Template:Cylindrical equations - Revision history

Meylan at 07:53, 25 August 2008

2008-08-25T07:53:47Z

← Older revision		Revision as of 07:53, 25 August 2008
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	<math>\frac{\partial \phi}{\partial z} = \alpha \phi , \quad		<math>\frac{\partial \phi}{\partial z} = \alpha \phi , \quad
	z=0		z=0
			</math>
	</center>		</center>
	<center>		<center>

Meylan at 07:53, 25 August 2008

2008-08-25T07:53:17Z

← Older revision		Revision as of 07:53, 25 August 2008
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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]~~		\quad (r,\theta,z) \in \Omega
	~~\times \mathbb{R}_{<0}, </math>~~
	~~</center>~~
	~~<center>~~
	~~<math>\frac{\partial \phi}{\partial z} - \alpha \phi = 0, \quad~~
	~~(r,\theta,z) \in \mathbb{R}_{>0}\,~~
	~~\times \, ]\!- \pi, \pi] \times \{ 0 \},</math>~~
	~~</center>~~
	~~as well as~~
	~~<center>~~
	~~<math>~~
	~~\frac{\partial \phi}{\partial z} = 0, \quad (r,\theta,z) \in~~
	~~\mathbb{R}_{>0}\, \times \,]\!- \pi, \pi] \times \{ -d \},~~
	</math>		</math>
	</center>		</center>
	~~in the case of constant finite water depth <math>d</math> and~~
	<center>		<center>
	<math>		<math>\frac{\partial \phi}{\partial z} = \alpha \phi , \quad
	~~\sup \big~~\{ \~~, \|~~\phi~~\| \ \big\| \ (r,~~\~~theta,~~z~~) \in \mathbb{R}_{>0~~}\,		z=0
	~~\times~~ \, ]\~~!- \pi, \pi] \times \mathbb{R}_{<~~0~~} \,\big\} < \infty~~
	~~</math>~~
	</center>		</center>
	~~in the case of infinite water depth. Moreover, the [[Sommerfeld Radiation Condition]]~~
	<center>		<center>
	<math>		<math>
	~~\lim_{r \rightarrow \infty} \sqrt{r} \, \Big(~~		\frac{\partial \phi}{\partial z} = 0, z=-h
	\frac{\partial}{\partial ~~r} - \mathrm{i~~} ~~k \Big) \phi~~ = 0
	</math>		</math>
	</center>		</center>
	~~with the wavenumber <math>k</math> also applies.~~

Meylan at 07:50, 25 August 2008

2008-08-25T07:50:44Z

New page

The problem for the complex water velocity potential in suitable non-dimensionalised
cylindrical coordinates, <math>\phi (r,\theta,z)</math>, is given by
<center>
<math> \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}_{>0} \, \times \ ]- \pi, \pi]
\times \mathbb{R}_{<0}, </math>
</center>
<center>
<math>\frac{\partial \phi}{\partial z} - \alpha \phi = 0, \quad
(r,\theta,z) \in \mathbb{R}_{>0}\,
\times \, ]\!- \pi, \pi] \times \{ 0 \},</math>
</center>
as well as
<center>
<math>
\frac{\partial \phi}{\partial z} = 0, \quad (r,\theta,z) \in
\mathbb{R}_{>0}\, \times \,]\!- \pi, \pi] \times \{ -d \},
</math>
</center>
in the case of constant finite water depth <math>d</math> and
<center>
<math>
\sup \big\{ \, |\phi| \ \big| \ (r,\theta,z) \in \mathbb{R}_{>0}\,
\times \, ]\!- \pi, \pi] \times \mathbb{R}_{<0} \,\big\} < \infty
</math>
</center>
in the case of infinite water depth. Moreover, the [[Sommerfeld Radiation Condition]]
<center>
<math>
\lim_{r \rightarrow \infty} \sqrt{r} \, \Big(
\frac{\partial}{\partial r} - \mathrm{i} k \Big) \phi = 0
</math>
</center>
with the wavenumber <math>k</math> also applies.