<blockquote>
  <p>I understand that the integral of the spectrum over all frequencies is the variance which means that the integral should have $m^2$ units? Please correct me if I am wrong.</p>
</blockquote>

<p>You are correct. If elevation $\eta(t)$ is the measured quantity (units of $m$), the Fourier transform of wave variance $\eta^2$ yields spectrum $S(f)$ with the units of $m^2/Hz$, or if you are working with angular frequency $\omega = 2\pi f$, it yields $S(\omega)$ with the units of $m^2/rad/Hz$.</p>

<p>More details can be found in the answer to <a href="https://earthscience.stackexchange.com/questions/4521/is-wave-spectrum-always-positive-and-why/4522">this question</a>. </p>

CC BY-SA 3.0
2017-01-26T14:23:10.413Z
2017-01-26T14:23:10.413Z
2017-04-13T12:50:30.540Z
milancurcic
<p>In ocean spectra such as the Pierson Moskowitz or the JONSWAP models, the units of $S(\omega)$ are $m^2/(rad/s)$ (or whatever unit of measurement you are working in). Where does the $m^2$ come from and what does it mean physically?</p>

<p>I understand that the integral of the spectrum over all frequencies, i.e. $\int S(\omega) \,d\omega $, is the variance which means that the integral should have $m^2$ units? Please correct me if I am wrong.</p>

CC BY-SA 3.0
2017-01-25T21:32:45.803Z
Not sure without more context, but does [this table of units](https://en.wikipedia.org/wiki/Radiance#SI_radiometry_units) help?
gerrit
CC BY-SA 3.0
2017-01-25T23:19:38.900Z
$m^2$ is "square meters" or "meters squared". You can't just "integrate" an equation, you have to state what the integration is in respect to. So is the integration over time, over area, or what?
MaxW
CC BY-SA 3.0
2017-01-26T09:58:41.530Z
By "integral of the spectrum over all frequencies", I meant integrate w.r.t. $\omega$ (angular frequency). So $\int S(\omega) \,d \omega$.
RH_data_maths
CC BY-SA 3.0
2017-01-27T10:46:26.217Z
Please [edit] that into your question
Jan Doggen
<oceanography><models><waves>
Units of wave spectrum
RH_data_maths
# Units of wave spectrum

(Asked by: username_0 on 2017-01-25)

In ocean spectra such as the Pierson Moskowitz or the JONSWAP models, the units of $S(\omega)$ are $m^2/(rad/s)$ (or whatever unit of measurement you are working in). Where does the $m^2$ come from and what does it mean physically?

I understand that the integral of the spectrum over all frequencies, i.e. $\int S(\omega) \,d\omega $, is the variance which means that the integral should have $m^2$ units? Please correct me if I am wrong.

**Tags:** oceanography, models, waves

> **Comments:**
> **username_1**: Not sure without more context, but does [this table of units](https://en.wikipedia.org/wiki/Radiance#SI_radiometry_units) help?  
> **username_2**: $m^2$ is "square meters" or "meters squared". You can't just "integrate" an equation, you have to state what the integration is in respect to. So is the integration over time, over area, or what?  
> **username_0**: By "integral of the spectrum over all frequencies", I meant integrate w.r.t. $\omega$ (angular frequency). So $\int S(\omega) \,d \omega$.  
> **username_3**: Please [edit] that into your question  

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## 1 answers

### Answer by username_4 on 2017-01-26. Score: 5 | Accepted answer

> I understand that the integral of the spectrum over all frequencies is the variance which means that the integral should have $m^2$ units? Please correct me if I am wrong.

You are correct. If elevation $\eta(t)$ is the measured quantity (units of $m$), the Fourier transform of wave variance $\eta^2$ yields spectrum $S(f)$ with the units of $m^2/Hz$, or if you are working with angular frequency $\omega = 2\pi f$, it yields $S(\omega)$ with the units of $m^2/rad/Hz$.

More details can be found in the answer to [this question](https://earthscience.stackexchange.com/questions/4521/is-wave-spectrum-always-positive-and-why/4522).


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