I don't understand the equation37 in simulate ocean water by Jerry Tessendorf.The result is all complex number, how to be the slope.Even if I compute the magnitude of it,the result is just positive which is obvious wrong.As There must be some points whose slope is negative.Who can help me.Thank you.
Complex results in inverse Fourier transform for simulating ocean water
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pde
fourier-analysis
mathematical-modeling
fluid-dynamics
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0Do you mean that they should be small?But I found that they are not small enough compare to the real part. – 2012-11-28
1 Answers
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If $h(x)$ is a real valued function we have that since $h(x) = \bar{h}(x)$ that its Fourier series
$ h(x) \approx \sum \tilde{h}(k) \exp ikx = \sum \bar{\tilde{h}}(k) \exp -ikx \approx \bar{h}(x)$
So $\tilde{h}(k) = \bar{\tilde{h}}(-k)$. This is a fundamental property of the Fourier transform of real valued functions.
Now if we write
$ \nabla h(x) \approx \sum i k \tilde{h}(k) \exp ikx = \sum \eta(k) \exp ikx $
we note that
$ \bar\eta(k) = \eta(-k) $
by a direct computation. And hence
$ \nabla h(x) = \overline{\nabla h}(x) $
is a real valued function.
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0If$n$is a even number,then h*(k) = h(n-k).If$n$is a odd number,then h*(k) = h(n-1-k).These can guarantee that the result of the IDFT is real number.I don't quite understand what you're saying,but thx. – 2012-12-05