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} }[/math]

This is the dispersion equation for a free surface.

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.