Frequency Domain Problem

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It is standard in linear water wave theory to take a Fourier Transform in time and assume that the solution for the real velocity potential [math]\displaystyle{ \Phi(x,y,z,t) }[/math] can be written as

[math]\displaystyle{ \Phi(x,y,z,t) = \phi(x,y,z) e^{-\mathrm{i}\omega t} \, }[/math]

where [math]\displaystyle{ \omega }[/math] is the real and [math]\displaystyle{ \phi(x,y,z) }[/math] is a complex function. This means that any time derivative can simply we replaced by multiplication by [math]\displaystyle{ i \omega }[/math] (this only works because of the linearity in time). Sometimes it is assumed that the exponential is positive(but this is not the convention used here.

The problem is now said to be a frequency domain problem.