Units of a wavenumber spectrum

Short definition of Units of a wavenumber spectrum
There are two commonly used units for a wavenumber and it is important to be clear about which one you are using because the level of a spectrum depends on the unit.

This is the common definition for Units of a wavenumber spectrum, but other definitions maybe discussed within the wiki.

Mathematicians and theoreticians usually use ‘angular’ units expressed in radians and this should be indicated by $\mathrm{rad}{\mathrm{m}}^{-1}$ -– radians per meter. It is the counterpart to frequency expressed in $\mathrm{rad}{\mathrm{s}}^{-1}$ -– radians per second. Never express the units as ${\mathrm{m}}^{-1}$ just because an angle technically has no units. This usage is ambiguous. The other unit, which is preferred by investigational scientists because it is derived naturally by a Fourier transform, among other reasons, is $\mathrm{cpm}$ -– cycles per meter. It is the counterpart of $\mathrm{Hz}$ -– cycles per second. The two measures of wavenumber differ by a factor of $2\pi$ which is not small compared to one.

Here we use the symbol $\hat{k}$ to indicate the angular wavenumber expressed in units of $\mathrm{rad}{\mathrm{m}}^{-1}$, and we use the symbol $k$ to indicate the cyclic wavenumber in units of $\mathrm{cpm}$. Their relationship is

$\hat{k}=2\pi k$

Regardless of the unit of wavenumber that you employ, the integral over a wavenumber band gives the variance within that band and this variance must be wavenumber-unit independent. Here are some examples that apply in the inertial subrange. For the velocity spectrum, we must have

$F_{22}({\hat{k}}_1)\mathrm{d}{\hat{k}}_1=F_{22}(k_1)\mathrm{d}{k}_1$

and substituting ( ) gives

$\begin{array}{rl}{\tilde{F}}_{22}({\hat{k}}_1)\mathrm{d}{\hat{k}}_1& =\frac{4}{3}{C}_1{\left(2\pi k_1\right)}^{-5/3}\mathrm{d}(2\pi k_1)\\ & ={\left(2\pi \right)}^{-2/3}\frac{4}{3}{C}_1{k}_1^{-5/3}\mathrm{d}{k}_1\end{array}$

which means that, in the inertial subrange, the cross-profile spectrum of velocity, ${\tilde{F}}_{22}(k_1)$, expressed in units of $\mathrm{cpm}$, is smaller than the same spectrum, ${\tilde{F}}_{22}({\hat{k}}_1)$, expressed in units of $\mathrm{rad}{\mathrm{m}}^{-1}$.

Similarly, the universal shear spectrum is

$\begin{array}{rl}{\tilde{G}}_{22}({\hat{k}}_1)\mathrm{d}{\hat{k}}_1& =\frac{4}{3}{C}_1{\left(2\pi k_1\right)}^{1/3}\mathrm{d}(2\pi k_1)\\ & ={\left(2\pi \right)}^{4/3}\frac{4}{3}{C}_1{k}_1^{1/3}\mathrm{d}{k}_1\end{array}$

which means that the shear spectrum, expressed in units of $\mathrm{cpm}$, is larger by a factor of $(2\pi {)}^{4/3}$ in the inertial subrange than the shear spectrum expressed in units of $\mathrm{rad}{\mathrm{m}}^{-1}$. Finally, the complete shear spectrum must integrate to 2/15 over all wavenumbers and, therefore, the peak of the shear spectrum expressed in units of $\mathrm{cpm}$ is larger than the shear spectrum expressed in units of $\mathrm{rad}{\mathrm{m}}^{-1}$ by a factor of $2\pi$.