# Converting a turbulence spectrum in frequency to wavelength

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?

• 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? Sep 25, 2020 at 3:09
• 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. Oct 28, 2020 at 15:26