View source for Template:Separation of variables for a free surface first part

The separation of variables equation for deriving free surface eigenfunctions is as follows:
<center>
<math>
Z^{\prime\prime} + k^2 Z =0.
</math>
</center>
subject to the boundary conditions
<center>
<math>
Z^{\prime}(-h) = 0
</math>
</center>
and
<center>
<math>
Z^{\prime}(0) = \alpha Z(0)
</math>
</center>
We can then use the boundary condition at <math>z=-h \, </math> to write
<center>
<math>
Z = \frac{\cos k(z+h)}{\cos kh}
</math>
</center>
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:
<center><math>
k\tan\left(  kh\right)  =-\alpha \,
</math></center>
which is the [[Dispersion Relation for a Free Surface]]