Template:Separation of variables for a dock

Separation of Variables for a Dock

The separation of variables equation for a 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}=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]