Elliptic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional elliptic coordinate system in theperpendicular
z
F1
F2
-a
+a
x
The most common definition of elliptic cylindrical coordinates
(\mu,\nu,z)
x=a \cosh\mu \cos\nu
y=a \sinh\mu \sin\nu
z=z
where
\mu
\nu\in[0,2\pi]
These definitions correspond to ellipses and hyperbolae. The trigonometric identity
x2 | |
a2\cosh2\mu |
+
y2 | |
a2\sinh2\mu |
=\cos2\nu+\sin2\nu=1
shows that curves of constant
\mu
x2 | |
a2\cos2\nu |
-
y2 | |
a2\sin2\nu |
=\cosh2\mu-\sinh2\mu=1
shows that curves of constant
\nu
The scale factors for the elliptic cylindrical coordinates
\mu
\nu
h\mu=h\nu=a\sqrt{\sinh2\mu+\sin2\nu}
whereas the remaining scale factor
hz=1
dV=a2\left(\sinh2\mu+\sin2\nu\right)d\mud\nudz
and the Laplacian equals
\nabla2\Phi=
1 | |
a2\left(\sinh2\mu+\sin2\nu\right) |
\left(
\partial2\Phi | |
\partial\mu2 |
+
\partial2\Phi | |
\partial\nu2 |
\right)+
\partial2\Phi | |
\partialz2 |
Other differential operators such as
\nabla ⋅ F
\nabla x F
(\mu,\nu,z)
An alternative and geometrically intuitive set of elliptic coordinates
(\sigma,\tau,z)
\sigma=\cosh\mu
\tau=\cos\nu
\sigma
\tau
\tau
\sigma
(\sigma,\tau,z)
F1
F2
d1+d2
2a\sigma
d1-d2
2a\tau
F1
a(\sigma+\tau)
F2
a(\sigma-\tau)
F1
F2
x=-a
x=+a
A drawback of these coordinates is that they do not have a 1-to-1 transformation to the Cartesian coordinates
x=a\sigma\tau
y2=a2\left(\sigma2-1\right)\left(1-\tau2\right)
The scale factors for the alternative elliptic coordinates
(\sigma,\tau,z)
h\sigma=a\sqrt{
\sigma2-\tau2 | |
\sigma2-1 |
h\tau=a\sqrt{
\sigma2-\tau2 | |
1-\tau2 |
and, of course,
hz=1
dV=a2
\sigma2-\tau2 | |
\sqrt{\left(\sigma2-1\right)\left(1-\tau2\right) |
and the Laplacian equals
\nabla2\Phi=
1 | |
a2\left(\sigma2-\tau2\right) |
\left[ \sqrt{\sigma2-1}
\partial | |
\partial\sigma |
\left(\sqrt{\sigma2-1}
\partial\Phi | |
\partial\sigma |
\right)+\sqrt{1-\tau2
Other differential operators such as
\nabla ⋅ F
\nabla x F
(\sigma,\tau)
The classic applications of elliptic cylindrical coordinates are in solving partial differential equations, e.g., Laplace's equation or the Helmholtz equation, for which elliptic cylindrical coordinates allow a separation of variables. A typical example would be the electric field surrounding a flat conducting plate of width
2a
The three-dimensional wave equation, when expressed in elliptic cylindrical coordinates, may be solved by separation of variables, leading to the Mathieu differential equations.
The geometric properties of elliptic coordinates can also be useful. A typical example might involve an integration over all pairs of vectors
p
q
r=p+q
\left|p\right|
\left|q\right|
r
x
r=2a\hat{x
r
p
q