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

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== Separation of variables==
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=== Separation of Variables for a Dock ===
  
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Applying Laplace's equation in the vertical direction assuming
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The separation of variables equation for a floating dock
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a separation constant <math>\mu</math> we obtain
 
 
<center>
 
<center>
 
<math>
 
<math>
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\zeta_{zz}+\mu^{2}\zeta=0.\,
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Z^{\prime\prime} + k^2 Z =0,
 
</math>
 
</math>
 
</center>
 
</center>
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We then use the boundary condition at <math>z=-h</math>, which is
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subject to the boundary conditions
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the same for all <math>x</math> to write
 
 
<center>
 
<center>
 
<math>
 
<math>
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\zeta=\cos\mu(z+h)\,
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Z^{\prime} (-h) = 0,
 
</math>
 
</math>
 
</center>
 
</center>
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where the separation constant <math>\mu^{2}</math> must
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and
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satisfy different equations depending on whether we are under
 
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the free-surface or the dock covered region. For the free surface
 
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<center><math>
 
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k\tan\left(  kh\right)  =-\alpha,
 
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</math></center>
 
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which is the [[Dispersion Relation for a Free Surface]]
 
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and for the dock covered region
 
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<center>
 
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<math>
 
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\kappa\tan(\kappa h)=0,
 
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</math>
 
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</center>
 
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Note that we have set <math>\mu=k</math> under the free
 
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surface and <math>\mu=\kappa</math> under the plate. We denote the
 
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positive imaginary solution of free-surface equation by <math>k_{0}</math> and
 
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the positive real solutions by <math>k_{m}</math>, <math>m\geq1</math>. The solutions of
 
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dock equation are
 
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<math>\kappa_{m}=m\pi/h</math>, <math>m\geq 0</math>. We define
 
 
<center>
 
<center>
 
<math>
 
<math>
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\phi_{m}\left( z\right) =\frac{\cos k_{m}(z+h)}{\cos k_{m}h},\quad m\geq0
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Z^{\prime} (0) = 0.
 
</math>
 
</math>
 
</center>
 
</center>
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as the vertical eigenfunction of the potential in the open
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The solution is
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water region and
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<math>k=\kappa_{m}= \frac{m\pi}{h} \,</math>, <math>m\geq 0</math> and
 
<center>
 
<center>
 
<math>
 
<math>
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\psi_{m}\left(  z\right)  = \cos\kappa_{m}(z+h),\quad
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Z = \psi_{m}\left(  z\right)  = \cos\kappa_{m}(z+h),\quad
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m\geq 0
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m\geq 0.
 
</math>
 
</math>
 
</center>
 
</center>
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as the vertical eigenfunction of the potential in the dock
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We note that
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covered region. For later reference, we note that:
 
 
<center>
 
<center>
 
<math>
 
<math>
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\int\nolimits_{-h}^{0}\phi_{m}(z)\phi_{n}(z) d z=A_{m}\delta_{mn}
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\int\nolimits_{-h}^{0}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
 
</math>
 
</math>
 
</center>
 
</center>
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<center>
 
<center>
 
<math>
 
<math>
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A_{m}=\frac{1}{2}\left(  \frac{\cos k_{m}h\sin k_{m}h+k_{m}h}{k_{m}\cos
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C_{m} =  
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^{2}k_{m}h}\right)
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\begin{cases}
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h,\quad m=0 \\
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\frac{1}{2}h,\,\,\,m\neq 0
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\end{cases}
 
</math>
 
</math>
 
</center>
 
</center>
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and
 
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<center>
 
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<math>
 
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\int\nolimits_{-h}^{0}\phi_{n}(z)\psi_{m}(z) d z=B_{mn}
 
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</math>
 
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</center>
 
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where
 
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<center><math>
 
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B_{mn}=\frac{k_{n}\sin k_{n}h\cos\kappa_{m}h-\kappa_{m}\cos k_{n}h\sin
 
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\kappa_{m}h}{\left(  \cos k_{n}h\right)  \left(  k_{n}
 
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^{2}-\kappa_{m}^{2}\right)  }
 
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</math></center>
 
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and
 
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<center>
 
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<math>
 
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\int\nolimits_{-h}^{0}\psi_{m}(z)\psi_{n}(z) d z=C_{m}\delta_{mn}
 
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</math>
 
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</center>
 
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where
 
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<center>
 
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<math>
 
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C_{m}=\frac{1}{2}h,\quad,m\neq 0 \quad \mathrm{and} \quad C_0 = h
 
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</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]

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