1,m+2,, 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 w1(ξ)=a1Snm(1)(iξ,γ)+b1Snm(2)(iξ,γ).

If b1=b2=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 b2=0, then this property holds outside the focal disk.

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

Equation (30.13.7) for ξξ0 together with the boundary condition w=0 on the ellipsoid given by ξ=ξ0, poses an eigenvalue problem with κ2 as spectral parameter. The eigenvalues are given by c2κ2=γ2, where γ2 is determined from the condition

30.14.9 Snm(1)(iξ0,γ)=0.

The corresponding eigenfunctions are then given by (30.13.8), (30.14.8), (30.13.13), (30.13.12), with b1=b2=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).