Difference between revisions of "Frequency Domain Problem"

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This is closely connected with the [[Fourier Transform in Time]]. Essentially after this
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{{complete pages}}
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we are left with a problem in which all time dependence is proportional to <math>\exp (i\omega t)</math>
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and the resultant problem (for complex valued potential, displacement, etc.)  
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It is standard in linear water wave theory to take a
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is said to be in the frequency domain (as opposed to the time domain).
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[http://en.wikipedia.org/wiki/Fourier_Transform Fourier Transform] in time and assume
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In many practical applications this is the only solution required, i.e. engineers simply want
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that the solution for the real velocity potential <math>\Phi(x,y,z,t)</math>
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a table of force as a function of frequency. Often the dependence is taken as
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can be written as
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<math>\exp (-i\omega t)</math>
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<center>
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<math>\Phi(x,y,z,t) = \phi(x,y,z) e^{-\mathrm{i}\omega t} \,</math>
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</center>
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where <math>\omega</math> is the real  and <math>\phi(x,y,z)</math>
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is a complex function. This means that any time derivative
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can simply we replaced by multiplication by <math>i \omega</math> (this only works
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because of the linearity in time).
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Sometimes it is assumed that the exponential is positive(but this is not the convention used here.
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The problem is now said to be a frequency domain problem.
  
 
[[Category:Linear Water-Wave Theory]]
 
[[Category:Linear Water-Wave Theory]]

Latest revision as of 09:30, 28 April 2010


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.

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