Difference between revisions of "Template:Separation of variables for a dock"

From WikiWaves
Jump to navigationJump to search
m (clarification and tidy up)
 
(2 intermediate revisions by one other user not shown)
Line 1: Line 1:
 
   
 
   
== 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>
Line 21: Line 21:
 
</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>
Line 31: Line 31:
 
<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>
Line 37: Line 37:
 
<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>

Latest revision as of 23:20, 8 August 2009

Separation of Variables for a Dock

The separation of variables equation for a floating dock

[math]\displaystyle{ Z^{\prime\prime} + k^2 Z =0, }[/math]

subject to the boundary conditions

[math]\displaystyle{ Z^{\prime} (-h) = 0, }[/math]

and

[math]\displaystyle{ Z^{\prime} (0) = 0. }[/math]

The solution is [math]\displaystyle{ k=\kappa_{m}= \frac{m\pi}{h} \, }[/math], [math]\displaystyle{ m\geq 0 }[/math] and

[math]\displaystyle{ Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad m\geq 0. }[/math]

We note that

[math]\displaystyle{ \int\nolimits_{-h}^{0}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn}, }[/math]

where

[math]\displaystyle{ C_{m} = \begin{cases} h,\quad m=0 \\ \frac{1}{2}h,\,\,\,m\neq 0 \end{cases} }[/math]