§30.13 Wave Equation in Prolate Spheroidal Coordinates

Keywords:: prolate spheroidal coordinates, spheroidal wave functions, wave equation
Referenced by:: ¶ ‣ §1.5(ii), §30.1, §30.2(i)
Permalink:: http://dlmf.nist.gov/30.13

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

§30.13(i) Prolate Spheroidal Coordinates

Notes:: See Erdélyi et al. (1955, §16.1.2) and Meixner and Schäfke (1954, §1.123).
Keywords:: coordinate systems, prolate spheroidal coordinates
Permalink:: http://dlmf.nist.gov/30.13.i

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

30.13.1

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

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

$z=c\xi\eta,$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $z$ : complex variable, $x$ : real variable, $y$ : real variable, $\xi$ : prolate spheroidal coordinate, $\eta$ : prolate spheroidal coordinate, $\phi$ : prolate spheroidal coordinate and $c$ : positive constant
A&S Ref:: 21.2.2 (in different form)
Permalink:: http://dlmf.nist.gov/30.13.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 corresponds to

30.13.2

$1<\xi<\infty,$

$-1<\eta<1,$

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

Symbols:: $\xi$ : prolate spheroidal coordinate, $\eta$ : prolate spheroidal coordinate and $\phi$ : prolate spheroidal coordinate
Permalink:: http://dlmf.nist.gov/30.13.E2
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

The coordinate surfaces $\xi=\mbox{const}.$ are prolate ellipsoids of revolution with foci at $x=y=0$ , $z=\pm c$ . The coordinate surfaces $\eta=\mbox{const}.$ are sheets of two-sheeted hyperboloids of revolution with the same foci. The focal line is given by $\xi=1$ , $-1\leq\eta\leq 1$ , and the rays $\pm z\geq c$ , $x=y=0$ are given by $\eta=\pm 1$ , $\xi\geq 1$ .

§30.13(ii) Metric Coefficients

Notes:: See Meixner and Schäfke (1954, §1.123).
Keywords:: metric coefficients, prolate spheroidal coordinates
Permalink:: http://dlmf.nist.gov/30.13.ii

30.13.3 $h_{\xi}^{2}=\left(\frac{\partial x}{\partial\xi}\right)^{2}+\left(\frac{\partial y}{\partial\xi}\right)^{2}+\left(\frac{\partial z}{\partial\xi}\right)^{2}=\frac{c^{2}(\xi^{2}-\eta^{2})}{\xi^{2}-1},$

Symbols:: $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ , $z$ : complex variable, $x$ : real variable, $y$ : real variable, $\xi$ : prolate spheroidal coordinate, $\eta$ : prolate spheroidal coordinate, $c$ : positive constant and $h_{\xi},h_{\eta},h_{\phi}$ : metric coefficients
A&S Ref:: 21.4.2
Permalink:: http://dlmf.nist.gov/30.13.E3
Encodings:: TeX, pMML, png

30.13.4 $h_{\eta}^{2}=\left(\frac{\partial x}{\partial\eta}\right)^{2}+\left(\frac{\partial y}{\partial\eta}\right)^{2}+\left(\frac{\partial z}{\partial\eta}\right)^{2}=\frac{c^{2}(\xi^{2}-\eta^{2})}{1-\eta^{2}},$

Symbols:: $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ , $z$ : complex variable, $x$ : real variable, $y$ : real variable, $\xi$ : prolate spheroidal coordinate, $\eta$ : prolate spheroidal coordinate, $c$ : positive constant and $h_{\xi},h_{\eta},h_{\phi}$ : metric coefficients
Permalink:: http://dlmf.nist.gov/30.13.E4
Encodings:: TeX, pMML, png

30.13.5 $h_{\phi}^{2}=\left(\frac{\partial x}{\partial\phi}\right)^{2}+\left(\frac{\partial y}{\partial\phi}\right)^{2}+\left(\frac{\partial z}{\partial\phi}\right)^{2}=c^{2}(\xi^{2}-1)(1-\eta^{2}).$

Symbols:: $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ , $z$ : complex variable, $x$ : real variable, $y$ : real variable, $\xi$ : prolate spheroidal coordinate, $\eta$ : prolate spheroidal coordinate, $\phi$ : prolate spheroidal coordinate, $c$ : positive constant and $h_{\xi},h_{\eta},h_{\phi}$ : metric coefficients
Permalink:: http://dlmf.nist.gov/30.13.E5
Encodings:: TeX, pMML, png

§30.13(iii) Laplacian

Notes:: See Meixner and Schäfke (1954, §1.123).
Keywords:: Laplacian, prolate spheroidal coordinates
Permalink:: http://dlmf.nist.gov/30.13.iii

30.13.6 $\nabla^{2}=\frac{1}{h_{\xi}h_{\eta}h_{\phi}}\left(\frac{\partial}{\partial\xi}\left(\frac{h_{\eta}h_{\phi}}{h_{\xi}}\frac{\partial}{\partial\xi}\right)+\frac{\partial}{\partial\eta}\left(\frac{h_{\xi}h_{\phi}}{h_{\eta}}\frac{\partial}{\partial\eta}\right)+\frac{\partial}{\partial\phi}\left(\frac{h_{\xi}h_{\eta}}{h_{\phi}}\frac{\partial}{\partial\phi}\right)\right)=\frac{1}{c^{2}(\xi^{2}-\eta^{2})}\left(\frac{\partial}{\partial\xi}\left((\xi^{2}-1)\frac{\partial}{\partial\xi}\right)+\frac{\partial}{\partial\eta}\left((1-\eta^{2})\frac{\partial}{\partial\eta}\right)+\frac{\xi^{2}-\eta^{2}}{(\xi^{2}-1)(1-\eta^{2})}\frac{{\partial}^{2}}{{\partial\phi}^{2}}\right).$

Symbols:: $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ , $\xi$ : prolate spheroidal coordinate, $\eta$ : prolate spheroidal coordinate, $\phi$ : prolate spheroidal coordinate, $c$ : positive constant and $h_{\xi},h_{\eta},h_{\phi}$ : metric coefficients
A&S Ref:: 21.5.1 (in corrected form)
Permalink:: http://dlmf.nist.gov/30.13.E6
Encodings:: TeX, pMML, png

§30.13(iv) Separation of Variables

Notes:: See Meixner and Schäfke (1954, §1.133).
Permalink:: http://dlmf.nist.gov/30.13.iv

The wave equation

30.13.7 $\nabla^{2}w+\kappa^{2}w=0,$

Symbols:: $\kappa$ : parameter and $w_{j}$ : solution to DE
Referenced by:: §30.13(iv), §30.13(v), §30.14(iv), §30.14(iv), §30.14(v)
Permalink:: http://dlmf.nist.gov/30.13.E7
Encodings:: TeX, pMML, png

transformed to prolate spheroidal coordinates $(\xi,\eta,\phi)$ , admits solutions

30.13.8 $w(\xi,\eta,\phi)=w_{1}(\xi)w_{2}(\eta)w_{3}(\phi),$

Symbols:: $\xi$ : prolate spheroidal coordinate, $\eta$ : prolate spheroidal coordinate, $\phi$ : prolate spheroidal coordinate and $w_{j}$ : solution to DE
Referenced by:: §30.13(iv), §30.13(v), §30.14(iv), §30.14(iv), §30.14(v)
Permalink:: http://dlmf.nist.gov/30.13.E8
Encodings:: TeX, pMML, png

where $w_{1}$ , $w_{2}$ , $w_{3}$ satisfy the differential equations

30.13.9 $\frac{d}{d\xi}\left((1-\xi^{2})\frac{dw_{1}}{d\xi}\right)+\left(\lambda+\gamma^{2}(1-\xi^{2})-\frac{\mu^{2}}{1-\xi^{2}}\right)w_{1}=0,$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\xi$ : prolate spheroidal coordinate, $w_{j}$ : solution to DE, $\gamma^{2}=\kappa^{2}c^{2}$ : parameter, $\lambda$ : real parameter and $\mu$ : real parameter
A&S Ref:: 21.6.1 (in different form)
Referenced by:: §30.13(iv), §30.13(iv)
Permalink:: http://dlmf.nist.gov/30.13.E9
Encodings:: TeX, pMML, png

30.13.10 $\frac{d}{d\eta}\left((1-\eta^{2})\frac{dw_{2}}{d\eta}\right)+\left(\lambda+\gamma^{2}(1-\eta^{2})-\frac{\mu^{2}}{1-\eta^{2}}\right)w_{2}=0,$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\eta$ : prolate spheroidal coordinate, $w_{j}$ : solution to DE, $\gamma^{2}=\kappa^{2}c^{2}$ : parameter, $\lambda$ : real parameter and $\mu$ : real parameter
A&S Ref:: 21.6.2 (in different form)
Referenced by:: §30.13(iv), §30.13(iv), §30.14(iv), §30.14(iv)
Permalink:: http://dlmf.nist.gov/30.13.E10
Encodings:: TeX, pMML, png

30.13.11 $\frac{{d}^{2}w_{3}}{{d\phi}^{2}}+\mu^{2}w_{3}=0,$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\phi$ : prolate spheroidal coordinate, $w_{j}$ : solution to DE and $\mu$ : real parameter
Referenced by:: §30.14(iv)
Permalink:: http://dlmf.nist.gov/30.13.E11
Encodings:: TeX, pMML, png

with $\gamma^{2}=\kappa^{2}c^{2}\geq 0$ and separation constants $\lambda$ and $\mu^{2}$ . 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 $\mu=m$ (a nonnegative integer) and

30.13.12 $w_{3}(\phi)=a_{3}\mathop{\cos\/}\nolimits\!\left(m\phi\right)+b_{3}\mathop{\sin\/}\nolimits\!\left(m\phi\right).$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $m$ : nonnegative integer, $b_{j}$ : coefficients, $\phi$ : prolate spheroidal coordinate, $w_{j}$ : solution to DE and $a_{j}$ : coefficients
Referenced by:: §30.13(v), §30.14(v)
Permalink:: http://dlmf.nist.gov/30.13.E12
Encodings:: TeX, pMML, png

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

30.13.13 $w_{2}(\eta)=a_{2}\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(\eta,\gamma^{2}\right)+b_{2}\mathop{\mathsf{Qs}^{{m}}_{{n}}\/}\nolimits\!\left(\eta,\gamma^{2}\right).$

Symbols:: $\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)$ : spheroidal wave function of the first kind, $\mathop{\mathsf{Qs}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)$ : spheroidal wave function of the second kind, $m$ : nonnegative integer, $n\geq m$ : integer degree, $b_{j}$ : coefficients, $\eta$ : prolate spheroidal coordinate, $w_{j}$ : solution to DE, $\gamma^{2}=\kappa^{2}c^{2}$ : parameter and $a_{j}$ : coefficients
Referenced by:: §30.13(v), §30.14(iv), §30.14(v)
Permalink:: http://dlmf.nist.gov/30.13.E13
Encodings:: TeX, pMML, png

The solution of (30.13.9) with $\mu=m$ is

30.13.14 $w_{1}(\xi)=a_{1}\mathop{S^{{m(1)}}_{{n}}\/}\nolimits\!\left(\xi,\gamma\right)+b_{1}\mathop{S^{{m(2)}}_{{n}}\/}\nolimits\!\left(\xi,\gamma\right).$

Symbols:: $\mathop{S^{{m(j)}}_{{n}}\/}\nolimits\!\left(z,\gamma\right)$ : radial spheroidal wave function, $m$ : nonnegative integer, $n\geq m$ : integer degree, $\xi$ : prolate spheroidal coordinate, $b_{j}$ : coefficients, $w_{j}$ : solution to DE, $\gamma^{2}=\kappa^{2}c^{2}$ : parameter and $a_{j}$ : coefficients
Referenced by:: §30.13(v)
Permalink:: http://dlmf.nist.gov/30.13.E14
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 line.

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

Notes:: See Meixner and Schäfke (1954, Chapter 4).
Keywords:: interior Dirichlet problem, prolate spheroidal coordinates, wave equation
Permalink:: http://dlmf.nist.gov/30.13.v

Equation (30.13.7) for $\xi\leq\xi _{0}$ , and subject to 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 $\gamma$ is determined from the condition

30.13.15 $\mathop{S^{{m(1)}}_{{n}}\/}\nolimits\!\left(\xi _{0},\gamma\right)=0.$

Symbols:: $\mathop{S^{{m(j)}}_{{n}}\/}\nolimits\!\left(z,\gamma\right)$ : radial spheroidal wave function, $m$ : nonnegative integer, $n\geq m$ : integer degree, $\xi$ : prolate spheroidal coordinate and $\gamma^{2}$ : real parameter
Permalink:: http://dlmf.nist.gov/30.13.E15
Encodings:: TeX, pMML, png

The corresponding eigenfunctions are given by (30.13.8), (30.13.14), (30.13.13), (30.13.12), with $b_{1}=b_{2}=0$ . For the Dirichlet boundary-value problem of the region $\xi _{1}\leq\xi\leq\xi _{2}$ between two ellipsoids, the eigenvalues are determined from

30.13.16 $w_{1}(\xi _{1})=w_{1}(\xi _{2})=0,$

Symbols:: $\xi$ : prolate spheroidal coordinate and $w_{j}$ : solution to DE
Permalink:: http://dlmf.nist.gov/30.13.E16
Encodings:: TeX, pMML, png

with $w_{1}$ as in (30.13.14). The corresponding eigenfunctions are given as before with $b_{2}=0$ .

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).