# Dispersion Relation for a Free Surface

## Introduction

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 $\displaystyle{ z\,, }$ which points vertically upwards, and the free surface is at $\displaystyle{ z=0\,. }$

We also assume that Frequency Domain Problem with frequency $\displaystyle{ \omega }$ and we assume that all variables are proportional to $\displaystyle{ \exp(-\mathrm{i}\omega t)\, }$

The water motion is represented by a velocity potential which is denoted by $\displaystyle{ \phi\, }$ so that

$\displaystyle{ \Phi(\mathbf{x},t) = \mathrm{Re} \left\{\phi(\mathbf{x},\omega)e^{-\mathrm{i} \omega t}\right\}. }$

The equations therefore become

\displaystyle{ \begin{align} \Delta\phi &=0, &-h\lt z\lt 0,\,\,\mathbf{x} \in \Omega \\ \partial_z\phi &= 0, &z=-h, \\ \partial_z \phi &= \alpha \phi, &z=0,\,\,\mathbf{x} \in \partial \Omega_{\mathrm{F}}, \end{align} }

(note that the last expression can be obtained from combining the expressions:

\displaystyle{ \begin{align} \partial_z \phi &= -\mathrm{i} \omega \zeta, &z=0,\,\,\mathbf{x} \in \partial \Omega_{\mathrm{F}}, \\ \mathrm{i} \omega \phi &= g\zeta, &z=0,\,\,\mathbf{x} \in \partial \Omega_{\mathrm{F}}, \end{align} }

where $\displaystyle{ \alpha = \omega^2/g \, }$)

We use separation of variables We express the potential as

$\displaystyle{ \phi(x,z) = X(x)Z(z)\, }$

and then Laplace's equation becomes

$\displaystyle{ \frac{X^{\prime\prime}}{X} = - \frac{Z^{\prime\prime}}{Z} = k^2 }$

### Separation of variables for a free surface

We use separation of variables

We express the potential as

$\displaystyle{ \phi(x,z) = X(x)Z(z)\, }$

and then Laplace's equation becomes

$\displaystyle{ \frac{X^{\prime\prime}}{X} = - \frac{Z^{\prime\prime}}{Z} = k^2 }$

The separation of variables equation for deriving free surface eigenfunctions is as follows:

$\displaystyle{ Z^{\prime\prime} + k^2 Z =0. }$

subject to the boundary conditions

$\displaystyle{ Z^{\prime}(-h) = 0 }$

and

$\displaystyle{ Z^{\prime}(0) = \alpha Z(0) }$

We can then use the boundary condition at $\displaystyle{ z=-h \, }$ to write

$\displaystyle{ Z = \frac{\cos k(z+h)}{\cos kh} }$

where we have chosen the value of the coefficent so we have unit value at $\displaystyle{ z=0 }$. The boundary condition at the free surface ($\displaystyle{ z=0 \, }$) gives rise to:

$\displaystyle{ k\tan\left( kh\right) =-\alpha \, }$

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 $\displaystyle{ k_{0}=\pm ik \, }$ and the positive real solutions by $\displaystyle{ k_{m} \, }$, $\displaystyle{ m\geq1 }$. The $\displaystyle{ k \, }$ of the imaginary solution is the wavenumber. We put the imaginary roots back into the equation above and use the hyperbolic relations

$\displaystyle{ \cos ix = \cosh x, \quad \sin ix = i\sinh x, }$

to arrive at the dispersion relation

$\displaystyle{ \alpha = k\tanh kh. }$

We note that for a specified frequency $\displaystyle{ \omega \, }$ the equation determines the wavenumber $\displaystyle{ k \, }$.

Finally we define the function $\displaystyle{ Z(z) \, }$ as

$\displaystyle{ \chi_{m}\left( z\right) =\frac{\cos k_{m}(z+h)}{\cos k_{m}h},\quad m\geq0 }$

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

$\displaystyle{ \int\nolimits_{-h}^{0}\chi_{m}(z)\chi_{n}(z) \mathrm{d} z=A_{n}\delta_{mn} }$

where

$\displaystyle{ A_{n}=\frac{1}{2}\left( \frac{\cos k_{n}h\sin k_{n}h+k_{n}h}{k_{n}\cos ^{2}k_{n}h}\right). }$

## Matlab Code

A program to calculate solutions to the dispersion relation for a free surface dispersion_free_surface.m