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.

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

<math> \hat{k} = 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} (k_1)\, \mathrm{d} k_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 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{split} \end{equation} </math>

which means that, in the inertial subrange, the cross-profile spectrum of velocity, <math>\tilde{F}_{22}(k_1)</math>, expressed in units of <math>\mathrm{cpm}</math>, is smaller than the same spectrum, <math>\tilde{F}_{22}(\hat{k}_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 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{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>.