Difference between revisions of "Cylindrical Eigenfunction Expansion"

From WikiWaves
Jump to navigationJump to search
Line 4: Line 4:
 
\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,
&& (r,\theta,z) \in \mathbb{R}_{>0} \, \times \ ]- \pi, \pi]  
+
\quad (r,\theta,z) \in \mathbb{R}_{>0} \, \times \ ]- \pi, \pi]  
\times  \mathds{R}_{<0}, </math>
+
\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 \mathds{R}_{>0}\,
+
(r,\theta,z) \in \mathbb{R}_{>0}\,
 
\times \, ]\!- \pi, \pi]  \times  \{ 0 \},</math>
 
\times \, ]\!- \pi, \pi]  \times  \{ 0 \},</math>
  
Line 14: Line 14:
  
 
<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 $d$ and
+
in the case of constant finite water depth <math>d</math> and
  
 
<math>
 
<math>
\sup \big\{ \, \abs{\phi} \ \big| \ (r,\theta,z) \in \mathbb{R}_{>0}\,
+
\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 $k$ also applies.
+
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.