# Template:Separation of variables for a free surface first part

From WikiWaves

The separation of variables equation for deriving free surface eigenfunctions is as follows:

[math] Z^{\prime\prime} + k^2 Z =0. [/math]

subject to the boundary conditions

[math] Z^{\prime}(-h) = 0 [/math]

and

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

We can then use the boundary condition at [math]z=-h \, [/math] to write

[math] 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]z=0[/math]. The boundary condition at the free surface ([math]z=0 \,[/math]) gives rise to:

which is the Dispersion Relation for a Free Surface