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30 Spheroidal Wave FunctionsApplications

§30.13 Wave Equation in Prolate Spheroidal Coordinates


§30.13(i) Prolate Spheroidal Coordinates

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

30.13.1 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 corresponds to

30.13.2 1 <ξ<,
-1 <η<1,
0 ϕ<2π.

The coordinate surfaces ξ=const. are prolate ellipsoids of revolution with foci at x=y=0, z=±c. The coordinate surfaces η=const. are sheets of two-sheeted hyperboloids of revolution with the same foci. The focal line is given by ξ=1, -1η1, and the rays ±zc, x=y=0 are given by η=±1, ξ1.

§30.13(ii) Metric Coefficients

30.13.3 hξ2 =(xξ)2+(yξ)2+(zξ)2=c2(ξ2-η2)ξ2-1,
30.13.4 hη2 =(xη)2+(yη)2+(zη)2=c2(ξ2-η2)1-η2,
30.13.5 hϕ2 =(xϕ)2+(yϕ)2+(zϕ)2=c2(ξ2-1)(1-η2).

§30.13(iii) Laplacian

30.13.6 2=1hξhηhϕ(ξ(hηhϕhξξ)+η(hξhϕhηη)+ϕ(hξhηhϕϕ))=1c2(ξ2-η2)(ξ((ξ2-1)ξ)+η((1-η2)η)+ξ2-η2(ξ2-1)(1-η2)2ϕ2).

§30.13(iv) Separation of Variables

The wave equation

30.13.7 2w+κ2w=0,

transformed to prolate spheroidal coordinates (ξ,η,ϕ), admits solutions

30.13.8 w(ξ,η,ϕ)=w1(ξ)w2(η)w3(ϕ),

where w1, w2, w3 satisfy the differential equations

30.13.9 ddξ((1-ξ2)dw1dξ)+(λ+γ2(1-ξ2)-μ21-ξ2)w1=0,
30.13.10 ddη((1-η2)dw2dη)+(λ+γ2(1-η2)-μ21-η2)w2=0,
30.13.11 d2w3dϕ2+μ2w3=0,

with γ2=κ2c20 and separation constants λ and μ2. 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 μ=m (a nonnegative integer) and

30.13.12 w3(ϕ)=a3cos(mϕ)+b3sin(mϕ).

Moreover, w has to be bounded along the z-axis away from the focal line: 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 general solution of (30.13.10) is

30.13.13 w2(η)=a2Psnm(η,γ2)+b2Qsnm(η,γ2).

The solution of (30.13.9) with μ=m is

30.13.14 w1(ξ)=a1Snm(1)(ξ,γ)+b1Snm(2)(ξ,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 line.

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

Equation (30.13.7) for ξξ0, and subject to 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 γ is determined from the condition

30.13.15 Snm(1)(ξ0,γ)=0.

The corresponding eigenfunctions are given by (30.13.8), (30.13.14), (30.13.13), (30.13.12), with b1=b2=0. For the Dirichlet boundary-value problem of the region ξ1ξξ2 between two ellipsoids, the eigenvalues are determined from

30.13.16 w1(ξ1)=w1(ξ2)=0,

with w1 as in (30.13.14). The corresponding eigenfunctions are given as before with b2=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).