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

Revision as of 04:27, 26 August 2008

The separation of variables equation for a dock

[math]\displaystyle{ \frac{\mathrm{d}^2 Z}{\mathrm{d} z^2} + k^2 Z =0. }[/math]

subject to the boundary conditions

[math]\displaystyle{ \frac{dZ}{dz}(-h) = 0 }[/math]

and

[math]\displaystyle{ \frac{dZ}{dz}(0) = 0 }[/math]

The solution is [math]\displaystyle{ k=\kappa_{m}=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) d z=C_{m}\delta_{mn} }[/math]

where

[math]\displaystyle{ C_{m}=\frac{1}{2}h,\quad,m\neq 0 \quad \mathrm{and} \quad C_0 = h }[/math]

@@ 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
-</math>
-</center>
-as the vertical eigenfunction of the potential in the open
-water region and
-<center>
-<math>
-\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
-covered region. For later reference, we note that:
-<center>
-<math>
-\int\nolimits_{-h}^{0}\phi_{m}(z)\phi_{n}(z) d z=A_{m}\delta_{mn}
-</math>
-</center>
-where
-<center>
-<math>
-A_{m}=\frac{1}{2}\left(  \frac{\cos k_{m}h\sin k_{m}h+k_{m}h}{k_{m}\cos
-^{2}k_{m}h}\right)
-</math>
-</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>