Template:Separation of variables for a free surface

Separation of variables for a free surface

We now separate variables and write the potential as

[math]\displaystyle{ \phi(x,z)=\zeta(z)\rho(x) }[/math]

Applying Laplace's equation we obtain

[math]\displaystyle{ \zeta_{zz}+k^{2}\zeta=0. }[/math]

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

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

.