Cylindrical Eigenfunction Expansion: Difference between revisions
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\phi}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 | \phi}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 | ||
\phi}{\partial \theta^2} + \frac{\partial^2 \phi}{\partial z^2} = 0, | \phi}{\partial \theta^2} + \frac{\partial^2 \phi}{\partial z^2} = 0, | ||
\quad (r,\theta,z) \in \mathbb{R}_{>0} \, \times \ ]- \pi, \pi] | |||
\times \ | \times \mathbb{R}_{<0}, </math> | ||
<math>\frac{\partial \phi}{\partial z} - \alpha \phi = 0, | <math>\frac{\partial \phi}{\partial z} - \alpha \phi = 0, \quad | ||
(r,\theta,z) \in \ | (r,\theta,z) \in \mathbb{R}_{>0}\, | ||
\times \, ]\!- \pi, \pi] \times \{ 0 \},</math> | \times \, ]\!- \pi, \pi] \times \{ 0 \},</math> | ||
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<math> | <math> | ||
\frac{\partial \phi}{\partial z} = 0, (r,\theta,z) \in | \frac{\partial \phi}{\partial z} = 0, \quad (r,\theta,z) \in | ||
\mathbb{R}_{>0}\, \times \,]\!- \pi, \pi] \times \{ -d \}, | \mathbb{R}_{>0}\, \times \,]\!- \pi, \pi] \times \{ -d \}, | ||
</math> | </math> | ||
in the case of constant finite depth | in the case of constant finite water depth <math>d</math> and | ||
<math> | <math> | ||
\sup \big\{ \, | \sup \big\{ \, |\phi| \ \big| \ (r,\theta,z) \in \mathbb{R}_{>0}\, | ||
\times \, ]\!- \pi, \pi] \times \mathbb{R}_{<0} \,\big\} < \infty | \times \, ]\!- \pi, \pi] \times \mathbb{R}_{<0} \,\big\} < \infty | ||
</math> | </math> | ||
in the case of infinite depth. Moreover, the radiation condition | in the case of infinite water depth. Moreover, the radiation condition | ||
<math> | <math> | ||
\lim_{r \rightarrow \infty} \sqrt{r} \, \Big( | \lim_{r \rightarrow \infty} \sqrt{r} \, \Big( | ||
\frac{\partial}{\partial r} - \i k \Big) \phi = 0 | \frac{\partial}{\partial r} - \mathrm{i} k \Big) \phi = 0 | ||
</math> | </math> | ||
with the wavenumber | with the wavenumber <math>k</math> also applies. | ||
Revision as of 09:24, 20 April 2006
The problem for the potential in cylindrical coordinates, [math]\displaystyle{ \phi (r,\theta,z) }[/math], is given by
[math]\displaystyle{ \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial \phi}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 \phi}{\partial \theta^2} + \frac{\partial^2 \phi}{\partial z^2} = 0, \quad (r,\theta,z) \in \mathbb{R}_{\gt 0} \, \times \ ]- \pi, \pi] \times \mathbb{R}_{\lt 0}, }[/math]
[math]\displaystyle{ \frac{\partial \phi}{\partial z} - \alpha \phi = 0, \quad (r,\theta,z) \in \mathbb{R}_{\gt 0}\, \times \, ]\!- \pi, \pi] \times \{ 0 \}, }[/math]
as well as
[math]\displaystyle{ \frac{\partial \phi}{\partial z} = 0, \quad (r,\theta,z) \in \mathbb{R}_{\gt 0}\, \times \,]\!- \pi, \pi] \times \{ -d \}, }[/math]
in the case of constant finite water depth [math]\displaystyle{ d }[/math] and
[math]\displaystyle{ \sup \big\{ \, |\phi| \ \big| \ (r,\theta,z) \in \mathbb{R}_{\gt 0}\, \times \, ]\!- \pi, \pi] \times \mathbb{R}_{\lt 0} \,\big\} \lt \infty }[/math]
in the case of infinite water depth. Moreover, the radiation condition
[math]\displaystyle{ \lim_{r \rightarrow \infty} \sqrt{r} \, \Big( \frac{\partial}{\partial r} - \mathrm{i} k \Big) \phi = 0 }[/math]
with the wavenumber [math]\displaystyle{ k }[/math] also applies.
