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

From WikiWaves
Jump to navigationJump to search
m (clarification and tidy up)
 
(8 intermediate revisions by 2 users not shown)
Line 1: Line 1:
 
   
 
   
−
== 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 floating dock
−
a separation constant <math>\mu</math> we obtain
 
 
<center>
 
<center>
 
<math>
 
<math>
−
\zeta_{zz}+\mu^{2}\zeta=0.\,
+
Z^{\prime\prime} + 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)\,
+
Z^{\prime} (-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>
 
−
<math>
 
−
\kappa\tan(\kappa h)=0,
 
−
</math>
 
−
</center>
 
−
Note that we have set <math>\mu=k</math> under the free
 
−
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
 
 
<center>
 
<center>
 
<math>
 
<math>
−
\phi_{m}\left( z\right) =\frac{\cos k_{m}(z+h)}{\cos k_{m}h},\quad m\geq0
+
Z^{\prime} (0) = 0.
 
</math>
 
</math>
 
</center>
 
</center>
−
as the vertical eigenfunction of the potential in the open
+
The solution is
−
water region and
+
<math>k=\kappa_{m}= \frac{m\pi}{h} \,</math>, <math>m\geq 0</math> and
 
<center>
 
<center>
 
<math>
 
<math>
−
\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>
−
as the vertical eigenfunction of the potential in the dock
+
We note that
−
covered region. For later reference, we note that:
 
 
<center>
 
<center>
 
<math>
 
<math>
−
\int\nolimits_{-h}^{0}\phi_{m}(z)\phi_{n}(z) d z=A_{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 58: Line 37:
 
<center>
 
<center>
 
<math>
 
<math>
−
A_{m}=\frac{1}{2}\left(  \frac{\cos k_{m}h\sin k_{m}h+k_{m}h}{k_{m}\cos
+
C_{m} =  
−
^{2}k_{m}h}\right)
+
\begin{cases}
 +
h,\quad m=0 \\
 +
\frac{1}{2}h,\,\,\,m\neq 0
 +
\end{cases}
 
</math>
 
</math>
 
</center>
 
</center>
−
and
 
−
<center>
 
−
<math>
 
−
\int\nolimits_{-h}^{0}\phi_{n}(z)\psi_{m}(z) d z=B_{mn}
 
−
</math>
 
−
</center>
 
−
where
 
−
<center><math>
 
−
B_{mn}=\frac{k_{n}\sin k_{n}h\cos\kappa_{m}h-\kappa_{m}\cos k_{n}h\sin
 
−
\kappa_{m}h}{\left(  \cos k_{n}h\right)  \left(  k_{n}
 
−
^{2}-\kappa_{m}^{2}\right)  }
 
−
</math></center>
 
−
and
 
−
<center>
 
−
<math>
 
−
\int\nolimits_{-h}^{0}\psi_{m}(z)\psi_{n}(z) d z=C_{m}\delta_{mn}
 
−
</math>
 
−
</center>
 
−
where
 
−
<center>
 
−
<math>
 
−
C_{m}=\frac{1}{2}h,\quad,m\neq 0 \quad \mathrm{and} \quad C_0 = h
 
−
</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]

Retrieved from "https://wikiwaves.org/wiki/index.php?title=Template:Separation_of_variables_for_a_dock&oldid=9544"