Template:Separation of variables for a free surface
Separation of variables for a free surface
We use separation of variables
We express the potential as
[math]\displaystyle{ \phi(x,z) = X(x)Z(z)\, }[/math]
and then Laplace's equation becomes
[math]\displaystyle{ \frac{X^{\prime\prime}}{X} = - \frac{Z^{\prime\prime}}{Z} = k^2 }[/math]
The separation of variables equation for deriving free surface eigenfunctions is as follows:
[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) = \alpha Z(0) }[/math]
We can then use the boundary condition at [math]\displaystyle{ z=-h \, }[/math] to write
[math]\displaystyle{ Z = \frac{\cos k(z+h)}{\cos kh} }[/math]
where we have chosen the value of the coefficent so we have unit value at [math]\displaystyle{ z=0 }[/math]. The boundary condition at the free surface ([math]\displaystyle{ z=0 \, }[/math]) gives rise to:
which is the Dispersion Relation for a Free Surface
We denote theee 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) \mathrm{d} z=A_{n}\delta_{mn} }[/math]
where
[math]\displaystyle{ A_{n}=\frac{1}{2}\left( \frac{\cos k_{n}h\sin k_{n}h+k_{n}h}{k_{n}\cos ^{2}k_{n}h}\right). }[/math]
