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 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} - \alpha \phi, \, z=0 }[/math]

where [math]\displaystyle{ \alpha }[/math] is the wavenumber in Infinite Depth which is given by [math]\displaystyle{ \alpha=\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 find a separation of variables solution to Laplace's Equation and apply the boundary condition at [math]\displaystyle{ z=-h }[/math] and we obtain the following expression for the velocity potential

[math]\displaystyle{ \phi(x,z) = e^{kx} \cos 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 \sin(kH) = \alpha \cos(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^{ikx} \cosh k(z+H) \, }[/math]

in which case the dispersion equation becomes

[math]\displaystyle{ k \tanh(kh) = \alpha\,\,\,(2) }[/math]

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]\displaystyle{ k }[/math] so that for every solution the negative is also a solution. The solutions of equation(2) are just [math]\displaystyle{ 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]\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).