Difference between revisions of "Frequency Domain Problem"

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{{complete pages}}
 
{{complete pages}}
  
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This is closely connected with the [[Fourier Transform in Time]]. Essentially after this
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It is standard in linear water wave theory to take a
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we are left with a problem in which all time dependence is proportional to
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[http://en.wikipedia.org/wiki/Fourier_Transform Fourier Transform] in time and assume
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<center><math>\exp (i\omega t)\,</math></center>
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that the solution for the real velocity potential <math>\Phi(x,y,z,t)</math>
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and the resultant problem (for complex valued potential, displacement, etc.)  
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can be written as
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is said to be in the frequency domain (as opposed to the time domain).
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<center>
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In many practical applications this is the only solution required, i.e. engineers simply want
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<math>\Phi(x,y,z,t) = \phi(x,y,z) e^{i\omega 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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</center>
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<center><math>\exp (-i\omega t)\,</math></center>
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where <math>\omega</math> is the real  and <math>\phi(x,y,z)</math>
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but we will not use this convention in this wiki.
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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 (possibly more than half the time)
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it is assumed that the exponential is negative.
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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]]

Revision as of 08:34, 9 September 2009


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^{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 (possibly more than half the time) it is assumed that the exponential is negative.

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

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