Elliptic Cylindrical Coordinates

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

$$\left[\begin{array} {c} x_1\\ x_2\\ x_3 \end{array}\rig... ...\,\rm{sinh} \, \xi_3 \, \rm{sin}\,\xi_1 \\ \end{array}\right],$$ (8)

where [x₁,x₂,x₃] are the underlying Cartesian coordinate variables, $[\xi_1,\xi_2,\xi_3]$ the RWE elliptic cylindrical coordinates, and parameter a a stretch parameter controlling the breadth of the coordinate system.

 

ECC
Figure 6 Four sample extrapolation steps of an elliptic cylindrical coordinate system.

The metric tensor describing the geometry of the elliptic coordinate system is given by,  

$$\left[g_{ij}\right] = \left[\begin{array} {ccc} A^2 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & A^2 \\ \end{array}\right],$$ (9)

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

$$\left[g^{ij}\right] = \left[\begin{array} {ccc} A^{-2} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & A^{-2} \\ \end{array}\right],$$ (10)

and weighted metric tensor ($m^{ij}=\sqrt{\left\vert \mathbf{g} \right\vert}\, g^{ij}$) is given by,  

$$\left[m^{ij}\right] = \left[\begin{array} {ccc} 1 & 0 & 0 \\ 0 & A^2 & 0 \\ 0 & 0 & 1 \\ \end{array}\right].$$ (11)

The corresponding extrapolation wavenumber is generated by using tensors g^ij and m^ij in the general wavenumber expression for 3D non-orthogonal coordinate system Shragge (2006). Note that even though the elliptic coordinate system varies spatially, the local curvature parameters remain constant: n₁=n₂=n₃=0. Thus, inserting the values of g^ij, m^ij and n_j leads to the following extrapolation wavenumber for stepping outward in concentric ellipses $k_{\xi_3}$,  

$$k_\xi_3= \pm \sqrt{ A^2 s^2 \omega^2 - k_\xi_1^2 - A k_\xi_2^2 }.$$ (12)

The wavenumber for 2D extrapolation in elliptic coordinates reduces to  

$$\left. k_\xi_3\right\vert _{k_\xi_2=0} = \pm \sqrt{ A^2 s^2 \omega^2 - k_\xi_1^2 }.$$ (13)

Note that equation A-5 does not contain a kinematic approximation of the extrapolation wavenumber.