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30 Spheroidal Wave FunctionsApplications

§30.13 Wave Equation in Prolate Spheroidal Coordinates

  1. §30.13(i) Prolate Spheroidal Coordinates
  2. §30.13(ii) Metric Coefficients
  3. §30.13(iii) Laplacian
  4. §30.13(iv) Separation of Variables
  5. §30.13(v) The Interior Dirichlet Problem for Prolate Ellipsoids

§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(ξ21)(1η2)cosϕ,
y =c(ξ21)(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 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(ξ2η2)ξ21,
30.13.4 hη2 =(xη)2+(yη)2+(zη)2=c2(ξ2η2)1η2,
30.13.5 hϕ2 =(xϕ)2+(yϕ)2+(zϕ)2=c2(ξ21)(1η2).

§30.13(iii) Laplacian

30.13.6 2=1hξhηhϕ(ξ(hηhϕhξξ)+η(hξhϕhηη)+ϕ(hξhηhϕϕ))=1c2(ξ2η2)(ξ((ξ21)ξ)+η((1η2)η)+ξ2η2(ξ21)(1η2)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((1ξ2)dw1dξ)+(λ+γ2(1ξ2)μ21ξ2)w1=0,
30.13.10 ddη((1η2)dw2dη)+(λ+γ2(1η2)μ21η2)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).