# Units of wave spectrum

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.

• Not sure without more context, but does this table of units help?
– gerrit
Commented Jan 25, 2017 at 21:32
• $m^2$ 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
Commented Jan 25, 2017 at 23:19
• By "integral of the spectrum over all frequencies", I meant integrate w.r.t. $\omega$ (angular frequency). So $\int S(\omega) \,d \omega$. Commented Jan 26, 2017 at 9:58
• Please edit that into your question Commented Jan 27, 2017 at 10:46

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 $m^2/rad/Hz$.