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I have the following problem that I would like to solve by using the 'fft' function in matlab.

Some background on the problem: The evolution of sea surface height is given, to first order, by

[tex]

\eta_o(x,t)=A\cos\left(kx-\omega t\right)

[/tex]

From this, we see that there are three parameters, A, k, \omega, that dictate \eta(x,t). According to linear theory, in general \eta will be a super position of these waves and will be given by

[tex]

\eta(x,t) = \sum_n A_n \cos\left(k_nx-\omega_nt \right)

[/tex]

Each of these waves obey the deep water dispersion relationship

[tex]

\omega_n^2=gk_n

[/tex]

Finally, to find the A_n, we note that we know \eta(0,t). The A_n are constants, therefore we can find them by inverting the relation

[tex]

\eta(0,t)= \sum_n A_n \cos\left(\omega_nt \right)

[/tex]

which is done in matlab by taking a fft of \eta(0,t). The last pieces of information I need are the \omega_n s. I'm not sure what sets the frequencies of the system and this is what is giving me trouble. Of course there might be a significantly easier way to do all of this.

Any help would be appreciated,

Nick