Dispersion Relation for a Free Surface
Separation of Variables
The dispersion equation arises when 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]\displaystyle{ z }[/math] which is point vertically up and the free surface is at [math]\displaystyle{ z=0. }[/math]. The equations for the Frequency Domain Problem with radial frequency [math]\displaystyle{ \,\omega }[/math] in terms of the potential alone which are
[math]\displaystyle{ \frac{\partial \phi}{\partial z} - k_{\infty} \phi, \, z=0 }[/math]
where [math]\displaystyle{ k_{\infty} }[/math] is the wavenumber in Infinite Depth which is given by [math]\displaystyle{ k_{\infty}=\omega^2/g }[/math] where [math]\displaystyle{ g }[/math] is gravity. We also have the equations within the fluid
[math]\displaystyle{ \nabla^2\phi =0 }[/math]
[math]\displaystyle{ \frac{\partial \phi}{\partial z} = 0, \, z=-H. }[/math]
We then look for a separation of variables solution to Laplace's Equation and obtain the following expression for the velocity potential
[math]\displaystyle{ \phi(x,z) = e^{ikx} \cosh k(z+H) \, }[/math]
If we then apply the condition at [math]\displaystyle{ z=0 }[/math] we see that the constant [math]\displaystyle{ \,k }[/math] (which corresponds to the wavenumber) is given by
[math]\displaystyle{ k \sinh(kH) = k_{\infty} \cosh(kH) }[/math]
or
[math]\displaystyle{ k \tanh(kH) = k_{\infty}\,\,\,(1) }[/math]
This is the dispersion equation for a free surface.
We can also write the separation of variables as
[math]\displaystyle{ \phi(x,z) = e^{kx} \cos k(z+H) \, }[/math]
in which case the dispersion equation becomes
[math]\displaystyle{ k \tan(kH) = -k_{\infty}\,\,\,(2) }[/math]
Equation (1) has one real positive solution (plus imaginary solutions) and equation (2) has an infinite number of positive real solutions (plus imaginary solutions). Sometimes (especially in older works) both equations are used so that only real solutions need to be considered. This separation certainly makes sense in numerical solutions but does adds unnecessarily to the notation.
Solution of the dispersion equation
The solution consists of one real and infinite number of imaginary roots with positive part plus their negatives. The vertical eigenfunctions form complete set for [math]\displaystyle{ L_2[-H,0]\, }[/math] and they are orthogonal.
