# Template:Separation of variables for a dock

### Separation of Variables for a Dock

The separation of variables equation for a floating dock

$Z^{\prime\prime} + k^2 Z =0,$

subject to the boundary conditions

$Z^{\prime} (-h) = 0,$

and

$Z^{\prime} (0) = 0.$

The solution is $k=\kappa_{m}= \frac{m\pi}{h} \,$, $m\geq 0$ and

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

We note that

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

where

$C_{m} = \begin{cases} h,\quad m=0 \\ \frac{1}{2}h,\,\,\,m\neq 0 \end{cases}$