§30.14 Wave Equation in Oblate Spheroidal Coordinates

Referenced by:: ¶ ‣ §1.5(ii), §30.1, §30.2(i)
Permalink:: http://dlmf.nist.gov/30.14

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

§30.14(i) Oblate Spheroidal Coordinates

Notes:: See Erdélyi et al. (1955, §16.1.3) and Meixner and Schäfke (1954, §1.124).
Keywords:: coordinate systems, oblate spheroidal coordinates, wave equation
Permalink:: http://dlmf.nist.gov/30.14.i

Oblate spheroidal coordinates $\xi,\eta,\phi$ are related to Cartesian coordinates $x,y,z$ by

30.14.1

$x=c\sqrt{(\xi^{2}+1)(1-\eta^{2})}\mathop{\cos\/}\nolimits\phi,$

$y=c\sqrt{(\xi^{2}+1)(1-\eta^{2})}\mathop{\sin\/}\nolimits\phi,$

$z=c\xi\eta,$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $z$ : complex variable, $x$ : real variable, $y$ : real variable, $\xi$ : oblate spheroidal coordinate, $\eta$ : oblate spheroidal coordinate, $\phi$ : oblate spheroidal coordinate and $c$ : positive constant
A&S Ref:: 21.3.2 (in different form)
Permalink:: http://dlmf.nist.gov/30.14.E1
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

where $c$ is a positive constant. The $(x,y,z)$ -space without the $z$ -axis and the disk $z=0$ , $x^{2}+y^{2}\leq c^{2}$ corresponds to

30.14.2

$0<\xi<\infty,$

$-1<\eta<1,$

$0\leq\phi<2\pi.$

Symbols:: $\xi$ : oblate spheroidal coordinate, $\eta$ : oblate spheroidal coordinate and $\phi$ : oblate spheroidal coordinate
Permalink:: http://dlmf.nist.gov/30.14.E2
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

The coordinate surfaces $\xi=\mbox{const}.$ are oblate ellipsoids of revolution with focal circle $z=0$ , $x^{2}+y^{2}=c^{2}$ . The coordinate surfaces $\eta=\mbox{const}.$ are halves of one-sheeted hyperboloids of revolution with the same focal circle. The disk $z=0$ , $x^{2}+y^{2}\leq c^{2}$ is given by $\xi=0$ , $-1\leq\eta\leq 1$ , and the rays $\pm z\geq 0$ , $x=y=0$ are given by $\eta=\pm 1$ , $\xi\geq 0$ .

§30.14(ii) Metric Coefficients

Notes:: See Meixner and Schäfke (1954, §1.124).
Keywords:: metric coefficients, oblate spheroidal coordinates
Permalink:: http://dlmf.nist.gov/30.14.ii

30.14.3 $h_{\xi}^{2}=\frac{c^{2}(\xi^{2}+\eta^{2})}{1+\xi^{2}},$

Symbols:: $h_{\xi},h_{\eta},h_{\phi}$ : metric coefficients, $\xi$ : oblate spheroidal coordinate, $\eta$ : oblate spheroidal coordinate and $c$ : positive constant
A&S Ref:: 21.4.3 (in different form)
Permalink:: http://dlmf.nist.gov/30.14.E3
Encodings:: TeX, pMML, png

30.14.4 $h_{\eta}^{2}=\frac{c^{2}(\xi^{2}+\eta^{2})}{1-\eta^{2}},$

Symbols:: $h_{\xi},h_{\eta},h_{\phi}$ : metric coefficients, $\xi$ : oblate spheroidal coordinate, $\eta$ : oblate spheroidal coordinate and $c$ : positive constant
Permalink:: http://dlmf.nist.gov/30.14.E4
Encodings:: TeX, pMML, png

30.14.5 $h_{\phi}^{2}=c^{2}(\xi^{2}+1)(1-\eta^{2}).$

Symbols:: $h_{\xi},h_{\eta},h_{\phi}$ : metric coefficients, $\xi$ : oblate spheroidal coordinate, $\eta$ : oblate spheroidal coordinate, $\phi$ : oblate spheroidal coordinate and $c$ : positive constant
Permalink:: http://dlmf.nist.gov/30.14.E5
Encodings:: TeX, pMML, png

§30.14(iii) Laplacian

Notes:: See Meixner and Schäfke (1954, §1.124).
Keywords:: Laplacian, oblate spheroidal coordinates
Permalink:: http://dlmf.nist.gov/30.14.iii

30.14.6 $\nabla^{2}=\frac{1}{c^{2}(\xi^{2}+\eta^{2})}\left(\frac{\partial}{\partial\xi}\left((\xi^{2}+1)\frac{\partial}{\partial\xi}\right)+\frac{\partial}{\partial\eta}\left((1-\eta^{2})\frac{\partial}{\partial\eta}\right)+\frac{\xi^{2}+\eta^{2}}{(\xi^{2}+1)(1-\eta^{2})}\frac{{\partial}^{2}}{{\partial\phi}^{2}}\right).$

Symbols:: $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ , $\xi$ : oblate spheroidal coordinate, $\eta$ : oblate spheroidal coordinate, $\phi$ : oblate spheroidal coordinate and $c$ : positive constant
A&S Ref:: 21.5.2 (in corrected form)
Permalink:: http://dlmf.nist.gov/30.14.E6
Encodings:: TeX, pMML, png

§30.14(iv) Separation of Variables

Notes:: See Meixner and Schäfke (1954, §1.134).
Permalink:: http://dlmf.nist.gov/30.14.iv

The wave equation (30.13.7), transformed to oblate spheroidal coordinates $(\xi,\eta,\phi)$ , admits solutions of the form (30.13.8), where $w_{1}$ satisfies the differential equation

30.14.7 $\frac{d}{d\xi}\left((1+\xi^{2})\frac{dw_{1}}{d\xi}\right)-\left(\lambda+\gamma^{2}(1+\xi^{2})-\frac{\mu^{2}}{1+\xi^{2}}\right)w_{1}=0,$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\xi$ : oblate spheroidal coordinate, $w_{j}$ : solution to DE, $\gamma^{2}=\kappa^{2}c^{2}$ : parameter, $\lambda$ : real parameter and $\mu$ : real parameter
A&S Ref:: 21.6.3 (in different form)
Referenced by:: §30.14(iv), §30.14(iv)
Permalink:: http://dlmf.nist.gov/30.14.E7
Encodings:: TeX, pMML, png

and $w_{2}$ , $w_{3}$ satisfy (30.13.10) and (30.13.11), respectively, with $\gamma^{2}=-\kappa^{2}c^{2}\leq 0$ and separation constants $\lambda$ and $\mu^{2}$ . Equation (30.14.7) can be transformed to equation (30.2.1) by the substitution $z=\pm i\xi$ .

In most applications the solution $w$ has to be a single-valued function of $(x,y,z)$ , which requires $\mu=m$ (a nonnegative integer). Moreover, the solution $w$ has to be bounded along the $z$ -axis: this requires $w_{2}(\eta)$ to be bounded when $-1<\eta<1$ . Then $\lambda=\mathop{\lambda^{{m}}_{{n}}\/}\nolimits\!\left(\gamma^{2}\right)$ for some $n=m,m+1,m+2,\dots$ , 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 $w_{1}(\xi)=a_{1}\mathop{S^{{m(1)}}_{{n}}\/}\nolimits\!\left(i\xi,\gamma\right)+b_{1}\mathop{S^{{m(2)}}_{{n}}\/}\nolimits\!\left(i\xi,\gamma\right).$

Symbols:: $\mathop{S^{{m(j)}}_{{n}}\/}\nolimits\!\left(z,\gamma\right)$ : radial spheroidal wave function, $m$ : nonnegative integer, $n\geq m$ : integer degree, $b_{j}$ : coefficients, $a_{j}$ : coefficients, $\xi$ : oblate spheroidal coordinate, $w_{j}$ : solution to DE and $\gamma^{2}=\kappa^{2}c^{2}$ : parameter
Referenced by:: §30.14(v)
Permalink:: http://dlmf.nist.gov/30.14.E8
Encodings:: TeX, pMML, png

If $b_{1}=b_{2}=0$ , then the function (30.13.8) is a twice-continuously differentiable solution of (30.13.7) in the entire $(x,y,z)$ -space. If $b_{2}=0$ , then this property holds outside the focal disk.

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

Notes:: See Meixner and Schäfke (1954, Chapter 4).
Keywords:: interior Dirichlet problem, oblate spheroidal coordinates, radial spheroidal wave functions, spheroidal wave functions, wave equation
Permalink:: http://dlmf.nist.gov/30.14.v

Equation (30.13.7) for $\xi\leq\xi _{0}$ together with the boundary condition $w=0$ on the ellipsoid given by $\xi=\xi _{0}$ , poses an eigenvalue problem with $\kappa^{2}$ as spectral parameter. The eigenvalues are given by $c^{2}\kappa^{2}=-\gamma^{2}$ , where $\gamma^{2}$ is determined from the condition

30.14.9 $\mathop{S^{{m(1)}}_{{n}}\/}\nolimits\!\left(i\xi _{0},\gamma\right)=0.$

Symbols:: $\mathop{S^{{m(j)}}_{{n}}\/}\nolimits\!\left(z,\gamma\right)$ : radial spheroidal wave function, $m$ : nonnegative integer, $n\geq m$ : integer degree, $\xi$ : oblate spheroidal coordinate and $\gamma^{2}$ : real parameter
Permalink:: http://dlmf.nist.gov/30.14.E9
Encodings:: TeX, pMML, png

The corresponding eigenfunctions are then given by (30.13.8), (30.14.8), (30.13.13), (30.13.12), with $b_{1}=b_{2}=0$ .

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).