Template:Cylindrical equations

The problem for the complex water velocity potential in suitable non-dimensionalised cylindrical coordinates, [math]\displaystyle{ \phi (r,\theta,z) }[/math], is given by

[math]\displaystyle{ \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial \phi}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 \phi}{\partial \theta^2} + \frac{\partial^2 \phi}{\partial z^2} = 0, \quad (r,\theta,z) \in \Omega }[/math]

[math]\displaystyle{ \frac{\partial \phi}{\partial z} = \alpha \phi , \quad z=0 }[/math]

[math]\displaystyle{ \frac{\partial \phi}{\partial z} = 0, z=-h }[/math]