So, I have a spectrum calculated from an underwater sensor, $S_{vv}(f)$, and want to convert it to $S_{vv}'(k)$, where $k$ is wavelength and $f$ frequency.

I assume Taylor hypothesis for frozen turbulence, i.e. turbulent structures are unchanged when advected by the mean flow. Thus: $$U = \frac{2 \pi f}{k} $$ and $$\frac{dk}{df} = \frac{2 \pi}{U} , $$ but it seems that to preserve the variance, I need to multiply $S_{vv}(f)$ with a factor so $$\int S_{vv}(f)df = \int S_{vv}'(k)dk$$

Can anyone explain how it's done?

  • 1
    $\begingroup$ What is $U$? Velocity? And is $k$ not perhaps the wavenumber instead of the wavelength? So then $k=2\pi/\lambda$, where $\lambda$ is the wavelength? $\endgroup$ Sep 25, 2020 at 3:09
  • $\begingroup$ Usually you convert the time series to a spatial series (i.e., measurement as function of position instead of as function of time) then compute the power spectrum. This is because the U you describe is not necessarily a constant, so each $\Delta t$ will correspond to a different $\Delta x$. That is, the conversion in the power spectrum is not necessarily linear. $\endgroup$ Oct 28, 2020 at 15:26


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