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 points vertically upwards, and the free surface is at [math]\displaystyle{ z=0\,. }[/math]
We also assume that Frequency Domain Problem with frequency [math]\displaystyle{ \omega }[/math] and we assume that all variables are proportional to [math]\displaystyle{ \exp(-\mathrm{i}\omega t)\, }[/math]
The water motion is represented by a velocity potential which is
denoted by [math]\displaystyle{ \phi\, }[/math] so that
[math]\displaystyle{ \Phi(\mathbf{x},t) = \mathrm{Re} \left\{\phi(\mathbf{x},\omega)e^{-\mathrm{i} \omega t}\right\}. }[/math]
The equations therefore become
(note that the last expression can be obtained from combining the expressions:
where [math]\displaystyle{ \alpha = \omega^2/g \, }[/math])
We use separation of variables We express the potential as
[math]\displaystyle{ \phi(x,z) = X(x)Z(z)\, }[/math]
and then Laplace's equation becomes
[math]\displaystyle{ \frac{X^{\prime\prime}}{X} = - \frac{Z^{\prime\prime}}{Z} = k^2 }[/math]
Separation of variables for a free surface
We use separation of variables
We express the potential as
[math]\displaystyle{ \phi(x,z) = X(x)Z(z)\, }[/math]
and then Laplace's equation becomes
[math]\displaystyle{ \frac{X^{\prime\prime}}{X} = - \frac{Z^{\prime\prime}}{Z} = k^2 }[/math]
The separation of variables equation for deriving free surface eigenfunctions is as follows:
[math]\displaystyle{ Z^{\prime\prime} + k^2 Z =0. }[/math]
subject to the boundary conditions
[math]\displaystyle{ Z^{\prime}(-h) = 0 }[/math]
and
[math]\displaystyle{ Z^{\prime}(0) = \alpha Z(0) }[/math]
We can then use the boundary condition at [math]\displaystyle{ z=-h \, }[/math] to write
[math]\displaystyle{ Z = \frac{\cos k(z+h)}{\cos kh} }[/math]
where we have chosen the value of the coefficent so we have unit value at [math]\displaystyle{ z=0 }[/math]. The boundary condition at the free surface ([math]\displaystyle{ z=0 \, }[/math]) gives rise to:
which is the Dispersion Relation for a Free Surface
The above equation is a transcendental equation. If we solve for all roots in the complex plane we find that the first root is a pair of imaginary roots. We denote the imaginary solutions of this equation by [math]\displaystyle{ k_{0}=\pm ik \, }[/math] and the positive real solutions by [math]\displaystyle{ k_{m} \, }[/math], [math]\displaystyle{ m\geq1 }[/math]. The [math]\displaystyle{ k \, }[/math] of the imaginary solution is the wavenumber. We put the imaginary roots back into the equation above and use the hyperbolic relations
[math]\displaystyle{ \cos ix = \cosh x, \quad \sin ix = i\sinh x, }[/math]
to arrive at the dispersion relation
[math]\displaystyle{ \alpha = k\tanh kh. }[/math]
We note that for a specified frequency [math]\displaystyle{ \omega \, }[/math] the equation determines the wavenumber [math]\displaystyle{ k \, }[/math].
Finally we define the function [math]\displaystyle{ Z(z) \, }[/math] as
[math]\displaystyle{ \chi_{m}\left( z\right) =\frac{\cos k_{m}(z+h)}{\cos k_{m}h},\quad m\geq0 }[/math]
as the vertical eigenfunction of the potential in the open water region. From Sturm-Liouville theory the vertical eigenfunctions are orthogonal. They can be normalised to be orthonormal, but this has no advantages for a numerical implementation. It can be shown that
[math]\displaystyle{ \int\nolimits_{-h}^{0}\chi_{m}(z)\chi_{n}(z) \mathrm{d} z=A_{n}\delta_{mn} }[/math]
where
[math]\displaystyle{ A_{n}=\frac{1}{2}\left( \frac{\cos k_{n}h\sin k_{n}h+k_{n}h}{k_{n}\cos ^{2}k_{n}h}\right). }[/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).
See Also
Matlab Code
A program to calculate solutions to the dispersion relation for a free surface dispersion_free_surface.m
