View source for Dispersion Relation for a Free Surface

The dispersion equation for a free surface is one of the most important equations in linear water wave theory. It arises when [http://en.wikipedia.org/wiki/Separation_of_Variables separating variables] subject to the boundary conditions for a free surface.

The same equation arises when separating variables in two or three dimensions and we present here the two-dimensional version. We denote the vertical coordinate by <math>z\,,</math> which points vertically upwards, and the free surface is at <math>z=0\,.</math>

The equations for the [[Frequency Domain Problem]] with radial frequency <math>\,\omega</math> in terms of the potential alone which are 
<center><math>\nabla^2\phi =0\,, </math></center>
<center><math>\frac{\partial \phi}{\partial z} = \alpha \phi, \quad z=0\,,</math></center>
<center><math>\frac{\partial \phi}{\partial z} = 0, \quad z=-h\,,</math></center>
where <math>\alpha\,</math> is the wavenumber in [[Infinite Depth]] which is given by 
<math>\alpha=\omega^2/g\,,</math> where <math>g\,</math> is gravity. We generally non-dimensionalize
so that gravitational constant is unity. 

We then find a separation of variables solution to [[Laplace's Equation]] and 
apply the boundary condition at <math>z=-h\,,</math> obtaining the
following expression for the velocity potential:
<center>
<math>\phi(x,z) = e^{kx} \cos k(z+h) \,.</math>
</center>
If we then apply the condition at <math>z=0\,,</math> we see that the constant <math>\,k</math>
(which corresponds to the wavenumber) is given by
<center><math>  -k \sin(kh) = \alpha \cos(kh)\,, </math></center>
or
<center>
<math>  k \tan(kh) = -\alpha\,.\quad(1)</math>
</center>
This is the dispersion equation for a free surface.

We can also write the separation of variables as 
<center>
<math>\phi(x,z) = e^{ikx} \cosh k(z+h) \,</math>  
</center>
in which case the dispersion equation becomes
<center>
<math>  k \tanh(kh) = \alpha\,\,\,(2)</math>
</center>

= Solution of the dispersion equation =

Equation (1) has two purely imaginary solutions plus a countable number
of positive and negative real solutions. Note that the equation is
even in <math>k</math> so that for every solution the negative is also a solution. 
The solutions of equation (2) are just <math>i</math> times the solutions
to equation (1). Sometimes (especially in older works) both equations
are used so that each equation needs only to be solved for real solutions. 
We denote the solutions to (1) by <math>k_n</math>
where <math>k_0</math> is the imaginary solution with
positive imaginary part and <math>k_n</math>
are the real solutions positive solutions ordered so that they are
increasing. 

The dispersion equation is a classical Sturm-Liouville equation.
The vertical eigenfunctions <math>\cos k_n (z+h)</math>
form complete set for <math>L_2[-h,0]\,</math> and they are orthogonal.
Also, as <math>n\to\infty</math> <math>k_n \to \ n\pi/h</math>
so that in the limit the vertical eigenfunctions become the same as
the Fourier cosine series for <math>L_2[-h,0]\,</math> (remembering
that the eigenfunctions satisfy the boundary conditions of zero
normal derivative at <math>z=h</math> which is why we have the 
cosine series).

= Matlab Code =

A program to calculate solutions to the dispersion relation for a free surface
[http://www.math.auckland.ac.nz/~meylan/code/dispersion/dispersion_free_surface.m dispersion_free_surface.m]
[[Category:Linear Water-Wave Theory]]
[[Category:Pages with Matlab Code]]