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 of equation (1) consists of one real and infinite number of imaginary roots with positive part plus their negatives. These solutions multiplied by [math]\displaystyle{ i }[/math] are the solutions to equation (2). We denote the solutions to (2) by [math]\displaystyle{ k_n }[/math] where [math]\displaystyle{ k_0 }[/math] is the imaginary solution with positive imaginary part and [math]\displaystyle{ 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]\displaystyle{ \cos k_n (z-H) }[/math] form complete set for [math]\displaystyle{ L_2[-H,0]\, }[/math] and they are orthogonal. Also, as [math]\displaystyle{ n\to\infty }[/math] [math]\displaystyle{ 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]\displaystyle{ L_2[-H,0]\, }[/math] (remembering that the eigenfunctions satisfy the boundary conditions of zero normal derivative at [math]\displaystyle{ z=H }[/math] which is why we have the cosine series).