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 [math]\displaystyle{ \mathrm{rad\, m^{-1}} }[/math] -– radians per meter. It is the counterpart to frequency expressed in [math]\displaystyle{ \mathrm{rad\, s^{-1}} }[/math] -– radians per second. Never express the units as [math]\displaystyle{ \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]\displaystyle{ \mathrm{cpm} }[/math] -– cycles per meter. It is the counterpart of [math]\displaystyle{ \mathrm{Hz} }[/math] -– cycles per second. The two measures of wavenumber differ by a factor of [math]\displaystyle{ 2\pi }[/math] which is not small compared to one.

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

[math]\displaystyle{ \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]\displaystyle{ F_{22}(\hat{k}_1) \mathrm{d}\hat{k}_1 = F_{22} (k_1)\, \mathrm{d} k_1 }[/math]

and substituting ( ) gives

[math]\displaystyle{ \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]\displaystyle{ \tilde{F}_{22}(k_1) }[/math], expressed in units of [math]\displaystyle{ \mathrm{cpm} }[/math], is smaller than the same spectrum, [math]\displaystyle{ \tilde{F}_{22}(\hat{k}_1) }[/math], expressed in units of [math]\displaystyle{ \mathrm{rad\, m^{-1}} }[/math].

Similarly, the universal shear spectrum is

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