Oblate Spheroidal Coordinates

The analytic transformation between the oblate spheroidal and Cartesian coordinate systems (see example in figure ). A elliptical coordinate system is specified by,  

$$\left[\begin{array} {c} x_1\\ x_2\\ x_3 \end{array}\rig... ...\,\rm{sinh} \, \xi_3 \, \rm{sin}\,\xi_1 \\ \end{array}\right].$$ (14)

where, again, a is a stretch parameter controlling the breadth of the coordinate system.

 

OSC
Figure 7 Three sample extrapolation steps of an oblate spheroidal coordinate system.

The metric tensor g_ij describing the geometry of oblate spheroidal coordinates is given by,  

$$\left[g_{ij}\right] = \left[\begin{array} {ccc} A^2 & 0 & 0 \\ 0 & B^2 & 0 \\ 0 & 0 & A^2 \\ \end{array}\right],$$ (15)

where $A=a \sqrt{ {\rm sinh}^2 \,\xi_3 + {\rm sin}^2 \,\xi_1 }$ and $B=a {\rm cosh}\,\xi_3 \, {\rm cos} \, \xi_1$. The determinant of the metric tensor is: $\left\vert \mathbf{g} \right\vert= A^4\,B^2$. The associated (inverse) metric tensor is given by,  

$$\left[g^{ij}\right] = \left[\begin{array} {ccc} A^{-2} & 0 & 0 \\ 0 & B^{-2} & 0 \\ 0 & 0 & A^{-2} \\ \end{array}\right].$$ (16)

and weighted metric tensor is given by,  

$$\left[m^{ij}\right] = \left[\begin{array} {ccc} B & 0 & 0 \\ 0 & \frac{A^2}{B} & 0 \\ 0 & 0 & B \\ \end{array}\right].$$ (17)

The corresponding extrapolation wavenumber is generated by inputting tensors g^ij and m^ij into the generalized wavenumber expression for 3D non-orthogonal coordinate systems Shragge (2006). Unlike in elliptic cylindrical coordinates, though, the oblate spheroidal system has non-stationary nⁱ coefficients: $n^1= a {\rm cosh}\, \xi_3 \,{\rm sin}\, \xi_1$, n²=0 and $n^3= a {\rm sinh}\, \xi$. The resulting extrapolation wavenumber is  

$$k_\xi_3= \frac{ {\rm i tanh} \,\xi_3}{2} \pm \sqrt{ A^2 s... ...^2 + {\rm i} k_\xi_1{\rm tan}\,\xi_1 - {\rm tanh}^2\, \xi_3 }.$$ (18)

The wavenumber for 2D extrapolation in elliptic coordinates reduces to  

$$\left. k_\xi_3\right\vert _{k_\xi_2=0} = \pm \sqrt{ A^2 s^2 \omega^2 - k_\xi_1^2 }.$$ (19)