Template:Separation of variables for a dock

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Separation of Variables for a Dock

The separation of variables equation for a floating dock

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

subject to the boundary conditions

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

and

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

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

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

We note that

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

where

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