Template:Separation of variables for a dock - Revision history

Mike smith: clarification and tidy up

2009-08-08T23:20:31Z

clarification and tidy up

← Older revision		Revision as of 23:20, 8 August 2009
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	=== Separation of Variables for a Dock ===		=== Separation of Variables for a Dock ===

	The separation of variables equation for a dock		The separation of variables equation for a floating dock
	<center>		<center>
	<math>		<math>
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	</center>		</center>
	The solution is		The solution is
	<math>k=\kappa_{m}=m\pi/h</math>, <math>m\geq 0</math> and		<math>k=\kappa_{m}= \frac{m\pi}{h} \,</math>, <math>m\geq 0</math> and
	<center>		<center>
	<math>		<math>
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	<center>		<center>
	<math>		<math>
	C_{m}=\frac{1}{2}h,\~~quad~~ m\neq 0 \~~quad \mathrm~~{~~and~~} ~~\quad C_0 = h.~~		C_{m} =
	</math></center>		\begin{cases}
			h,\quad m=0 \\
			\frac{1}{2}h,\,\,\,m\neq 0
			\end{cases}
			</math>
			</center>

Adi Kurniawan: /* Separation of Variables for a Dock */

2008-12-23T05:32:49Z

Separation of Variables for a Dock

← Older revision		Revision as of 05:32, 23 December 2008
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	<center>		<center>
	<math>		<math>
	\int\nolimits_{-h}^{0}\psi_{m}(z)\psi_{n}(z) d z=C_{m}\delta_{mn},		\int\nolimits_{-h}^{0}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
	</math>		</math>
	</center>		</center>

Adi Kurniawan: /* Separation of Variables for a Dock */

2008-09-23T04:20:56Z

Separation of Variables for a Dock

← Older revision		Revision as of 04:20, 23 September 2008
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	== Separation of Variables for a Dock ==		=== Separation of Variables for a Dock ===

	The separation of variables equation for a dock		The separation of variables equation for a dock

Meylan at 10:51, 11 September 2008

2008-09-11T10:51:45Z

← Older revision		Revision as of 10:51, 11 September 2008
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	<center>		<center>
	<math>		<math>
	\int\nolimits_{-h}^{0}\psi_{m}(z)\psi_{n}(z) d z=C_{m}\delta_{mn}		\int\nolimits_{-h}^{0}\psi_{m}(z)\psi_{n}(z) d z=C_{m}\delta_{mn},
	</math>		</math>
	</center>		</center>

Meylan at 10:51, 11 September 2008

2008-09-11T10:51:23Z

← Older revision		Revision as of 10:51, 11 September 2008
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	<center>		<center>
	<math>		<math>
	Z^{\prime\prime} + k^2 Z =0.		Z^{\prime\prime} + k^2 Z =0,
	</math>		</math>
	</center>		</center>
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	<center>		<center>
	<math>		<math>
	Z^{\prime} (-h) = 0		Z^{\prime} (-h) = 0,
	</math>		</math>
	</center>		</center>
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	<center>		<center>
	<math>		<math>
	Z^{\prime} (0) = 0		Z^{\prime} (0) = 0.
	</math>		</math>
	</center>		</center>
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	<math>		<math>
	Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad		Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
	m\geq 0		m\geq 0.
	</math>		</math>
	</center>		</center>
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	<center>		<center>
	<math>		<math>
	C_{m}=\frac{1}{2}h,\quad m\neq 0 \quad \mathrm{and} \quad C_0 = h		C_{m}=\frac{1}{2}h,\quad m\neq 0 \quad \mathrm{and} \quad C_0 = h.
	</math></center>		</math></center>

Meylan: /* Separation of Variables for a Dock */

2008-09-11T10:50:39Z

Separation of Variables for a Dock

← Older revision		Revision as of 10:50, 11 September 2008
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	<center>		<center>
	<math>		<math>
	C_{m}=\frac{1}{2}h,\quad,m\neq 0 \quad \mathrm{and} \quad C_0 = h		C_{m}=\frac{1}{2}h,\quad m\neq 0 \quad \mathrm{and} \quad C_0 = h
	</math></center>		</math></center>

Meylan at 04:31, 26 August 2008

2008-08-26T04:31:00Z

← Older revision		Revision as of 04:31, 26 August 2008
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	<center>		<center>
	<math>		<math>
	~~\frac~~{\~~mathrm{d}^2 Z}{~~\~~mathrm{d} z^2~~} + k^2 Z =0.		Z^{\prime\prime} + k^2 Z =0.
	</math>		</math>
	</center>		</center>
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	<center>		<center>
	<math>		<math>
	\~~frac{dZ}{dz~~}(-h) = 0		Z^{\prime} (-h) = 0
	</math>		</math>
	</center>		</center>
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	<center>		<center>
	<math>		<math>
	\~~frac{dZ}{dz~~}(0) = 0		Z^{\prime} (0) = 0
	</math>		</math>
	</center>		</center>

Meylan at 04:27, 26 August 2008

2008-08-26T04:27:40Z

@@ Line 21: / Line 21: @@
 </center>
 The solution is
-<math>\kappa_{m}=m\pi/h</math>, <math>m\geq 0</math>. We define
+<math>k=\kappa_{m}=m\pi/h</math>, <math>m\geq 0</math> and
 <center>
 <math>
-\phi_{m}\left(  z\right)  =\frac{\cos k_{m}(z+h)}{\cos k_{m}h},\quad m\geq0
+Z = \psi_{m}\left(  z\right)  = \cos\kappa_{m}(z+h),\quad
 m\geq 0
 </math>
 </center>
-as the vertical eigenfunction of the potential in the dock
+We note that
 <center>
 <math>

Meylan: /* Separation of Variables for a Dock */

2008-08-26T04:25:50Z

Separation of Variables for a Dock

← Older revision		Revision as of 04:25, 26 August 2008
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	== Separation of Variables for a Dock ==		== Separation of Variables for a Dock ==

	~~Applying Laplace's~~ equation ~~in the vertical direction assuming~~		The separation of variables equation for a dock
	a ~~separation constant <math>\mu</math> we obtain~~
	<center>		<center>
	<math>		<math>
	\~~zeta_~~{zz}+\mu^{2}~~\zeta~~=0.\,		\frac{\mathrm{d}^2 Z}{\mathrm{d} z^2} + k^2 Z =0.
	</math>		</math>
	</center>		</center>
	~~We then use~~ the boundary ~~condition at <math>z=-h</math>, which is~~		subject to the boundary conditions
	~~the same for all <math>x</math> to write~~
	<center>		<center>
	<math>		<math>
	\~~zeta=\cos\mu~~(z+h)\,		\frac{dZ}{dz}(-h) = 0
	</math>		</math>
	</center>		</center>
	~~where the separation constant <math>\mu^{2}</math> must~~		and
	~~satisfy different equations depending on whether we are under~~
	~~the free-surface or the dock covered region. For the free surface~~
	~~<center><math>~~
	~~k\tan\left( kh\right) =-\alpha,\,~~
	~~</math></center>~~
	~~which is the [[Dispersion Relation for a Free Surface]]~~
	and ~~for the dock covered region~~
	<center>		<center>
	<math>		<math>
	\~~kappa\tan~~(~~\kappa h~~)=0,		\frac{dZ}{dz}(0) = 0
	</math>		</math>
	</center>		</center>
	~~Note that we have set <math>\mu=k</math> under the free~~		The solution is
	~~surface and <math>\mu=\kappa</math> under the plate. We denote the~~
	~~positive imaginary~~ solution ~~of free-surface equation by <math>k_{0}</math> and~~
	~~the positive real solutions by <math>k_{m}</math>, <math>m\geq1</math>. The solutions of~~
	~~dock equation are~~
	<math>\kappa_{m}=m\pi/h</math>, <math>m\geq 0</math>. We define		<math>\kappa_{m}=m\pi/h</math>, <math>m\geq 0</math>. We define
	<center>		<center>

Meylan: /* Separation of Variables in the Vertical Direction */

2008-08-26T03:01:11Z

Separation of Variables in the Vertical Direction

← Older revision		Revision as of 03:01, 26 August 2008
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	== Separation of Variables ~~in the Vertical Direction~~==		== Separation of Variables for a Dock ==

	Applying Laplace's equation in the vertical direction assuming		Applying Laplace's equation in the vertical direction assuming