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30 Spheroidal Wave FunctionsApplications

§30.14 Wave Equation in Oblate Spheroidal Coordinates


§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)(1-η2)cosϕ,
y =c(ξ2+1)(1-η2)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)1-η2,
30.14.5 hϕ2 =c2(ξ2+1)(1-η2).

§30.14(iii) Laplacian

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