Template:Separation of variables for a free surface

Separation of variables for a free surface

The separation of variables equation for a free surface is

[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) = \alpha Z(0) }[/math]

We then use the boundary condition at [math]\displaystyle{ z=-h }[/math] to write

[math]\displaystyle{ Z = \frac{\cos k(z+h)}{\cos kh} }[/math]

The boundary condition at the free surface ([math]\displaystyle{ z=0 }[/math]) is

[math]\displaystyle{ k\tan\left( kh\right) =-\alpha, }[/math]

which is the Dispersion Relation for a Free Surface We denote the positive imaginary solution of this equation by [math]\displaystyle{ k_{0} }[/math] and the positive real solutions by [math]\displaystyle{ k_{m} }[/math], [math]\displaystyle{ m\geq1 }[/math]. We define

[math]\displaystyle{ \phi_{m}\left( z\right) =\frac{\cos k_{m}(z+h)}{\cos k_{m}h},\quad m\geq0 }[/math]

as the vertical eigenfunction of the potential in the open water region. From Sturm-Liouville theory the vertical eigenfunctions are orthogonal. They can be normalised to be orthonormal, but this has no advantages for a numerical implementation. It can be shown that

[math]\displaystyle{ \int\nolimits_{-h}^{0}\phi_{m}(z)\phi_{n}(z) d z=A_{n}\delta_{mn} }[/math]

where

[math]\displaystyle{ A_{n}=\frac{1}{2}\left( \frac{\cos k_{n}h\sin k_{m}h+k_{n}h}{k_{n}\cos ^{2}k_{n}h}\right) }[/math]

.