Units of a wavenumber spectrum - Revision history

Ilker at 22:57, 29 November 2021

2021-11-29T22:57:08Z

← Older revision		Revision as of 22:57, 29 November 2021
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	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>.		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		Similarly, the universal shear spectrum is

	<math>		<math>

Ilker at 22:56, 29 November 2021

2021-11-29T22:56:41Z

← Older revision		Revision as of 22:56, 29 November 2021
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	\begin{equation}		\begin{equation}
	\begin{split}		\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~~ ) \\		\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 \,~~\hat{k}_1~~^{1/3}\, \mathrm{d}~~\hat{k}_1~~		&= \left(2\pi\right)^{4/3}\, \frac{4}{3} C_1 \,k_1^{1/3}\, \mathrm{d}k_1
	\end{split}		\end{split}
	\end{equation}		\end{equation}

Ilker at 22:53, 29 November 2021

2021-11-29T22:53:57Z

← Older revision		Revision as of 22:53, 29 November 2021
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	\begin{equation}		\begin{equation}
	\begin{split}		\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~~ ) \\		\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 \,~~\hat{k}_1~~^{-5/3}\, \mathrm{d}~~\hat{k}_1~~		&= \left(2\pi\right)^{-2/3}\, \frac{4}{3} C_1 \,k_1^{-5/3}\, \mathrm{d}k_1
	\end{split}		\end{split}
	\end{equation}		\end{equation}
	</math>		</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>.		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		Similarly, the universal shear spectrum, using ( ) is

Ilker at 22:50, 29 November 2021

2021-11-29T22:50:46Z

← Older revision		Revision as of 22:50, 29 November 2021
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	Never express the units as <math>\mathrm{m^{-1}}</math> just because an angle technically has no units.		Never express the units as <math>\mathrm{m^{-1}}</math> just because an angle technically has no units.
	This usage is ambiguous.		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.		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.
	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>		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.		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.
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	For the velocity spectrum, we must have		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>		<math> F_{22}(\hat{k}_1) \mathrm{d}\hat{k}_1 = F_{22} (k_1)\, \mathrm{d} k_1 </math>

	and substituting ( ) gives		and substituting ( ) gives
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	</math>		</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>.		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		Similarly, the universal shear spectrum, using ( ) is

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{{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>.