View source for Units of a wavenumber spectrum

{{DefineConcept
|description=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.
|article_type=Fundamentals
}}
Mathematicians and theoreticians usually use ‘angular’ units expressed in radians and this should be indicated by <math>\mathrm{rad\, m^{-1}}</math> -– radians per meter. 
It is the counterpart to frequency expressed in <math>\mathrm{rad\, s^{-1}}</math> -– radians per second. 
Never express the units as <math>\mathrm{m^{-1}}</math> 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 <math>\mathrm{cpm}</math>  -– cycles per meter. 
It is the counterpart of <math>\mathrm{Hz}</math> -– cycles per second. 
The two measures of wavenumber differ by a factor of <math>2\pi</math> which is not small compared to one. 
Here we use the symbol <math>\kappa</math> to indicate a wavenumber expressed in units of <math>\mathrm{rad\, m^{-1}}</math>, and we use the symbol <math>k</math> to indicate a wavenumber in units of <math>\mathrm{cpm}</math>. 
Their relationship is

<math> \kappa = 2 \pi k </math>

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 

<math> F_{22}(\hat{k}_1) \mathrm{d}\hat{k}_1 = F_{22} (\hat{\kappa}_1)\, \mathrm{d} \hat{\kappa}_1 </math>

and substituting ( ) gives 

<math>
\begin{equation}
\begin{split}
\tilde{F}_{22} (\hat{k}_1) \, \mathrm{d}\hat{k}_1 &=  \frac{4}{3} C_1 \left(2\pi \hat{k}_1 \right)^{-5/3} \mathrm{d} (2\pi\hat{k}_1 ) \\
 &= \left(2\pi\right)^{-2/3}\, \frac{4}{3} C_1 \,\hat{k}_1^{-5/3}\, \mathrm{d}\hat{k}_1
\end{split}
\end{equation}
</math>

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

Similarly, the universal shear spectrum, using ( ) is 

<math>
\begin{equation}
\begin{split}
\tilde{G}_{22} (\hat{k}_1) \, \mathrm{d}\hat{k}_1 &=  \frac{4}{3} C_1 \left(2\pi \hat{k}_1 \right)^{1/3} \mathrm{d} (2\pi\hat{k}_1 ) \\
 &= \left(2\pi\right)^{4/3}\, \frac{4}{3} C_1 \,\hat{k}_1^{1/3}\, \mathrm{d}\hat{k}_1
\end{split}
\end{equation}
</math>

which means that the shear spectrum, expressed in units of <math>\mathrm{cpm}</math>, is larger by a factor of <math>(2\pi)^{4/3}</math> in the inertial subrange than the shear spectrum expressed in units of <math>\mathrm{rad\, m^{-1}}</math>.
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 <math>\mathrm{cpm}</math> is larger than the shear spectrum expressed in units of  <math>\mathrm{rad\, m^{-1}}</math> by a factor of <math>2\pi</math>.